ABSTRACT SOME SEMANTICAL PROBLEMS IN DEONTIC LOGIC AND IMPERATIVE LOGIC BY Hsiu-hwang Ho This dissertation deals with the problem of whether current systems of deontic and imperative logic formalize satisfactorily our intuitive dentic and imperative notions, and the problem of whether those systems can be used to justify normative reasonings. Some semantical inadequacies of the current systems are noted and suggestions are made for their removal. Chapter One contains reconstructions of certain well-known systems of deontic logic: von Wright's system vW, Fisher-Aqvist's system FA, 3 family of systems called OT*, 054* and 055*, and Anderson's systems OM, OM' and OM". The systems OT*, 054* and 055* are developed quite thoroughly: a number of important theorems are proved, the problem of irreducible deontic modalities is solved for each system and containment relations among the systems are studied. Certain familiar problems and difficulties of current deontic logic are examined, for instance, the paradoxes of "derived obligation", contrary-to-duty imperativesand Chisholm's dilemma, the Kantian Principle and the paradox of the Good Samaritan. Hsiu-hwang Ho In Chapter Two certain amendments are made to the systems OT*, 054* and 055*. The deontic Operator '0' (obligation) is explained in terms of, and hence relativized to, a set of moral rules, Deontic variables are taken to range over propositions that we call circumstantialized act-propositions, or simply CM-act-propositions, in which the elements of agent, time and location of endeavoring are specified. An attempt‘is made to justify these amendments by meta- ethical observations. Hintikka-style semantics is furnished. Quanti- fiers are readily introducible into the amended systems, and it is argued that quantified deontic logic is necessary to eXpress some moral codes or moral principles. Some suggestions about solving in the amended systems the paradoxes and difficulties listed above are advanced. Finally, the problem of introducing alethic modalities into deontic logic is raised. The Kantian Principle and the ”law" that what is necessary is obligatory and what is impossible is forbidden are discussed. In the last chapter the relation between the evaluative and directive uses of language in a moral context is examined. An attempt is made to show that a deontic logic and the corresponding imperative logic are isomorphic models of a related normative logic. An attempt is also made to explicate the notion of normative validity. A partial characterization of the truth conditions for deontic (or imperative) sentences is proposed, and it is argued that the usual definition of (as sertoric) validity is applicable to normative Hsiu-hwang Ho arguments. Finally, the problem of the possibility of imperative logic is raised. Jérgensen's dilemma and related problems are examined. Two unorthodox imperative operators, which are imperative counter- parts of 'You are permitted to do. . . ' and 'It is indifferent that you do. . . are introduced. Attention to these operators seems to contribute to a correct understanding of certain diSputed argument forms. Three appendices are included: a list of axioms and rules, a list of definitions, and a list of theorems. There is also a compre- hensive bibliography that lists most of the important works in deontic and imperative logic through 196 8. SOME SEMAN TIC AL PROB LEMS IN DEONTIC LOGIC AND IMPERATIVE LOGIC BY Hsiu-hwang Ho A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of PhilosoPhy 1969 To the memory of PROFESSOR HENRY S. LEONARD ii a QUE? 3’/)"7l AC KNOWLEDGMENTS The author wishes to express his deep gratitude to Professor Gerald J. Massey for his untiring and time-consuming careful reading of this dissertation. His detailed criticism and helpful suggestions have led to many improvements both in formulation and in argumentation. Thanks are also due to Professor Herbert E. Hendry and Professor George C. Kerner with whom discussions always turned out to be fruitful and beneficial to the author. iii TAB LE OF CONTENTS DEDICATION 11 ACKNOWLEDGMENTS iii 3 1. INTRODUCTION 1 CHAPTER 1: SOME DEONTIC SYSTEMS 7 S 2. VON WRIGHT'S SYSTEM vW 8 § 3. THE PARADOXES OF ”DERIVED OBLIGATION" 25 2'34. 35. E6. §7. fie. 3 9. 310. E11. § 12. § 13. 3 14. CONTRARY-TOI-DUTY IMPERATIVES AND CHISHOLM'S DILEMMA 31 DEONTIC LOGIC AND MODAL LOGIC 37 DEONTIC LOGICS AS ”SUBSYSTEMS" OF MODAL LOGICS 51 SYSTEMS OT*, vW and FA 66 DEON TIC MODALITIES AND ITERATION OF MODALITIES 76 SYSTEM 054* AND ITS FOURTEEN MODALITIES 81 SYSTEM 055* AND ITS SIx MODALITIES 91 ANDERSONIAN CONSTANT AND DEONTIC SYSTEMS OM, OM' AND OM" 92 THE RELATION BETWEEN OT*, 054*, 055* AND OM, OM’, OM" 96 FURTHER THEOREMS AND FURTHER PROBLEMS IN OM-OM". KANTIAN PRINCIPLE AND THE PARADOX OF THE GOOD SAMARITAN lOl VON WRIGHT'S TENSE-DEONTIC SYSTEMS 104 iv CHAPTER 11: S15. § 16. En. §18. E19. § 20. 321. § 22. CHAPTER III: S 23. 3 24. 325. § 26. META- ETHICS AND SOME MODIFIED SYSTEMS OF DEONTIC LOGIC . . . . TOWARD A SOUND SYSTEM OF DEONTIC LOGIC. DEONTIC VARIABLES RANGE OVER CM-ACT-PROPOSITIONS .......... THE INTERPRETATION OF '0' IN TERMS OF MORAL RULES ................ META-ETHICS AND ETHICO-SOCIOLOGY: SOME OBSERVATIONS ......... THREE SYSTEMS OF DEONTIC LOGIC: CMO,T*, CMO,S4* AND CMO,sz°‘ . . HINTIKKA-KRIPKE SEMANTICS FOR CMO,T* - CMO,55* .............. CHISHOLM'S DILEMMA, THE DILEMMA OF CONFLICTING OBLIGATIONS AND THE PARADOX OF THE GOOD SAMARITAN REVISITED ..................... QUANTIFIERS AND ALETHIC MODALITIES IN DEONTIC LOGIC ........... . . MORAL USES OF LANGUAGE AND IMPERATIVE LOGIC. . . . DUAL FUNCTIONS OF LANGUAGE IN A MORAL CONTEXT: EVALUATION AND DIRECTION . . . ........ NORMATIVE SENTENCES: DEONTIC AND IMPERATIVE ........... THREE CORRESPONDING SYSTEMS OF IMPERATIVE LOGIC: CMI,T*, CMI,S4* AND CMI,55*. ..... PURE NORMATIVE LOGICS: THE LOGICS OF REQUIREMENT CMRT*, CMRS4* AND CMRSS*. . ~116 .117 .123 -l32 - 140 - 151 .161 168 -182 .192 .193 .210 ~216 .222 § 27. § 28. § 29. § 30. § 31. § 32. APPENDICES BIBLIOGRAPHY ............ . . . . .......... NORMATIVE ARGUMENTS: PURE AND MIXED - - TOWARD A DEFINITION OF NORMATIVE VALIDITY . . . THE CRITERION OF NORMATIVE VALIDITY. . . SOUNDNESS OF NORMATIVE ARGUMENTS . . . SOME ADDITIONAL REMARKS ON THE EXPLICATION OF NORMATIVE VALIDITY. UNDERSTANDING IMPERATIVE LOGIC A. LIST OF AXIOMS AND RULES . B. LIST OF DEFINITIONS . C. LIST OF THEOREMS ...... vi 225 - 230 . 242 . 250 - 251 . 256 - 267 - 268 - 272 . 274 3 1. INTRODUCTION The behavior of deontic predicates such as 'obligatory', 'permissible' and ‘forbidden' has long been receiving philOSOphers' attention. As early as in the Middle Ages, it was observed that there exists a similarity between the concept obligation and the concept necessity on the one hand, and the behavior of the concept permission and of the concept possibility on the other. However, the philOSOphical treatment of these deontic concepts had been largely peripheral and made in passing until early this century when Ernst Mally tried to formalize systematically the deontic concepts. 1 It was he who first used the word 'deontik' and called his study of these concepts ‘deontik logik'. Subsequently, a remarkable number of efforts have been made either directly in deontic logic or in fields closely related to it, e. g. , in the logic of imperatives or in the logic of commands. Examples Of these efforts made before 1950 can be found most significantly in the following literature: Kurt Grelling [1939], Karel Reach [1939], Karl Menger [1939], Albert Hofstadter and John Charles Chinoweth McKinsey [1939], Alf Ross [1941] and Herbert Gaylord Bohnert [1945]. lSee Mally [1926]. The author-cum-date reference is made to the bibliography at the end Of this dissertation. l It is perhaps sound to say, however, that the ice of modern deontic study was not really broken until the late 1950's when the Finnish logician Georg Henrik von Wright published his earliest studies in deontic logic with an effort to formalize the deontic con- cepts of permission, obligation, prohibition and commitment. Since the publication of the earliest papers by von Wright, the study of deontic concepts has received wide Spread philOSOphical attention both in the English-Speaking world and in Scandinavian countries. Although hardly twenty years have elapsed, we find a wide range of deontic logics on display. Among them some systems are based upon standard pr0positional logic,4 others take alethic modal logics as their cornerstones.5 There are still others in which quantifiers play an indi5pensable role. 6 Besides, of all the varieties some systems are two-valued,7 others three-valued;8 some systems . . . 9 . formalize the relatIVIzed deontic concepts, others Incorporate 2In English, the word 'deontic' was coined, according to von Wright, by Charles Dunbar Broad. See von Wright [1951a] . 3See, especially, von Wright [1951a] and [1951b]. 4For example, von Wright [1951a], [1951b], [1956]. [1965a] and Fisher [1961b]. 5E. G. Anderson [1956] and Prior [1957]. 6See Hintikka [1957]. 7Von Wright [1951b]. 8Fisher [1961b] and Aqvist [1963b]. 9Von Wright [1956] and Rescher [1958]. tense-logical notions as their basic concepts.10 Furthermore, some philosophers discuss deontic logic in the context of, or in coordination with, imperative logic or directive logic; 11 others base their deontic logics on another formalized or formalizable system, such as the ”logic of better", 12 and so on. This list of variety in deontic logic can be extended considerably, and all of the deontic systems are devised to capture the formal structure of deontic concepts. In the course of deve10pment of these various deontic systems, different types of procedure to single out the "deontic truths" have also been advanced. Among them, axiomatics is hardly a new tech- nique as one may expect. The truth table or matrix method and the normal form method are also commonly used. In addition, Quine's truth-value analysis, Hintikka's model-set method, Kripkean model structure together with Beth's semantical tableaux, and Fitch's subordinate proof, all have found their ways into deontic logic. This brief description of deontic logic may lead one to con- clude that the modern development of deontic logic has now reached a mature and advanced stage. This conclusion, however, is too hasty if not totally unjustifiable. For one thing, logic may not be just a game of manipulating symbols. We usually intend a logic 10Von Wright [1965b] and Aqvist [1966]. 11E. g., Geach [1958], Castafieda [1958], [1968], and Ross [1968]. leqvist [1963c]. 4 to be a formalization or systematization Of a set Of concepts of which the underlying "logic" is intuitively conceived. In our present case, this set of concepts is the so-called deontic concepts: Obligation, permission, prohibition (forbiddance) and commitment. A deontic logic is meant to explicate, these concepts. Hence, the success of a deontic logician depends not only on whether he has a syntactically well-built system, but also on whether his system admits of a sound semantical interpretation which is genuinely deontic. From this point Of view, it is not without good reason that some phiIOSOphers also call deontic logic the logic 9_f Obligation. This reminds us from the very beginning what deontic logic aims at, and provides us with an intuitive ground to justify its degree of success. It is a common belief, and a usual practice, too, among deontic logicians that the concepts of obligation, permission and prohibition are interdefinable with the help of some logical constants (e. g. , the negation and conjunction connectives). 13 It follows immediately that the logic of obligation, the logic of permission, and the logic of prohibition are, or could be, one and the same logic. But what about the logic of commitment? IS the concept "commitment" definable in terms of one or several of the other deontic concepts with perhaps the help of certain logical constants? The answer is far less definite. 13See, for example (D2. 1)--(D2. 3) in next section. Von Wright fir st tried to formalize the concept of commitment in terms of obligation and the material conditional. 14 Since that proposal was put forward, criticism and new proposals have been mounting in the literature. But until now there seems to be no single satisfactory formulation which is commonly accepted by deontic logicians. To make the situation even worse, Roderick M. Chisholm introduced the so-called contrary-tO-duty imperative into deontic studies, 15 thereby adding to the already puzzling problem a new dimension of difficulty. This is just an indication of the semantic difficulties which a deontic logician encounters. In addition, the problems of deontic logic come from pragmatic considerations, too. Some philOSOPhers tend to think that a sound deontic system should be able to function as a logic of imperative (or directive) inference which can be used to justify imperative reasoning just as ordinary logic has been used to justify descriptive or indicative reasoning. Indeed, some philoso- phers, notably Ross [1968], even call a logic of imperatives deontic logic. And the problems of imperative logic have often been treated as the problems of deontic logic. There are, then, two classes of problems we have so far mentioned. On the one hand, there are semantic problems of how to interpret a deontic logic as a sound logic of obligation and other 14Von Wright [1951a]. 15Chisholm [196 3a] . deontic concepts. Or, what turns out to be the same thing, the problem of how to "correctly” formalize our intuitive deontic concepts. And, on the other hand, we have the pragmatic problems of how our logic can be used as a logic of imperatives. These are, indeed, two sets of problems we want to consider in this discussion. But before we set out to discuss these problems, we shall first try to present some systems of deontic logic. We Shall treat three systems OT*, 054* and 055* quite thoroughly, and compare them with von Wright‘s system vW, Fisher and Aqvist's system FA and Ander son's systems OM, OM' and OM". This will help us to locate our problems precisely in their prOper contexts, and make us under- stand more adequately the nature of the problems. CHAPTER I SOME DEONTIC SYSTEMS § 2. VON WRIGHT'S SYSTEM vW G. H. von Wright may be properly thought of as a pioneer in the modern study of deontic logic. His early works in the 1950's have been the principal sources of inSpiration and guidance in the develOp- ment of deontic logic. It was mainly from his works that current systems of deontic logic Sprang and received their present shapes. Moreover, during these past seventeen years (1951-1968), deontic logicians or deontically minded philosophers have been greatly indebted to his constant introduction of original ideas and his steady contribution of new results. For instance, his idea of relativizing deontic concepts, in particular, the concept Of relative or conditional permission, led to the develOpment of the system of conditional permission. And his introduction of a special type of tense-logical connectives gave rise to another type of deontic logic, namely tense- deontic logic. In order to appreciate the problems of deontic logic mentioned in the last section, let us fir st of all reconstruct the earliest system 1See von Wright [1956]. [1964] and Rescher [1958], [1962]. 2Von Wright [1965b], [1966] and Aqvist [1966]. See § 14. 9 of modern deontic logic prOposed by von Wright in 1951, namely, his system of the logic of permission. System vW has the following vocabulary and formation rules: I. Vocabulary i) Individual variables: '3', 'b', 'c', 'a ', 'bl', 'gl', '32" . .. These variables range over act-types or "act-qualifying prOperties”. 4 ii) Deontic connectives: '~', ‘&', 'v', '3', ' iii) Deontic predicate: 'P' (which may be read as "It is permissible that, . . "). . . . 4 . 1v) Sentential connectives: '~'. '&' , 'v', '3', '5' (Wlth the usual semantics). v) Grouping indicators: '[', 'J ' II. Formation rules i) An individual variable standing alone is a (deontic) name (an atomic name). :- -1 r' ‘1 t' fl ii) If a and B are (deontic) names, then ~a , [or 8: B] , [avB] , r '1 r 1 . [a 3 B] and [a E B] are (deontic) names (molecular names). . I" 1. . 111) If or 13 a name, than Par 15 a sentence (an atomic sentence). We call it a P-sentence. For the sake of uniformity in our discussion, we do not try to be in strict conformity with von Wright's original symbolism and formulation. The same set of symbols are used both as deontic connectives and as sentential connectives. But these two classes of objects can always be effectively distinguished one from the other by the forma- tion rules. Hence, this ambiguity is superficial and harmless. 5Von Wright uses 'name' and 'sentence' in [1951a], some logician may prefer to use 'term' and 'well—formed formula', respectively. 10 iv) If A and B are sentences, then I’~A-', r[A 8: B] , tIAVB], , [TA :3 BT’ and rEAE— B]! are sentences (molecular sentences). v) Only those formulas which can be formed by i) and ii) are names, and only those formulas which can be formed by iii) and iv) are sentences. The concepts of obligation, forbiddance and (moral) indiffer- ence can, then, be defined in terms of the concept of permission in the following way: r 1 _ r~ ‘1 (D2.1) OCY -Df P~a r '1 __ I; ‘I (D2.2) Fa -Df Pa ., (D2.3) 'Ia“ =13}f 'IPaser/J l'Oar‘, rFa? and r10}! may be called, reSpectively, an O-sentence, an F-sentence, and an I-sentence. Von Wright, then, asks the question: What are the "things" which are pronounced obligatory, permissible, forbidden, etc. ? The answer, according to him, is straightforward, namely, ”acts", or more precisely Speaking, act-qualifying prOperties. However, it should be pointed out that the concept of an act (or act-type) is diffi- cult to define. Actually in von Wright's earliest writings, the mean- ing of that term is not precisely construed. For instance, we may ask the following question: Is there any difference between saying that act_a_ is permissible and the performance of act a is permissible? Some people may be inclined to say no, having in mind that to say 11 "the performance of act a" is just a redundant way of saying "act 3", because the very concept of act implies the concept of performance. It seems that von Wright himself shares this view, for we find that he Speaks indiscriminately of an act and the performance of an act. For example, in [1951a] he uses, among other things, the following expressions: (2. 1) If an act is obligatory, (2. 2) If doing what we ought to do commits us to . . . (2. 3) If failure to perform an act commits us to . . . Thus in our reproduction Of his system vW, we are going to interpret the individual variables in the same manner. We will read 'Pg' either as "a is permissible" or as "the performance Ofa is per- missible", and leave open, for the time being, the question of how we should properly analyze a deontic sentence. Let us then agree with von Wright in saying that an act is either performed or not performed. This is called the performance value of an act. If we let '1' and '0' denote the performance values "performance" and ”non-performance", respectively, and let 'f' denote the performance function, namely, the mapping from the set of act-names to { 1, O] , then we may put down the following seman- tical rules: When expressions are listed in separate lines, no single quotes will be used. For example, (2. 1) should otherwise be written as: (2. 1) 'If an act is obligatory, . . . '. 12 III. Semantical rules i) An act a is either performed or not performed, that is f(a)=l or f(a)=0. ii) f('~a1)=l if f(a)=0; otherwise f("~a1)=O. iii) i('[a & B]1):1 if f(a)=l and may—.1; otherwise, i("[.: &511)=o. iv) f(r[avB]fl):l if either f(a)=1 or f(B)=1; otherwise, f('[evai‘)=o. v) i(r[o:a]")=1 if either f(a):0 or i(a)=1; otherwise, f(“[a:a]‘>=o. vi) £("[ase]")=1 if either f(a)=1 and f(B)=1 or £(o)=o and £(s)=o; otherwise, £("[as 53‘ )-_-o. The list may be regarded as a set of semantical rules for the deontic connectives. To avoid the difficulty in talking about acts, let us associate the performance value (which is originally associated with acts) from now on with act-names. 7 It is easy to see that the performance value of a molecular act name is uniquely determined by the performance values Of its constituent atomic act names. We may call the former a performance function of the latter. This is a departure from von Wright's formulation, for he says that "An gt will be called a performance-function of certain other acts, if its 7This is hardly a new practice. We find philosophers, notably Leonard, apply the predicate 'true' both to a sentence (Leonard calls it a statement) and a proposition. Thus, a sentence may be defined as true if and only if the prOposition it indicates is true. Cf. Leonard [1967], §§5.3£. l3 performance-value. . .uniquely depends upon the performance-values of those other acts . . . " Here we have an analogue of truth function in prOpositional logic. Von Wright even pushes this analogy a little further by calling r 1 r 'I r' ‘I r 1 r~aj [0&5] , [Q’VB] , [0'38] and [OF—B] , anegation-act, a conjunction-act, a distinction-act, an implication-act, and an equivalence-act, reSpectively. We shall, however, prefer to call the last two a conditional-act and a biconditional-act, respectively. Let Q be a deontic evaluation function from the set Of act- names to the set {1*, 0*} where '1*' denotes permission, and '0‘"I denotes non-permission, i. e. , forbiddance. These two are called the deontic values. If the deontic value of a molecular act name is uniquely determined by the deontic values of its component act names, then the former is said to be a deontic function of the latter. It should be noted that not every molecular act name is a deontic function of its component names. 9 In other words, deontic function does not in general coincide with performance function. For instance, from the fact that a is permissible, it is not determined whether r~a1 is permissible or not permissible. It may be the case that both an act and its negation-act are permissible, or it may be the case that only 8Von Wright [1951a], p. 2. My italics. Again, von Wright defines a deontic function as a relation on acts rather than on act names. We find him saying: "An act will be called a deontic function of certain other acts, if the deontic value of the former uniquely depends upon the deontic values of the latter" and ". . .not any act which is a performance-function of certain acts is also a deontic function of them. " 111111., p. 6. 14 one of them is. However, just as a disjunction-act name has the performance value performance ("1") if and only if at least one of its component act names has the performance value performance, a disjunction-act has the deontic value permission ("1*") if and only if at least one of its component act names has the deontic value per- mission. We are thus able to establish the following: IV. 0 (Ta v B :11) = 1* if 0(a) = 1* or 0(8) = 1*; otherwise, m'tav 81‘) = 0* This means that in the case of a disjunction-act name, deontic function and performance function coincide. This provides us with a way of establishing a decision procedure for deontic (logical) truth in system vW. In order to outline this decision procedure, let us fir st put down von Wright's two basic principles in this system. One of them can be inferred from what we have just said above. Another principle is justifiable offhand by our intuitive notion of permission. (2.4) Principle £f permission: "Any given act is either itself permitted (permissible) or its negation is permitted (.permissible)".lo Hence, for every 3, either 'P3' is true or 'P~_a' is true. 10Ibid., p. 9. l5 (2. 5) Principle o_f deontic distribution for Ermission: If a is 11 '1 '(a1 v a2 v . . . v an)', then [Pay is true if and only if '(Pal v Pazv . . .v Pan)' is true. Von Wright's version: "If an act is the disjunction of two other acts, then the proposition that the disjunc- tion is permitted is the disjunction of the proposition that the fir st act is permitted and the proposition that the second act is permitted (This principle can, naturally, be extended to disjunctions with any number n of members)". Let us now proceed to explain our decision procedure. Let A be any arbitrary P-sentence, atomic or molecular, in system vW; let A , A ., A be a complete list of atomic (P-) sentences 1 2' k appearing in A. Clearly A is a truth function of Al' A2, . . , Ak. . . ' . r ‘I 13 Now, each A1 in the list A1, A2, .. . , Ak 18 of the form Pari , where a, may be an atomic or a molecular name. Let us assume 1 further that 'al', 'az', . . . , 'an' is a complete list of atomic names occurring in A, that is, in either Al or A2 or . .. or Ak. As a first step toward the decision procedure, we shall construct a com- plete disjunctive normal form 61 of each ai relative to the complete list of atomic names 'al', 'az', . . . , ‘an' appearing in A, except when 11We shall follow the usual convention on the omission of grouping indicators, and we shall use parentheses as substitutes for brackets, thinking of the former as poorly-drawn versions of the latter. 12.1312. , p. 7 Von Wright's own parenthetical remarks. 13The use of the same subscript 'i' in both Ai and Pcri is harmless. We do this for the sake of facilitating the subsequent pre sentation. 16 a. is contradictory. In that case we write oi as '_a_ &~_a'. This 1 special case will be treated later. Meanwhile let us suppose that no ai is contradictory. Thus, each 51 is adisjunction r- i 6 v 612v... v6 11 1 r .1 (1 s r s Zn) in which each 6; is a conjunction r51 8: 82 &- - - 8: an where each am (1 s m s n) is either 'am' or '~am'. Now, just as in propositional logic a formula has the same truth value as its complete disjunctive normal form, in system vW a name has the same performance value as its complete disjunctive normal form. This comes from the fact that performance function, as we saw above, resembles truth function in a straightforward way. Consequently, each Ai' that is Poi can be written as (2.6) P(611v6;v... voi) But the truth value of (2. 6) is, according to (2. 5) above, the same as i i i (2.7) P61vP62v... vpar Let us call each P6; (1 sj s r) in (2.7) a P-constituent of (2. 6), and hence of Pan, that is, Ai. And we shall call (2. 7) a P-disfiinction of 1 _. A,. 1 When each Ai (i: 1, 2,. . . , k) of A has been written as a P-disjunction, we may establish a complete list of (distinct) P-con- stituents rP¢1-', l’PgSZ‘ , ..., "Pg; (1 s s s 2n) of A as the set consist- ing of those and only those P-constituents which appear in the P-dis- junction of any of the A1 of A. Now, since A is a truth function of A1, A2, . . . , Ak’ and each Ai of A is, among other things, a truth 17 function of its P-constituents. It follows that A is a truth function of its complete list of P-constituents. A decision procedure is now immediately visible. Let A be an arbitrary P-sentence we want to examine, let rP¢11 , rP¢z1, . , r1395: be a complete list of P-constituents of A. We shall con- struct a truth table for A in the usual fashion except that on the upper part (row) of the(1eft-side) assignment columns we list the complete list of P-constituents rP¢l1 , IP952“ , ..., rPgs: of A instead of the complete list of variables as we commonly do in prOpositional logic. On the upper row of (right-side) evaluation columns, we put down A* (which is exactly like A except each Ai thereof is replaced by its P-disjunction P51 v P512 v . . . v Pbi. It is easy to see that A* is truth functionally equivalent to A. We then proceed to put down all the possible combination of truth values, 1. e. , 't' 's amd 'f' 's, r under the list rP¢11 , I”1395.! Pas; in the assignment columns, 2, . . . , with a restriction we shall mention later. After the assignment columns are thus furnished with 't' 's and 'f' 's, we shall evaluate each row of the truth table in the familiar way by appealing to the semantical rules Of propositional logic until finally we come up with the truth values for the main connective of A*. If we have all 't' '3 under it, then A*, and hence A, expresses a deontic (logical) truth. We shall call it a deontic tautology of system vW, or a vW-tautology. Otherwise, A is not a vW-tautology. 18 Now, the Special case we mention earli’er.. If any 011 is contradictory, and thus we write it as '3 8: ~a', then the decision procedure goes essentially the same as above said except that we now have some occurrence(s) of 'P(a 8: ~_a_)' in A* which is to be evaluated. In a truth table we simply put an 'f' under 'P@&~a)' in every row in the evaluation column, and go ahead to further evaluation. The rationale of doing this is that a P-sentence is true if and only if at least one of its P-constituents is true. But now 'PLa 8r ~_a_)' has no P-constituent, hence it is always false. Of course, what we have outlined above is only a decision procedure applicable to a P-sentence. However, since other deontic sentences, i.e., O-sentences, F-sentences, and I-sentences, are all definable in terms of P-sentences, we have mutatis mutandis a decision procedure applicable to any deontic sentence. Let us give an example to show what a truth table looks like. Suppose we want to examine the following sentence, (2.8) F3 D 0 (ash) we fir st rewrite it as a P-sentence of vW, that is (2.9) ~ Pa 2) ~P ~(33Q) Let us call it A which consists of two atomic (P-)sentences, namely, 'Pa' and 'P~ (a Db)‘. The name '_a' appearing in the first atomic sentence can be put into its complete disjunctive normal form relative to the complete list of atomic names occurring in A, namely, '3' and 19 'b', as 'Q 8: b) v (a 8: ~19); and the name '~(a D _l_3_)' appearing in the second atomic sentence can be similarly rendered as '~~ Q 8: ~b)‘ which is the same as ‘(a 8: ~b)‘. Now, rewrite 'Pa' and 'P~(§._ 3 b)‘ as their P-disjunctions. We have '[PQ 8: h) v Pg 8: ~2)]' and 'P(§ 8: ~_b_)', re5pectively. Thus, A* of A is the following sentence (2.10) ~[P(_a a: _b_) v Pg 8: ~13]: ~P(§ a: ~h). The following truth table constructed in the above-described manner shows that (2. 10) is a vW-tautology. (In this case, we have 'P(_a 8: b)’ and 'PQ 8: ~_b)‘ as a complete list of P-constituents of A). P12 8: 2) Fe 8: ~2) ~(P(2&h) v Pie&~2)) :2 was: ~21 t t f t t t t f t f t f f t t t f t t f f t t f t t f f t f f f t t f Since we have 't' 's and only 't' '5 under the main connective '3' of A*, A* is a vW—tautology. Consequently, A is a vW-tautology. That is to say, (2. 8) abbreviates a sentence which is a vW-tautology. However, in using this truth-tabular method, one restriction must be Observed. This restriction can be laid down as follows. Suppose, 'gl', ’_a_z', . . . , 'an' is a complete list of distinct atomic names appearing in A, then we might have less than or equal to Zn distinct P-constituents "Pol‘ , "P9521, rings; of A. If the number of P-constituents of A is 2n, i.e., 5:2n then, in the assignment column, the row in which all rP¢i1 's are assigned falsehood 20 should be deleted from the truth table. For example, if A is 'P_a_ v P~_a_._ v P]; v P~_b', then we have two distinct atomic names, '_a_' and '_b'. After the above-mentioned manipulation, A can be shown as being equivalent to (2.11) PB&2)VP(2&~b_)vP(~é&h)VP(~§.&~h) Here we have 22: 4 distinct P-constituents, hence the foregoing restriction applies. The truth table for (2. 11) then looks like P(_a.&‘t_>) P(a. &~h) P(~a&h)P(~a&~h) . . . . . t t t t f t t t f t t f f t t t t f t f t f t t f f t f f f t t t t f f t t £- t f t f f f t f t t f f t f f t f f i There is no additional row in which we have straight 'f' '8 across the evaluatio n c olurrm s . 21 The rationale behind this restriction is easily seen from the principle of permission, i. e. , (2. 4) together with the principle of deontic distribution for permission, i. e. , (2. 5). When all the 2n P-constituents are false, we have (2.12) ~P(318:gz8:_a_38:...8:an)8:~P(~§._18:_a_.28:§_38:... &an)8:... &~P(&1&§2&"'&%1-1&"§-n) Thatis (2.13) ~[P(21&22&2:3&~~&2n)VP(~21&éz&é3&-~ .8zan)v...v PQ1&§_2&... &é'n-l&~§n)] This by (2. 5) is equivalent to (2.14) ~[P(al&a_2&_a_38:...8:_ah)v(~al8:gz8:_a_38:... .8:%1)v(_a_18:~328:g38:... &_an)v...v(_a18:az8:.. '&'a‘n-l 8:~_a_n)] But (2. 14) by propositional logic (henceforth: PL) is the same as (2.15) ~[P(§_lv~_al)8:(gzv~gz)8:... 8:(_ahv~§n)] which, by (2. 5) again, is (2. l6) ~[(P§1 v P~§_1) 8: (P3,,2 v Pivéz) 8: . . . 8: (P9,r1 v P~_a_n)] That is, (2. 17) ~(P_a_1v P~_a_1)v ~ (P__a_2 v Prop-.2) v . .. v~(P_a._n v P ~_§_n) 22 This contradicts (2.4) which reads: for every 31' either P21 or P~_a_i. This restriction can also be intuitively justified as follows. Suppose the restriction is violated, then from what we have just demonstrated, it follows that an act and its negation-act are both forbidden (not permissible). This, by definitions (D1. 1) and (D1. 2) above, means in turn that both an act and it's negation-act are obliga- tory, an assertion at direct variance with our intuition. The following are some of the deontic tautologies in system vW. We use ' l- A‘ to mean that A is a deontic tautology. (Th. 1) [- P_a_ '5 ~O~§ (a is permissible if and only if not-9., is not obligatory. (Th. 2) (- P_a_ ‘-—= ~F _a_ (a is permissible if and only if a is not forbidden. (Th. 3) (- OaDPg. (If a is obligatory, then a is permissible) (Th. 4) )- F~a3 P; (If not-g is forbidden, then a is permissible). (Th. 5) )- O(_a_v~_§) (a_-or-not-_a is obligatory). (Th. 6) )- F(§8:~a) (g-and-not-g is forbidden). (Th. 7) )- ~(Og 8: O~_a) (It is not the case that _a_ is obligatory and not-a is also obligatory). (Th. 8) (- O(§_8:_b_) 5. O; 8: Oh L-and-b is obligatory if and only if _a_ is obligatory and _b_ is obligatory). 23 (Th. 9) |- O_a_ v Oh. I) O(a v_b_) (If either _a_ is obligatory or b is obligatory, then _a_-or-b is obligatory. It may be remarked in passing that 'v' here is to be understood as strictly analogous to the 'v' in propositional logic. From 'p v q' we are not entitled to infer 'p'. Likewise, from 'O(§ v b)’ we cannot jump to the conclusion 'Oa' . Otherwise, (Th. 9) above and (Th. 10) below become very curious deontic laws. 14 (Th. 10) )- O(§8:~_a_) D O_lg (If g-and-not-g is obligatory, then any _b_ is obligatory). (Th. 11) l- ngb) E . P; v Pb (a-or-l; is permissible if and only if either a is permissible or b is permissible). (Th. 12) [- O_a_._ 8: O(gsb) . 3 02 (If a is obligatory and _a_-only-if-b is obligatory, then b is obligatory). (Th. 13) [- Pa 8: O(_a_ 31;). D P_b_ (If _a_ is permissible and a-only-if-ll is obligatory, then b is permissible. (Th. 14) )- F2 8: O(_a_ 231.2). :3 F3 (If 2 is forbidden and _a-only-if-b is obligatory, then _a is forbidden. (Th. 15) (- (F1; 8: F_c_) 8: O[a:: (_bvgfl. 3 Fa (Similarly) (Th. 16) 1- ~[O(a v_l_)_) 8: (P_a_ 8: Erin (Similar to (Th. 7)) (Th. 17) ) Q; 8: on; 8: b.) 3 _c_]. soups) (similar to (Th. 12)) — ’ 14 For the meaning of 'v' or 'or' in deontic logic or imperative logic, see§ 31. 24 (Th. 18) |- F3 D 0(3 :32) (If _a_ is forbidden, then a-only-if—b is obligatory). (Th. 19) (- Ob: OE :12) (If l_3_ is obligatory, then _a_-only-if-jg is obligatory). 3 3. THE PARADOXES OF "DERIVED OBLIGATION" In [1951a] and [1951b], von Wright prOposed to formulate in system vW another moral concept which seems important and far- reaching in a moral discussion, namely the concept of moral commitment or derived obligation. According to him, the concept of doing something that commits us to do something else could be analyzed into the concept of Obligation and that of a conditional-act. He made the analysis in the following way: (3. 1) Doing one act _a_ commits us to do another act _b if and only if the conditional-act _a_ D b is obligatory. Thus moral commitment is defined in terms of obligation and condi- tional acts. For example, if making a promise commits us to keep it then the conditional-act if-promise-making-then-promise-keeping is obligatory. This analysis, at first sight, seems natural and sound enough. For it says that it is obligatory that either a promise is not made or else it is kept, thus answering very closely to our intuitive notion of a moral commitment. However, if we read this notion of commitment uniformly into the theorems of system vW, then some strange and even counter-intuitive results appear. 25 26 Consider the following two theorems: (Th. 18) 1.__ Faaogzih) (Th.19) (- O_h_:>O(_a_:>1_o_) (Th. 18) says, without going into the concept of commitment, that if the act _a is forbidden, then it is obligatory that either a is not done or _b_ is done. And (Th. 19) says, likewise, that if b is obligatory, then it is obligatory either not to do a or to do _b. Both theorems, under this interpretation, appear to be harmless and indeed plausible and are thus welcomed by our intuition. But once we read 'O(_a_ Db)‘ as "doing 3, commits us to do _b_" into them, then these two theorems become, reSpectively, (3. 2) If a is forbidden, then doing it commits us to do any act _13. and (3. 3) If h is obligatory, then doing any act _a commits us to do b. That is to say, doing a forbidden act commits us to do any arbitrary act, and any arbitrary act commits us to do an obligatory act. These statements, e3pecially the first one, seem counter-intuitive or "paradoxical". For, according to (Th. 18), breaking a promise (presumably a forbidden act) commits us, among other things, to murder (of course, also commits us not to murder). This certainly 27 worries us very much. (Th. 18) and (Th. 19) have thus been called the "paradoxes" of derived obligation or the paradoxes of commitment. G. H. Hughes suggests that we formalize "doing 5;: commits us to do b" not as "O(§._ D _b_)' but as '_a D O_b_'. 2 Of course, this formula, as it stands, is not well-formed in von Wright's original version of vW. However, we can reformulate the formation rules of it to accommodate Hughes' formula. Under this new formulation, the first paradox dis- appears, for 'F_a :> (a D Ob)’ is no longer a theorem in this widened system vW+. 3 But the second one abides, because 'Oh D (_a_ 3 Oh)' is simply a theorem of propositional logic. 1 These two theorems, which resemble the so-called paradoxes of strict implication in alethic modal logic, were first pointed out by Prior [1954]. 2See Prior [1962], p. 224. We may construct a mixed truth table to show that 'Fa D (a D Ob)‘, that is, '~P§_ D Q I) ~P~h)', or '~[P(.a 8: 12) v P(_a8:~h)] D {a 3 ~ [PE 8:~_1_)_) v P(~g 8:~l)_)]}', is not a deontic tautology in system v . H3882) Ha8:~h) Hva&~b) a ~[P(a_&h) v Pta&~_b.)13 {aD~[P(s&~l>)vH~a&~b)ll HHHHHHHGMHMHHHMC‘P HHC‘FHHHWHHHWHHHflfi HHHHFPHHWHHHMWRFPW HHHHMHHHfifirt-fififififi 28 Von Wright himself made a more dramatic proposal. He thought that just as modal logic is inadequate to formalize the con- cept of entailment, the formalization of the concept of moral commit- ment cannot be accomplished within system vW. 4 He thus proposed a new system of deontic logic in which the concept of permission is relativized in the following way. We now Speak not of some act's being permissible or not, but rather, of its being permissible or not under certain condition. We let ‘PQ, _<_:_)' mean that act a is permissible under condition 5;. This is a relative notion of permission, or as it is sometimes called, conditional permission. And 'P‘ is used here as a binary deontic predicate rather than a singulary one as we saw in system vW. Other deontic concepts, 1. e. , obligation, forbiddance and indifference can be similarly relativized. The relation among them can be depicted by definitions strictly analogous to (D2. 1) -- (D2. 3) in last section, namely (D3.1) "O(o, 8)1 =D I'~P(~c1r. is)" f '1 _ r ‘1 (D3..2) rF(a,6) -Df ~P(a.6) (D3. 3) "I(a, 15)” :Df "(P(a, 6) 8: P(~a. 6))" This deontic system of conditional permission of von Wright will be called 'CvW'. 5 4See von Wright [1951a], p. 9, and [1956], p. 509. There are other deontic systems based on conditional per- mission. For example, the system develOped in Rescher [1958]. 29 The following are two additional axioms of CvW given by Von Wright himself:6 (113.2) Ptasrh. g) a Pie. 2) at P02. 2) Von Wright made the following proposal for the formulation of the concept of commitment. He said that a necessary condition for saying that doing an act commits us to do another act is that the latter is obligatory under the condition that the former is done. 7 That is to say, if doing 3 commits us to do _b_, then O(_a, h) is the case. Now, since the following two formulas, as he claimed, are not theorems of CvW8 (3~4) Fe. 9mg) D 0(2. _a_) (3. 5) 0(2. 2V~_c_) :> O(_a. h) the paradoxes are thus avoided in the system CvW of conditional per- mission. Of course, the question still remains whether we are able to avoid these paradoxes without going into the relativization of deontic concepts or whether similar paradoxes arise within CvW. 6Von Wright [1956], p. 509. He had not given a complete list of axioms for system CvW. 7Von Wright, ibid. We are not in a position to prove von Wright's claim, because the complete primitive basis of system CvW was not given. 30 It may be noted in passing that the notion of "absolute per- mission", that is, the notion of permission formalized in system vW, can be reformulated in terms of the conditional permission of system CvW. For we may write 'P(§, gv~g)' to stand for 'Pa' where gv~g_ may be called a tautologous condition, that is, a condition which is ever-present. In other words, to say that something is absolutely permissible is to say that it is permissible under any condition what- soever . S 4. CONTRARY-TO-DUTY IMPERATIVES AND CHISHOLM'S DILEMMA Deontic logic, as we pointed out earlier, is meant to capture certain deontic concepts and formalize them in a systematic way. The success of a certain deontic logician may thus be judged, at least partially, by how well his system will accommodate these concepts. For example, it is reasonable for us to demand that a deontic logician must try to develop a deontic system which will accommodate as thoroughly and as successfully as possible all the concepts which we intuitively conceive as deontic. If a system of deontic logic fails to incorporate a certain deontic concept satisfactorily, we may think it defective in that respect. One example is the case of commitment we have mentioned earlier in conncection with system vW in which von Wright's original prOposal of formulating the notion of commit- ment leads to paradoxical, and hence unsatisfactory, results. The following is another example. Roderick M. Chisholm [1963a] first pointed out that deontic logics formulated in the manner of system vW suffer another drawback. This fly in the ointment, which can also be found in systems 0T* -— 055* and Anderson's 0M -—- 0M", 1 can be described as follows. In system 1see as 6-12. 31 32 vW and other similar systems we cannot formulate the concept which Chisholm called the contrary-_tngu—ty imperative adequately without getting into undesirable results. To see the problem, let us proceed as follows. Fir st of all, it is easily observed that human beings are not morally perfect. A man violates from time to time certain moral codes, or neglects consciously or unconsciously certain ethical duties. Hence, our moral rules may often be couched in such a way that will both allow and dictate a reparative course of action. For instance, one of our moral rules, when rendered explicit, may have the following schema: (4. 1) One ought to do so and so; but if one, for some reason or other, fails to do it, one ought, by all means, to do such and such. This schema may even be generalized to read: (4. 2) One ought to do so and so; but if one, for some reason or other, fails to do it; one ought, by all means, to do such and such; but if one again, for some reason or other, fails to do it, one ought, by all means, to do . . . ; but if one again, for some reason or other, fails to do it, one ought, by all means, to do thus and thus. ZChisholm, in [1963a], p. 33. put forward the following fOI‘mulation: "You ought to do _a, but if you do not do a, then you must, by all means, do 2. " 33 However, for some practical reasons which are easily conceived, this chain of restorative clauses will not extend very long. It must soon stop somewhere. But exactly at what point our moral rules cease to prescribe any further reparative course Of action seems largely to depend upon what kind of morality is involved and what situation we are in. But we are not going to discuss these issues any further as they will certainly lead us too far away from our principal concern of this section. The desirability, if not indiSpensability, of including reparative clauses in our moral rules is easily understandable if not immediately obvious. For one thing, human beings are far from being morally incorruptible, as we mentioned above. One may, at sometime or other, do something erng in full awareness. If our moral rules admit no restitution of any kind, they leave no room for a man to make a compensation for his erng—doing. This type of unreparable morality may sometimes even drive a man who has done something wrong into a corner and make him lead an evil life henceforth. Another reason for welcoming the type of morality that has room for reparation is this. Most of our moral rules, unlike the jus scrigtum, are not explicitly put down. Quite Often, if not as a rule, we have to work on a trial-and-error basis in order to see whether or not our course of action complies with the vaguely conceived moral rules. In short, to act morally requires not only the will to be moral but also the intelligence and insight. And it is needless to say that to err is 34 human. Therefore, it seems rather unreasonable to advocate an absolute morality which allows no violations and offers no chance of restitution of any kind. These are, then, some of the reasons for the admittance of the reparative course of action in our moral life expressed by contrary- to-duty imperatives. 3 Hence, it is no wonder that we want to see a deontic system accommodate this concept well and to our expectation. But, as we mentioned earlier, Chisholm has shown that system vW and other similar systems fail in this reSpect. To see this, let us use Chisholm's example. Let us suppose that, according to our moral rules, it is obliga- tory for a man to go to the assistance of. his neighbors, and it is also obligatory that if he goes he tells them he is coming. Now, suppose further that the moral rules also prescribe that if he does not go, then he is forbidden to tell them that he is coming (i. e. , that it is obligatory that he does not tell them that he is coming), and, furthermore, let it be the case that the man, at variance with his duty, does not go. In order to see how this situation introduces a difficulty into a deontic system such as vW or vW+, let us express the foregoing assumptions in terms of the language of vW+. Let '_a_' denote the act of going to his neighbor's assistance, 'b‘ denote that of telling them he is coming. 3There are other reasons for reparative morality. For instance, a man may not be in a position adequate for the fulfillment of his duty. This relates itself to the question whether "ought" always implies "can" in every sense of this word. We shall come across this question later. 4Ibid., pp. 34—35. 35 Then we have the following: (4. 3) O_a (_a_ is obligatory) (4. 4) O(a 31;) (It is Obligatory that if—a-then-b) (4. 5) ~51 D O~b_ (If not—a, then it is obligatory not-_b_) and (4.6) ~§ (Not-_a_, or a is not done) It is known that formulas like (4. 5) are not well-formed in system vW. Hence, what it purports to describe cannot be expressed in system vW. That is to say, system vW cannot handle this particular kind of state of affairs which (4. 5) describes from the very beginning. However, it has also been indicated that this system can be reformulated into system vW+ which will admit formulas such as (4. 5) as well-formed. Let us assume that we are talking about system vW+. In system vW+, it can be easily shown that (4.7) Oa8:0(9,:>_13). 30h 5 That is, (Th. 12) of system vW, remains a theorem. And since modus ponens is supposed to be desirable rule of inference in ordinary deontic systems, we can infer from (4. 3), (4.4) and (4. 7) the following (4. 8) 0h (b is obligatory) 5Von Wright's truth-tabular method suffices to show that it is a theorem. 36 But from (4. 5) and (4. 6) we have (4. 9) O~b (Not-b is obligatory) Hence we have, by rule of adjunction, the following: (4. 10) 011 8: 0~1_)_ (_b_ is obligatory and not-_b_ is obligatory) However, the following formula is provable in vW+, indeed it is a theorem, i. e. , (Th. 7), Of vW; (4. ll) ~(O_b_ 8: 0~b_) (It is not the case that _b_ is obligatory and not-b is Obligatory) We have a patent contradiction. We have, therefore, the following dilemma. Either the system, as in the case of vW, cannot formulate the concept of contrary-to-duty imperative, or else, it contains a contradiction. We shall call it Chisholm's dilemma. ‘H(. F. I '3. § 5. DEONTIC LOGIC AND MODAL LOGICS In von Wright‘s original treatment ([1951a] and [1951b]), deontic concepts such as "obligation", "permission", "forbiddance", and "indifference" are said to be modal concepts quite on a par with alethic concepts like "necessity", "possibility"; epistemic concepts "verification", "falsification", "undecidedness" and existential con- cepts "universality", "existence" and "emptiness". Indeed he has pointed out from time to time the similarity and dissimilarity among these four different groups of modal concepts. For instance, com- paring deontic concepts with alethic concepts we find the following striking resemblance. We observe, on the one hand, that what is necessary is possible and what is impossible is necessarily not the case; we have, on the other hand, what is obligatory is permissible, and what is not permissible is obligatory not to do. However, this happy similarity cannot be pushed very far. For instance, although it is commonly held that what is necessary is the case, we hardly need to point out the falsity that what is Obligatory is done. It is because of this not very pervasive resemblance that we find you Wright saying that "there are essential similarities but also characteristic differences between the various groups of modalities. They all deserve, therefore, 37 38 a special treatment. "1 Indeed he treated alethic modalities, epistemic modalities and deontic modalities separately and each to a fairly great extent in [1951a] and [1951b]. Later when he made further contributions to deontic logic, he conceived it as a quite independent discipline without making further efforts to bring other branches of "modal logics" into considera- tion. However, he did look into the possibility that deontic logic might find its basis on another theory, e. g. , on a certain type of tense logic. But other logicians and philOSOpherS are evidently deeply impressed by the above-mentioned similarity between deontic con- cepts and alethic concepts. They think it deserves far more than our passing attention. They reason that if we reinterpret certain Operators and variables in alethic modal logic in such a way that a well-formed formula in this system becomes a well-formed formula of deontic logic, and then take away from the alethic modal logic certain theorems which do not hold in deontic logic, then what we obtain is a system of deontic logic based upon alethic modal logic. So far much effort has been made in this direction, and we have now in the literature quite a few systems of deontic logic built up in this way. Among the logicians who have made contributions in this reSpect we find the well-known 1Von Wright [1951a], p. 1. 2See, for example, von Wright [1965b] and [1966]; also Aqvist [1966]. Compare with Halldén [1951] and Aqvist [1963c] . Cf. 8 l4. 39 names: Anderson, Castafieda, Fisher, Fitch, Kripke, Lemon, and Prior. In what follows, we shall try to make a preliminary remark on how to construct a system of deontic logic in the above-mentioned fashion. In the next few sections three systems of deontic logic OT*, 054* and 055* based, reSpectively, on modal logics T, 54 and 55 will be examined. Comparison will be made between these systems and von Wright's vW or, rather, a Special version vW* of vW, Fisher- Aqvist's system FA andAnderson's systems OM—OM". While we are making a survey of these systems we shall remind ourselves of those deontic problems we have so far mentioned and indicate some further problems. First of all, let us suppose that we have a system of standard propositional logic, say Church's system P2.3 We will call it "PL". A (propositional) modal logic may then be built up by adding to PL certain vocabulary, formation rules, axioms and/or rules of inference. For instance, the following primitive basis depicts a modal logic which is commonly called system T. 35ee Church [1956], p. 119ff, or Massey [1969]. part 11. 4See Feys [1937-1938]. This system is deductively equivalent to, or has the same theorems as, von Wright's system M in [1951b]. This result is proved by Sobocinski [.1953]. 40 I. Vocabulary 1) An infinite supply Of propositional variables: 'p', 'q', 'r', I I I I I I I I P1 9 ql 9 r1 3 It 3 ii) Two singulary, sentential connective: '~', '0'. iii) A binary sentential connective: '3'. iv) Two grouping indicators: '[ ', '] ' . II. Formation rules i) A propositional variable standing alone is a wff (well-formed formula). ii) If A and B are wffs, so are r~A1, r[A I) B] and rDA:1 iii) Nothing else is a wff. III. Rules of inference (R l ) Sub stitution (R2) modus ponens. (R3) Necessitation. Other sentential connectives including the modal ones are defined in the usual way. For instance, (D5. 1) '<>A" =Df r~El~A1 Again, we will follow the common practice in the omission of grouping indicators, and later the use of parentheses as poorly drawn brackets. 41 IV. Axioms (A1) t-pD [q3p] (A2) l—[p3[qu]]D[[qu]D[pDr]] (A3) l-[~q 3 ~p131p 3 9] (A4) +— DLp sq] 2 [Up 2 DqJ (A5) I— DpD P 5 It is well known that by adding (A8) l—DP 3C] D p to the above primitive basis for system T, we obtain a system called 54. And if we add instead the following axiom (A9) I— ~Dp D D~Dp then, the result is a system known as 55. Let us denote the primitive basis of system T as {(Rl) - (R3), (A1) - (A5)} leaving its vocabulary and formation rules understood. By the same token, {(Rl) - (R3), (Al) - (A5), (A8)} is the primitive basis of S4, and so on. Due to the similarity between our intuitive concept of necessity and that of obligation on the one hand, and between our concept of possibility and that of permission on the other, it is quite natural that people try to reinterpret the two alethic modalities ' Up' and 'Op‘ as "P is obligatory" and "p is permissible", reSpectively. Indeed, this deontic reinterpretation of alethic modalities seems so interesting 5(Al) - (A3) for propositional logic is due to Lukasiewicz. See Lukasiewicz and Tarski [1930], p. 35 (note 9). This is a simpli- fication of Frege's six axioms in Begriffsschrift, 1879. 42 and far-reaching that it has thus far received much attention from logicians. A small example may help indicate our preliminary con- fidence in this reinterpretation. Suppose that we take (D5. 1) and reinterpret accordingly the two modalities it contains. Then we get a deontic counterpart I" '1 l' 'I (D5.2) PA st ~O~A which is certainly welcomed by our intuition. Strictly speaking, however, (D5. 2) cannot be said to express the same idea of defining one deontic modality in terms of another as what is expressed by, say (D2. 1) - (D2. 3) of 3 2. The deontic modali- ties which (D5. 2) and (D2. 1) talk about are really two different kinds. In (D2. 1), a deontic modality under consideration is an expression consisting of a deontic predicate followed by a name for act, while in (D5. 2), a deontic modality, on the other hand, is an expression com- posed of a deontic Operator but followed rather by a name for a prop- osition. In general, if we try to reinterpret the primitive basis of a modal logic, say system T, in the manner stated above, we are con- structing a deontic system of a different type from the one we discussed in S 2, namely von Wright's system vW. In vW, as we recall, a deontic predicate applies to an act-name. But now in a deontically reinterpreted modal system, instead of act-names we have proposition-names which are prOper Operands of the deontic Operators. Thus what are now said to be obligatory, permissible and the like are no longer acts but as»: h 4.» .fd ! a a! .Ii nib 111'- n,- no «A. .Nln 43 propositions. This deviation from von Wright's original conception of deontic modality need not give us too much trouble. For we can readily understand a deontic formula like (5. 1) Op as meaning (5. 2) The realization Of p is Obligatory. or, perhaps more conventionally (5. 3) To bring it about that p is obligatory. where 'p' denotes a prOposition, or if we like, a state of affairs. This type Of deontic logic is much favored in the recent development. Von Wright has himself adopted this general line of practice, too, although he does it in a Slightly more Specific form. This renewed conception of deontic modality is not immune to every difficulty. But, for the time being, we shall simply assume its work- ability until later in the next chapter. When we reinterpret the modal system T, 54 and 55 deontically in the manner explained above, it involves the change of 'D' to '0' throughout the axioms and rules of inference in the foregoing systems. Consequently, we have: (A1). - (A3) (same) (Ad4) 1'0“) 3 <1) 3 (Q3 30q) 6He lets the individual variables denote. so-called "generic states of affairs". See, for example, [1963b], [1964] and [1967b]. 44 (Ad5) )- OP 3 p (Ad8) l-Op 300p (Ad9) l—~OPDO~Op (R1) - (R2) (same) (Rd3) Deontic necessitation: from A we may infer r OA-1 . Now, corresponding to modal system T, 54 and 55, we may define three deontic systems OT, 054 and 055 which have, respectively, the following primitive basis: {(Al) - (A3), (Ad4) — (Ad5), (R1) - (R2), (Rd3)}, {(Al) - (A3), (Ad4) — (Ad5), (Ad8), (R1) - (R2), (Rd3)},and {(Al) - (A3), (Ad4) - (Ad5), (Ad9), (R1) - (R2), (Rd3)}. However, these deontic systems are intolerably counter-intui- tive (from our point of view, of course) because of the presence of (Ad5) which, when rendered into our everyday language, reads (5. 4) If something ought to be done, then it is done. This certainly does not describe the moral phenomena we daily experience, because in our world it seems quite obvious that the following is true: (5. 5) 1 There is something which ought to be done but, as a matter of fact, is not done. Note that (5. 4) and (5. 5), when carefully reformulated, are directly in contradiction to each other. Hence, any deontic system which includes (Ad5) and thus asserts (5. 4) must be rejected as totally unnatural and inadequate to our world. Indeed, any system of deontic 1 1 -1 9% ‘1 -( U ac 1) 'u 45 logic containing (Ad5) may be objected from another point of view. Suppose that (Ad5) can be asserted, then by (R1) we have (5.6) O~p3~p This, by a usual definition, i. e. , (D5. 2), becomes (5. 8) p 3 Pp That is, (5. 9) If something is done, then it is permissible. (5. 4) and (5. 8) then entail that no duties are neglected and nothing forbidden is done. A world with this prOperty may be called a morally perfect world because in such a world there is no sin but there is every virtue provided that to) sin is forbidden and to be virtuous is obligatory. But in a world like this, peoP1e will not have much interest in deontic logic because of its triviality. For instance, OT, 054 and 055 are nothing but Special interpretations (or models) of T, 54 and 55, respectively. Therefore, deontic concepts reveal no Special charac- teristics of themselves. They are only COpy images of alethic modal concepts. In this case, deontic laws coincide with alethic modal laws. It may be noted in passing that although we call a world in which (5. 4) and (5. 9) hold a morally perfect world, this should not lead us to think that in such a world it is also true that 7It may be the case that, under some moral rules, not every virtue is obligatory. Cf. Chisholm [1963b]. 46 (5. 10) A non-Obligatory thing is not done. To assert (5.10) is to assert or, equivalently, (Ale) p 30p And since we know that (Ale) cannot be proved in CT, 054 or 055, 8 we have to add it as an axiom. But by adding this axiom to these sys- tems, the following becomes provable. (5.12) Op sp A world having this prOperty may be called a strictly virtuous world in that not only is every virtue present but also every non-virtue is absent provided that, according to the morality of this world, it is obligatory to live virtuously. Note also that the morality of such a world may be thought of as logically trivial. Because when (5. 12) is provable in a deontic system, then that System simply collapses into the usual propositional logic. Thus, the deontic laws strictly coincide with the propositional logical laws. This, again, makes the construction Of deontic logic superfluous. Since (Ad5) is unwelcome in deontic logic as we have just shown, let us delete it from the primitive bases of OT, 054 and 055. 8This can be Shown by any of the decision procedures known for T, 54 and 55. 47 Instead we add to each of these systems the following two weaker axioms: (Ad6) )- Op 3 ~Ovp (Ad?) +- OEOp 3p] The resulting systems will be called OT*, 054* and 055*“, reSpec- tively. And we define a system 055* as the system which results from adding (Ad8), i. e. , 'Op DOOp' , to 055*'. 10 Axiom (Ad6) is a highly desirable one, because it is, by (D5. 2), equivalent to (5. l3) 0p 3 Pp That is, (5. 14) If something is obligatory, then it is permissible. AS to (Ad7), there may be someone who will frown on it mainly because of the iteration of deontic modality. It is not clear, for example, what we mean when we say that it is obligatory that it is 9Similar axiom-set can be found, for example, in Hanson [1965] . 10The reason for defining 055* in this manner rather than calling 055*' 055* is to preserve the nice prOperty that 055* contains 054* just as 054* contains OT*. Another way of obtaining 055* would be the addition of the following axiom to 055*' (Adll) 1-0 [P DQJ] However, to do this would mean to explicitly postulate a morally excessive rigorism, so to Speak, in that what is done ought to be that which ought to be done. This leaves no room for morally indifferent acts. iv an 5.... .46 ..v» 1? I u. kgl n ’4. i (H 'n\ , 1F, ‘ . .J‘ 48 l obligatory to do such and such. 1 However, there are logicians, notably Prior, who contend that (Ad7) is harmless, or indeed welcome, 12 in deontic logic. For it reads intuitively (5. 15) It ought to be the case that what ought to be done be done. which seems not only innocent but also desirable. Another reason against the acceptance of (Ad7) is eXpressed by Feys. 13 He observes that in some systems of modal logic, particu- larly Lewis‘ modal systems 51 - 55, (A5), i. e. 'Clp D p‘ can be proved from the alethic counterpart of (Ad7), namely, from (Ad?) 1- DEDP D p] But the deontic counterpart of (A5), that is (Ad5), is to be rejected. Hence, Feys suggests that we delete, among other things, (A7) or its deductive equivalent from Lewis' systems and replace it with certain other axioms in order to obtain intuitively satisfactory deontic logics. It is true that from (A7) we can prove (A5) in any of Lewis' systems provided that we use the following two rules: 1It may be recalled that deontic formulas containing interated deontic modalities are not well-formed in system vW. 1ZSee, for example, Prior [1956], p. 86f. 3In "EXpression modale du « devoir-Stre». " An abstract from the Amsterdam Meeting of the Association for Symbolic Logic. Journal of Symbolic Logic, vol. 20, 1955, pp. 91-92. 49 (R6) Strict detachment: From A and T] [A3 B]: to infer B. (R7) Replacement of strictly equivalents. As Prior notes the proof can be carried out in any of Lewis' systems. Since system T is known to contain 52, 15 the proof can also be obtained in system T. However, some logicians, Prior is one of them, contend that it is unsatisfactory to exclude (Ad7) from deontic logic. 16 They lay the blame of the deducibility of (A5) from (A7) on (R6) and (R7) rather than on (A7). Indeed, it is easily seen that the deontic counterparts of these two rules—call them (Rd6) and (Rd7), reSpectively—are not sound rules in deontic logic. For from so and so is the case and it is obligatory that if so and so is the case then such and such is the case, we cannot correctly infer that such and such is the case. Likewise, from it ought to be the case that so and so is the case if and only if such and such is the case, there is no warranty that 'so and so' and 'such' and such are then interchangeable in any deontic formula. Prior offhand rejects these two rules. As to the latter rule, i. e. , (Rd7), he makes the following remarks: "There seems to be the same intuitive Objectionableness about the rule permitting substitution of strict equivalents, 14For the proof, see Prior, ibid. , p. 87. 15See, for example, Feys [1965], p. 124. 6Prior, ibid., p. 87. Also, Anderson: "Review of A. N. Prior's A Note on the Logic of Obligation and Feys' Reply", Journal of Symbolic Logic, vol. 21, 1956, p. 379. 50 when this is interpreted deontically, that is, when it is taken to mean that if or Bligh—ti? imply and be implied by B, then a and B _a_r_e_ interchangeable in any formula. This is optimism, too, is it not? I cannot think immediately of 17 a counter-example to it. . " The following simple example may be of the same kind that he has in mind. Suppose, according to a set of moral rules, that it is the ca so that (5. 16) It ought to be the case that if one violates amoral rule, then one is punished. (5. 17) It ought to be the case that one is punished only if one violates a moral rule. and (5. 18) It is permissible that one be punished. From (5. l6) - (5. 18) one may not be able to infer that (5. 19) It is permissible that one violates a moral rule. 17Prior, ibid., p. 87. His italics. 3 o. DEONTIC LOGICS AS "SUBSYSTEMS" OF MODAL LOGICS For the sake of clarity, let us, first of all, summarize the primitive bases of three deontic systems we just arrived at, namely, OT*, 054* and 055*. Again, we shall omit the vocabulary and formation rules. The following rules of inference are laid down for all three systems: (R 1) Sub stitution. (R2) modus ponens . (R3) Deontic necessitation. For system 0T* we put down the following axioms: (Al) I— p D Equ] (A2) )- CPDEqull D EEPDqJDEPDrll (A3) )- [~q:>~p] 31p D q] (Ad4) )- OEPD q] 3 [0p 3 Oql (Ad6) )— Op D~0~p (Ad7) I— O[Op : p] If we add: (Ad8) )— Op 3 00p 51 52 we have system 054*. And if we add to system 054* the following (Ad9) )-~0p:>0~0p the result is system 055*. Other sentential connectives are defined as usual. And the following definitions are common to all three systems: (D81) (D82) (D83) 1" ‘I I' 1 PA "Df "0"A F' ‘1 _ I" '1 FA _Df PA 1" 1 _ r 71 1A ..Df PA 8: P~A Here t-.IA-' may be rendered as "A is (morally) indifferent". It may be pointed out that by adding (Ad5), that is, 'Op 3 p', to OT*. 054* and 055* we obtain CT, 054 and 055, reSpectively. This can be seen by showing that (Ad6) and (Ad7) are derivable in CT, 054, and 055. 1 This means that OT*, 054* and 055* are, 1The proof of (Ad7) is trivial, since we have (Ad5) and(Rd3). It is also straightforward to Show that (Ad6) is provable in those sys- tems. We may sketch the proof as follows: (1) (2) (3) (4) (5) (6) (7) (8) )— Op 3 p (Ad5) 1—[q:3~r]:>[r3~q] PL I— [O ~p D ~p] 1') [p3 ~O~p] Substitution (Henceforth: Sub.) (2) I— O~pD~p Sub. (1) (— p D~O~p (3) (4) modus ponens (hereafter: MP) l—[qu] DEIqu] D[p=>r1] PL +- [OP 3 p] D [[p D ~O~p1 3 [Op D~0~p]] Sub. (6) 1- Op D~O~p (1) (5) (7) Two uses of MP Note: in our proofs we Shall make free use of simultaneous substitu- tion and replacement of equivalents in propositional logic. They are, as well—known, derived rules on the basis [(Al) -- (A3), (R1) -- (R2)}. 53 reSpectively, subsystems of CT, 054 and 055. But CT, 054 and 055, as constructed in the last section, are exactly the same as T, 54 and 55, respectively, except where there is an '0' in the former we have a '['_'_]' in the latter. We shall say that the former are, reSpectively, the deontic variants of the latter, and the latter, the alethic variants of the former. The following chart shows the containing relation among some of the systems we have just mentioned: 5.5 2 5'4 —— 2 T 3. :. ;. I g E 0S5 2 OS'4 _2 OT I l '1” " 1’ 055* 2 044* ._ 2 __ OT* where 'Sl——2 — S'2 means that system 51 contains system 52, and 'Sr—z—S'Z means 51 contains and is contained by 52, i. e. , S1 and 52 are mutually containing or equivalent. 0n the other hand, we let 'Sl ----- 2 ----- 52' and 'S1 ----- = ----- S ' mean, reSpectively, the same as ’SI——2—52' and ‘51—— = —52' except that in the former 51 and 52 may contain different designs of symbols. For example, one is the deontic variant (which contains. '0') of the other which is an alethic system (that contains ‘0‘ instead). We shall next exhibit some of the theorems in our deontic systems. First of all, the theorems of system OT*. Needless to say, all theorems 54 of 0T* are also theorems of 054* , and all theorems of 054* are also theorems of 055*. The following are theorems which Show that"material equivalence" between certain O-Sentences, P-Sentences, F-sentences and I-SentenceS: (OT*l) (OT*2) (OT*3) (OT*4) )— Op 5 ~P~p (p is Obligatory if and only if not-p is not permissible) Pf. (l) )— p '=' ~~p PL (2) )- Op 5 ~~Op Sub. (1) (3) t- 0 ~~p '5 ~~O~~p Sub. (2) (4) r0p2~P~p (3). (Ddl). PL )- Op =2 F~p (p is obligatory if and only if not-p is forbidden) Pf. (1) f—Op (2) l—Op F~p (1), (D82) )— Pp 5 ~Fp (p is permissible if and only if p is not forbidden) Pf. (l) )— Pp :-_-~~Pp PL (2) i—Pp-—=~Fp (1). (Dd2) )— Pp E ~O~p (p is permissible if and only if not-p is not obligatory) Pf. Similarly, (Ddl) (OT*5) (OT*6) (OT*7) (OT*8) (OT*9) 55 )— Fp s ~Pp (p is forbidden if and only if p is not permissible) Pf. (OT*3), PL )— Fp E 0 ~ p (p is forbidden if and only if not-p is obligatory) Pf. (OT*4). (D82), PL )— 0 .0 p --_= ~Pp (Not-p is obligatory if and only if p is not permissible) Pf. (OT*5), (OT*6), PL [— P .0 p -=- ~Op (Not-p is permissible if and only if p is not obligatory) Pf. (OT*l ), PL I— F ~ p 5 ~P~p (Not-p is forbidden if and only if not-p is not permissible) Pf. (OT*S), PL From the above nine theorems, we may list the following twelve mutual transformations between different deontic sentences. We denote each of them by (T1), (T2), . .. , (T12) so that we can refer to them later in our proofs of other theorems. These transformations are: (T1) (T2) [OA'1 « ------ » r~P~A1 'LOA'« ------ » r‘P A] '-O~1":I «- ----- >> "EPA-1 (T4) (T5) (T6) (T7) (T8) (T9) (T10) (T11) (T12) (OT*IO) r~O~A «. ----- » rPA.’ '_PA1 « ------ » CPA EPAq « ----- » rFA" rP~A1 <4 ----- >> £F~A1 r~ 4.1114: ----- » [EVA] r’FA] « ------ »"O~1-‘:I ’LFA-I « ------ » 5.0.619 [FAA «. ----- » rCA" I14734.1? << ------ > r~OA-' )— Ip '5 . Pp 8: P~p (p is indifferent if and only if p is permissible and not-p is also permissible) (2) )- Ip 5. Pp 8: P ~p (l), (Dd3) The following theorems are those that Show the "material implication" between deontic sentences: (OT*ll) )- OpDPp (If p is obligatory, then p is permissible) Pf. (Ad6), (Ddl) There follow from (OT*ll) togehter with (T1) -—- (T12) the following five additional theorems: (OT*lZ) |.- O ~p D~Op (If not-p is obligatory, then p is not obligatory) 57 (OT*13) l- ~Pp :3 P~p (If p is not permissible, then not-p is permissible) (OT*14) )— Fp:>~0p (If p is forbidden, then p is not obligatory) (OT*15) )- Fp :3 ~F ~p (If p is forbidden, then not-p is not forbidden) (OT*16) )— F .0 p :3 Pp (If not-p is forbidden, then p is permissible) It is easily seen that the above Sixteen theorems are common to every system of deontic logic in which (Ddl) -—- (Dd3) are incor- porated and in which classical propositional logic is presupposed. (OT*17) l— 0(p V~p) (Lp-or-not-p is obligatory) Pf. (l) |- p V~p PL (2) 1-0(P V~P) (1). (Rd3) In general, we can prove (6. 4) [- O__t_ where '_t_' stands for a tautology. That is to say, to realize a tautology is obligatory. (OT*18) (— F(p 8: ~p) (p-and-not-p is forbidden) Pf. (l) )- O(p V~p) (OT*17) (2) )- 0~(P & ~p) (1). PL (3) )- F(P & ~P) (2). (T9) 58 Likewise, we have, in general, the following as a theorem: (6. 5) l- F_f where '1’ stands for a contradiction. That is to say, to actualize a contradiction is forbidden. Some philosophers may frown at (6. 4) and (6.5) and hence at (OT*l7) and (OT*18). Because these theorems imply that our moral code makes a prescription, among other things, concerning the act that people logically cannot help doing, 1. e. , the "tautologous act" and the act that people logically cannot do. i. e. , the "contradictory act". It seems more natural to have our moral code say nothing on these "logical acts", because they are not intentional acts, and someone may claim that only intentional acts are subject to moral judgment. The same point may be viewed, perhaps more convincingly, from another angle. That is, to consider the corresponding examples in imperative logic. Suppose a person is requested to do a contradictory act, he will never be able to fulfill it no matter how hard he tries; and if he is ordered to carry out a tautologous act, he needs to do nothing and the order is automatically executed. Now, we may ask: IS the request a genuine request, and the order a real one? We shall leave this question aside without making any attempt to answer it. Meanwhile, we might think of the unnaturalness of (OT*17) and (OT*18) as one of the expenses wepay for our deontic systems. 59 (OT*19) below is not new to us. It seems indeed a truism in deontic logic. (OT*19) )— ~(0p 8: O~p) (It is not the case that p is obligatory and not-p is also obligatory) Pf. (1) [— 0p 3 ~O ~p (Ad6) (2) r ~0p v ~0~p (1). PL (3) )-~(0P & O~P) (2). PL However, this theorem reminds us of Chisholm's dilemma which we dealt with earlier. That is, this dilemma again threatens us in sys- tems OT*, 054* and 055* just as it does in system vW or vW+. From (OT*19) and (T1) — (T12) the following two theorems come as immediate consequences: (OT*20) )- Pp v P ~p (Either p is permissible or not-p is permis sible) (OT*21) [- ~(0p 8: Fp) (It is not the case that p is both obligator y and forbidden) Let us prove, at this point, a derived rule to the following effect: (Rd4) If )- rA3B1 , then (— t.0A 3 0B1 Pf. (1) [- ADB Assumption (2) )- 0(ADB) (1). (Rd3) (3) (- 0(p3q) 2). Op 3 Oq (Ad4) 60 (4) )— O(A DB) :3. 0A 3 OB Sub. (3) (5) (— 0A ‘3 OB (2)(4), MP As we shall witness soon, this rule facilitates from time to time the proofs of many subsequent theorems. The following are theorems in which molecular names are involved. (OT*22) [- O(p 8: ~p) :3 Oq (If p-and-not-p is obligatory, then any q is obligatory) Pf. (l) )- p 8: ~p. Dq PL (2) )- O(p & ~p) :> Oq (1). (Rd4) (OT*23) )- Op 3 O(p v q) (If p is obligatory, then p-or-q is obligatory) Pf. Similar to that of (OT*22) [ Hereafter: Similar: (0T*22)] (OT*24) )— Pp :> P(p v q) (If p is permissible, then p-Or-q is permissible) Pf. (1) l- ~(p V q) 3 ~19 PL (2) 1- 0 ~(P V (1): O ~P (1): (Rd4) (3) l—~0~PD~0~(PVq) (2). PL (4) )- PP D P(P V <1) (3), (Ddl) (OT*25) )- Fp D F(p 8: q) (If p is forbidden, then p-and-q is forbidden) Pf. Similar: (OT*24) (OT*26) (OT*27) (OT*28) (OT*29) 61 )— F( p v q ) 3 Fp (If p-or-q is forbidden, then p is forbidden) Pf. (OT*24), PL, (T6) )— O(p 8: q) 5. Op 8: Oq (p-and-qis obligatory if and only if p is obligatory and q is obligatory) Pf. (1) |-—p8:q.3p PL (2) )— O(p & q) 3 0p (1). (Rd4) (3) )— O(p 8: q) : Oq Similarly (4) )— 0(p 8: q) 3. 0p & Oq (2)(3). PL (5) Fp3(q:.p&q) PL (6) )- Op 30(q3- p & q) (5). (Rd4) (7) l- 0(p 3 q) 3. 0p 3 Oq (Ad4) (8) )— O(qD. p 8: q) :5. Oq DO(p 8: q) Sub. (7) (9) )- OP 3 (0q30(p & q)) (6K8). PL (10) (— Op 8: Oq. 3 O(p 8: q) (9), PL (11) l— 0(p 8: q) a Op & Oq (4)00). PL [- P(p v q) 5. Pp v Pq (p-Or-q is permissible if and only if p is permissible or q is permissible) Pf. (OT*27), PL, (T3) )— F(p v q) a . Fp 8: Fq (p-or-q is forbidden if and only if p is forbidden and q is forbidden) Pf. (OT*28), PL, (T6) 62 (OT*30) (- O(p v q) 3. Pp v Pq (If p-or-q is obligatory, then either p is permissible or q is permissible) Pf. (1) )- 0p :3 Pp (OT*11) (2) )- 0(p v q) : P(p v q) Sub. (1) (3) (- P(p v q) 5. Pp v Pq (OT*28) (4) 1- 00.9 V q) 3- PP V Pq (2)(3). PL (OT*31) )— P(p 8: q) 3. Pp 8: Pq (p-and-q is permissible only if p is permissible and q is permissible) Pf. (1) |-— Pp D P(p v q) (OT*24) (2) +- P(p & q) :1 P((p & q) V (p &~q)) Sub. (1) (3) (- P(p 8: q) 3 Pp (2), PL (4) [- P(p 8: q) :3 Pq Similarly (5) l- P(P 8! q) 3. PP & Pq (3X4). PL (OT*32) )— 0p v Oq. D O(p v q) (If either p is obligatory or q is obligatory, then p-or-q is obligatory) Pf. (OT*23), PL (OT*33) I- Fp 3 O(p 3 q) (If p is forbidden, then if-p-then-q is obligatory) Pf. PL, (Rd4), (T9) (OT*34) )- Oq D O(p :3 q) (If q is obligatory, then if-p-then-q is obligatory) Pf. PL, (Rd4) 63 These two theorems, according to the reading quoted above, are quite innocuous. (OT*33) says that if to bring it about that p is forbidden, then either-not-to-bring-it-about-that-p-or-to-bring-it- about-that-q is obligatory. 0r, roughly Speaking, if doing one thing is forbidden, then it is obligatory either not to do it or to do something else. And this seems quite acceptable to our intuition. Likewise, (OT*34) may be similarly rendered. And the result is again unobjec- tionable. However, if we use rO(A D B]. to express commitment as we saw in § 3, then the paradoxical results are immediately seen. Actually, (OT*33) and (OT*34) are exactly the same, reSpectively, as (Th. 18) and (Th. 19) of system vW except the difference in individual variables. And the latter theorems have been called the paradoxes of derived obligation. That means that our systems do not escape the defects we found in in system vW in association with these paradoxes. (OT*35) )- Op 8: O(qu). 3 Oq (If p is obligatory and if-p-then-q is obligatory, then q is obligatory) Pf. (l) I-(p8:.p:>q):q PL (2) )- O(p 8: (qu)) 2 Oq (1), (R84) (3) )- O(p 8: q) 5. Op 8: Oq (OT*27) (4) 1— 0(p 8: (p a q)) :2, 0p 8: O(p : q) Sub. (3) (5) )— Op 8: O(p 3 q). DOq (2)(4), PL (OT*36) )- Pp 8: O(p Dq). D Pq (If p is permissible and if-p-then-q is obligatory, then q is permissible) Pf. PL, (R84), (OT*30), (T3) 64 (OT*37) (— Fq 8: O(p 3 q). 3 Fp (If q is forbidden and if-p-then-q is obligatory, then p is forbidden) Pf. (OT*36), PL, (T6) (OT*38) )- (Fq 8: Fr) 8: O(p Z) (q v r)). 3 Fp (If q is forbidden and r is forbidden and if-p-then-(q-or-r) is obliga- tory, then p is forbidden) Pf. (OT*37), PL, (OT*29) (OT*39) )- ~(O(p v q) 8: (Fp 8: Fq)) (It is not the case that p-or-q is obligatory, yet p is forbidden and q is forbidden) Pf. (1) (— ~(Op 8: Fp) (OT*21) (2) (- ~(O(P V q) & F(P V q)) Sub- (1) (3) (— F(p v q) '=". Fp 8: Fq (OT*29) (4) )- ~(O(p V q) 8: (Fp 8: Fq)) (2)(3). PL (OT*40) (— 0P 8: 0((p 8: q) Dr). 3 O(q : r) (Similar to (OT*38)) Pf. (OT*35), PL It may be observed that many of the theorems listed here are variant theorems of system vW. They differ only in style of variables. Indeed, this coincidence is not accidental. We shall, in the next section, prove that every theorem of system vW is a variant theorem of system OT*. Of course, the converse is not true. Before closing this section, let us prove another derived rule of our system(s). 65 (R85) Ii (— "ADB", then (— r'PA: P13"I The proof is straightforward: Pi. (1) (— A213 Assumption (2) )—~B:~A (1), PL (3) (— 0~BDO~A (2), (R84) (4) )—~O~A3~O~B (3), PL (5) )— PA 3 PB (4), (T4) Again, this rule will be used on certain occasions to facilitate our proofs. § 7. SYSTEMS OT*. vW and FA To show that system vW of section 2 is a variant subsystem of OT*, we shall fir st axiomatize a variant system vW* of vW as follows . I. Vocabulary Same as vW except that we now have as individual variables I I I I I I I I I I I I I ' p, q, r, pl,q1, r1, p2,... whichrangeover prOpositions and that we now include not only 'P' but also '0', 'F', and 'I' as deontic predicates. II. Formation rules Similar to that of vW. Here we have not only P-sentences, but also 0-sentences, F-sentences, and I-sentences. III.Axioms (A1) (— PDIqu] (A2) )- IPDEqull D [[quJDIpDrj] (A3) )- [~qD~p1:>Ip=>q1 (A86) )- op D~O~p (OT*28) )— P(p v q) 5. Pp v Pq 66 67 IV. Rules of inference (R1 ) Substitution. (R2) modus ponens. (Rd6) P-extensionality: From '-A --_= B.1 we may infer I-PA :5 PB ‘1. This axiomatization is essentially Prior's restatement of principles listed by von Wright in [1951a] and [1951b], 1 except that we have here a different type of individual variables and that some parts of this systematization are not explicitly mentioned in Prior's formulation. It is readily seen that all the axioms of vW* are either axioms or theorems of OT*. Besides, the first two rules of inference are exactly the same as the first two rules of OT*. The only thing that remains to be seen is that (Rd6), which is the only thing new in this system, can be derived in OT*. But this is Obvious, because the rule is entailed by, and indeed is a weaker version of, (RdS) of OT*. It follows that every theorem of vW* is also a theorem of OT*. To show that the converse does not hold, we need only to present an example which iS a theorem of 0T* but not a theorem of vW*. As we know, trivially (Ad7), i. e. , '0(0p :> p)‘ is a theorem of OT*. Nevertheless, it is not a theorem, indeed not even a well-formed formula, of vW*. This concludes the proof that system vW* is a (proper) subsystem of OT*. lSee Prior [1962], p. 221. 68 Now, system vW* and system vW are otherwise the same except in two respects. First, these two systems have different individual variables as we have seen above. Secondly, in vW* we have '0', 'P', 'F' and 'I' as deontic predicates, while in vW we have only 'P'. (This is also true for system 0T*). It follows that what is a theorem of vW* might only be an abbreviation of a theorem of vW. For example, 'Op 3 Pp' is a theorem of vW*, but only '~P~p::> Pp' is a variant theorem of vW (or OT*). If we keep this latter difference in mind, and are willing to tolerate the discrepancy as we did above, we might as well say that system vW, like system vW*, is a variant (proper) subsystem of system OT*. But since this discrepancy is sometimes unnegligibly important as we shall see in section 12 we might want to say that vW, like vW*, is a variant quasi-subsystem of OT*, meaning that what is a theorem of the former may only be an abbreviation of a theorem of the latter, or vice versa (what is an abbreviation of a theorem of the former is a theorem of the latter). It has been mentioned that (OT*1) —- (OT*16) are theorems common to every deontic logic 0f the common type, namely, one in which we have (Ddl) - (Dd3), (Al) -- (A3) and (R1) -— (R2). And now we have shown that vW* is a proper subsystem of OT*. But these remarks are not to be taken as in the least suggesting that OT* is a system comprehensive enough to accommodate every deontic theorem one might expect, even if we agree to let the deontic logic be developed along the lines set out by von Wright in system vW. 69 In what follows we Shall reconstruct and examine a deontic system FA which was first prOposed by M. Fisher and later amended by L. Aqvist. 2 This system, like vW, takes the set of acts as the range of the individual variables. But, unlike the latter, it possesses the following new feature found neither in vW nor in OT*. 1. e. , FA is a three-valued system. This system, Similar to vW but unlike OT*, has a finite characteristic matrix. First, the primitive basis of system FA. 1. Vocabulary i) Deontic variables: 'a', 'b', 'c', 'a ' ii) Deontic connectives: 'N', 'K', 'A', 'C', 'E' iii) Sentential connectives: '~', '8:', 'v', '3', '2' iv) Deontic predicates: 'P‘, '0', 'F', 'I' II. Formation rules i) Every deontic variable is a deontic name. ii) If a and B are deontic names, so are rNafl , l”Kara." , rAaB“, rCaB-I and 'Eae‘. iii) Nothing else is a deontic name. iv) If a is a deontic name, then FPO? , r0011, 'Fa... and '10-. are wffs (atomic). v) If A and B are wffs, so are I17A" , r[A 8: B]: ([A V B? , '[A D B]. , and '[A :-: B]... (molecular). vi) Nothing else is a wff. 2Fisher [1961b] and Aqvist [1963b]. 70 . . . . . 3 The axiomatization of this system is avallable. However, since the value-table type of decision procedure is handy,4 we Shall follow this approach. III. Semantics Instead of putting down a set of rather lengthy semantical rules, we Shall present, as Fisher and Aqvist did, a set of matrices to capture the semantics of this system. We let '0 ', '1' and '2' stand for the deontic values "obligatoriness", "indifference" and "forbiddance", respectively. The set of matrices can then be given as follows: A) For deontic names B B or N K O 1 2 A 0 1 2 0 2 0 0 1 2 0 0 0 0 l 1 a l l 1 2 a{l 0 l 1 2 0 2 2 2 2 2 0 l 2 B B W A“ C | o 1 2 E o 1 2 0 0 1 2 0 0 1 Z a l 0 1 l a l 1 l 1 2 0 0 0 2 2 l 0 ibid. 3This has been worked out by Aqvist, ibid. See Appendix I. 4Fir st presented by Fisher, ibid. and revised by Aqvist, 71 B) For atomic wffs a Ca Pa F a Id 0 t t f f l f t f t 2 f f t f C) For molecular wffs The usual fundamental truth tables for ' ~', A deontic tautology of FA, or an FA-tautology, may be defined in the usual way, namely, that a wff is a tautology if it is true under every value assignment. Since the consistence and completeness of FA have been proved by Aqvist, we have the following result: A is a theorem of FA if and only if A is an FA-tautology. To serve as an illustration, we shall use the above matrices to Show that (Th. 18) of system vW is also a theorem of FA. In order to shorten the calculation, we follow Fisher in the employment of Quine's truth value analysis . (Th. 18) 1- F_a_ D OCER (If _a is forbidden, then if-g-then-b is obligatory) Pi. _azl Fl :JOc1_‘t_> f pOClE Similarly when a = 0, 2 5Ouine [1950], S 5, pp. 22ii. 72 Hence, (Th. 18) is an FA-tautology. By (7. 1) it is a theorem of FA. Likewise, we can Show that (Th. 19) and (Th. 7) of vW are again theorems of FA. It follows that system FA, just as 0T* and vW, is plagued by the nightmare of the paradoxes of derived obligation and Chisholm's dilemma, if we want to formulate commitment in terms of obligation and the conditional. Systems FA and 0T* share many theorems. Or, more strictly Speaking, many theorems of this system are variant theorems of that system. For instance, the variants of the following theorems of OT* are all FA-tautologies, and hence theorems of FA. (0T*41) :- Fp 8: Fq. : F(p 8: q) (Ii p is forbidden and q is forbidden, “the p-and-q is forbidden) Pf. (1) (2) (3) (4) (5) (6) (7) l-~1>3~(1>&q) PL F0~PDO~(p&q) (1). (Rd4) I—p:q.3.p:>(r3q) PL I—O~p30~(p8:q).:>.0~p3(0~q3 O~(p&q)) Sub-(3) F0~p3(0~q30~(p&q)) (2)(4). PL |"(0~P&0~C1)DO~(P&<1) (5). PL )— Fp 8: Fq .:> F(p 8: q) (6), (T9) (0T*42) :- Fp 8: Oq. : F(p 8: q) (If p is forbidden and q is obligatory, then p-and-q is forbidden) Pf. (OT*25), PL 73 (0T*43) I- Fp 8: Iq. D F(p 8: q) (If p is forbidden and q is indifferent, then p-and-q is forbidden) Pf. Similar: (0T*43) (0T*44) [— Fp v Op v Ip (Either p is forbidden or p is obligatory or p is indifferent). Pf. (1) l—p3p PL (2) 1— (PP & P ~p) 3 (Pp & P~p) Sub- (l) (3) )— ~Fp 8: ~0p. D Ip (2), (T5), (T2), (Dd3) (4) )— ~(Fp v 0p) 31p (3), PL (5) |-— Fp v 0p v Ip (4), PL (0T*45) — Ip D I ~ p (If p is indifferent, then not-p is also indifferent) Pf. (1) [— (Pp 8: P~p) 3 (Pp 8: P~p) PL (2) |- Ip D (P ~~p 8: P~p) (1), (Dd3), PL (3) )— Ip 3 (P~p 8: P~~p) (2), PL (4) )— Ip 2 I ~p (3). (D83) Indeed, most of (0T*1)--(0T*40) are also variant theorems of FA. DeSpite the fact that FA shares many theorems of OT*, there are notable differences between these two Systems which deserve our atten- tion. Among them, the following are the most significant ones. (1) It is no longer the case that a "tautologous-act" is obligatory. Neither is it true that a "contradictory-act" is forbidden. That is, the variant theorems (0T*l7)‘ and (0T*l8)‘ below of (OT*17) and (OT*18) are not theorems of FA. 74 (OT*17)‘ 0A_a_N_a (a-or-not-g is obligatory) (OT*18)‘ FKgNg (.p—and-not-p_is forbidden) This can be easily verified. This feature may be looked upon by some deontic logicians as a desirable feature. (2) System FA like system vW contains no theorems in which iterated modalities are involved. But unlike vW, FA is not a variant subsystem of OT*. For instance, the following rather counter-intuitive thesis (7. 2) is a theorem of FA. (7. 2) (Pg 8: P_b_) D PK_a_1_)_ (If _a_. is permissible and _b is permissible then grand-1; is permissible) However, its variant (7. 2)‘ (Pp & Pq) D P(p 8: q) 6 is not a theorem of OT*. Hence, (7. 2) is not a theorem of vW, either. In addition to (7. 2), system FA also contains other counter- intuitive theorems. For instance (7. 3) I§_._ 8: lb. 3 Ith (If g is indifferent and b is indifferent, then _a_- and-_b_ is indiffer ent) (3) Another significant difference between system FA and system 0T* is that while the former has a finite characteristic 6We defer the proof that (7. 2)‘ is not a theorem of OT* until the end of section 9. 75 matrix, 7 the latter does not have this prOperty. To Show that system 0T* does not have a finite characteristic matrix, we first recall Dugundji's proof that there is no finite characteristic matrix for any one of Lewis systems. Now, his proof can be easily modified to Show that system M of von Wright (which is system T of Feys) also has no ‘ finite charac- teristic matrix, and to Show that system 0T*, being a variant sub- system of system T, also lacks this prOperty. 7This has been shown by Aqvist, ibid. S 8. DEONTIC MODALITIES AND ITERATION OF MODALITIES Exactly in parallel to the usual definition of alethic modalities, we may put down a definition of deontic modalities as follows. A deontic modality is a well-formed formula that is constructed in terms of a propositional variable, '~' and '0'. Or, more precisely, we may define a deontic modality recursively: i) Any propositional variable is a deontic modality. ii) If a is adeontic modality, then reed" and l’08!" are deontic modalitie S. iii) Nothing else is a deontic modality. A deontic modality will be said to be of degree k(k 2 0) if it contains k occurrences of '0'. A deontic modality of degree zero is an improper modality; otherwise, a prOper one. In what follows we shall prove some theorems in which iterated (prOper) modalities are involved. None of these theorems, as we indicated before, are theorems of vW* or FA, since they are not even wffs of those systems. 76 77 (0T*46) )- 00p 2 Op (If it is obligatory that p is obligatory, then p is obligatory; or, if p is obligatorily obligatory, then p is obligatory) Pf. (l) :- O(Opr) (Ad7) (2) )- OOPDOP (1). (Ad4) (0T*47) - (0T*49) below are immediate consequences of (0T*46) and (T1) - (T12). (0T*47) [- OFp D Fp (If p is obligatorily forbidden, then p is forbidden) (0T*48) )- FPp D Fp (If p is foribddenly permissible, then p is forbidden; or if it is forbidden that p is per- missible, then p is forbidden) (OT*49) t FP ~ p 3 Op (If not-p is forbiddenly permissible, then p is obligatory) It is easily felt that to read |‘FPA-I as A is forbiddenly permissible seems extremely strange. But is it more strange than to read 1' (ZIOA1 as A is necessarily possible? The following theorem is another way to express what is said in (0T*47). (0T*50) I- 0(Fp D ~p) (It ought to be the case that if p is forbidden, then not-p is the case, or, then p is not done) Pf. (Ad7), PL, (T9) (0T*51) (OT*SZ) (OT*53) (OT*54) (OT*55) 78 )- O(p :3 Pp) (It ought to be the case that if p is done then it is permissible) Pf. (OT*SO), PL, (T5) )- Op 3 OPp (If p is obligatory, then p is obligatorily permissible) Pf. (0T*51), (Ad4), PL )- FpD FOp (If p is forbidden, then p is forbiddenly obligatory). Another strange English rendering. Pf. (1) :- Op 3 OPp (OT*SZ) (2) )- 0 ~pDOP~p Sub. (l) (3) )- Fp DO~0p (2). (T9). (T2) (4) I-FPDFOP (3), (T9) )- PFp D P ~ p (If p is permissibly forbidden, then not-p is permissible) Pf. (1) )- OpDOPp (0T*52) (2) t ~0Pp3~0p (1). PL (3) I- P~Pp : P~p (2). (T2) (4) fPFpDP~p (3), (T5) |- POp 3 Pp (If p is permissibly obligatory, then p is permissible) Pf. (l) )- Fp D FOp (0T*53) (2) )- ~F0p:>~Fp (1), PL (3) I- POP'D PP (2). (T5) 79 (OT*56) t PpD PPp (If p is permissible, then p is permissibly permissible) Pf. (1) I—OOpD 0p (OT*46) (2) |-~Op :>~00p (1), PL (3) (~0~p 3 ~OO~p sub. (2) (4) F Pp D P ~ 0 ~ p (3). (T4). (T2) (5) I-PPDPPP (4). (T4) In view of the rules (Rd4) and (Rd5), we know that we can establish theorems containing deontic modalities of greater and greater degree. For example, from (OT*56) by five applications of (Rd4), we have (8. 1) )- OOOOOPp D OOOOOPPp And then by three additional applications of (Rd5), we obtain (8. 2) )— PPPOOOOOPp 3 PPPOOOOOPPp which is an abbreviation of (8. 3) )- ~O~~O~~0~00000~0~p 2 ~0~~0~~0~00000~0~~0~p which is equivalent to (8.4) (- ~OOO~OOOOO~O~pD ~OOO~OOOOO~OO~p1 Thus we have a theorem involving modalities of degree nine and degree ten, r e Spectively. 1We shall not try to render this formula into everyday language. 80 Although the number of modalities can be multiplied indefinitely, we may, as in the case of alethic modal logic, ask the question: How many irreducible deontic modalities do we have in system 0T*? The answer is easily obtained. It has infinitely many modalities. To see this. First we recall the well-known result, due to Sobocir’iski,2 that system T possesses an infinite number of (alethic) modalities. Now, Since system 0T* is a deontic variant subsystem of T, it is obvious that 0T* possesses at least as many deontic modalities as T contains alethic modalities. This follows from the fact that there can be no 0T* theorems other than those whose variants already appeared in T that can be used to reduce the number of modalities. Therefore, system 0T* possesses an infinite number of deontic modalities. As is easily seen, this infinity is of course denumerable. 2Sobociflski, ibid. 8 9. SYSTEM 054* AND ITS FOURTEEN MODALITIES System 054*, as we know, is a system of deontic logic result- ing by adding to system 0T* the following axiom: (Ad8) )— Op 300p (If p is obligatory, then p is obligatorily obligatory) Hence, every theorem of OT* is also a theorem of 054*. Because of this relationship and in view of continuity, we Shall let (054*1) - (054*56) denote the same theorems,reSpectively, as (OT*1) - (OT*56). Thus, the next theorem Of 054* will be (054*57). (054*57) (- o ~p 2 00 ~p (If not-p is obligatory, then it is obligatorily obligatory) Pf. (Ad8), PL Another way of expressing the same thing is this: (9. l) )— Fp D OFp (If p is forbidden, then p is obligatorily forbidden) which is the converse of (0T*48). (054*58) )- 0 ~ p 5 00~p (Not-p is obligatory if and only if it is obligatorily obligatory) Pf. (054*57), (0T*46), PL 81 82 Another way of expressing this theorem is the following theorem. (054*59) (OS4*60) (054*61) (054*62 ) [- PPp "=- Pp (p is permissibly permissible if and only if it is permissible) Pf. (054*58), PL, (T2), (T4) (— 00p '5 0p (p is obligatorily obligatory if and only if it is obligatory) Pf. (Ad8), (OT*46) )— 0 ~ 0p D~0p (If p is obligatorily not obligatory, then it is not obligatory) Pf. (l) |- Op 3 00p (Ad8) (2) [- ~00p D~Op (1), PL (3) I- P~0P3P~P (2): (T2) (4) )— Op 3 Pp (OT*11) (5) I—O~0pDP~0p Sub.(4) (6) [— 0 ~ Op 3 P~ p (3)(5), PL (7) I-0~0P D~0P (6). (T2) |- 0 ~ Op 5 0 ~ 0 ~ 0 ~ Op (If p is obligatorily not obligatory, then p is obligatorily not obligatorily not obligatorily not obligatory) Pf.(1) (- 0 ~Op D~0p (OS4*61) (2) )— P0~OpDP~0p (1), (Rd5) (3) )— ~O~O~0p D~O~~0p (2). (T4) (4) |—~O~O~OpD~OOp (3), PL (5) (6) (7) (8) (9) (10) (11) 83 (— ~O~O~Op : ~Op (4), (084*60) kO~O~O~OpDO~Op6LHMM (— 0p : ~O~Op (1), PL )- 00p 3 0 ~O ~ 0p (7), (Rd4) )— op : 0 ~ 0 ~ Op (8), (0S4*60) (— O~0p D 0~0~0~0p Sub.(9) (— 0~0p=_= O~O~O~Op (6)(10), PL Theorem (054*60) and (054*62) may be regarded as "reduc- tion theorems" of system 054*. Since 054* is a deontic subsystem of 54, it follows, by the reasons we mentioned earlier, that 054* contains at least as many deontic modalities as 54 contains alethic modalities. Hence, system 054* possesses at least fourteen distinct deontic modalities. However, in the light of the above reduction theorems, we are able to Show that 054* possesses exactly fourteen distinct modalities. They are: (9.2) (1) (2) (3) (4) (5) (6) (7) (8) degree 0 —impr0per ~P 0p 0~p(Fp) degree 1 0~MPp) O~ Op(FOp) O~O~p(OPp) degree 2 g prOper 84 (9) ~0~0p(P0p) degree 2 proper (10) ~0~0~p(PFp) (11) 0~0~0p(0P0p) (12) 0~0~0~p(0PFp) degree 3 (14) ~O~O~O~p(POPp) These fourteen modalities are distinct. This follows from the well-known fact that the correSponding fourteen alethic modalities in 54 are distinct. To show that (9. 2) is a complete list of distinct (irreducible) deontic modalities in 054*, let us, fir st of all, make the following observation: (9. 3) (9. 2) is a complete list of distinct irreducible deontic modalities (in 054*) of degree equal to or less than three. To prove this, let us consider four different cases: (Case 1) Modalities of degree zero. That is, a propositional variable prefixed by a string of '~' '3 of any length. (Other candidates will turn out to be either not deontic modalities or not of degree zero). In this case, (1) and (2) of (9.2) can be seen to constitute a complete list in view of the following elementary logical laws: 85 2k m (9.4) l-~----”~c1r-‘E 0’ 2k+l rm 1 r -) (95) ~oooooooo~aE Na (Case 2) Modalities of degree one. That is, a modality in which there is exactly one occurrence of '0' prefixed or suffixed by a string of ‘~' ‘s (maybe an empty string) and followed by a proposi- tional variable. Or, in Short, a modality of the following form: (9. 6) O O O O _.._ p Where '. . -' and '---' are strings (maybe empty) of '~' '5. Again, by (9. 4) and (9. 5), it is easily seen that (3) - (6) in (9. 2) exhaust all the possibilities of distinct deontic modalities of degree one. (Case 3) Modalities of degree two. That is, a modality with two occurrences of '0' '5 each of which may be prefixed or suffixed by a Str ing (perhaps empty) of '~' '5 and finally followed by a propo- sitional variable. In other words, modalities of the following form: (9'7) oooO—-—O-—-oo.—p Suppose that the number of '~' '5 in the string between the two oCcurrences of '0‘ 'S is even, then, by (9. 4) and PL, (9. 7) becomes (9. 8) oooOO—oo—p which by (054*60) is the same as M 86 (9.10) ..oO—..-p That is to say, in this case, (9. 6) is reduced to a modality of a lesser degree. Hence, we need only consider the case in which there is a string of odd number of '~' '5 between '0' 's . But this, by (9. 5) means that we need only to consider modalities of the following form: (9.11) °°°O~O-oo-p Again» by (9.4) and (9.5), it is easily seen that (7) - (9) in (9, 2) are the only distinct modalities of degree two. (Case 4) Modalities of degree three. The proof is exactly analogous to the proof in (Case 3) except here we need to consider modalities of the following form: (9-12) .--0~0~0--- p Again, (11) - (14) of (9. 2) exhaust all the possibilities. This concludes the proof that (9. 2) is a complete list of distinct deontic modalities of degree equal to or less than three. We now turn to the following thesis: (9' 12) There are no irreducible deontic modalities (in 054*) of degree greater than three. We shall prove this thesis by verifying the following claim. If . M ls a modality of degree equal to or greater than three, then M is . eq-‘llvalent, i. e. , reducible, to one of the fourteen modalities in ( 9 . 2'): The proof is carried out by mathematical induction on the degree k 87 of M. When the degree of M is four, then, for the reason we indicated and proved above, we only need to consider the following four cases: i) M: 0~O~O~0p ii) M: O~0~O~0~p iii) M: ~O~O~0~0p But these modalities, by (054*62), are, respectively, the same as v) 0~Op which are, in fact, (7), (8), (9) and (10) of (9. 2), reSpectively. Hence, they are not new modalities. And our thesis holds for the base of the induction. Now, as the hypothesis of the induction, suppose that our thesis is true when the degree of M is k (k 2 4), we want to Show that our thesis is still true when the degree of M is k + 1. Four cases to be considered (other cases are dismissed for the same reason stated above): i) M = OM' ii) M = ~OM' iii) M = 0 ~ M' 11V) M=~O~M' 88 where M' is a modality of degree k and thus, by the hypothesis, can be reduced to one of the fourteen modalities of (9. 2). Again, by (9. 4), (9. 5), (054*60), (054*62) and PL, and by checking through each of the fourteen possibilities for M', we will see that in each case M turns out to be equivalent to one of (l) — (14) in (9.2). Therefore, by the principle of mathematical induction, we conclude that there are no modalities of degree greater than three which cannot be reduced to one of the fourteen modalities in (9.2). Next, by combining (9. 3) and (9. 12) we obtain immediately the result that (9. 2) is a complete list of irreducible deontic modalities in 054*. That is to say, system 054*, just as system 54, possesses exactly fourteen irreducible modalities. The implication relations holding among the deontic modalities are again exactly parallel to those holding among their corresponding alethic modalities. They can be summarized in the following chart:1 Op => O~O~Op =5 { }=~O~O~o~p a ~O~p 0~O~p where ' =-.~' denotes logical implication. From this chart, we can draw some further theorems of 054* which have not yet appeared above. However, we will not do so here. le. Feys [1965], p. 95. The imprOper modalities are not among them, because (A5) fails in deontic logic. 89 Before closing this section, let us prove a claim we made in a note, i. e. , note 6, of section 7. That is, we want to Show that (7.2)' Pp 8: Pq. D P(p 8: q) is 3191 a theorem of OT*. Let us prove this by showing a stronger result: that (7. 2)‘ is not even a theorem of 054*. And since 054* is a super system of OT*. the desired result follows. The proof is as follows. Suppose, for a contradiction, that (7. 2)‘ were a theorem of 054*, then we could establish the following: (1) (— Pp 8: Pq .-—.-. P(p 8: q) (7.2)', (054*31) (Z) )- ~(Pp & Pq) E ~P(p & q) (1). PL (3) (- ~(Pp 8: P ~ p) *5 ~P(p & ~p) Sub- (2) (4) )— ~(Pp 8: P ~p) e F(p 8: ~p) (3), (T6), PL (5) I- F(p & ~p) 3. ~Pp V ~ P ~p (4), PL (6) I- F(p 8: ~p) (054*18) (7) )" ~Pp V ~P~p (5H6). PL (8) l- Pp D 0p (7). PL. (T1) (9) (— Op 3 Pp (054*11) (10) I-Op Pp (8). (9). PL (11) )- 0p 5 ~P~p (054*1) (12) |—OpE~O~p (10)(11), PL 90 And since 'Op 5 ~0~p' is provable, it follows together with (054*60) that the fourteen modalities of 054* reduce to Six. But this is absurd, because we know that 054* possesses at least fourteen modalities. Hence, we have a contradiction. That means (7. 2)' canpgi be a theorem in 054*. A similar proof can be used to show that (7. 2)' is not a theorem of 055*, either. 3 3Another way of obtaining the same result is by showing that the deontic variant of (7. 2)', namely, 'Op 8: 0 q. . D O(p 8: q)‘, is not a theorem of SS. Q 10. SYSTEM 055* AND ITS 51x MODALITIES Next comes the strongest of our deontic systems, 055*, which is the result of adding (Ad9), i. e. , as an axiom to 054* above. Thus, every theorem of 054* is a theorem of 055*. And again we will denote by '(055*1)' - '(055*62)' the theorems (054*1) - (054*62), reSpectively. The following is an additional theorem of 055*: (055*63) )— 0 ~ 0p 2 ~0p (p is obligatorily not obligatory if and only if p I is not obligatory) Pi. (055*61), (Ad9) This is the reduction theorem of 055*. In view of it, the fOurteen modalities in 054* are immediately reduced to the following six: (10.1) (1) p (2) ~p (3) 0p (4) 0~p (Fp) (5) ~Op (6) ~0~p (Pp) The only implication relatiai holding among these modalities is depicted by (Ad6), or equivalently, by (055*11). 91 W)- t S 11. ANDERSONIAN CONSTANT AND DEONTIC SYSTEMS OM, OM', OM" Instead of following the manner in which we constructed the deontic logics 0T*, 054* and 055*, Anderson [1956] took three systems of alethic modal logic M, M' and M" of von Wright --— which are equivalent, reSpectively, to T of Feys, S4 and 55 of Lewis —-— let them remain intact, but added to the vocabulary of each system a propositional constant '_B'. He then defined the notions of obligation, permission and forbiddance in terms of this constant. The results turn out to be three deontic logics, OM, OM' and OM". To reconstruct these three systems, let us, fir st of all, put down the primitive basis of the weakest one, that is, system OM. I. Vocabulary i) PrOpositional variables:'p', 'q', 'r', 'pl' ... ii) Propositional constant: 'WB." iii) Sentential connectives: '~', '8:', 'v', '3', '2', ‘0' iv) Grouping indicators: '[', ']' II. Formation rules i) A propositional variable or a propositional constant standing alone is a wff. 92 93 ii) If A and B are wffs, so are "‘~A_l , l-<>A_1 , '—[A 8: B]1, '_[A v 131'1 , I—[A :5 13]", and [-[A s 13]". iii) Nothing else is a wff. III. Axioms (A1) (A2) (Same) (A3) (AN4) l:- p 3 Op (AN5) I- O [P V q] '5 [Op v Oq] IV. Rules of inference (R1) (Same) (R2) (RN3) ExtensiOnality: From FA 3 13" we may infer 1 '_<>A : 03". (RN4) Necessitation: From A we may infer t-~<>~A*l . V. Definitions I" ‘l _ (DNl) DA .13 (DN2;) "A-—is'1 =D£FDEA 313]" 'B 8: O~B' W! m I" '1 f ~0~A (DN3) g,“ em (DN4) "PA" =Df"<>[A 8: ~WSWII‘1 1This is stronger than von Wright's original version, see von Wright [1951b], p. 85. 94 I— 1 _ '~~ "I (DN5) 0A .1), P~A I' 'I __ I" 'l (DN6) FA _Df ...PA 2 (DN7) "IA" =DfrPA 8: P~A" System OM' is the result of adding the following (AN6) to OM. (AN6) (- <><>p 3 Op And if we add (AN7) to OM', we have system 0M" . (AN7) )- <>~<>p : ~<>p3 Intuitively Speaking, ..R' stands for some unspecified bad thing, and '5“: denotes "sanction" of a certain kind which is defined as "the bad thing that can be avoided happens" ('g 8: 03"). And the permiss- ibilityofbringing it about that A is defined as "A can be done without the sanction" ('0[A 8: ~§]'). Thus, in Andersonian systems, we are not reading '[3' as obligatory and '0' as permissible. When we say, for example, that (OT*56) Pp 3 PPp is provable in CM, what we mean is that there is a proof of (11.1) <>[p 8: ~53 DO[O[P 8: ~§J 8: ~§]4 2t-IA‘I is not defined in Anderson, ibid. , but we add it here for completeness. 3(AN6) and (AN7) are called by von Wright "the first reduction axiom" and "the second reduction axiom", reSpectively. See, ibid. , p. 84. 4 For a proof of (11.1), see Anderson, ibid. , p. 188. 95 In what follows we shall make free use of any theorem which has already been established in Anderson [1956]. No proofs in this case will be given, and the theorem numbering prefixed by 'OM', 'OM' ' or 'OM" ' is Anderson's. 3 12. THE RELATION BETWEEN 0T*, 054*, 055* AND OM, OM', OM" It turns out that 0T*, 054* and 055* bear a certain relation, reSpectively, to OM, OM' and 0M". Namely, that every theorem of 0T*(054* or 055*) is an abbreviation of a certain theorem of OM (OM' or OM"). That is to say, the former are quasi-subsystems of the latter. We shall also say that the latter quasi-contain the former. To make a careful distinction between a theorem and an abbr evia- tion of a theorem is all-important when inter-system inquiry is under way. This is esPecially the case when different systems under considera- tion possess different symbols. For example, we may ask the question: whether or not the following (0M8) is a theor em of OT*? (0M8) (— Pp 30[p 8: ~,§_l The answer cannot be given as easily as one might expect. In one sense, (0M8) i_s_ a theorem of OT*, because one might say that it is nothing but (12.1) Pp 3 Pp having in mind that (0M8) can be abbreviated as (12. 1). However, in another equally, if not more significant, sense, (0M8) is Bit a theorem 96 97 of OT*. Indeed, it is not even a well-formed formula of OT*. If we are willing to take the relation of quasi-containing seriously, we may Show that 0T*, 054* and 055* are quasi-contained, reSpectively, by OM, OM' and 0M". To see this, we shall first Show that 0T* is a quasi-subsystem of 0M. In order to prove this, it suffices to demonstrate that all axioms of 0T* are either axioms or theorems, or their abbreviations, of 0M, and that every rule of 0T* is either a rule or a derived rule of OM. The axioms (A1) - (A3) are the same. (Ad4), (Ad6) and (Ad7) can be proved in OM as follows (as we have said above, we shall make use of the theorems already established in OM). (Ad4) :— O(p :q) 2 (0p 3 Oq) Pf. (l) (- (Op 8: O(p :>q)) 2 Oq (OM29) (2) (— O(qu) D(Op : Oq) (1), PL (Ad6) )— Op 3 ~O~p Pf. (1) )—Op 3 Pp (OM10) (ZN-Pp ~0~p (0M5) (3) )- Op 3 ~0~p (1)(2), PL (AD?) )— O(Op D p) Pf. (1) (— O(Op 2 p) (OM45) Again, the rules (R1) - (R2) are the same. It suffices to Show that (Rd3), from A to infer '- 0A1, is a derived rule in CM. 98 The proof is as follows. Suppose A is a theorem (or axiom) of 0M, we want to Show that from this it follows that l‘OAfl, or rather, its abbreviation, is also a theorem of OM. We may establish the proof as follows: Pf. (l) I—p3(~p3q) PL (2) )— AD(~A D5”) Sub.(l) (3) (- ~(~A 35) D~A (2), PL (4) )- <>~(~A 2,5,) : <>~A (3). (RN3) (5) (- ~O~A D~<>~(~A 3‘5") (4). PL (6) )— A Assumption (7) )— ~<>~A (6), (RN4) (8) I- ~<>~(~A 3§J (5)(7), (R2) (9) [— ~<>(~A 8: ~§) (8), PL (10) I:- ~P~A (9), (DN4) (11) [-0A (10), (DN5) This completes the proof that 0T* is a quasi-subsystem of OM. To Show that 054 is a quasi-subsystem of OM', we need only to Show that (Ad8) of 054 is provable in OM'. The proof: (Ad8) (— Op 3 00p Pf. (1) )— PPp : Pp (OM'60) (2) (- PP~p DP~p Sub.(l) (3), (- ~P~p3 ~PP~p (2). PL (4) (- 0p ‘55 ~P~p (0M1) 99 (5) )— Op : FP ~p (3)(4). PL. (DN6) (6) (- Fp 0 ~p (0M4) (7) [—0p 3 0~P~p (5)(6), PL (8) )— Op 3 00p (7)(4). PL And to Show that 055"< is a quasi-subsystem of 0M", we prove that (Ad9) of 055* is an (abbreviated) theorem of OM". (Ad9) |- ~0p D 0 ~ Op Pf. (1) )— P~Pp ~Pp (OM"64) (2) )— P~Pp D~Pp (1), PL (3) (— P~P~p 3 ~P~p Sub.(2) (4) )— Op 5 ~P~p (0M1) (5) )- POp 2 0p (3)(4). PL (6) (— ~Op D ~P0p (5), PL (7) (—'~0p amp (6). (DN6) (8) I—Fp'é 0~p (0M4) (9) I- ~0P 3 0~ 0P (7N8). PL This concludes our proof that 0T*, 054* and 055* are, reSpectively, quasi-subsystems of OM, OM' and 0M". The following chart shows the containing relations holding among the systems we have so far discussed. We make no effort to distinguish between containing, quasi-containing, or the containing with reSpect to a certain variation of symbols, as the distinctions are now quite obvious. The broken line Shows that the two systems connected 100 by it share some theorems but neither contains or is contained by the other . OM"----2 —---- OM'----2 ----- 0M ' I : E E N N N . 1 : 55 ---- 2 un 54 --—-2_--- T (My) 04") (N?) I , I II II I‘I ; E ; 055--- 2--- -- 054 2---- 0T 0 . a 2 e 3 IV N N ’FA 0555*" 2 ---- 054*-- 2---- OT*-- 2-— vW (vW*) Of course, it goes without saying that the containing relation is, among other things, reflexive, transitive but non-symmetric. S 13. FURTHER THEOREMS AND FURTHER PROBLEMS IN 0M - 0M", KANTIAN PRINCIPLE AND THE PARADOX OF THE GOOD SAMARITAN The interest of the Andersonian approach to deontic logic lies mainly in the simplicity of constructing deontic modalities. Thus, in system 0M - 0M" we are able not only to present deontic concepts but also to express alethic modal concepts. Moreover, we can systematize these two groups of concepts in combination. That is to say, we will have well-formed formulas of 0M - OM" in which both deontic and alethic modalities appear. This feature iS interesting because it seems clear that alethic concepts do find a way into our moral discussion. For instance, someone may want to follow Kant in saying that what is obligatory must be possible. ("ought" implies "can"). And here "possible" or "can" is clearly an alethic modal concept. However, this great gain has not been made with no pains. As is too familiar in philOSOphy, when we widen the deontic horizon by allowing the alethic modalities, some additional problems immediately develop themselves in the newly acquired territory. In what follows, we shall present some of the interesting deontic theorems which are provable in 0M" (hence, in 0M and OM') but not 101 102 in 055* (hence, not in 054*, nor in 0T* and vW*). Most of them are theorems in which both deontic and alethic modalities present themselves. (0M7) )- 0p “=- . ~P—3.§. (p is obligatory if and only if not-p strictly implies the sanction) (OMZI) [- Dp :3 0p (If p is necessary, then p is obligatory) (OM23) )— ~Op 3 Pp (If p is impossible, then p is forbidden) These two theorems state something stronger than what are eXpressed in (0T*17) and (0T*18). According to (0M21) and (0M23), for instance, not only that to go and not to go to fight in Vietnam is forbidden, it is also forbidden to draw a round square or to trisect an arbitrary angle with only a ruler and a compass. (OM24) (— Op :3 0p (If p is obligatory, then p is possible) This is the so-called Kantian principle that what one ought to do one can do. But is this a sound principle? As we recall, the formulation of commitment in terms of obligation and the conditional leads to paradoxical results. Now, in OM - OM", we may want to propose another candidate for that concept, we might want to let strict implication together with obligation capture the essential idea of commitment. That is, we may want to read 'p—30q' as "p commits us to do q". 1Anderson [1956] read. '—3' as logical entailment. Later in [1967] he preferred "relevant implication". 103 Under this new formulation, the paradoxes of derived obligation disappear. Because the following are no longer theorems of OM - 0M". (13.1) Fp D. p—-30q (13.2) OpD. q—30p However, it seems questionable whether peOple want to make moral commitment as strong as a strict implication. The following theorems are also of Special interest: (OM27) [Pp 8:. p-—-3q] D Pq (If p is permissible and p strictly implies q, then q is permissible) (OM28) [Fp 8:. q—3p] : Fq (Ii p is forbidden and q strictly implies p, then q is forbidden) Now, if we follow Anderson to read '--a' as entailment, another "paradox" follows from (0M28). Consider the following Situadflon. Suppose that robbery is forbidden. Since to help someone Who is the victim of robbery entails, among other things, that robbery occur 5, it follows, by (OM28), that the good Samaritan who helps the man Who becomes the victim of robbery on his way to Jericho also doe 8 something forbidden. This unhappy result is called the paradox 0f \ % Good Samaritan. g 14. VON WRIGHT'S TENSE-DEONTIC SYSTEMS In 5 5 we indicated that von Wright proposed one type of deontic logic which may be called tense-deontic logic. It is a kind of deontic I logic that is based upon a certain system of tense logic. But since Von Wright has outlined several different systems of tense logic,l we may have several different systems of tense-deontic logic. In this section we shall briefly explain what a tense-deontic logic (of von Wright's type) is and indicate the advantage of this kind 0f deontic logic. First of all, we shall reconstruct here a system of von Wright's tense logic which we shall call system th. This system may be regarded as a formalization of our intuitive tense connective '0! and then 5' as in "The door of Rm 14 Morrill Hall is open §_n_d_t_hgl_ the door of Rm l4 Morrill Hall is closed" where 'a' and '5' stand for the propositions that the door of Rm l4 Morrill Hall is Open and that file door of Rm l4 Morrill Hall is closed, reSpectively. We shall wlize this binary tense connective as '0! >—-—> B ‘. 1See von Wright [1965b] and [1966]. v0 2Von Wright symbolizes 'a and then B' as 'cyTfi'. See n W- . . . right, lbld. 104 105 The following is an axiomatization of von Wright's system th. I. Vocabulary i) Propositional variables: 'p', 'q', 'r', 'pl', . . . ii) PrOpositional connectives: '~', '8:', 'v', '3', '=-.:', '>-—>'. (The first five are the usual truth-functional connectives, the last one is the tense connective 'and then') iii) Grouping indicators: '[', ']'. II. Formation rules i)- A prOpositional variable standing alcne is a wff (We shall call it a Brigg-wff or T-wff). ii) If A and B are (T-)wffs, so are r-~A_l , I—I:A 8: B31, r—[A v B]_', r[A D B]-1 , r.[A a B]-I and I-[A >—+—>B]1.4 iii) Nothing else is a (T-)wff. HI: Axioms (A1) - (A3) Same as in system OT. (T4) I- [qul >——->[rv P11-5[P>——>1‘]V[P>—->p1] v [q >-—>r] v [q >—> p1] (T5) )- [p>—->q3 8: [r >-—->p11'-[[p&r] >——>[q&p1]v [q>—->p11v[pl >—> qll (T6) |- p a [p>-—>[qv ~q]] \(TU )— ~ [p >—-> [q &~q]] 3Cf. von Wright [1966]. I . 4We rank these connectives as follows: '~'. '8:', 'V'. '3': the. ’ ' >—->' where a connective overranks (but is not overranked by) s 13 Standing on the its left. l/l 106 IV. Rules of inference (R1) Substitution (R2) modus ponens It may be noted that certain uses of the usual "temporal quantifier 5" such as 'always', 'sometimes' and 'never' can be defined in terms of the tense connective '>—->' together with the usual propositional connec- tive s. This has been shown by von Wright in [1966]. For instance, the sentence (14:. 1) The sun [now] rises in the east and will alwaxs rise in the east. can be paraphrased as (14. 2 ) The sun [now] rises in the east and it is not the case that whatever the circumstances may be, and then the sun does not rise in the east. or in s ymbols (14— 3) p&~[(qV~q)>—>~p1 Whe 1‘ e 'p' stands for 'The sun rises in the east' and 'q v ~q' depicts a <2 - . . ondltion or a state of affairs which is ever present. Let us use '0' to \rnean 'always' in the above-mentioned sense, namely, ' 0 p' means to 1: Von Wright adds a rule which he calls the "rule of extensionality" E 'I‘ he following effect: ". . . that provably equivalent T-expressions ad; wffs] are interchangeable salva veritate". However, we Shall not as 111115 rule here, because the rule seems helpful only when we try, on Wright does, to explore the semantical problems of system th. 107 'It is the case that p now and it will be the case that p at all later Then the following definition may be given: times'. I' 'I 1'" "I (D14.1) 49A =Df A8:~(3 >——>~A.) where 'i' is a tautology in PL. We are of course aware that 'always' is often used in a more extensive sense. 'It is always the case that p' usually means that it was the case that p and it is the case that p and it will be the case that p. Hence, we may say that ' *p' captures only a partial meaning of the commonsensical 'always', namely 'now and hereafter' or 'from now on'. But this partial meaning is of Special interest to a deontic or imperative logician, because when we utter a command in which 'always' appears, the 'always' is used, as a rule, in the above less eXtensive sense. For instance, when we order: (14- 4) Always keep the door closed! we u S ually mean (14 - 5) Keep the door closed now and keep the door closed at all later times! Hence '0' correSponds very closely to the use of 'always' in the us . . . ual deontlc and/or Imperative contexts. Likewise, 'sometimes' and 'never' also have correSponding 1e s I O I I 0 s extenSlve meanings. We can always make the qualificatlon "from no \ “.011” in their meanings. But we do not claim that past-tense commands are impossible. W e l eave the possibility Open. 108 If we say Sometimes [now or at a later time] the sun does not (14. 6) rise in the east. this presumably is the contradiction of (14. 3). Thus, (14. 6) can be symbolized as ~(p & ~((q V ~q) >—>~p)) that is (14. 7) ~pV ((qV~q)>-—>~p) It follows that (14. 8) Sometimes the sun rises in the east. can be symbolized as (14- 9) pv((qv~q)>——>p) If We use '0' to mean 'sometimes', then we have the following definition ‘D1 4 I" ‘l _ I'— ~ ~ '7 - 2) 0A -Df 49 A Again, '0‘ represents only a partial meaning of the everyday use of I S I D 1'Tlesitlmes'. The meaning of 'never' can be likewise explained. When we Say (1 4 ‘ 1 0) The sun never rises in the east. W 3 mean 109 It is always the case that the sun does not rise in (14.11) the~ east. 01' (14.12) 45 ...p that is (14. 13) ~p &~(£>—>P) Let us use 'Bf' to mean 'never' in this sense, then we have the following definition: (D14- r .1 Z '— ~ .1 3) :QA Df 89 A In what follows we shall list some of the theorems of system th We Shall omit the proofs. Most of the theorems listed here coincide With the theorems given by von Wright, and the proofs can be found in “”1 Wright [1966]. l- (p >-> q) V (p >->~q) V (~p>——>q) V (~p>—>~q) (It is. the case that p and then q, or it is the case that p and then not-q, or it is the case that not-p and then q, or it is the case that not-p and then not-q) (’I‘ 11111 2) )— (p >-—>q) 3 p (If it is the case that (p and then q), then it is the case that p) ('3:- 1111—1 3) '— ~((p 8: ~p) >—9 q) (It is not the case that (it is the case that (p and not-p) and then q)) (Thm 4) (Thrn 5) (Thm 6) (Thm 7) (Thrn 8) (Thrn 9) (Thm 10) (Thm ll) (Thm 12) (Th-m 15) (Thm 16) (Thrn 17) 110 r-(p&(q>—->r))-=- ((p&q)>—>r) )— ((p8:q) >—>r) D (p>——>r) l—(p>—> (q&r))D (p>—>q) (.((p>_>q) 8: (p>—>r))-=-((p>——>q) >—->r) t-(p>—->(q>——>r)) D (PHI) (.491 (It is always the case that _t_) (- Mi (It is never the case thati) (— $p :3 p (If it is always the case that p, then it is the case that p) (- p D 0p (If it is the case that p, then it is sometimes the case that p) '— 0-(p 8: q) 5. 0p 8: 0q (It is always the case that p-and-q if and only if it is always the case that p and it is always the case that q) I- 181 (p V q) E. E. p & ¢q(Similar1V) |—©(qu)5- ®p V ®q (— 0p 3 ~ 0 p (If it is sometimes the case that p, then it is not the case that it is never the case that p) The following theorems involve the iteration of temporal (1L1 e.’]51t:if'ier s: 111 (Thm 18) |—'¢p : 45-4} p (If it is always the case that p, then it is always the case that it is always the case that p) (Thm 19) I- 00- p D 0 0p (If it is sometimes the case that it is always the case that p, then it is always the case that it is sometimes the case that p) Here we sense an unnaturalness or, rather, an unfamiliarity in the English renderings of “$00 , '0 ‘0- p' and '-¢-Op'. Tense logic like most other branches of logic goes beyond the SCOpe of our bare intuition. Let us now indicate how to extend the tense logical system th to a. tense-deontic system. There are many ways to accomplish this goal- We shall however give only an example. Suppose we add to the primitive basis of th the following ing]? edients, the resulting system will be called a tense-deontic system 0T*(th). First, to the vocabulary of th, we add iv) Deontic Operator: '0' Secondly, to the formation rules of th we add II- . iv) A T-wff IS a wff. ., v) If A and B are wffs, so are I’OA‘, "~A", r[A 8: B] , "I r[AvB]_I, r'[ADBI1 and '_[AEB] . vi) Nothing else is a wff. 112 Thirdly, we add the following axioms: 111. (A84) (- O[p 3 q] 3 [0p 3 Oq] (Ad6) I- OP 3 ~O ~p (Ad7) (- O[Op 3p] And finally, to the rules of inference, we add IV. (Rd3) Deontic necessitation. It is immediately seen that the system OT*(th) just constructed 13, as the name suggests, a "combination" of our earlier (deontic) System OT* and tense system th. We may in the same manner combine VWt with 054*and obtain a tense-deontic system 054*(vW), and so on. One of the leading characteristics of this type of tense-deontic logic is that the deontic Operator may take tense expressions (T-wffs) as its Operands. This is a feature which is interesting enough to receive our closer attention, because it helps us, among other things, make pre Cise what iS the exact content of an obligation, an imperative, and What not. Let us first give an example to illustrate this point. If a certain Mr. A gives Mr. B the following command: (1 4 ~ 14) Open the door! Suppose further that the door is already open. Then Mr. B need not do anYthing in order to fulfill the command. The order is automatically carried out. Only in the case that the door is closed should Mr. B “flake the necessary effort of opening it to accomplish the command given 113 by Mr. A in (14.14). However, these two different cases -—- in one of which Mr. B needs to make a genuine effort and in the other he does not need to —-— are not distinguished when (14. 14) is uttered. But suppose the command is issued in the following manner: (14. 15) Bring it about that (the door is closed and then the door is Open)! Then the order is clear. If the door is closed now, (14. 15) Simply Orders one to Open it. Suppose, on the other hand, that the door is Open now, (14.15) orders one to first close it and then Open it. In both cases, genuine efforts are demanded in order to fulfill the command 1n Question. Similar remarks can be made about the deontic "counterpart" 0f ( 14. 14). For instance, correSponding to (l4. 14) we may put down a C1eontic sentence like: (14 - l6) 0p Wllezt-e 'p' stands for 'The door is Open', and corresponding to (1 4 - 1 5) the following one: (14- 17) O(q>—>p) Wh-ere 'q' stands for 'The door is closed'. Likewise one may issue a command to the following effect: Keep the door open! \ 7For the meaning of 'deontic counterpart', see 5 24. 114 This may be symbolized in its deontic counterpart as (14.18) 049p These examples indicate that when we let the deontic operator apply to tense-wffs, we might make commands or obligations, and so on, more precisely stated. We could be more sure what is the exact content of a command, and hence what constitutes the fulfillment of that command. However, only to let the tense wffs be candidates for the Operands Of a deontic Operator seems not sufficient to cope with the problem we just pointed out. For example, suppose we say: (14- 19) O(p >——>r) Where 'p' stands for the same sentence as above, and 'r' stands for 'The light comes in'. That is, (l4. 19) means: (14 - 20) It ought to be the case that (the door is Open and then the light comes in). NOW, suppose the Opening of the door "naturally" brings in the light, men in order to fulfill (l4. 19), one need only Open the door (if it is not already Open). One need not do anything else to w the light Come in. Hence, it seems necessary that in order to issue a command, We rIlust take into account the factor whether or not that command is fulfilled by the recipient of the cormnand rather than by some other sour(:eS. If a man does not make any effort to bring about a certain 115 state of affairs, then the occurrence of that state of affairs should not be accredited to him. However, we Shall not pursue this matter any further as it will certainly lead us too far from our main topics of this dissertation. CHAPTER II META-ETHICS AND SOME MODIFIED SYSTEMS 0F DEONTIC LOGIC E 15. TOWARD A SOUND SYSTEM OF DEONTIC LOGIC No sooner had the deontic logics gradually taken their shapes than the criticism and misgivings began to multiply in the literature, Mo st of the mistrust of deontic logic comes from the fact that current systems of deontic logic do not provide us with "sound" systems of the 10 gic of obligation (or permission). That is to say, the systems that We have on hand do not yield semantical interpretations which are intuitively satisfactory as logics which formalize the intuitive notions 0f obligation, permission, and the like. People want a deontic logician ‘30 see to it that his system can be used to formalize or to justify our deOntic or normative arguments just as our indicative (or assertoric) 108105, such as propositional logic, can be used to formalize and juBtify our indicative arguments. In the literature, the criticism of deontic logic has been fr equently made in the context of imperative logic. The reason for doing so seems twofold. First, imperative logic has by far a longer history than deontic logic. People are more familiar with, and have a. c-‘-1earer idea of, the much discussed problems and the oft-cited e"la-triples in imperative logic than in deontic logic. Indeed, since V011 Wright constructed his earliest system of deontic logic in the 117 118 early 1950's, it has not been made immediately clear what the "status" of a deontic sentence is. For instance, is a deontic sentence a report of moral code, or is it used rather to prescribe or to evaluate? Con- sequently it is not clear whether a deontic sentence is capable of being true or false. 0n the other hand, it seems unquestionable that an imperative sentence is normally or typically used to issue a command (in the broadest sense), and it seems commonly, if not unanimously, agreed upon that imperative sentences are neither true nor false. 1 Another reason for treating the problems of deontic logic in imperative logic is this. Many people, notably A. Ross, have taken it to be the case that deontic logic and imperative logic are, to say the least, so closely related to each other that a problem in one logic is automatically or mutatis mutandis a problem in the other. This con- ViCtion, to be sure, is not groundless. In the next chapter, we want to go even further to say that these two logics are two isomorphic rt1<>dels of the same (normative) logic. Consequently, the problems of imperative logic become the problems of deontic logic, and 3i_c<3 m. Thus when one wants to argue that the current systems of deontic logic are inadequate, one may take an example from imperative logic to support one's criticism. Indeed, this is what Williams [1963] , Keene [1966] and Kenny [1966] have done. A well-known example of a. Criticism along this line was originally given by A. Ross in connection \ 1There are philOSOphers, notably Leonard, who hold a different View. 119 2 with imperative logic, but has later been frequently cited in the context of deontic logic. Ross observed that the following (imperative) argument seemed extremely problematic if not totally invalid: (1 5.1) Post the letter! Post the letter or burn it! However, in our deontic logics, we have Shown that (OT*23) Op 3 O(qu) is a theorem. And, according to propositional logic, (15.2) Op 3. Ovaq is a. theorem. Now, it seems that either (OT*23) or (15. 2) can be us ed to justify (15. 1). But in the opinion of many people, the validity of (15. l) is highly questionable. Therefore, deontic logic, perhaps together with the usual indicative logics, seems to shed no light on, or at least gives no practical guide to, the validity of imperative r ea-soning. That is, deontic logic is inadequate so far as imperative a‘rguments are concerned. This may be called the problem of the W of deontic systems. \ —. 2See Ross [1941], p. 62 or [1944], p. 38. . 3We use '~'-' to indicate the alleged conclusion drawn from the llne a standing immediately prior to it. The original wording of Ross' .exarnple is this: 'Slip the letter into the letter-boxl' -'-Slip the letter Into the letter-box or burn it!' [£931.] 120 We shall not, for the time being, discuss the problems related to arguments like the above one. We leave it until the next chapter when we come to examine the relation between deontic logic and imperative logic. Another line of attack against present systems of deontic logic is to question whether they can express or formulate certain alleged deontic n0tions:succes.sfull,y.4 That is, the (expressive) completeness 95 geontic systems. As we mentioned in section 4, Chisholm has pointed out the fact that certain deontic systems, particularly system VW of von Wright, either cannot formulate the no'tion of contr ary-to-duty imperative, or else they contain a contradiction. We have also shown that none of the systems we investigated in Chapter One is immune to this defect. Besides, we also know that we have been so far unable to for mulate the concept of commitment in those systems. Now, to this list of formulational or expressive inadequacy, we may add another 01'16: the notion of "conflict of duties". This is a notion closely related, and parallel in many respects, to the notion of contrary-to-duty impera- tive. It has often been pointed out that, under certain circumstances, it J112:.ay be obligatory for us to do a certain act and also obligatory for us not to do it, i. e. , to do the negation-act of it. For instance, a certain American youth of today may find himself in the following moral preC‘iicament. It is, on the one hand, obligatory for him to answer 4 See 3 4. 121 the summons of his country to go to fight in Vietnam. But, on the other hand, it is also obligatory for him to listen to his conscience and not to go to fight there. Obviously, this situation cannot be ade quately expressed in our deontic systems in the last chapter with- out generating a contradiction, because in each of those systems, it can be shown that (1 5 - 3) It is not the case that a certain act and its negation are both obligatory. In short, the following, or its variant, is a theorem of those systems: (0T*19) ~(Op & O~p) Therefore, we have another dilemma very similar to Chisholm's. 1\Tailitlrlely, either we are unable to express the notion of conflicting duties in our systems of deontic logic, or else they contain a contra- dic tion. The classic solution to the problem of conflicting duties, or the common explanation of this problem, is to appeal to the well- kn<>an distinction, first made by William David Ross, between so- called prima-facie duties and actual ones. Many people today are Still fond of following Ross in saying that for the American young man. for instance, it is his Erima-facie duty to go to fight in Vietnam, and it is also his prima-facie duty not to go to fight there. However, he W111, under the particular circumstance in question, find out or "see" 122 which course of action he really ought to take, namely, to arrive at his actual duty in that particular situation. This solution, although attractive and convenient at first sight, seems on second glance less than satisfactory. The very idea of draw- ing a distinction between prima-facie duties and actual duties in the case of conflicting obligations lies on a rather strong presupposition —-—- that whenever two sets of moral rules are incompatible, there is a meta-rule (explicitly or otherwise) which directs that one of these sets overrules the other, or that both sets are overruled by a third set of rules --- which is problematic and questionable. That this is indeed the Case will be seen in the latter sections along with our discussion of the formal theory of ethics and our presentation of an alternative approach 5 130 the problem. The alternative we shall adopt, and thus one of the main purposes Of this chapter, is to develop an alternative semantic theory of deontic 1Ogic in which both Chisholm's dilemma and the dilemma of conflicting Obligations, and hopefully, even the paradox of the Good Samaritan can be satisfactorily accounted for. 5See especially 3 l7. § 16. DEONTIC VARIABLES RANGE OVER CM -AC T - PROPOSITIONS The first step toward a semantic interpretation of deontic logic is to Spell out the range of the deontic variables. We have, on two previous occasions, familiarized ourselves with two different readings of deontic variables. In von Wright's system vW and Fisher and Aqvist's System FA, we let a deontic variable range over act-types. In these Systems, it is an act—type that is said to be obligatory, permissible or what not. Later, we indicated that deontic logicians nowadays have much favored another practice, namely, to let a deontic variable take not act-types but propositions or states of affairs as its values. Accord- ing to them, it is a prOposition or a state of affairs that is said to be Obligatory, permissible, and so on. Neither of these two readings 0f deontic variables proved to be totally satisfactory. First of all, it should be kept in mind that what is performed by any person at any time in any place is an act-instance, namely, an instance of an act-type, rather than an act-type. It would be a category mistake if we should allow ourselves to talk about the performance of act-types. No one can by any means perform an act-type which is an abstract entity, or as von Wright calls it, a set of "act qualifying 123 124 properties". To say that a certain act-type, e. g. , smoking, is permissible is perhaps only an abbreviated way of saying that every act-instance which falls into the category of smoking, i. e. , the individual act of smoking, is permissible. Or, any individual act which has the characteristic properties of smoking is permissible. It is clear that what we need in this connection are three different types of symbols rather than just deontic variables in order to successfully symbolize sentences like (1 6. 1) Smoking is permissible. To be specific, we need variables for individual acts, variables for aCt-types and finally a symbol for a predicate which means that " ® falls into the category of Q) " or " Q) has the characteristic prop- erties of ® ”, or the like. 1 But this symbolism seems only to point 2 to another problem: quantifiers become indispensable in deontic logic. Or, in other words, no satisfactory deontic logic is possible other than a quantified theory if we want deontic logic to be a logic of obligation Which formalizes our intuitive deontic notions. For (16. 1) will then me an (16.2) For every act-instance, if it falls into the category of smoking, then it is permissible. The circled numeral notation is borrowed from Quine. See Quine [1950], p. 131. Hintikka is the first philos0pher who pointed out the indispen- sability of quantifiers in deontic logics similar to system vW. See Hintikka [1957]. 125 It may be pointed out, however, that more often than not we do find moral codes expressed in such an unspecified manner as exemplified by the following examples: (1 6 . 3) Thou shalt not kill. (Ten Commandments) (16.4) Do not do to others what you do not want others to do to you. (Confucius) (1 6 . 5) An unexamined life is not worth living. (Socrates) (1 6. 6) Be prepared. (A motto for Boy Scouts) and (16. 7) Honesty is the best policy. (English saying) These examples show that moral codes are usually expressed in an obscure manner. They resemble (16.1) more than (16. 2). In Particular, they do not mention, or even explicitly presuppose the eXistence of, the individual acts to which these codes may apply. How- ever, attention must also be called to the fact that a moral code would fail to direct pe0p1e‘s action should there be no "bridge rules" to Ctonnect particular act-instances to a certain moral code. Another difficulty in the usual conception of a deontic variable has already been pointed out earlier in section 2. It is a problem associated with the notion of acts and the notion of performance. Does the notion of an act, as we asked before, entail or presuppose the notion of performance? The answer seems to be this. If we take 126 'act' to mean act-instances, then the answer is "yes". But if by 'act' we mean act-types, then the answer is quite uncertain. We may even incline to say that the answer is in the negative. Or, at least we have in this case come up with a different notion of performance. Consider tlle following example. If we say that an act-type, e. g. , smoking, is performed, this seems at most a disguised way of saying that an act— instance, i. e. , a smoking-instance or an individual smoking, which falls into a certain category, i. e. , smoking, is performed. Since there exists such a discrepancy between these two concepts of act, a deontic logicianmust decide which one of these two notions of act, or both of them, he wants to incorporate into his system. Otherwise, he Will find himself talking about act-qualifying prOperties and the actuali- zation of certain act-instances which have these properties indiscrimin- ately. Lack of such discrimination is responsible for the awkward Situation in system vW where we found ourselves saying that if the Performance of an act(-type) is so and 0, then . . . . is such and such. But no one can ever perform-an act-type! The difficulty is avoided, of Course, if we let a deontic variable range over individual acts, and Construct deontic logic as a quantified theory. A similar difficulty may appear when we have not carefully enough specified what we mean when we say that a deontic variable ranges over propositions. For instance, we found ourselves reading 'Op D Pp‘ as "What is obligatory is permissible” or "If something is obligatory, then it is permissible“. However, these latter expressions 127 seem rather to be legitimate renderings of '(p)(Op 3 Pp)‘. But this difficulty seems negligible, because, as in other well-established br anches of logic, for instance, propositional logic, when we assert an Open sentence, we have a convention that the sentence holds for all values of its (free) variables. One of the advantages of taking propositions as the range of deontic variables lies, however, in the fact that this practice makes deontic logic more akin to other branches of logic, so that what is already known in other branches of logic will automatically shed light on the development of deontic logic. We saw this in the last chapter When we constructed certain deontic logics as subsystems of the corres- ponding alethic modal logics. But, insofar as we want to talk about Our actions in deontic logic, the notion of an act of a certain type must be at least implicitly, if not otherwise, contained in the notion of the PI‘Opositions we have in mind as the range of the deontic variables. Indeed this is clearly so as can be seen in the way we managed to interpret a deontic sentence 0T* — 055*. As we recall, a formula like 0p Where 'p' stands for a proposition, was understood as saying that to bring it about that p is obligatory. The notion of bringing-it-about that Certainly entails endeavorings, or in short, acts. In what follows, we shall also take propositions as values of a deontic variable. But we shall do so in a more Specific manner. We shall not only maintain that it is the class of act-instances that concerns 128 us in deontic logic, but also agree that when, where, and by whom an act is performed must be Specified. In other words, ye shall have in our newly devised and interpreted systems of deontic logic sentences of the following form: (1 6. 8) It is obligatory that @ at the time © in the place ® brings it about that @ . That is, (1 6. 9) It is obligatory that someone at a certain time in a certain place brings about a certain act. Here, the ' @ ' in (16. 8) stands for an act-type. For instance, we may say that (1 6.10) It is obligatory that John on July 10, 1968, in Room 14, Morrill Hall, brings it about that cleaning and putting in order. Here cleaning and putting in order is an act-type. It goes without Saying that (16.10) means the same as (16.11) John ought to clean and put in order Room 14 of Morrill Hall on July 10, 1968. Only that in (16. 10) we have a sentence of the following form: (16. 12) John . . . brings it about that cleaning and putting in order. 129 while in (16. 11) we have a sentence of the following form: (1 6 . 13) John... cleans and puts in order .. We shall call sentences like (16. 12) or (16. 13) which designate someone's doing something act—sentences, and call the prOpositions they stand for act-propositions. In particular, we shall say that an act-sentence like (1 6. 12) is an act-sentence in explicit form if it exhibits the following g r amrnatic a1 form (1 6. 14) _x_ brings it about that p. where 'x' stands for a certain agent or actor, and 'p' for a certain act—type. When there is no danger of confusion, we Shall use either form of an act-sentence. But when we need to be precise, an act- sentence in explicit form is called for. An act-sentence may be put in a more specific manner by adding Several types of modification. In particular, we might, as we suggested above,manifestly express when and where the bringing-about depicted by an act-sentence takes place. That is, we may write an act-sentence in the following form (16.15) x at time _t_ in place _w__ brings it about that p. In (16. 15) we specified at what time and in what place the bring- ing about endeavored by 35 takes place. The two modifications, namely Spatial and temporal ones, say, in short, under what circumstances 1‘. brings it about that p. Thus, we shall say that when time and location 130 are Specified, an act-sentence is circumstantialized. Circumstantialized act-sentences will be called CM-act-—sentences. They are standard s entences we Shall encounter in the latter discussion. -A Special remark may be in order. Earlier we criticized the use of act-types as the values of a deontic variable. But now we employ act-types again. However, it should be clear that act-types are now used in a rather Special way. We, so to Speak, circumstantialize an act-type and hence individualize it. Thus what we have obtained are individual acts rather than act-types. Let us say that a CM-act-sentence stands for a CM-act-proposi- tion, and let us declare that deontic variables range over CM-act-propo- sitions. Or, as some logicians may want to say, that deontic variables take CM-act-sentences as their substituents. For the sake of brevity we Shall symbolize (6. 15) as (16.16) (x,_t_, 3);) Where '(x, t, w)‘ may be called the CM-parameters of (16.16). It is Often convenient to hold the CM-parameters constant, hence we intro- duce the following abbreviation: (C16. 1) When the CM-parameters 135i, _w_)' is constant in a formula or in a discourse, we Shall omit the parameter and write a starred sentence. 3 Cf. (16. 18) below. For an effective way of making this abbreviation, see section 19. 131 An act is not done if and only if its negation-act is done. Hence we have the following ”equivalence": (16.17) LELEM" 5'95,_t,_v1)~A" It follows that although expressions like (16.18) ~p* are ambiguous, the ambiguity is indeed harmless. Likewise, (16.19) ’(5. s. y )A 8: (22.1.. arm“ has the same truth condition as (16.20) '(a. _t. a )(A & 1?»)1 Hence, the following equivalence holds: (16.21) “a. 1.. 31A & (a, 1:. m3" 2 “(as .t. a. )(A at B)" Other sentential connectives are defined in the usual way. Again we 1'ealize that connectives like '~' '8:', 'v' , . . 3 . are used in two different ways. But this ambiguity is trivial and innocuous. § 17. THE INTERPRETATION OF '0' IN TERMS OF MORAL RULES The second step toward a sound system of deontic logic is to find a suitable meaning or interpretation for the deontic Operator '0' (or 'P' as the case may be). This task is generally neglected by deontic logicians. As we saw in the last chapter, all the systems we examined take either '0' or 'P' as a deontic operator standing for a certain primitive deontic notion. Although these primitive deontic operators have been intuitively rendered as "obligatory" for 'O' or "permissible‘' for 'P', these notions have never been precisely defined. It happens that to leave these primitive notions vaguely or ambiguously understood accounts indeed for part of the reason why the resulting deontic systems are so unsatisfactory and why the previously mentioned dilemmas seem always to plague those systems. The main purpose of this section is thus to assign a precise meaninghto, or make an exact interpretation of, the deontic Operator '0'. We shall no longer tacitly understand '0' as standing for a roughly conceived notion of obligation. We shall explain obligation in terms of other concepts. The need to make the meaning of '0' clear and precise can be easily appreciated. Simply try to answer the following questions: 132 133 How is it that we are obliged to do so and so? What obliges us to do such and such? What makes it the case that thus and thus is obligatory? It is easy to see that different authorities may oblige us to per- form different acts under different circumstances. So that acts are obligatory or not for different reasons under different conditions. In short, the notion of obligation is not an absolute notion. An act is obligatory or not relative to a certain authority which obliges, or fails to oblige, us to perform that act. Different authorities may issue different types of obligation. That is to say, when there are different authorities which oblige us to do this or that, we have different notions of obligation: obligationl, obligationz, obligation3, and so on. ‘Obliga- tion' is a blanket term, there are a great many different notions under this name. We shall say that it is a set of moral rules which obliges us to do so and so. Or, in our terminology, we shall say that r (x, _t_, w)Aml or A* is obligatory if and only if a set of rules R requires that A be brought about by x at L in 3!. Different sets of rules may direct us differently, that is, issue to us different obligations. Let us, hence relativize an obligation On as that which is directed or required by rules R. In short, we will put down the following definition: (1317- 1) r—O“A’i‘n|= R requires that A* Df So defined, the notion of obligation is no longer an absolute notion. It becomes relative to a set of rules. 134 Other deontic notions can be likewise defined. For instance, r‘P',A*.' is the case if and only if R does not require that r-~A* '1 , i. e. , if and only if R allows that A*. And FFR A)!” is the case if and only if R requires that r-~A* 1 , or, if and only if R prohibits 1' that A*. Finally, IR A)!"1 is the case if and only if R does not require that A* nor does it require that r~ *1. That is to say, R allows that A* and also allows that r‘~A’V'. That is, I" *‘l l" '1 (D17. 2) p,A = ~O, ~A* Df (D17.3) rFRA*-1 2Df r19,1381"I (Dl7.4) r1,A*" = '_P,A* 8: p,~A*" Df Henceforth, when we talk about obligation, we always mean an . . . . . 1 . . 2 . . 3 indexed one, 1. e. , either obligation , or obligation , or obligation , and so on, which are, respectively, derived from, or backed up by, R R , and so forth. moral rules R1, 2, 3 We shall say that a CM-act-proposition p* is determinate under a set of (moral) rules R, if R requires that p*, or it prohibits that p*, or it allows that p* and also allows that ~p* _ That is to say, p* is determinate under R if and only if the following (17. 1) holds: (17.1) O,p* v F..p* V IRP* We shall also say that a set of rules is used to determine the ”deontic value” of a CM-act-prOposition. A set of rules Ri determines that p* 135 is obligatoryi if the set of rules requires that p*. It determines that p* is permissiblei if the set of rules allows that p*, and so on. Let us call "obligatorinessi", "forbiddennessi" and "indifferencei" deontic values of a CM-act-prOposition under R1 and denote them by '0', '2' and '1', respectively. Thus, when Ri determines that p* is obligatoryi, we shall write 'Ri(p*) = 0', and when Ri determines that >1: p is indifferentl, we write 'Ri(p*) : 1', and so on. Hence, (1?. 1) may also be written as (17. 2) Ri(p*) : 0 v Ri(p*) : 2 v Ri(p*) = 1 The notion of determinateness under a set of rules can also be extended to a class of CM-act-propositions. Let us say that a set A of CM-act-prOpositionS is determinate under Ri if every member of A is determinate under Ri' And we shall use 'Ai' to denote the set of all CM-act-proposition determinate under the set of rules Ri. It turns out that, for each i, A1 is the set of all CM-act-prOpositionS. A set of rules Ri is said to be consistent if and only if there is no CM-act-pr0position p* such that (17.3) below holds: (17.3) o,p* & O, ~p* otherwise, Ri is inconsistent. In other words, a set of rules is said to be consistent if it does not require that a certain proposition and its negation both be brought about by someone at a certain time in a certain place. Otherwise, the set of rules is said to be inconsistent. 136 Two sets of rules, Ri and Rj, are said to be compatible if and only if Ri U Rj is consistent. Otherwise, they are incompatible. Let us, from now on, talk only about consistent sets of rules. We shall say that a set Ri of rules overrules another set Rj _w1_th_ respect to, or under, a set of meta-rules Q if and only if Q directs that the determination of R1 has precedence over that of Rj. Roughly Speaking, a set of rules overrules another set under a set of meta-rules if the meta-rules dictate that the first set of rules takes the place of the second in making moral judgment or evaluation. And we Shall say that Ri prOperly overrules Rj if Ri overrules Rj and Ri :6 Rj. If we let 'Over' stand for the overruling relation, and read 'Over(R1,R2,Q)' as "R1 overrules R2 under Q", then it is easy to see that the following (17.4) holds: (17.4) Over(R1, R2, Q) 8: Over(R2, R3, Q). I) Over(R1, R3, Q) We shall say that the overruling relation is quasi-transitive in the sense of (17. 4). It is also quasi-nonsymmetric defined Similarly, as can be easily seen. A set 2 of sets of rules is said to be closed under (meta-rules) Q if and only if there is a member Ri of 2 such that Ri prOperly over- rules every other member of 2 under Q. Otherwise, Z is Open under Q. Further, a set 2', of sets of rules is said to be hierarchical under Q if some member of 2. is prOperly overrules by some other member of 2. Otherwise, Z is insular under Q. To be more precise, we shall 137 say that a set 2 of sets of rules is h_i_e_1;_achhiLa_l under Q _atth_e 9121.15. Rj(Rj E 2) if and only if there is a R1 6 2 such that Ri =¥ Rj and Over(Ri, Rj' Q). Hence, a hierarchical set (of sets of rules) under Q is a set of sets (of rules) which is hierarchical under Q at some points. And an insular set under Q is a set of sets which is hierarchical under Q at no point. Just as the concept of hierarchy under Q may be further rela- tivized, the concept of openness of a set of sets under Q can also be so Specified. We shall say that a set 2 of more than one set is 9293 Eggs}: Q a_t 1:113 po_ir£ Rj’ if and only if there is no Ri E 2 such that Ri 4: Rj and Over(Ri, Rj, Q). Hence, a set of (more than one) set(s) is Open under Q if it is open at some points. Of course, the concept of closure of a set of sets may also be likewise rzlativized. And a set of sets is closed under Q if it is closed under Q at every point except one which stands for the set that overrules every other set. We may draw a chart to illustrate the (prOper) overruling relation under Q between sets of rules. Let circles denote sets of rules. A blank circle '0' means that the set is not overruled (under Q) and a darkened circle '0' means that the set is overruled. And let Q denote the overruling relation under Q where the set represented by a circle (must be a darkened one) standing at the point of the arrow is overruled under Q by the set represented by a circle (may be a 138 blank one) standing at the tail of the arrow. Now, the following chart 1 shows that the illustrated set of rules is Open ./\°. Q (m O<———O O O+————-O (Chart 1) because it is Open at more than one point. The following chart 2, how- ever, illustrates a set of sets which is closed. It is Open at exactly one point representing the set that overrules every other set. 0 / \Q Q o o Q o o 0 fi\0 0 o Q (Chart 2) Finally, let us define the consistency of a set of sets of rules. A set z = {R1, R R3, . .. } of sets of rules is said to be absolutely 2’ consistent if and only if R1 i=1,2,3,... 139 is consistent (in the sense defined above). Otherwise, I: is absolutely inconsistent. And 2 is said to be practically consistent under Q if and only if U Rkj j=1,2,3, .. is nonempty and consistent, where each Rkj is a point at which 2 is open under Q. Otherwise, z is practically inconsistent under Q. Absolute consistency entails practical consistency. § 18. META-ETHICS AND ETHICO-SOCIOLOGY: SOME OBSERVATIONS In a formal theory of ethics or meta-ethics, we study the basic principles and/or the basic structure of an ethical theory or the study of morality. By the phrase 'ethical theory' or 'study of morality' we mean the results of reflecting on and idealization of the moral pheno- mena we have been experiencing. To theorize is to construct and reconstruct. And in the process of construction and reconstruction, a certain degree of idealization is always unavoidable. Hence, it is no wonder that certain features in our reconstruction of morality do not completely mirror their counterparts in the actual morality which we vaguely eXperience in our daily life. For instance, a certain distinction may be very hard to draw in actual morality, because the difference is so slight and delicate as to easily escape our attention. But for the sake of our discussion we might find it necessary to emphasize the distinction and bring it into the focus of our attention. Therefore, we do not claim that what we shall have said below is a faithful description of our morality; we only intend that our theory serves as a ”model" which bears the basic and important features, perhaps in a somewhat artificial manner, of our morality. 140 141 Another remark is also called for. We are not going to con- struct a complete theory of meta-ethics. What we shall try to do is to develop the theory to such an extent that is sufficient for explaining or solving the deontic problems we set forth for ourselves. We shall develop the theory of meta-ethics in a set of "assump- tions" and "consequences" making free use of the concepts and notations we established in the last section. Fir st, a list of assumptions. Each of them will be followed by some explanation or intuitive rendering. (Assumption 18. 1) All ethical systemgf moralities ( S, Q) E _a_n ordered pair consistingpigggt S 9_f sets _o_f moral rules and _a__s_e_t Q 2f meta-rules. (We shall call S a morality). Basically we think of a morality as what can be summarized in a set of moral rules. Hence we identify a set of moral rules with a morality. In an actual situation, however, this is not so clear. Althoughwe may, for instance, think of the Ten Commandments as forming a set of moral rules, i. e. , a morality, which many peOple take as their guiding principles of life, in most other cases we do not have a set of moral rules so explicitly spelt out. For instance, we might ask what is the set of rules constituting the morality to which American college students of today submit themselves. The answer is quite unclear. We may even doubt whether there exists such a set of moral rules or rule-like things. In general, a morality is only very vaguely "circumscribed" by a certain moral ideal which is more or less shared by people 142 in a certain community. What one ought to do under certain circum- stances is often left open and remains to be interpreted by established moral authority in that community. Nevertheless, we Shall assume here that a set of rules can always be, at least in principle, written down. We also take it to be the case that there are different moralities in the world or in our society. Each of them is depicted by a set of moral rules. Hence, we have a set of sets of moral rules. (We do not claim that this system is a finite set or an infinite set. We leave this question open. ) But a set of sets of moral rules is not by itself a system of morality or an ethical system. To have a system of morality, we need also a set of meta-rules which directs, for example, whether one set of moral rules overrules another when the two are in conflict. Again, it must be pointed out that we do not experience such a clear-cut distinction between moral rules and meta-rules in our daily life . (Assumption 18. 2) Every morality _i__s_ consistent. That is to say, no set of moral rules will oblige us to do some- thing and at the same time do its negation. In the symbolism we develOped in the last section, this can be put down as follows: For every set Ri of moral rules and for every CM-act-proposition p*, it is not the case that Ri(P*) = O and Ri(~P*) : 0. To assume this consistency property of a set of moral rules is indiSpensable. For, 143 otherwise, we will be unable to act morally and consistently. An inconsistent set of moral rules, for instance, may oblige one to go and not to go fight in Vietnam. (Assumption 18. 3) _I_n__a_ system ( S, Q ) o_f moralities, some (moral- ities properly overrule others under Q. That is to say, there are occasions when CM-act-prOpositions are put under the determination of different mutually incompatible. sets of moral rules, and some of them give way to others. (Assumption 18.4) The set of all CM-act-grgositions determinate under everJ set o_f moral rules _i_s_ the universal set V 9_f_ CM-act- proiositions. In other words, every CM-act-proposition is subject to moral judgment and becomes determinate under every set of moral rules. Or, as we may also say, any of our acts is submitted to morality. This assumption can be derived from our definition of the set of all CM-propositions determinate under a set of rule. But for the sake of emphasis we list it here. From the above assumptions, we may draw the following con- sequences: (Consequence l8. 1) There are _a_t least two sets_o_f moral rules_i_n a system f moralities. Pf. By (Assumption 18. 3) 144 (Consequence 18. 2) There are _a_t_ least two incompatible sets 93 moral rules ing. system of moralities. Pf. Similar: (Consequence 18.1) (Consequence 18. 3) A system_o_f_ moralities i_s_ hierarchical. Pf. Similar: (Consequence 18.1) (Consequence 18.4) A system _qf moralities i_s absolutely inconsistent. Pf. As a corollary of (Consequence 18. 2) This theorem which sounds surprising at first glance does, nevertheless, reflect our moral phenomena very well. We do experi- ence the fact that we have submitted ourselves to different moralities which are not compatible with one another. The morality of being a good parent, for example, may not be compatible with the morality of being a good husband. The morality of the battle field in Vietnam may be at great variance with the morality of a university campus. And the morality for the man in the street may be quite different from the morality for the justices of the Supreme Court, and so on. Indeed, occasions may often arise when we find ourselves directed by two sets of incompatible rules simultaneously, and no meta-rules dictate which of these sets of rules overrules the other, or dictate that they are both overruled by still another set of rules. Hence, it seems that what this theorem says is nothing other than a simple truth in our morality. This fact should be recognized in order to prevent certain false convictions about our morality. 145 It seems that we Should not make an unreflecting use of Ross' distinction between actual duties and prima-facie ones to solve every case of conflicting obligations. Although the distinction may be drawn theoretically, it is naive to think that we can apply this distinction to solve every problem of conflicting obligations. For this conviction presupposes that when there are incompatible sets of rules which gener- ate conflicting duties there is a meta-rule (explicit or otherwise) which directs that one of these sets overrules the other, or that a third set of rules overrules these two sets, and, besides, that this set of over- ruling rules will finally give us the answer what is our actual duty. This, however, is too strong a presupposition as we suggested. in § 15. Hence, the above-mentioned conviction seems far too oversimplified a solution to the problem of conflicting duties. It leaves us with a false idealistic view of morality that, under given circumstances, there is always a set of rules which will tell us what we really should do under those circumstances. Had this been the case, our moral life would be much easier to lead; and many more people would go to bed every night happier and with a much more peaceful mind. Unfortunately, the truth is that our moral system is inconsistent absolutely or practically and so we may find ourselves simultaneously submitted to some incompatible sets of moral rules. All the tragedies and tears are not always super- ficial. There are the genuine moral dilemmas, moral predicaments or moral perplexities! They are insolvable by any means, not to mention the verbal distinction between actual and prima-facie duties, so far as 146 our system of moralities remains inconsistent and so far as we try to solve the problem not by cutting the Gordian knot but by carefully untying it. However, to include (Consequence 18.4) in our theory is by no means to claim that a system of moralities Mbe inconsistent. What we claim is only that our system of moralities i_s_ inconsistent. There is no reason why there should not be, and indeed it is welcome that there is, a consistent system of moralities in our world. (And, hence, our (Consequence 18.4) becomes false of such a system. ) Many national and international problems come from the fact that we tolerate or are prepared to tolerate inconsistent systems of moralities. A consistent system of moralities may well emerge in the remote future. But this question is beyond our concern here. The above assumptions and consequences may be thought of as some observations in meta-ethics. Some additional remarks. Although we proved that a system of moralities is absolutely inconsistent, we are unable either to prove or to disprove that it is practically consistent. That is a question we leave Open. We also leave it open whether or not a system of morali- ties is infinitely hierarchical. Ethico-sociology studies the organization and structure of morality in a society. It stresses the relation between man as member of society and morality as a social institution. The following are some observations, again, arranged as assumptions and consequences. 147 (Assumption 18. 5) Asociety institutionalizes various roles for its members tgpl_ay. For example, a man and a woman become husband and wife because of the social institution of marriage. One person is a teacher of another owing to the social institution of schooling. A man becomes the comrnander-in-chief of his fellow men because there is a social institution of a nation. And a man becomes a priest due to the fact that there is a certain religious body, and so On. All the political, religious, economic, and educational organizations, and so on, are social institu- tions in the wider sense. Even the natural relation between mother and child is institutionalized in a civilized society like ours. Motherhood becomes a socially institutionalized role which a woman plays in a society. It is a role quite different from, for example, that which a mother duck plays to her ducklings along the banks of the Red Cedar River. (Assumption l8. 6) _A_t arg time, every man plays 3.1: least one role. (Assumption 18. 7) Some men play different roles simultaneously. For instance, one may at the same time be a husband and a father, a mathematician, a soldier, and a philOSOpher, and so on. (Assumption 18. 8) _A_ sociceiy institutionalizes with each role exactly one morali_ty. (We shall call this morality the morality associated with this role. ) 148 (Assumption l8. 9) Different roles have different moralities. (Assumption 18. 10) I_fa man plays a role, then the society which Elsa puts him under, 93 submits him t_q, the morality associated with that role. (We shall also say that a man submits himself _t_g the morality. ) Let us now define the notion of one morality overruling another as meaning that the set of rules of the fir st morality overrules the set of rules of the second morality. Here we let a certain set Q of meta- rules tacitly under stood. (Assumption 18. l l) I_f__a man submits himself 3) incompatible moralities, and there _i_s_ another morality which overrules these moralities, then the man submits himself _t_q the overruling morality. If there are many overruling moralities, mutually compatible or not, then he submits himself to all of them. From these assumptions the following consequences can be inferred. (Consequence 18. 5) A society institutionalizes a system if moralities. Pf. From (Assumption 18. 5) and (Assumption 18. 8) (Consequence 18.6) Every man (in a sooim) submits himself t_o _a_t least one morality _a_t a time. Pf. From (Assumption 18. 6), (Assumption 18. 8), and (As sumption 1 8. 10). 149 (Consequence 18. 7) There are_a_t least two different moralities i_n_a_ society. Pf. From (Assumption 18. 7), (Assumption 18. 8), (Assumption 18. 9), and (Assumption 18. 10) (Consequence 18. 8) Some men submit themselves to different moralitie 3 simultaneously. Pf. Similar: (Consequence 18.7) It may be mentioned that from the results we stated above, we can neither prove nor disProve the following thesis: (18. 1) Some men do submit themselves to incompatible moralities simultaneously. Hence, we leave Open the following "corollary" of (18. 1) (18.2) The conflict of obligation is possible. A man may find that he ought to do something and at the same time ought not to do it. Or, he may find that he ought to perform a certain act and at the same time ought to perform its negation. A moral predicament. A final remark may be in order before we come to the end of this section. It should also be realized that in our theory we can neither prove nor disProve the following: 1The truth of this statement should be determined in a meta- theory of morality rather than in a system of morality. 150 (18.3) For any two incompatible moralities, it is always the case that one of them overrules the other. or (18. 4) For any two incompatible moralities, there is always another morality which overrules them. Should (18. 3) or (18.4) be true in a society, then we could prove that the system of morality in that society is practically consistent provided that the system is finitely hierarchical. However, as we mentioned in the last section, the practical . consistency of a system of morality is left Open. This should be so, because of the undecidability of (18. 3) and (1 8. 4). S 19. THREE SYSTEMS OF DEONTIC LOGIC: CMo,T*, CMORS4* AND CMORSS* In accordance with what we have developed in 33 15—16, we shall outline here three new systems CMORT*, CMOR 54* and CMORSS* of deontic logic which are, reSpectively, variant systems of OT*, 054* and 055* of the last chapter. First, the vocabulary (common to all three systems): (I) Vocabulary i) Variables for agents or actors: 'xl', 'xz', 'x ' °‘ ° ' o I I I I I I 11) Variables for time. t1 , t2 , t3 , . . . iii) Variables for location: 'wl', w ', 'w3', iv) Act(-type)variables: 'p', 'q', 'r', 'pl', v) Deontic connectives: '~', '8:', 'v', 'D', '2' vi) Sentential connectives: '~', '&', 'v', '3', "5' viii) Parameter delimiters: '(', I)! ix) Grouping indicators: ‘[', ']' Next come the formation rules. They are again cormnon to all three systems. 151 152 (II) Formation rules i) An act(-type) variable standing alone is a deontic term. ii) If 91 and 92 are deontic terms, so are r~el1’ "[91 & 92:11, ,. 1". [61 :> 92]" and l'[e -=- 9 11. r. [91v 1 2 92 iii) If a. B.'Y are, respectively, variables for agent, time and location and 93 is a deontic term, then F(a, B, v);I is a CM-act- sentence (form). iv) If X and Y are CM-act-sentences, so are r'~X-1 , r[X 8: YT, rtx v YT, r[x :3 Y? , and r[x s Yl'. v) A CM-act-sentence is a _vyf_f (well-formed formula). vi) If A is a wff, so is rORAC'. vii) If A and B are wffs, so are l-~A1 , l”[A 8: B]-1 , rfA v B]_', '_[Aa B].l and rIA 5 13]". viii) An expression is a deontic term only if it is so formed by i)-ii); it is a CM-act-sentence only if it is so formed by iii)-iv); and it is a wff only if it is so formed by v)-vii). A remark is called for. We have declared above that a deontic Operator like 'Oa' takes CM-act-sentences as its operands. Accord- ing to this proclamation, it seems that certain deontic expressions such as (19.1) 0,0,Lx._t, Mp in Which iterated deontic operators occur, are not well-formed, because 153 (19-2) OMS. L EDP which is the Operand of the first occurrence of 'OR' in (19. l) is no_t a CM-act-sentence, although (19.3) (35.13. 39p which is a sub-sentence of (19. 2) but is _n_9_1_: the operand of the fir st occurrence of ‘OR‘ in (1 9. l), is. Let us say that the fir st occurrence of 'On' in (19. 1) has (19.2) as its scope, and it refers, indirectly perhpas, to (19. 3). Let us use the following chart to convey the general idea of this (indirect) reference. (19.4) ’,/”'",3 ------ be 9 OR 0,0,(§._t. wp ( l ““‘“‘ i where the broken arrows indicate reference, the ”corner ed” lines indicate, re Spectively, the scopes of the Operators standing imrnedi- ately left of them. It may be noted that we may have tildes in addition to deontic operators among the Operators we mentioned above. We Shall call any occurrence of a deontic Operator except the right-most one in an iterated series of deontic Operators an iterated operator. For example, all the occurrences of 'On' in (19. 4) except the last one are iterated. In addition, we shall count the iterated deontic operators from right to left as first-degree iterated deontic Operator, second-degree deontic Operator, third-degree deontic Operator, and so on, while an uniterated deontic operator may be thought of as a 154 zero-degree iterated Operator. For example, the first (left-most) occurrence of 'OR' in (19.5) below is a fourth-degree iterated deontic Operator in that expression, and the last (right-most) occurrence of 'OR' in the same expression is a zero-degree iterated operator. (l9. 5) ~Oa O,,O,,~OI,Ofl (35, 1:, w)p Now, our proclamation about the Operands of a deontic Operator may be restated as follows: an uniterated or isolated deontic operator, as we said before, takes as its Operand CM—act-propositions; a first- degree iterated deontic Operator takes as its operand a deontic expres- sion in which there is an uniterated deontic operator Operating over an CM-act-pr0position‘ which the fir st-degree iterated deontic operator indirectly refers to. Likewise, we can specify the Operands of a deontic operator of a higher degree. In fact, a set of recursive definitions can be given to Specify more precisely the operand of any deontic Operator of any degree. However, we shall not do it here, because the main interest of setting up those definitions is to characterize the set of well-formed formulas in our new system(s) of deontic logic. This job is done effectively by the formation rules we put down above. Deontic operators other than '0: ', namely, 'PR' 'Fa' and “3‘ are defined in the usual way. Before we can see in what sense we shall make our present systems of deontic logic CMO, T* - CMORSS’k, reSpectively, variant systems of OT* - 085*, we shall first set up a set of rules which can 155 be used to effectively enumerate CM-act-sentences of the following form: (19.5) (1:14.. )p w J ""rn where p is a deontic variable. We have on several earlier occasions used expressions like (>2. L mp in which CM-parameters are not subscripted. This should be regarded as only an informal way of writing down a CM-act-sentence. Indeed we listed at the beginning of this section as variables for these parameters only ones with subscripts. The reason for doing so should become obvious in a few moments when we have set down the rule for enumerat- ing the above-mentioned CM-act-sentences. Now, the rules: i) Let Q be a function which maps the set of CM-act-sentences {B | B = (—i' tj, v_vm)A} to the set of superscripted starred sentences {A*k I k is a natural number} in the following manner: (zi .zj .zm) A* l 2 3 ' k- z1 zj zm {>(B)— i.e., .. 1. 2. 3 where 'zs' denotes the 3th prime in the natural order, namely, 2, 3, 5, For example, M (£1, _tl, 31.1)P) = P = p . Due to the unique factorization theorem, ii) iii) (19.6) 156 this enumeration is unique and unambiguous. We shall call k the index of the starred sentence p*. Take p*'s, arrange them according to the magnitude of their indices. We then have an enumeration Do the same as prescribed in ii) to q*'s, r*'s, pl’k‘s, q’lk's ri" 's, p; 's, and so on, we have the following set of sentences: #1772 3194* 1//// 7:7/7/7 / 7/73/7377// .:/7::./ / 17/77/ 3/.:.:3.// / /‘ 137/37437/47/3/77/ / 1.7:7// :733://:7/ ./ / .37/ 3/ .3:// 7:7/ ::/‘/ 177/ 277// 777///7 157 iv) Use diagonal method as indicated by the arrows in (19. 6) to enumerate the sentences in (19. 6). Denote this enumera- tion by (19.7) C This is the desired enumeration of all (atomic) CM-act- sentences we mentioned above. It may be noted in passing that this set of rules also provide an effective way to abbreviate any atomic CM-act-sentence to a starred 2250 * 1 sentence. For example, '(lgl, _tz, _v_v3)plz' is abbreviated as ‘plz . So far we have taken into consideration only the CM—act-sentences in which the deontic terms are atomic. However, it is readily seen that every CM-act-sentence can be reduced to a CM-act-sentence with atomic deontic terms by (16. 17) and (16.21) in 3 16 together with the usual transformations among the truth-functional connectives. Hence, the procedure outlined above applies to any CM-act-proposition. Next, we set up another mapping Y from the set of sentential variables p, q, r, pl' as given in the vocabulary for systems OT*-— 055* to the sentences in (19.7) in the following way up) = c1 . ‘Y(q) = c2. Hr) = c3, Y(pl):c4 no... and so on. 158 We then transform a theorem, including the axioms, of * * * * * * . OT (084 or 055 ) to that of CMORT (CMORS4 or CMORSS ) 1n the following way: 8 (mm '03" "*’(p)', 'Y(q)'. 'Wr)‘. "Y(p1)', is a theorem of CMo,T*(CMo,s4* or CM0,55*) provided that '(Thm)' is a theorem of OT*(OS4* or 055*). Furthermore, we may use the same numbering for theorems of these new CMOR ~systems as we did for O-systems. For example, (0T*19) ~(op & O~p) is a theorem of OT*, then (cmo,'r*19) ~(o,w(p) & o,~?(p>) or “40,01 & o,~cl) 0r *1 *1 ”(Cap 8‘ 0a NP ) or 30 3O ~(ORP* 8‘ on ~P* ) or finally Motel, ._v_v1>p & 0.~(§1. 12.14pm Ll DOI bee YE! 011 th [1 L1 159 is a theorem of CMORT*, and so on. The results, then are three alternative deontic systems which are mapping images of our three previous systems. Let us now compare these two different types of deontic logic. They are different in two significant respects: (1) the range of "deontic variables" or the operands of a deontic operator, and (2) the deontic notions standing behind the symbols '0' and 'OR '. Although enough has been said about them in section 16 and section 17, some additional remarks on the second feature of difference seem desirable. We may say that in the previous deontic systems, '0' stands for our £9311 notion of obligation which is backed up by the whole system of moralities of our society. However, it has been said in the last section that the system of moralities in our society is absolutely inconsistent. Whether it is practically consistent remains to be seen. Hence, it may well be the case that our total notion of obligation is not a ”consistent'' notion. Thus, it is no wonder if we find that sometimes holds. This indeed is the core of trouble which accounts for the difficulties in Chisholm‘s dilemma and the dilemma of conflicting obligation s . Our new systems of deontic logic, on the other hand, formalize only a Specific partial notion of obligation 0a which has a set R of moral rules standing in the background. And since we have agreed 160 that R is a consistent set of moral rules, the following cannot be the case: >3 * ORP & OR ~p This fact simply follows from the definition of the consistency of a set of moral rules. Of course, if our whole system of moralities is consistent, our previous systems of deontic logic will be as good as the present systems in this reSpect. They may even be more desirable ones if we provide those systems with CM-act-sentences, for they formalize the total notion of obligation rather than a partial one. E 20. HINTIKKA-KRIPKE SEMANTICS FOR CMOR T*—-CMo,ss* We have so far outlined three alternative systems of deontic logic CMOR T*, CMORS4* and CMORSS’k and shown that they are, reSpectively, the mapping images of our earlier systems 0T*, 054* and 085*. But, as we recall, we developed 0T* - 095* as axiomatic systems, and what we have for CMOR T* - CMORSS’k are likewise syntactical characterizations. In this section, we shall supplement them with semantical theories. The semantics we shall present is Hintikka's model set and model system with a minor modification in the manner of Kripke. 1 First, we shall call a deontic model s_et any set u. of CM-act- formulas (sentence forms) which satisfies the following conditions: * r *1 ((320.1) If A G g., then ~A 6' (c20.2) If I'[M‘ & 13*]_I e u, then A* e u and B* e u (czo.3) If F[A* v 13*)" e u, then A* e u or 13* e u, 1See, for example, Hintikka [1963] and Kripke [1963]. Hintikka's method is applied to deontic logic in Hintikka [1957] , Esper son [1967] and Aqvist [1967] , while Kripke's method is applied to deontic logic in Hanson [1965] . 161 162 Intuitively Speaking, a deontic model set is a partial description of a possible world. It depicts only the portion of a world which has to do with human endeavorings (which are what CM-act-sentences are designed for). Further, we let 0 be a non-empty set of deontic model sets defined above, and g be a binary relation defined on 0. And we let v be a member of O . We shall say that the orderd triple ( 0, v. _Ii) is a deontic model system if and only if the following conditions are satisfied: (C20. 4) It is not the case that Vliv (C20. 5) For every Lb E 0, there is at leat one p,+ such that Lit-1.13.11. (czo.6) If r'o,,A"‘_'e u. 6 0, then A* 6 up“ for every u,+ e a such that Lit-EU: (C20.?) If r15>,,A”"' e n e a, then A* 6 11+ for some 11* 6 n such that [’_ng The relation 3 above is called "copermissibility" by Hintikka and “deontic alternativeness" by Aqvist. 2 We shall follow Aqvist in saying that LL+ is a deontic alternative to u, if “+3“, holds. Intuitively, we may think of ”21' as a morally ideal world with respect to u, in the sense that what is obligatory in u. is actually the case in (1+. And we may informally think of n as a set of possible worlds, and v our actual world. To postulate the condition (C20. 4) above is to say 2Hintikka [1957] and Aqvist [1967]. 163 that the actual world is not a morally ideal world, because (C20. 4) implies that there may be "some things" which ought to be done but are not actually done in v . Now we say that a deontic model system is an 0T* (deontic) model system if the following condition holds: (C20. 8) B is reflexive on 0* where 0* is {fl-v} . It is an 054* (deontic) model system if in addition to being an 0T* model system, it also satisfies the following condition: (C20. 9) R is transitive on Q And finally a model system is an 055* model system if it satisfies (C20. 9) above and the following: + + + + + + (C20.lO) If “.13“, and (1,211“. then “'13“? for every u. lull: uz Next, we shall introduce the concepts of satisfiability and validity. We shall say that a set A of CM-act-formulas is OT*(OS4*, 055*) satisfiable if and only if there is an 0T* (054*, 055*) model system < 0, v, 3 ) such that 7t is a subset of some member of O. 3 Now, a CM-act-formula A* is 0T* (054*, 035*) satisfiable if and only if the set {A*} having A”: as its sole element is satisfiable. We say that A* is 0T* (054*, 055*) eontradictog if and only if A* is not 0T* (054*, 055*) satisfiable. And, finally, Vie say that a CM-act-formula A* is 0T* (054*, 055*) 311331 if and only if r ~A* '7 is 0T* (054*, 085*) contradictory. 3Hintikka will say that A is imbeddable _i_n fl. 164 We shall not try to show that CMOR T* - CMORSS* are consistent and complete with reSpect to 0T* - 055* validity, reSpectively. How- ever, as we shall apply this semantics in the next section to cope with a certain puzzle in deontic logic, namely Chisholm‘s dilemma, it seems desirable that we show how to determine that a formula is valid or contradictory by means of the model system. Let us do this by showing that the deontic axioms in CMOR T* - CMORS5* are, reSpectively, 0T* - 085* valid. First, for the sake of simplicity, we shall put down the deontic axioms of CMO.T* as follows: (Ad*4) l- o,(p* 3 q*) :> (o,p* 3 o, (1*)4 (Ad*6) }— o,p* 3 ~o,~p* (Ad*7> :— 0,((o.p*> a p*) To show that (Ad*4) is 0T* valid, we show that is negation is OT”< contradictory. Suppose it is not contradictory, then there is at least one member u. of Q in (0, v, 3) such that O,(p* 3 q*) 8: ~(O,p* 3 0..C1*) 4 This is an informal way to write the axioms. According to what we. have said in 319, we should have written (Ad*4) as 30 30 3o 30 t0n[P* Dq* 32>[0sp* Donq* J or better as *- OJBI, £1» 31):) : (£1.L1.2V.1)q]:> [Okay £1: 31)}; D ORLEI: £1: KIM] 165 that is On(~P* v q*) 8: ~(~0Rp* v 0,, q*) is a member of a subset of )1. By appealing to (020. 2) above, we have and ~(~o,p* v o,q*) e n 1.8., (o,p* & w,q*) e u or (0.p* & P. ~q*) E ii That means, by (C20. 2) again, we have the following: (20-1) O..(~p* V q*), 0.. 9*. P. ~q* E u But according to (C20. 7), since P.~q* 6 u, there is at least one LL+ E 0 such that n+3“, and (20. 2) ~q* 6 J And by (czo.o) and (20.1), since o,(~p* v q*), Onp* e u, we have the result that (~p* v q*) and p* are members of every pf; where LL13“. In particular, we have the following: But by (C20. 3) and (20.3) either ~p* e u+ or q* 6 n. If the first possibility, then this together with (20. 3) implies that (20.4) p* e n+ and ~p* e n+ 166 On the other hand, suppose that the second possibility holds, then this and (20. 2) implies that (20.5) q* 6 u,+ and ~q* E 1.1+ In either case, we conclude that u.+ is not a model set which contra- dicts our assumption that it is one (because 0+ 6 0). Therefore, the negation of (Ad*4) cannot be a member of (a subset of)an 0T* model set. This in turn implies that the negation of (Ad*4) is 0T* contra- dictory. Hence, (Ad*4) is 0T* valid. Likewise, we can easily show that (Ad*6) and (Ad*7) are 0T* valid. To prove that (Ad*o) is 0T* valid, use (C20. 1)-(c20. 3) and (C320. 5)-(20. 7). To prove (Ad*7) is 0T* valid, use (020. l)-(C20. 3) (C20. 5)-(C20. 8). It is easy to see that the above axioms are also 054* valid. We may then show that the following axiom—which when added to the axioms above forms the axioms of CMORS4* — is 084* valid; (Ad*8) I- 0.p* D 0.0.p* To see this, we apply the same procedure, showing that (20.6) ~(o,p* 3 o,o,p*) cannot be a member of a subset of a model set in an 054* model system. It follows then that (20. 6) is 0S4* contradictory. That is, (Ad*8) is 054* valid. The proof involves the essential uses of (CZO. 9). 167 Similarly, we may show that all the above axioms are 0S5* valid. And it is also readily seen, by using (020. 10) among other things, that the following axiom which is the "characteristic axiom" of CMORSS": is also 0S5* valid: (Ad*9) )— ~03 p* I) 0,, ~0R p* § 21. CHISHOLM'S DILEMMA, THE DILEMMA 0F CONFLICTING OBLIGATIONS AND THE PARADOX OF THE GOOD SAMARITAN REVISITED Now that we have exhibited alternative systems of deontic logic and pinpointed the troubles from which some of the deontic problems associated with the earlier systems of deontic logic come into being, let us now pay a renewed visit to what we have called Chisholm's dilemma, the dilemma of conflicting obligation, and the paradox of the Good Samaritan. Chisholm's dilemma, as we recall, is a predicament to the following effect:1 Certain systems of deontic logic are such that either they are unable to formulate contrary-to-duty imperative, or they contain a contradiction. Now, in our new systems, we want to show that the notion of contrary-to-duty imperative can be accounted for without running into a contradiction. A contrary-to-duty imperative eXpressed in the earlier systems takes the following form: (21 . 1) One ought to do q. (Oq) and lot. 3 4. 168 1's," . 169 (21. 2) One ought to do if-q-then-p. (O(q 3p)) But (21. 3) If one does not do q, one ought to do not-p. (~q :3 0~p) Now, (21.4) One does not do q. (~q) Therefore, (21- 5) One ought to do not-p. (0~p) But, from (20.1) and (20. 2) we have (21. 6) One ought to do p. (Op) Therefore, from (21.5) and (21.6), we get (21. 7) One ought to do p and one ought to do not-p. (Op & 0~p) However, there is a theorem in our previous system of deontic logic which reads: (0T*l9) It is not the case that one ought to do p and one ought to do not-p. (~(Op 8: 0~P)) Finally, from (21. 7) and (OT*19) we have a patent contradiction. Now, in our present systems of deontic logic, we shall, for the time being, read the above argument in the following way: (CM21.1) One, say 951, ought to bring it about that q at L1 in 3511 (0,651. _tl. E1)q) 170 and (CM21. 2) One ought to bring it about that if-q-then-p at _t1 in w . O.(§1,_tl, ElHq 3 p)) (CM21. 3) If one does not bring it about that q at _t1 in v_v_1, then one ought to bring it about that not-p at__t_1 in _v_v.1. (4&1: _tl: Elm 3 Dual. _t_l. E1)~P) Now, (CM21.4) One does not bring it about that q at _t in w . (~(a1,_t1._vy1)q) Therefore, (CM21. 5) One ought to bring it about that not-p at_j;1 in El. Indeed, from (CM21. 1) and (CM21. 2), we can derive (CM21. 6) One ought to bring it about that p at_tl in 121. (0.. (a1. 1.1. 3011);?) Because '(xl, _t_l, 31) (q 3 p)‘, as we mentioned before, has the same truth condition as'(xl, _tl’ 351m 2) (x1, _t'l’ _w_1)p', and because D . OR (£1, _t.l’ El)p is a variant theorem of (CMO.T*35). 171 And from (CM21. 5) and (CM21. 6) we have (CM21. 7) One ought to bring it about that p at _t1 in v_v_1, and one ought to bring it about that not-p at “£1 in 311. which when abbreviated is *30 *30 (CM21.8) o,p at o, ~p But 30 *30 (CMO,T*19) ~(0.p* & 0s~p ) is a theorem of our new system(s). Hence, a contradiction. Let us now look into the trouble by examining some of the possible exits out of this difficulty. First, it may be argued that there is some undesirable logical reasoning involved in the generating of the above contradiction. In particular, pe0p1e might want to argue against the derivation from 0.4%, ._t.1. .vyl)q and 0.(@1»_tl. ylm D (£1._tl.y1)p) to 0 (51, _tl, _vgl )p, having in mind that a set of moral rules R may require (_1, t1, w1)q and (x1, 1_t,___w1)q 2) (__1,1, w1)p but fail to require (x1, 11, E )p, Indeed, given that B is a logical conse- quence of A, R may require A without requiring B. Hence A may be obligatory but B may not. However, we shall not take this as a prOper way to escape our present difficulty, because this will upset our deontic systems—to be sure, any deontic system we have seen so 172 far—to a great extent, and force us to construct a deontic system anew. A second alternative to escape the above difficulty could be this. Since the assumption that our morality admits of the restorative course of action leads to a contradiction, hence, by reductio _a_d absurdum, our assumption cannnot be true. That is, our morality allows no such restitution. Again, this approach will be dismissed without any further ado, because this seems only to throw away the baby with the bath water, for we have said early in the last chapter that reparative course of action is desirable. The third and last alternative we shall consider, and indeed the one we favor, is to differentiate several levels of obligation within a morality. When we say rule R requires that p* , we mean, accord- 3O ing to our earlier definition, that p* is obligatory, or in short a): ORp We shall call this obligation a primary obligation. After we fail to fulfill our primary obligation, the moral rule R may require that we do ~p*3o as restitution or reparation. This will be called a secondary obligation, and will be symbolized as Indeed we may have, as we suggested in Chapter One, further levels of restoration, and hence, further levels of obligation: tertiary obliga- tion, quaternary obligation,. . .,n-ary obligation, each is introduced 173 only after the preceding one fails to be accomplished. We shall symbolize them, reSpectively, by After different levels of obligation are introduced, we shall stipulate a meta-moral rule to prescribe how different levels of obliga- tion are in force. The meta-fule is: (M21. 1) If a primary obligation and a secondary obligation are in conflict, the primary obligation is in force. If the primary obligation, for some reason or other, cannot be fulfilled, the secondary obligation is in effect. In this case, the primary obligation ceases to be in force. This seems a plausible rule indeed, because, as we mentioned above, it is only after the primary obligation fails to be accomplished that the secondary obligation goes into effect. Thus, at the time the secondary obligation takes effect the fulfillment of the primary obliga- tion is already out of question. The primary obligation is no longer in effect. Now, the difficulty which Chisholm pointed out does not threaten our present system of deontic logic in the same way as it did the former system. For we now only have 30 30 (CM21.8') o,p* at o;~p* 174 rather than (CM21. 8) above. Therefore, we do not have a contra- diction. We may note that in the case of (CM21. 8'), p’”‘30 .is a primary obligation prescribed by R , (but since it fails to be fulfilled for some reason or other), R then prescribes ~p*30 as a secondary obligation. According to (M21. 1), only this secondary obligation, not together with the primary one, is now in effect. Of course this does not mean that a man can escape his (primary) duty by shifting to an obligation of a lower level. An unfulfilled primary duty is still left unfulfilled. A reparative secondary obligation, for example, gives one a chance for compensation, it does not offer one a moral escape. This seems indeed a proper way, if not the prOper way, to get our selves out of the contradiction. For the primary obligation and (different levels of) the so-called contrary-to-duty imperatives, or, contrary-to-duty obligations, as we prefer to call them in this case, are not in force at the same time. A secondary contrary-to-duty obligation is in force only if the accomplishment of the primary obliga- tion is no longer to be considered; in general, an n—ary contrary-to- duty obligation is not in effect unless the primary obligation, the secondary obligation, and . . . and the n-l-ary contrary-to-duty obligation are all out of the question. However, it is immediately seen that (CM21. 8') above is not expressible in our present systems of deontic logic. But we may try 175 to extend our systems to incorporate different levels of obligations in such a way that (CMOR T*19) and 3 3o (CM21.9) ~(O,',p* O h 0:, ~ p* ) and, in general, are all theorems, but (CM21. 8) above is not. At this point we find that in one of his recent articles, Aqvist expresses the same idea of differentiating levels of obligation. 2 His solution to Chisholm‘s puzzle is similar to what we proposed above and thus can be readily transferred to our systems. In the last section we Spelt out the semantics for our present systems of deontic logic CMO.T* - CMORSS’k. But, as we recall, we have COped only with a deontic Operator 'OR' (and the corres- ponding 'Pn'1- Now, since we want to talk about a lower level of obligation, we shall introduce another deontic Operator '0;' (and the corresponding 'Pl',‘). Here we may take 'Onp’k' as saying that p* is a primary obligation, or that p* is primarily obligatory; and 'O,‘,p*' as saying that p* is a secondary obligation, or that p* is secondarily obligatory. With this introduction of an additional deontic Operator '0" we need then to make some correSponding adjustments in the semantic 8. 2See Aqvist [1967] . 176 Let us call the systems resulting from adding '0; ' to CMOa T* - CMORSS’“ (and also the other necessary amendments) CMORT’H" - CMORS5*+, reSpectively. Now for these ”extended" systems, we have the same notion of model set as we defined in the last section. But we want to Specify another relation in addition to R (the deontic alternativeness) of the last section in order to bring to light the relationship between the primary and the secondary Obligations. We denote this new relation by 'R°' and call it compensatory deontic alternativeness. We Shall read 'u’Rou' as 'u,’ is compensatorily alternative to u, '. 3 We Shall further stipulate that 0;, P;, and 3° have the correSponding properties postulated in (C20. 4)-(C20. 7). An OT*+(OS.4*+, 055’”) model system will then be defined as an ordered quadruple ( 0, v, 3, 5°) (where '0', 'v' and '_I_{_' are the same as defined in the last section) such that the following additional condition holds: (c21.1) If O,A*, 3* 6 n e O and u’R°u, where u. e 0 and 3* is truth functionally incompatible with A*, then B* 6 u,’ In addition, OT*+(OS4*+, oss*+) satisfiability and validity are defined in the manner parallel to the definitions in the last section. 3Aqvist calls R° “perfect deontic alternativeness'l and read 'u,’ Rou' as 'u.’ is an ideal extension of u,‘ . Ibid. 4Aqvist has an ordered triple < 0, R, R} ) in this case. But this model system fails to further distinguish between an OT*+ model system, an OS4*+ model system and an OSS*+ model system. 177 It may be observed that the condition (C21. 1) warrants, roughly Speaking, that the violation of a (primary) Obligation can be compensated in a compensatory deontic world as a (secondary) obligation. Chisholm' s puzzle. We can see that this immediately gives us a way out of As we have said above, what is said to be the Chisholm's puzzle is not a genuine contradiction, but one which looks like the following: (CM21. 8') O,p* 30 3O 8: 0:, ~p* But now it is easily seen that (CM21. 8') is not a contradiction in either of the systems OMO,T*+ - OMO,55*+. (CM21. 8') is OT*+ (also 054’“ and 055’”) satisfiable. model system 0 suffices to Show the desired result: That is, we can Show that The following + Q = {Hit U! 0 L1,} 3: {} 3° = {1 ”14.-..-- 5° ----------- u, -------- 3""’""">LL+ 30 *30 *30 *30 30 0013* 178 Next, we Shall observe how our new systems shed fresh light on the dilemma of conflicting obligations. This dilemma can be briefly stated as follows. In some systems of deontic logic, either it is impossible to express the notion Of ”conflicting Obligations" adequately, or else, they contain a contradiction. "Conflicting Obligations" is a Situation in which we are obliged to do several mutually incompatible acts, in particular, a certain act and its negation. Or, in symbols, Op & O~p It is evident from what we have arrived at in the last section that no Single set of moral rules R will oblige one to do something and its negation, because every set of moral rules is consistent. That is to say, (21.9). ORp* & O, ~p* 6 is always false for every OR. A conflict of duties can only happen when one submits oneself to incompatible moralities simultaneously, and hence there comes the moral predicament. That is, the following might be true: (21.11) O,p* & 0,! ~p* 5Cf. section 15. 6We Shall again use the unsuperscripted 'p*' etc. on the occasion when no confusion seems likely. 179 where R .11: R'. But this is not a contradiction in our systems Of deontic logic. Of course, (21. 11) is again not even expressible in our new systems. However one might again try to think of extending our systems in such a way—adding, among other things, an additional deontic Operator '0“: ' to joint the 'OR‘ we already have—that will accommo- date expressions like (21. 11). If this extension is carried out, then whether the resulting systems contain a contradiction or not, depends upon (i) whether we have meta-rules which direct overruling relation among different moralities, and (ii) what is the actual direction these overruling rules give. It might be the case that we Shall have a rule to the following effect: (21.12) If 0,, and OR: are incompatible and Onp* and ORr~p*, then there is an Or which overrules OR and 03’ , and Oqu* (or F..q* or IR.q*). where q* may be either p* or ~p*. In this case, the conflicting obligations (i. e. , between obliga- tion and Obligation' ) is settled. They are replaced by another new Obligation (i. e. , Obligation"). This is a Situation which comes very close to Ross' conception of finding the actual duty when there are conflicting prima-facie obligations. But it might also happen that there is no other On" which will overrule OR and On' ; but these two are still incompatible and are 180 no longer overruled by any other morality. In this case, our system is practically inconsistent. Conflicting Obligation is a genuine unsolvable problem in this case. However, it should be realized that this problem comes, as we showed in the last section, from the inconsistency of our system of moralities. It does not come from any inadequacy in our deontic logic. We come now to the paradox of the Good Samaritan. The follow- ing may be thought of as one of the possible solutions. First, we know that this paradox comes from the fact that we allow the following "principle" to hold: (21. 13) What entails a forbidden act is itself forbidden. or 7 (OM28) (Fp & q—3P) D Fq However, (21. 13) in our present systems demands a careful reformula- tion. We shall prOpose the following: (21. 14) What entails a Subsequent forbidden act is itself forbidden. Or, in our terminology, D (21.15) [F, L152. —tz' _sz)p &- (31.11. 3.1)q—a(_>22,_t2, E2)P] 7See 2 l3. 181 where x , _w and 1 1 'GW' means (2) comes after CD. It is clear, then, that the Good Samaritan did not do something may be the same, respectively, as x , _w 2 2’ forbidden by helping the victim of robbery, because to help a victim of robbery does not entail that a subsequent robbery occurs. That is to say, if we extend our systems by including alethic modalities and some predicate symbols, in particular 'G@®' we Shall reject (OM28) Of Anderson in favor of (21. 15). Of course, we do not claim that this is the only way to over- come or bypass the paradox of the Good Samaritan. Perhaps the con- cept of "entailment" is simply too strong for deontic logic in that Situation. 3 22. QUANTIFIERS AND ALETHICMODALITIES IN DEONTIC LOGIC Since we have confined ourselves to propositional deontic logic, the problem of quantifiers lies outside the scope of our discussion. However, after we have Observed the present type Of deontic logic, the introduction Of quantifiers seems a natural step of further develop- ment. Hence, this section seems a reasonable place for us to briefly discuss the matter. We Shall also briefly review, in the latter half of this section, the problems which arise when we incorporate alethic modal- ities into deontic logic. As we recall, a CM—act-sentence in the new systems of deontic logic has the following logical form: (22.1) 1 x at L in yv_ brings it about that p. or, in our symbolism, (22- 2) (a, _t, EDP which contains four different kinds of ”individual" variables. It is easily seen that each of these variables can be quantified. For instance, we may write down a quantified sentence like (22. 3) For any man at any time in any place there is some act which the man brings about. 182 183 This is an awkward logical version Of the following more natural sentence. (22. 4) Every man brings about some act at any time in any place. which is, in our symbolism, (22. 5) (221mm!) (3 p)( (a. _t_. Mp) Hence, we have at least four different types of variables for a quantifier to bind. But this is hardly the end of the story. Consider the following formula: (22. 6) (Emu/Hp) [0,3, _t, 2)./)1) 3 “'0. ~13.» L lb] or (22. 7) (5)(£)(_w)(p)10s mt. Mp 3 Fur. .3 mp] That is, (22. 8) At any time in any place, if anyone ought to bring about any act, then he is permitted to bring about that act at that time in that place. In (22.6) we see that 'OR' occurs as a free variable. But it can also be quantified. Thus, we may write That is, 184 (22. 10) In any sense Of 'ought', if anyone at any time in any place ought to bring about an act, then in the related sense of 'permission', he is permitted tO bring about that act at that time in that place. Here, 'related sense of 'permission' ' can be defined as "the same sense of not-ought-not". Because the following formula holds: (22. 11) (o, )(P, Mp*)(P.p* s ~0. ~p*> Of course (22. 11) is only an abbreviation for a formula in our language. It is not itself a formula in that language. Let us call a quantified deontic system which puts only those four kinds of individual variables under the scope of a quantifier, fi_r_S_t- order deontic logic. If a deontic Operator, in particular '0', ', is also brought into the range Of quantication, then we call the system second- order deontic logic. The Significance of quantifiers in deontic logic is readily appreciated. Deontic logic may be used to formalize ethical norms. Now, norms must be eXpressed in general terms which Speak of all cases or at least some cases. For instance, one of the Islamic norms might be this. (22. 12) One ought to go to Mecca to worship God at a particular time of the year. This certainly can only be satisfactorily expressed in fir st-order deontic logic. We may prOpose to write it as follows: 185 (22.13) (amongst 2px;». = Mecca & p = to worship the God a 0(3 _t, Mp) Another example. We may wish to express in our deontic logic the following statement: (22.14) If a set Ri of rules obliges a manfi atil in yv_ to l 1 bring about p, and another set Rj of rules obliges him at _t_l in yv_ to bring about ~p, then, there is a 1 further set Rk of rules which overrules R1 and R. and which obliges him at _t in yv_ to bring about q 1 1 (where q is either p or ~p). This may be expressed as a formula in second-order deontic logic as follows. (22.15) (o.i)(o,j)a1>(_tl>(y_1>(p)[(o.ital, _t,, 1E1)P 8: ~ 3 ORiO—{l'il’ _Vgl) p) ( ORk) (0,,k overrules 0R1 and O,J. 8: O,k@1._tl._w_1)q) 8: (q = p V q= ~P)] It seems obvious that an investigation into second-order deontic logic is not only desirable but essential if we want really to COpe with certain important deontic problems such as the dilemma of conflicting obligations. The reason is that the morality which we intuitively conceived might not be a clear-cut and homogeneous thing. It seems more appr0priate to think of our system of moralities as a mixture of moral rules of different levels. That is, moralities mingle with 186 meta-moralities. We seem not only to have moral rules but also to have meta-moral rules in our moralities. For instance, the following looks more like a meta-moral rule than a moral rule. (22. 16) When different goods are compared, have the greatest; when several evils are present, take the smallest. (Motze) Likewise, we may find the following rule being enforced in our morality. (22. 17) Abandon your ”partisan” duties, give way to altruistic Obligations. This, again, seems to be a meta-moral rule rather than a moral rule. These rules, roughly Speaking, tell us which morality overrules which others. Therefore, if it is true that our system of moralities is one in which moralities are mixed up with meta-moralities, then it is indiSpensable to make a survey of second-order deontic logic provided that we want to get a complete view of the whole story of our morality. Only in second-order deontic logic can the problem of meta-moralities be fully formulated and satsifactorily handled. Next we come to the problems of alethic modalities in deontic logic. We Shall discuss the problems by concentrating our attention on certain particular examples. First, the so-called Kantian principle that what I ought to do I can do. 187 AS we have seen in § 13, this principle appears as a theorem of Anderson‘s systems OM - OM" in the following form: (OM24) Op 3 Op A natural question now arises: What kind Of possibility do we have in mind when we use ‘0' to refer to it? Can it be logical possibility ( '0"), empirical possibility ( '0“), technical possibility ('ot') or personal possibility ( 'OP') ? We will not try to give a precise definition of each of the above four notions Of possibility. Suffice it to say that personal possibility entails technical possibility, technical possibility entails empirical possibility and empirical possibility, in turn, entails logical possi- bility. Hence, (OM24) will be a strongest thesis if '0' means personal possibility; and weakest, if '0‘ means logical possibility, with the other two alternatives standing in between in the order of strength if. '0' means technical possibility and empirical possibility, respectively. The difficulty associated with the Kantian principle is twofold. Fir st, the distinction between the possible and the impossible is not clear-cut. The demarcation line is subject to change. This is true even in the case in which what we mean by possibility is the logical one. Simply reflect on the fact that the class of logical laws does not seem to be well-defined and determinable once and for all. Fortunately, this difficulty seems to be a minor one. Once we realize 188 that the distinction is not clear and distinct, we can proceed with the necessary precaution to draw a most desirable line. The second difficulty, on the other hand, seems more formidable. The difficulty is this: it seems that what we expect as the Kantian principle is not the weakest version of (OM24), namely, to interpret '0' as logical possibility. When we say that what one ought to do one can do, we seem to mean at least a stronger version intending '0' to refer to empirical possibility, or technical possibility, or even personal possibility. Indeed it is completely natural if a person claims that what he means by the Kantian principle is this: (22. 8) What one ought to do one can do within one's power, i. e. , in one's personal possibility. That is P (22.9) Op 3 O p However, when we allow the Kantian principle to take a stronger form, it may no longer hold. Let us consider here the strongest version Of this principle, namely (22. 9). The other two versions can be dis- cussed in a similar vein. We may ask the following question: Does it ever happen that a person ought to do something which he has no ability to do? Obviously, there are lots of examples. I may be obliged to pay my debt but at the time when the payment is due I have no money to pay. Or, I might be under an obligation to keep my Office hours, but I was delayed 189 by an unexpected traffic jam, and SO on. The examples can easily be multiplied _a__c_1 nauseam. (22. 19) seems, in a word, greatly to go against our intuition. The uncertainty about the Kantian principle also accounts for part of the reason why it seems highly desirable to allow reparative efforts in our morality. Next, we are going to examine the following two theorems of Ander son’ 5 OM-OM". (OM21) Up 3 Op and (OM23) ~Op D Fp Again there is all the ambiguity concerning the meaning of '0' (or '8'), again there are different versions of these theorems, each Of them having different degrees of strength. And again, when we take the stronger versions of them, they seem at best dubious, if not flatly false. And to make things even worse, let us remind ourselves that there might be a principle or a meta-rule in our morality to the following effect: (22. 20) What one ought to do, one ought to 3y; what one is forbidden to do, one must not try. Indeed this seems to be a reasonable principle, because it is not only 1 Cf. § 4 and see note 3 of that section. 190 the actual performance, but also the attempt to perform which counts in a moral evaluation. For example, there is a great difference between total non-performance and having attempted but failed. How- ever, once (22. 20) is agreed upon, we immediately find ourselves drived to the wall. Consider (OM23). It says that what is impossible must not be done (is forbidden to be done). By (22.20), we have: (22. 21) What is impossible must not be attempted. This, of course, is intolerable as a thesis in deontic logic even if we take 'possibility' to mean the most unproblematic one, namely, logical possibility. For (22. 21) implies, among other things, that those mathematicians who tried to prove the dependnece of the parallel postulate on the other Euclidean postulates, or those who tried to trisect an arbitrary angle with only a straightedge and compass; or the logicians who attempted to find a general decision procedure for first-order logic, or those who attempted to Show the consistency and completeness of fir st-order arithmetic, cormnitted a certain kind of moral error!2 It follows that those who claim that (OM21) and (OM23) are harmlessly acceptable in deontic logic, because of the fact that logically necessary acts are automatically done and logically impossible acts cannot be performed anyway, seem to make a mistake. For while And things become even more astonishing if we allow '0' to designate other kinds of possibility. 191 they make a detour to avoid the original difficulty they will find them- selves unwittingly stepping into another pitfall nearby. CHAPTER III MORAL USES OF LANGUAGE AND IMPERATIVE LOGIC 2 23. DUAL FUNCTIONS OF LANGUAGE IN A MORAL CONTEXT: EVALUATION AND DIRECTION As we indicated in S 15, some criticisms Of imperative logic have been taken to be applicable mutatis mutandis to deontic logic. Consciously or not, this is done however with good reason. As we Shall see in this chapter, a system of deontic logic and the corres- ponding system of imperative logic may be regarded as isomorphic models of the same theory. 1 The justification of the relationship between the deontic model and the imperative model lies in the fact that in a moral context a language, as a rule, assumes two functions: evaluation and direction. A sentence in such a context can be used, on the one hand, to grade, to set up maxims, or to evaluate; it can, on the other hand, be used to order, to command, or to direct action. Let us call the former uses of language evaluative, and the latter uses directive. In a moral context, these two uses of language have a close relationship to each other. What this relationship is exactly, we Shall try to Spell out later. The phrase 'the correSponding system of imperative logic' will become clear as we proceed. See, eSpecially, §§ 25-26 below. 193 194 Meanwhile, some preliminary remarks are called for. First, by ‘imperative logic' we mean, in this discussion, the logic of commands in its more restricted sense. We may call it the logic of moral commands. Just as deontic logic may be understood either in a wider or less restricted sense as the logic Of Obligation in general, or, as the common prabtice goes, in a narrower or more restricted sense as the logic of moral obligation; so too imperative logic can be construed either in a less restricted sense as the logic of cormnands in general, or in a restricted sense, as we proposed above. It is in the narrower sense that we are going to use the terms 'deontic logic' and 'imperative log ic'. It is also in this sense that we maintain that deontic logic and imperative logic are really two sides of the same coin. They are mirror images of each other. 'The second point we want to make clear before we take up the discussion of pragmatics is this. We shall take it to be the case that there are two distinguishable, though not necessarily separable, theories of morality. We have, on the one hand, a theory Of (moral) v_al_u_e_, and, on the other hand, a theory of (moral) M. The notions which we commonly encounter in the former are "goodness", "badness", "ideal", “value", and so on; while that which we usually find in the latter are “obligation", "permission", "duty", and the like. For the sake of 2Although we have devoted ourselves only to the restricted sense of deontic logic and imperative logic, it is easy to see that general deontic logic and imperative logic are nothing but simple semantical extensions of the former. 195 facilitating our discussion, we Shall reconstruct these two theories in the following way. We shall consider a theory of value as a depiction of a morally ideal world or a description of a morally best possible world. Let Ti be such a theory of value, and Wi be the ideal world it depicts. We shall regard as a basic moral sentence in a theory of value any expres- sion of the following form: (23. 1) It is desirable1 that p. or, in symbols (23.2) 13,1,i p where p is a certain proposition (or state of affairs), not necessarily an act-proposition. And 'desirablel' means "desirable according to the theory of value Ti' ” Indeed, (23. 1) can, then, be defined as (23. 3) p is a true proposition (or state of affairs) in Wi That is, (D22.4) DTipsz p 6 Wi provided that we let the definiens mean (23. 3). A world is considered Simply as a collection of propositions or states of affairs. Similarly, we may, in agreement with what we have said in Chapter Two (5 17 in particular), think of a theory of duty as a collec- tion of (moral) rules. Let R‘- be a particular theory of duty, that is, 196 a set of (moral) rules. The following, then, is an example of a typical moral sentence in it. It is required1 that p*. 3 (23.5) Ri requires that p*. or, in symbols (23.6) Rip* where 'p*', as in the last chapter, stands for a CM-act-proposition. It may be mentioned that there is a certain relation holding between a theory of value and a theory of duty. What exactly this relation is, we shall, nevertheless, not try to investigate or Specify here; it lies beyond the scope of our discussion. Suffice it to say that, more Often than not, a theory of value "yields" at least one theory of duty. On the contrary, a theory of duty usually "presupposes" a certain theory of value. Besides, a theory of value may be thought of as expressing a certain maximum or ultimate expectation, while a theory of duty, in contrast, may be thought of as voicing a certain minimum or least requirement. Of course, all these statements are vague. But Since we are not in a position to make a closer scrutiny of this topic, we let them stand as vaguely as they are, serving not as a precise characterization, but merely as a rough indication or suggestion. 3Cf. (317.1) in § 17. 197 We now turn our attention to the main concern of this section: pragmatics of moral uses of language. We Shall maintain, from the beginning, that when we utter (23.1) It is desirable1 that p. we intend it to serve the same purpose(s) as the utterance Of (23. 7) p is goodi. together with, although perhaps not with the same degree of emphasis as, the utterance of (23. 8) DO your best to bring p about!1 Here, again, the superscript 'i' in (23.7) and(23. 8), as it does in (23. 1), refers us back to the theory Ti. If we use 'GTp' to symbolize i 'p is goodl', and '(01T )p' to symbolize 'DO your best to bring p . i about!1', then what we try to maintain is that the purpose(s) we try to achieve by uttering (23.2) DTip is the same as the combined (in a certain manner) purpose(s) we try to achieve by uttering (23.9) Grp i and (23.10) (OlTi)p 198 We may note that (23. 7) or (23. 9) is used characteristically to evaluate and (23. 8) or (23. 10) is used characteristically to direct action. It is our proposal that (23. 1) or (23. 2) be employed to fulfill both of these two purposes. In short, they perform a dual-function of language: evaluation and direction, making value judgments and directing action. The issue in question may be explained more clearly in terms of the pragmatic theory of meaning proposed and develOped by Henry S. Leonard. 4 According to Leonard, the purpose of an author in uttering a sentence can be analyzed into a concern and a topic of concern. Roughly Speaking, the concern of a purpose is what the author wishes to accomplish, and the t0pic of concern is "that proposition relative to which he has this concern. "5 Here, the word 'prOposition' is under- stood in the same sense as we understood it in the last chapter, as a state of affairs. Before we set out to give examples, let us remind ourselves that moral value judgments are impersonal evaluations, and moral commands are impersonal imperatives. When x, addressing y, says: 'It is obligatory1 that you go to fight in Vietnam', it is the set of moral rules R,, not x, that are claimed to prescribe that y goes to 1 4See Leonard [1957] or [1967], unit 14. However, we do not mean to say, as Leonard seems to imply, that pragmatic considera- tions can be used to fully characterize meaning. (See Ibid. , § 14. 3: Meaning as purpose). On this point, see Hsiu-hwang Ho, "The Prag- matic Concept of Translation" (to appear) 5Leonard, Ibid. 199 fight in Vietnam. The utterance made by x serves to remind y or to let y know that Ri makes such a prescription. Similar remarks hold for moral commands. However, if we like, we might think of 3g in the above Situation as a mouthpiece of, or as a Spokesman for, or a locum tenens of, the moral rules Ri when he makes moral value- judgments or issues moral commands. Thus, the distinction between personal and impersonal evaluations or imperatives becomes rather unnecessary. 6 In either case, it is all-important to keep in mind that, in our discussion, moral judgments and moral commands are always backed up by a set of moral rules. Our uses of 'moral Obligation' and 'moral imperative' are under stood similarly. It is with regard to this kind of moral Obligations that our deontic logic is said to be a logic of (moral) Obligation; likewise, it is this kind of moral imperatives that we try to formalize in our imperative logic. Let us now examine briefly how the Leonardian analysis can apply to moral sentences. First, moral sentences in the so-called theory of value. Suppose a person x says: (23.11) It is good1 for y to be friendly. In uttering this sentence, 3, acting as a mouthpiece of a certain ethical theory Ti’ grades or evaluates as good that y be friendly. Here, "to grade or evaluate as good (that. . . )" is the concern, 6 Cf. Leonard [1959], p. 185. 200 and "that y be friendly" is the tOpic of concern, of 5's uttering the sentence (23.11). Since x's concern in this case is to grade or to evaluate, we shall call the concern an evaluative one. And since the purpose of uttering a sentence (by a person), according to Leonard, consists of a concern and a tOpic of concern, we Shall also call the purpose of uttering (23. 11) an evaluative purpose. Language when used for an evaluative purpose is said to have an evaluative function. (23. 11) then, exemplifies the evaluative function of language. Similarly, when 3; in addressing y says: (23. 12) Do your best to be friendlyll. the concern is (again acting as a mouthpiece of Ti) "to command, or to direct, y to do his best to bring it about or to make it true (that. . . )", and the tOpic of concern is, again, "that y be friendly". likewise, Since the concern is to command or to direct someone to bring about some state of affairs, we shall call it a directive concern. We shall also say that (23. 12) serves a directive purpose, or that the language used in (23. 12) has a directive function. (23. 12) above exemplifies the directive function of language. It may be remarked that the same sentence may be used to serve different purposes. For example, when x says: (23. 13) It is raining. He may want to report a "fact", or he may want to tell a lie. He may use (23. 13) to suggest to someone who is going out to bring an 201 umbrella with him. Or he may even utter this sentence as a joke when.he sees someone polishing his car. These, and conceivably many others, are different functions which (23. 13) may be used to serve. However, a sentence standing in a certain context usually has a characteristic use or function. Under normal conditions, for example, in the radio weather broadcast, (23. 13) is used, as a rule, to report a fact to, rather than to play a joke on, the listeners. In the same manner, when we say that (23. 11) above has an evaluative function, we mean that that sentence has evaluative function as its characteristic use in a normal moral Situation. As a matter of fact, (23. 11) may have other uses, e. g. , directive use or other practical uses, provided the Speaker SO intends. A Similar remark holds for (23.12). The sentence (23.12) exemplifies, aS its charac- teristic use in a normal moral Situation, a directive function of language, but it could well be used otherwise under other conditions. In what follows, when we talk about the function of a sentence, we mean its characteristic use in a certain context explicitly Spelt out or otherwise understood. We shall next come to see the relationship between the evalua- tive function and the directive function of language used in a moral context. In particular, we shall exhibit the relationship between a moral sentence like (23. 11) serving an evaluative function and another moral sentence like (23. 12) serving a directive function. This relation- ship can be put in the following way. 202 Suppose _x_, acting as the mouthpiece of theory Ti (henceforth the same condition is assumed without explicit mention), Speaks to y thus: (23. 14) It is good for you to be friendly. and goes on saying that he does not command (or advise, etc. ) y to do his best to be friendly, that is, denies that he would make the following command: (23. 15) Do your best to be friendly! Then, there is an l'inconsistency” of a certain kind involved. This kind of inconsistency is to be explained as follows. In a moral context, when a man, speaking for a theory of value Ti’ has an evaluative concern with reSpect to a tOpic of concern, then [and only then—but this part will come later] he cannot but have a directive concern with reSpect to the same tOpic Of concern if he does not want to do some- thing odd or bizarre. This oddity, which may be thought Of as a pragmatic fallacy, resembles the so-called "Moore's paradox" in doxastic logic when one says, "Bertrand Russell is a philOSOpher, but I do not believe it. " Indeed the relationship we tried to characterize above and the oddity we just mentioned can be examined from a different angle. Instead of talking about a person acting as the mouthpiece of a theory of value in making evaluations and issuing commands, we may Simply Speak of the theory itself. We may think Ofa theory (of value) aS 203 capable Of making value judgments and capable Of issuing conunands. Now, since ethics is cormnonly taken to be not only a theoretical “science” but also a practical discipline, an ethical doctrine is devised not merely to differentiate the good from the bad, or the right from the wrong, but also to tell pe0ple what they ought to do, or what they should try their best to do. Hence, there is no wonder that when a theory (of value) evaluates to a person that a certain state of affairs is good, the theory also issues to him a command that he does his best to bring about that state of affairs. To claim that a theory evaluates a state of affairs as good but also claim that the theory decline to order, or refrains from cormnanding, the bringing-about Of that state of affairs seems to betray a rather odd conception of a value theory, because this conception, among other things, would deprive a value theory of any practical significance and working relevance as a guide of life. We have, in the above paragraphs, tried to establish a relation- ship between the evaluative concern (with reSpect to a tOpic of concern) and the directive concern (with reSpect to the sametopic of concern) which a person has when he utters sentences in a moral context. We Shall now give this relationship a name. We Shall say that in a moral context, with reSpect to a proposition (state of affairs) the evaluative concern pragmatically entails the directive concern, meaning that to admit the existence of the evaluative concern and at the same time deny the existence of the directive concern is to run into the 204 inconsistency we described above. Now, since the evaluative concern with reSpect to a prOposition and the directive concern with reSpect to the same proposition are characteristically expressed (in the sense in which Leonard uses the word) by two different sentences indicating (again in Leonard's sense) that proposition, we Shall extend the use of 'pragmatic entailment' and say that the Sentence eXpressing the evaluative concern pragmatically entails the sentence expressing the directive concern. For example, we shall say that the sentence (23. 14) It is good for you to be friendly. pragmatically entails the sentence (23. 15) Do your best to be friendly! This however, is only half of the story. We shall also maintain the converse. That is, that (23. 15) also pragmatically entails (23. 14) in a moral context. A Similar argument applies. First, consider a theory of value Ti which stipulates, among other things, what is good and/or what is bad. Can it be the case that Ti commands a certain person to do his best to bring about a certain proposition, yet does not claim that it evaluates that prOposition as good? The answer seems definitely in the negative. To put the matter in a different way, there would be the samekind Of oddity that we noted above if Ti commands someone to do his best to do something but does not evaluate that something as good. This seems to be an intolerable oddity. Consequently, a man Speaking as the mouthpiece 205 of a value theory, will certainly have the evaluative concern unless he does not have the corresponding directive concern. For instance, when 9; orders y to be friendly, and _y asks for a reason, it Shall have the ready answer: "Because it is good". The answer is w not in the sense that _x can immediately Speak it out, but rather in the sense that when x issues the command, he already "implies" that it is good. If x commands y to do his best to bring it about that p, but declares that p is not good, 3; either makes a mistake or fails to be a "faithful" mouthpiece of the value theory in question. Indeed, when we say, for example, that honesty is a good policy (evaluation), we want our auditors to be honest (direction); likewise, when we advise or tell someone to be patriotic (direction), we "imply" that patriotism is a good thing (evaluation). Hence, we Shall say that in a moral context the directive concern also pragmatically entails the corresponding evaluative concern. Again, we shall apply the notion of pragmatical entailment to sentences. We shall say, for example, (23. 15) pragmatically entails (23. 14). Let us say that two sentences are pragmaticalll mutually entailing or pgginatically equivalent if the fir st pragmatically entails, and is pragmatically entailed by, the second sentence. Let us also say that two concerns with reSpect to a certain tOpic of concern are prag- matically mutually entailing or equivalent if they pragmatically entail. each other. 206 Due to this pragmatically mutually entailing relation between the evaluative concern and the directive concern in a moral context, a sentence which characteristically expresses the evaluative concern becomes functionally equivalent with a sentence which characteristically expresses the directive concern, because each sentence pragmatically entails the other. They both have the same dual functions, namely, evaluation and direction. For example, due to this pragmatical relationship between evaluative concern and directive concern, (23. 14) and (23. 15) can be used _b_o_’c_h_ to evaluate and at the same time to direct action. Of course, this is not to deny that (23. 14) and (23. 15) may have different characteristic functions, one of them emphasizes the evalua- tive Side, the other has a stronger directive ring to it. In order not to go into the different aSpects of emphasis of the above-discussed two types of sentences in a value theory, we propose that we use It is desirable1 that p to perform the two functions, evaluation and direction, which are, reSpectively, the characteristic functions of It is good1 that p and DO your best to bring it about that p! 1 The relation between deontic logic and imperative logic will be justified on the same ground. That is, we Shall provide a 207 pragmatical justification. AS we mentioned above, the following expression is to be treated as a basic moral sentence in a theory of duty: (23. 5) It is required1 that p*. or (23.6) Rip)": In the same manner as we maintained above in the case of a theory of value, we shall say that (23. 5) is uttered in order to accom- plish what the following two sentences will jointly accomplish: (23. 16) It is Obligatory1 that p* or (23.17) O, .p* 1 together with 123-18) Bring it about that p’i‘!i which we want to symbolize as (23.19) (! Ri)p* Again, the concern of uttering (23. 16) is an evaluative one, and the topic Of concern is p*. On the other hand, the concern of uttering (23. 18) is a directive one, but again, the topic of concern is the same p*. Moreover, just as in the case of a theory of value, these two concerns, evaluation and direction, are pragmatically mutually 208 entailing. Consequently, (23. 17) and (23. 19) are pragmatically equivalent. One example: when we say that it is obligatoryi for someone to help his neighbors (at a certain time in a certain place), we also want him to help his neighbors (at that time in that place). The reverse is also true. These two concerns are pragmatically mutually entailing as we have repeatedly said. It would be extremely odd for us to pro- claim, for example, that it is everyone's dutyi to be patriotic but to feel completely all right when we see someone act as a traitor. By the same token, it seems that we would be committing nothing short of the aforementioned pragmatic fallacy should we morally command someone to go to fight in Vietnam but at the same time announce that it is not his duty (Obligation) to do so. At this point we shall again stress the fact that when a person makes a (moral) evaluation or issues a (moral) command, he works as if he were the mouthpiece of a set of (moral) rules. It is only under this assumption that we claim his evaluative concern prag- matically entails, and is pragmatically entailed by, his correSponding directive concern. And consequently, a sentence like (23. 17) is prag- matically equivalent to another sentence like (23. 18). We Shall here- after call sentences of the former type deontic sentences and sentence of the latter type imperative sentences. What we have so far tried to establish, then, is that a deontic sentence is pragmatically equiva- lent to the correSponding imperative sentence. They have the same 209 functions or uses, namely, evaluation and direction. Of course, we do not deny that a deontic sentence has evaluation as its characteristic function, and an imperative sentence, on the other hand, has direction as its characteristic function. Hence, a man when making a choice between these two sorts of sentences may have this or that kind of emphasis in Inind. It is on the ground of pragmatic equivalence that we try to maintain that a system of deontic logic and the corresponding system of imperative logic are but isomorphic models of the same normative logic. § 24. NORMATIVE SENTENCES: DEONTIC AND IMPERATIVE Let us now Specify what we mean by a deontic sentence and an imperative sentence which we have already mentioned but have only roughly exemplified and ambiguously identified in the last section. Let us say that an expression iS a deontic formula (or a deontic sent- ence form) if and only if it can be formed by a finite application of the following rules: i) If A* is a CM-act-formula, then l'O,,.A.""‘ is a (CM-) deontic formula. 1 ii) If B and C are (CM-) deontic formulas, so are r' ~B , r[3 sol-1, r[3vc]_', '_[33cl' and r[Be-OT. where a CM-act-formula and 'OR' are under stood as they were characterized in the last chapter. As we call 'Oa' a deontic Operator, we may analogously call '(! I,)' an imperative operator. We then proceed to define an imperative formula (or an imperative sentence form) as an expression which is formed by a finite application of the following rules: iii) If A* is a CM-act-formula, then r(!,)A*" is a (CM-) impe r ative for mula. 210 211 iv) If B and C are (CM-) imperative formulas, SO are r~B1, I. r[3&c]_', rEBvCT, [BDCT and T350? It may be noted that in imperative logic, it is of Special interest to write an imperative formula in the following Special form: (24. 1) (You, B. Y )A in which 'you' denotes as usual the person(s) addressed, and 'B' and 'y' are variables, as Specified in g 19, for time and location, reSpec- tively. 'And, finally, 'A' denotes an act. For instance, the following addressed to John (24. 2) (You, this coming Friday, Rm. l4, Morrill Hall) coming-to-take-the-fina1-examination. may be rendered in everyday English as the following sentence: (24.3) John, come to take the final examination this coming Friday in room 14, Morrill Hall! After we have defined a deontic formula and an imperative formula, the definition of a deontic sentence and that of an imperative sentence become straightforward. We may simply say that a sentence is deontic (imperative) if it instantiates a deontic (imperative) formula. And a sentence is normative if it is either deontic or imperative. For example, (24.4) below is a deontic sentence. (24. 4) It is obligatory1 for John to go to help his neighbors in their houses when they need him. 212 Because it instantiates the following deontic formula: OR.(Q’: B: Y )A 1 or, in short, 0,, p"< (24. 5) below, on the other hand, is an imperative sentence. (24. 5) You, close the door right now, and go to tell John it is raining!1 which exhibits the following imperative formula: (24.6) (1,.) Won, 3. Y)A1 8: (you. 8'. v')A2] 1 Of course, to identify a particular deontic or imperative sentence often requires insight. Sometimes paraphrasing is needed before we can tell whether or not a sentence is deontic or imperative or something else. Since it is sometimes desirable to make the agent, time, and location constant as in the case Of deontic logic, we Shall, again,in imperative logic, use ‘p*' to abbreviate a CM-act-formula. We Shall thus write (24. 6) as, e. g. , (24- 7) (1,.)(P* & (1*) 1 The difference between the exPreSSion abbreviated by 'p*' and that in which 'you' appears is that in the latter the variable for agents always takes as its substituent expression the second person pronoun, 213 namely, 'you'. But when we render an imperative sentence into every day language, the 'you', as appeared in (24. 5), is as a rule omitted. We Simply say 'Close the door!‘ instead of 'You, close the door!‘ except for emphasis. On several earlier occasions we noted that there are other forms of deontic formulas which can be defined interms of '’OR A* 1 together with sentential connectives, particularly, '~' and '&'. For instance, the following definitions show that at least three other forms of deontic formulas are available, namely, FPRA’k—I , l-F,, A*-‘ and rIRA’k-I. (317.2) rPRA*1 zfif r~OR~A*-' (317.3) rFRA’k-I =Df "~P,A*—‘ (317.4) l_1,,A’°‘_1 =3i rPRA" 8t 1.3R ~A"‘1 where 'A*' is any CM-act-formula, say, 'p*' Now, in a close analogy, we may try to set up in imperative logic the following definitions correSponding to the above ones, to characterize three other forms of imperative formulas. These definitions are: (324.1) F(JR)A*1 r~(!,)~A*" (324. 2) r (x, )A* " (324.3) rnights)“ fi "(./R)A* a. (JR)~A*-' 214 A note is immediately needed here to explain how to read the definitions (D24. 1) - (D24. 3). We shall, in particular, explain how it is meaningful to prefix a tilde to an imperative sentence like I‘~(!,, )~A*‘l in the definiens of (D24. 1). First, we may think of an imperative sentence. (24. 8) Bring it about that A* ! 01' * (24. 9) (1,, )A as true (where A* is a CM-act-prOposition), if and only if a certain mouthpiece, say _x, of moral rules R commands that A* be brought about. Hence, the negation of (24. 8) or (24. 9) is true if and only if 35 does po_t command that A be brought about. According to this rendering, (D24. 1) can be read as: "( JR)A* is commanded by x if and only if, by definition, (!R)&A* is not commanded by 35." But x is nothing more than the mouthpiece of R, hence, we may have a more straightforward way to read (D24. 1) - (D24. 3). We may simply read (D24. 1) as rules R makes the command (J)A* if and only if R does not require that ~A* be brought about. (D24. 2) and (D24. 3) can be similarly understood. Just as 'P‘, ', 'Fa ', and ‘IR' may also be thought of as deontic Operators, '(~4)', '(XR)', and '(#R)' may be regarded as three new imperative Operators in addition to the usual imperative operator '(!n)" 215 Of these three new imperative Operators, '(XR )‘ is hardly new to our intuition, it answers quite closely to our everyday notion of "DO not . . . 1" in "DO not bring it about that p*!" In contrast with our intuitive familiarity of '(XR )', we find no counterparts for '(,‘/R )' and '(#R)' in English. However, non-existence in our natural language is not in the least a diSproof of the possibility, or even plausibility, of these imperative operators. The lack Of unconventionality of these two imperative operators must be considered, from a logical point of view, as purely accidental, although the phenomenon is psychologically and practically eXplainable. In our daily life, when we come near to using the imperative notions correSponding to '(JR)' and '(#R)', we Simply use their deontic counterparts 'Pn' and '11:" namely, 'You may . . . ' or 'It is permissible that you . . . ' and "It is indifferent that you . . . ". This is another place to appreciate the pragmatic relationship between deontic operators and imperative Operators, and, in general, between deontic logic and imper ative logic. § 25. THREE CORRESPONDING SYSTEMS OF IMPERATIVE LOGIC: CM1,T*, CMI.S4* AND CM1,ss* Our conviction that a deontic logic and its corresponding imperative logic are but two models of a certain normative theory finds a justification in what we have said above concerning the close relation between evaluation and direction in the moral use Of language. This conviction immediately gives rise to the following result: to each system of deontic logic we have Specified in 3 19, namely, CMORT* , CMORS4* and CMORSS’k, there corresponds a system of imperative logic closely related to it. Let us call the imperative systems corres- ponding to the above-mentioned deontic systems CMIR T*, CMIRS4* and CMIRSS*, reSpectively. The prefix 'CM', again, reminds us that imperative Operators take CM-act-formulas as their Operands. These three systems of imperative logic may now be outlined as follows: I. Vocabulary (for all three systems alike): Same as the vocabulary of CMOR T* in § 19 except that iii) now reads: vii) Imperative Operator: '(!R)' 216 217 II. Formation rules (again, for each of those three systems): Same as that of CMORT* except that each occurrence Of 'deontic term' in the rules is now replaced by an occurrence of 'imperative term'. Furthermore, vii) is cancelled in favor Of the following: vii‘) If A isawff, so is r~(!,)A". We may, as Shown in the last section, introduce other impera- tive Operators by means Of the definitions (D24. 1) - (D24. 3). III. Theoremhood (including axiornhood): We Shall say that IO ' S 11,? (THM) | is a theorem of CMIR T* (or CMIRS4*, or CMI'SS*) provided that (THM) is a theorem of CMORT* (or OMO,S4*, or CMO,55*). Let us agree to use the same numbering for the theorems of CM1,T* - CMl,ss* as for the theorems of CMO,T* - CMO,ss* except that we prefix 'CMIR T*' and so on rather than 'CMOR T*' and so forth, to the numerals. For example, the following iS a theorem of CMIR T* (CM1,T*19) ~[(!..)p’”< 8: (!.)~p*3 which is also (OMI,S4*19) and (OM1,55*19). AS we recall, systems CMO. T* - CMO.SS* of deontic logic are designed only to formalize a certain unspecified partial notion of 218 obligation. Thus, we defined in 5 17 'ORA*‘ as "R requires that A*". In the same manner, systems CMIRT’k - CMIRSS* of imperative logic are meant only to systematize a certain unspecified partial notion of "do!", or "Bring it about that ... !" That is to say, a certain mouth- piece Or, if we like, a certain authority, of a certain set Of moral rules is always assumed as standing behind this partial imperative notion. If we let 'AR' stand for a certain mouthpiece of the set Of moral rules R, or a certain moral authority created to enforce the set of moral rules R, we may try to characterize the partial imperative notion of "DO!" by the following definition. (D25. 1) r‘(!" )A*1 AR commands that A* be brought about. :Df We use 'A*' instead of 'A' to emphasize that the Operand of '(!R )' is a CM-act-formula. But a moral authority or mouthpiece can either be identified with, or else be thought of as, nothing but an instrument or an agent created, as we said above, to enforce a set Of moral rules. That is, a moral authority must satisfy the following condition: (25. 2) AR commands that A* be brought about if and only if R requires that A* Hence, instead of (D25. 1) we may set up the following definition. (D25. 3) r(!R)A*1fif R requires that A* 219 Compare (D25. 3) with (D17. 1), i. e. , with r *1 . * (D17. 1) ORA R requires that A ="Df It may, at first sight, seem very strange that an imperative Operator and a deontic Operator are defined in exactly the same terms. That is, by means Of the requirement of R. A closer examination, however, will remove any misgivings. First, as we have repeatedly stressed, the two functions Of language in a moral context are pragmatically equi- valent. Thus, we have said in § 23 that when we utter (23. 5) It is required1 that p>z< or (25.4) Ri requires that p* we want to accomplish what the following two utterances may jointly accomplish: (23. 12 ) 0,, p* and (23.14) (1,)p* That is to say, on the deontic side, (25. 4) "means" the same as (23.12) but on the imperative Side, it has the "same meaning" as (23.14). We have then explained '(!R)' in terms of the requirement of a particular set of moral rules R. Consequently, what we have said in 3 18 applies, without further ado, to the case of imperative 220 logic. In particular, we Shall remind ourselves Of the following: that a set of moral rules is always consistent, hence (CMIR T*19) above is always true. But the union set of all the sets of rules may not be consistent; it may be absolutely inconsistent or practically inconsistent. Furthermore, because a person may submit himself to incompatible orders, the imperative version of conflicting duties, i. e. , conflicting orders, is possible. AS in the case of CM-deontic systems, it iS easy to see that quantifiers are readily introducible into CM-imperative systems. For example, the following saying of Confucius can be satisfactorily formu- lated only in quantified imperative logic. (25. 5) DO not do to others what you do not want others to do to you! Even Such a Simple command as (25. 6) Keep your promises! demands the use of quantifiers to be symbolized adequately. Hence, it seems beyond question that quantifiers are indiSpensable in an impera- tive logic comprehensive enough to COpe with our ordinary imperative arguments. The reason for this indiSpensability can be further appreciated if we recall our remarks in § 22 on the indiSpensability of quantifiers in deontic logic. Let us, again, call an imperative system a first-order impera- tive logic, if we allow the quantifiers to bind only individual variables. 221 When a system admits of quantifiers which bind imperative Operator S, it will be called a second-order imperative logic. A closer investiga- tion Of quantified imperative logic is beyond the scope of our present discussion. The difficulties and "paradoxes" which puzzle people in the SO- called "old" systems of deontic logic can be reformulated in a revised form in imperative logic. Moreover, our prOposed solutions apply equally well in imperative logic. § 26. PURE NORMATIVE LOGICS: THE LOGICS OF REQUIREMENT CMRT*, CMRS4* AN3 CMRss* Having Observed that an imperative system can be constructed as an isomorphic model of a correSponding deontic system, and that these two different logics have common bases in pragmatics and the formal theory of ethics, we might naturally expect, as we anticipated earlier, there to be a pure normative logic of which a deontic system and its corresponding imperative system are nothing but two Specific models. The primitive notion of this pure normative logic is (26. 1) R requires that . . . or, in symbols (26.2) R Hence, the resulting system may be called the logic 2: repuirement. But again, we formalize only a partial notion of requirement. It suffices to mention that we are able to easily construct three systems of pure normative logic CMRT*, CMRS4* and CMRSS’!‘ of which OMO,T*, CMO,S4* and CMO,ss* on the one hand, and CMIRT’k, CMIRS4* and CMIRSS’k on the other hand are, reSpectively, deontic and imperative models. After we have seen the primitive 222 223 bases Of CMOR T*, etc. , it is trivial to set down the primitive bases for CMRT* - CMRSS*. We shall also use the same numbering to list the theorems. For example, the following is a theorem Of CMRT*: (CMRT*19) ~[Rp* as R ~p*] of which the following is its deontic counterpart: (CMO,T*19) ~[O,p* & OR ~p*] and the following, its imperative one: (CM1,T*19) ~[(!..)p* & (1s) ~p*] The assumptions and consequences of meta-ethics in g 18 are now directly related to the concept of requirement which the pure normative logic is set up to formalize. Again, quantifiers can be introduced. And we may talk about first-order and second-order pure normative logic. Consequently, we may want to put more attention directly to pure normative logic, or the logic of requirement. However, Since the names 'deontic logic' and, especially, 'imperative logic' have long been well established in the literature, and since the expressions which are usually incorporated in these logics are far more conventional than the eXpressionS we might find in the logic of requirement, we Shall continue to talk about deontic logic and imperative logic. But 224 we understand that what we have already said and Shall have said about these logics can be readily mapped onto the logic of requirement. 1In a similar manner, we might construct a pure logic O_f value (which formalizes the concept of desirability) of which the logic of good- ness (which formalizes the concept of goodness) and the logic of moral injunction (which formalizes the concept of ''Do your best to bring it about that . . . !) are two Specific models. a 27. NORMATIVE ARGUMENTS: PURE AND MIXED B An argument, as usual, is understood as a sequence B1, 2, B B C of sentences, the last sentence C is called the 390-0: k: B B are called the conclusion, the other sentences B , B 3, ..., k 1 2’ premisses. A premiss is a premiss of an argument, and a conclusion is a conclusion of an argument. We Shall say that an argument is assertoric if its conclusion 1 is an assertoric sentence; it is deontic if its concluSlon 18 a deontlc sentence; and it is imperative if its conclusion is an imperative sentence. Deontic arguments and imperative arguments are said to be normative arguments. If all the premisses and the conclusion of an argument are made up of the same type of sentences, the argument is Said to be plrg, that is, either a pure assertoric argument or a pure deontic argument or a pure imperative argument. Otherwise, it is mixed. When an argument is a normative one, it will be called a uniform argument provided that its normative premisses are of the same type of sentence as its conclu- sion. It will be said to be non-uniform, if otherwise. A uniform 'By an assertoric sentence we mean either a factual or empirical sentence like "Paris is the capital of the United States" (which i_s false) or a logical or analytical sentence like "A black raven is black" (which is necessarily true). 225 226 (normative) argument is either a uniform deontic argument or a uniform imperative argument. For example, the following arguments are all pure. (27.1) John keeps all his promises. This is his promise. John keeps it. (27. 2) John oughti to keep his promises. John ought1 to help his neighbors. .'. John ought1 to keep his promises and help his neighbors. (27.3) Be honest!1 -- Do not cheat!1 where (27. 1) is a pure assertoric argument; (27.2), a pure deontic argument; and (27. 3), a pure imperative argument. On the other hand, the following arguments are mixed ones: (27. 4) All and only logicians ought1 to teach logic. John is a logician. .. John is permitted1 to teach logic. (27. 5) It is your duty:l to help John. .’. Help John! 1 (27. 6) Only senior members are permitted1 to Speak at the club meeting. 227 John Speaks at the club meeting. 0 . o co John 18 a senior member. (27. 7) Help your neighbors and only your neighbors!1 Help Johnll . . John is your neighbor. (27.4) is a mixed deontic argument, (27. 5), a mixed imperative one. Both are mixed normative arguments. On the other hand, (27. 6) and (27. 7) are mixed as sertoric arguments. Among the normative arguments we have listed above, i. e. , (27.2), (27.3), (27.4) and (27.5), two are uniform, namely (27.2) and (27.3), while (27.4) and (27.5) are not. (We regard (27.4) as uniform, because there is only one sort of norma- tive sentence involved in the whole argument. ) An immediate remark is in order. In our discussion, a norma- tive sentence is one in which the normative Operator—a deontic Operator or an imperative Operator—take CM-act-sentences and only CM-act- sentences as its Operands. In the above examples, we have, however, for the sake of brevity and naturalness, made use of normative sentences that are at most incomplete forms of certain full-fledged normative sentences. We acted nevertheless on the tacit assumption that the resumption to full-fledged normative sentences is always possible. We shall in the same manner, restrict our attention, for the time being, to assertoric sentences of a Special type, namely, CM-act- sentences. Again, we assume that every assertoric sentence we used 228 or shall use can be translated into a full-fledged CM-act-sentence. Up to this point, we have exhibited arguments of a relatively Simple type. The simplicity lies in the fact that each premiss and conclusion of each argument we dealt with is either an assertoric sentence, a deontic sentence, or an imperative sentence, exclusively. We did not examine, for example, arguments such as the one below: (27. 8) If one ever feels sad, read Psalm 23!1 John is sad. .3 John ought1 to read Psalm 23. in which there are some component sentences which are further made up Of different types Of sentences. In the case Of (27. 8) the first premiss is conditional sentence having an assertoric sentence as its antecedent and an imperative sentence as its consequent. We may call a sentence of this type a multi-natured or a multiplex sentence. In general, 'Ok(A1, A2, . , Ak)' is a k-place multiplex sentence if 'Ok' is a k-place sentential connective and there exist i and j (i, j = l, 2, . . . ,k) such that i :1: j and Ai and Aj are of different sentence type. That is, 'Ok(Al, A . , Ak)' is a k-place multiplex sentence, if in the 2’ list Al’ A2, . . . , Ak of sentence, at least two are not of the same type. More Specifically, we shall say that a sentence iS assertoric-deontic (or deontic-assertoric) if it is composed of assertoric sentences and deontic sentences. Analogously, we may have an assertoric-impera- tive sentence, a deontic—imperative sentence, an assertoric-imperative- deontic sentence, and so on. 229 In what follows, we shall confine ourselves mainly to the discus- sion of normative arguments, uniform or non-uniform, multiplex or otherwise, together with as sertoric arguments with normative premisses. Pure assertoric arguments will be treated merely in passing. § 28. TOWARD A DEFINITION OF NORMATIVE VALIDITY When we talk about arguments, one of the most important problems is the problem Of validity. Roughly Speaking, an argument (28.1) A1,A2,A3. k. is said to be valid if and only if its conclusion 'C' follows logically from its premisses (28.2) A A 1’ 2’ 3’ H" k Or, we may say, in other words, that (28. 1) is valid if and only if (28. 2) logicallj entails 'C'. The problem, however, is how to characterize the relation of "logical following" or ”logical entailing" holding between the premisses and the conclusion of an argument. But before we go to the very core of the discussion of validity of normative arguments, certain remarks and preliminary explanations are necessary. They will facilitate our later presentation and prevent misunderstanding. First Of all, it may be recalled that we have, upon the basis of the close relation that exists between a deontic sentence and its corres- ponding imperative sentence, defined a deontic sentence and its impera- tive couterpart in terms of the same definiens. In particular, '0, p*' 230 231 and '(!R)p*' are both defined as 'R requires that p*‘. Moreover, we have indicated in § 24 that the truth condition of '(!R)p*' can also be Specified in terms of the requirement of R. We said that '(!R )p*' is true if and only if R requires that p*. The same remarks are applicable to 'On p*'. We may, for instance, say—and this will be explicitly put down later—that 'OR p*' is true, again, if and only if R requires that p*. It is then clear that, according to our construing, a deontic sentence and its corresponding imperative sentence both have the same truth conditions. It follows that they are mutually replaceable pal—ve veritate in any formula in an extensional context. In addition, since we shall try to define the validity of a normative argument in terms of truth, those two sentences are also inferentially interchangeable _sglv_e_ validitate. Consequently, the validity of a deontic argument and the validity of an imperative argument are judged on the same footing. They have, so to Speak, the same "logic". For instance, what can be used to justify the following imperative argument (28. 3) If one ever feels sad, one ought1 to read Psalm 23. John is sad. .°. John, read Psalm 23!1 will be regarded as equally apprOpriately employed to justify the following deontic one: (27. 8) If one ever feels sad, read Psalm 23!1 232 John is sad. .°. John ought1 to read Psalm 23. They will be regarded, syntactically or inferentially, as the same argument. Secondly, in classical (assertoric) logic, we have, more often than not, used the concept of truth to explicate the concept of validity. For instance, we frequently characterize a valid argument as one of which the conclusion cannot but be true provided that all the premisses are true. Now, this definition is transferrable, with little substantial further ado, to the case of normative arguments, since, as indicated above, we regard a deontic sentence and an imperative sentence as being true or false, 1 and, as we shall see later, the usual definition of validity provides us with a very plausible criterion of "good" or "correct" normative arguments. A final remark is now in order. It may be argued that, in one sense, the definition Of validity within our normative systems outlined in earlier sections is trivial, or, at least, straightforward. We may simply say, for example, that a normative argument. ,A,.'.c (28.1) Al’ A2, A3, k 1Henry S. Leonard is among those philosophers who would say that normative sentences are on an equal footing as assertoric sentences insofar as their truth values are concerned. See Leonard [1959a] and [1961]. For a criticism of Leonard's position, see, e. g. Wheatley, J. M. 0., "Note on Professor Leonard's Analysis of Interrogatives, etc. ", PhiIOSOphyO_fScience, vol. 28, pp. 52-54, 1961; and Stahl, G., Review of Leonard [1959a], [1961], etc. , Journalpgsjmbolic Logic, vol. 31, pp. 666-668, 1966. 233 is valid if and only if (28.9) A1 &A2&A3&...&Ak.DC is an instance of a theorem of our system. 2 Thus, we may differen- tiate between CMRT*-validity. CMRS4*-validity and CMRSS’k-validity. 3 Of course, it goes without saying that the first sense of validity entails the second one; and the second, the third. However, what we try to accomplish in this section and the following ones is something quite different. We want to outline a theory which may be called the intuitive theory 9_f normative validity, or perhaps more adequately, the gpneral theory 9_f normative validity. Roughly speaking, such a theory is a stipulation of a criterion or a definition which distinguish the correct normative arguments from others which are not correct. Such a general theory of validity is of Special interest because of the fact that, up to the present, there is no normative logic which is universally accepted and may be called _th_e_ logic of normative arguments. Our theory may be thought of as a preliminary explication of our intuitive concept of normative validity. But before we attempt such a definition or criterion of validity for a normative argument, let us talk a little about our motivation in setting down such a definition. First, let us say that a normative sentence is used to make evaluations and/or to issue commands, just as an assertoric sentence is typically used to describe states of affairs 2 . . . . 'C', in this case, is a normative sentence, of course. 3Just as we have 0T* validity, OS4* validity and 055* validity in § 20. 234 or to give information. Now, a good or correct (pure) assertoric argument can be intuitively conceived as one of which the conclusion describes states Of affairs that are already described or contained in the premisses. Or, we may say that a correct assertoric argument is one of which the conclusion gives information that is already given in the premisses. For example, if we assert: (28.12) John and Mary have blue eyes. we may infer that (28.13) John has blue eyes. for the information given in (28. 13) is already contained in the infor- mation given in (28. 12). Thus, the argument of which (28. 12) is the premiss and (28. 13) the conclusion is a correct or good argument. On the other hand, when one asserts (28. 14) I feel pain when I kick what I believe to be a stone. one cannot correctly infer, as Samuel Johnson seemed to want to infer, that (28.15) A stone exists. Indeed, one cannot even correctly infer, as Bertrand Russell points out, that (28.16) My foot exists. That is, the reasoning from (28. 14) to (28.15) or from (28.14) to (28.16) 235 is not a good argument. The information given in (28. 15) or in (28. 16) is not already contained in (28. 14). Here we may think of making a logical reasoning as an effort to bring up some information which is contained in the information already given in the first place. Logical reasoning can never provide us with genuinely new information. Thus, we find Carl G. Hempel saying that logical reasoning may make explicit what is already contained implicitly in the premisses, but it cannot yield something which is really new (Hempel calls it theoretically new). In a close analogy, we may think of a (pure) normative argument as one by means of which people want to reason from certain evaluations and/or commands to certain (other) evaluations and/or commands. If the evaluations made, or the commands issued, in the conclusion of a (normative) argument are already contained, perhaps implicitly, in the evaluations or commandsmade or issued in the premisses, then we say that this argument is good or correct. Otherwise, it is not a correct argument. For example, the evaluation made in (28. 17) It is obligatory for John to help his neighbors. is contained in the evaluation made in (28. 18) It is obligatory for John to help his neighbors and keep all his promises. 4 See Hempel, "On the Nature of Mathematical Truth", reprinted in P. Benac‘erraf and H. Putnam (ed. ), PhilOSOphy p_f_ Mathematics, p. 379. 236 Hence, the (deontic) argument which consists of (28. 18) as its premiss and (28. 17) as its conclusion is a correct one. Likewise, the command which is issued in (28. 19) DO not do to others what you do not want other to do to you! is contained in the command issued in (28. 20) Do not do to others what you do not want others to dO to you, [but] do unto others as you would that they dO unto you ! Hence, the (imperative) argument from (28.20) to (28. 19) is a good one. On the contrary, the argument from (28. 17) to (28. 18) or the argument from (28. 19) to (28. 20) is not a good argument, because in each case the conclusion makes an evaluation (or issues a command) which is not already contained in the premiss. The utility Of distinguishing the good normative arguments from those that are bad can also be better appreciated if we first make a comparison by looking into the usefulness of distinguishing good as ser- toric arguments from others which are not. Let (28.1) A1, A2, A3, ..., Ak' .. C as Specified before be a (pure) assertoric argument, and suppose that it is a good one. Since it is good, the information contained in 'C' is already contained in the conjunction of (28.2) A1, A 2, 3, . . . , k Now, let us think of a piece of information as either true or false i_n a world W, Since (28. l) is a good argument, we know that 'C' conveys true information in a world W provided that what (28. 2) jointly conveys is true information in W. This follows from the fact that the information conveyed in 'C' is contained in the information conveyed jointly in (28. 2). Thus, we see that by recognizing that a certain argument, say (28. l) is a good argument, we know that a certain piece of information, say that which is conveyed in 'C', can be "extracted" from another piece (or other pieces) of information, for instance, that (those) which is (are) conveyed jointly in (28. 2). This seems to answer cogently to our intellec- tual curiosity and practical utility, too, of finding out whether a piece of information lggically follows from another piece (or other pieces) of information when we use such a locution as 'from . . . , it follows (logic- ally) that . . . ' . To be more precise, instead of talking about a piece of infor- mation or pieces of information, let us Speak of a proposition or propo- sitions understanding tacitly that the piece of information conveyed in a sentence—we know that what are the premisses and conclusion of an argument are certain sentences—is the prOposition denoted or indicated by that sentence. After this transference, it is easily seen that by recognizing a certain argument as a good one, we know that a certain proposition, i. e. , the one which is indicated by the conclusion 238 of this argument, can be "extracted" from another proposition or other propositions, i. e. , one(s) indicated by the premiSS(es). Now, a proposi- tion is thought of as true or false 1.11 a world W, and sentence is true or false in W if and only if the proposition it indicates is true or false in W. Thus, the relation of "logical following", now to be defined on the set of sentences, can be reconstructed, as we commonly see, in the following way: (C28. 21) 'C' logically follows from (28. 2) if and only if, in every world W, 'C‘ would be true in W if 'Al', I I I I I I ' A2 , A3 , . . . , Ak should all be true in W. i We shall say that (28. 2) logically entails 'c' if and only if 'C' logically follows from (28. 2). Now, a good or correct argument can be identified as one of which the premisses logically entail the conclusion. A good argument is also called a 331E argument. Let us now turn our attention to the case of normative arguments. The utility of a definition or criterion of normative validity can be understOod in a Similar way as we understand the usefulness of a cri- terion of validity for as sertoric arguments. Just as by recognizing a valid assertoric argument, we are able to "extract" some proposition from other propositions because of their "containing" relation, by recognizing a normative argument as being good or valid, we should be, or want to be, able to extract some evaluation from other evaluations and/or some command from other commands. Or, in other words, if 239 we should know that a (normative) argument is good, then we would be able to tell that the evaluation conveyed by the conclusion is made (by a certain set of rules), provided that the evaluations conveyed by the premisses (jointly) are made (by that set of rules), or that the command expressed by the conclusion is issued (by a certain set of rules) if the commands expressed by the premisses are issued (by that set Of rules). If the relationship we just depicted holds between the premisses and the conclusion of a (pure) normative argument, then we Shall say that the argument is ya_:_l_ir(!)~q:| However, it is commonly accepted, and indeed is a theorem, i. e. , (CM1,T*19), that (32.10) ~[(!)q 8: (!)~q] Consequently, by PL again, we conclude (32.11) (!)p That is to say, what we should do is to post the latter rather than do something else, in particular, we Should not burn it! Here we treated (15. 1) as if it had the form (32. 1). In case it has the form (32. 2), similar argument can be put forth. This can be seen as follows. In order to fulfill (32. 11), we have to fulfill 14We write 'not-burn it!' for '~(burn it)‘ which, as we indicated above, is not equivalent to 'Do not burn it!' but rather to the imperative counterpart of 'you are permitted not to burn it'. 264 (32-12) (!)(p V q) because we maintain that (32. 2) is valid. Now, for the same reason, we must also fulfill (32.13) (!)(p V ~q) Hence, we must fulfill (32.14) (!)(p v q) & (!)(p V~q) It follows that we must fulfill (32-15) (!)[(P V q) & (P V ~Q)] because (32.16) (!)(p v q) & (!)(p v ~q) E (H [(p v q) & (p v ~q)] is an instance of a theorem, i. e. , an instance of (CMIR T*27). But (32.15) is, by PL, equivalent to (32.11) (!)p Again, what we should do is to post the mail, there is no way to see that we may burn it. Another question which has often been discussed in the liter ature is whether or not the imperative counterpart of modus tollens is a valid 15 principle or valid argument form. The argument form in question is: 15See, for example, Castefieda, ibid., and Geach, ibid. 265 (32.17) p 3 (!)q16 ~(!)q P Castefieda argues for the validity of modus tollens, but he gives the following example: (32.18) If he comes, leave the files Open! Do not leave the files Open! .. He does not come. This argument, valid or not, is not an instance of (32. 17). It instantiates another quite different argument form, namely, (32. 19) p 3 (!)q17 (X)q .'. P A further argument form which has received considerable atten- tion is the imperative counterpart of the so-called disjunctive syllogism, viz., (32.20) (!)p v (!)q ~(!)q .'. (! )p This argument form is valid according to our criterion. 7It is easy to see that (31. 19) is a valid argument form. To show this, use (CMIR T*21), i. e. , ~((!)P & (X)~P) and PL. 1 8See, for example, Williams, ibid. 266 or, in another form: (32-21) (”(1) V q) (! )~q (! )1) It should be mentioned again that the negation of '(!)q' is '(J)~q' not '(X)q'. Hence, (32.20) is equivalent to (32.22) (!)p V (!)q (J)~q (! )p rather than to (32.23) (!)p V (!)q (X)q (!)p In giving examples, peOple tend to confuse (32.22) with (32. 23). This seems a place to see the merit of bringingto our attention the two imperative operators which do not exist in English. In concluding this section, let us mention, and it is easy to show, that all the argument forms (32. 20)-(32. 23) are valid. APPENDICES A. AXIOMS AND RULES System vW* Axioms: (A1) I- P 3 [q 3p] (A2) I-[p3[q3r]] D[[p3q13[p3r]] (A3) I— E~q D~pl 3 [p 3 q] (Ad6) !— PEp V q] '='. Pp V Pq Rules: (R 1 ) Substitution (R2) £55193 ponens (Rd6) P-extensionality: From I-A E B1 we may infer rPA e- P13" System FA Axioms: (FAl) [- O_a_ 2) P3 (FAZ) )— 03 & op. : oxgg (FAB) )- P_a_ & P_b. 3 PK_a_b Rules: (RFAl) Substitution (RFAZ) modus ponens (RFA3) Extensionality: if r X 3 Y1 is any of the PL-theorems PLl-PL8 listed below, and 268 II II II II II X':Sp’ q! ~’ &’V x and l I I I I I I l I I p: q: “’9 8K: V Y' _S 'a', lbl, INI, 1K1, IAIYI and the deontic connectives in X' and Y' are placed "correctly", then I”PX'D PY'1 is a theorem of FA. (PLl) P~[p8rq] 3 ~[q&p] (PLZ) f—[p&q]Dp (PL3) )— [p at q] 3 q (PL4) )— [p at q] 2 [p v q] (PL5) I- p=>~~p (PL6) )— ~~p D p (PL?) I- ~f~~p & ~~q] D~[p & q] (PL8) #- p 3 [qu] (RFA4) Replacement in PL-theorems: If x is any theorem of PL, and 5 is a propositional variable, and C a deontic formula, then 5 scxl is a theorem of FA. III . IV. 270 Systems 0T*-055* Axioms: i) for 0T* (A1) - (A3) Same. (Ad4) )— o[p 3 q] 3 [0p 3 Oq] (Ad6) [— Op 3 ~O~p (Ad7) )- o[op :2 pl ii) for 054* Axioms for OT):< plus (Ad8) (Ad8) )- Op 3 OOp iii) for 055* Axioms for 054* plus (Ad9) (Ad9) [— ~Op :3 O ~Op Rules (for 0T*-035*): (R1)-(R2) Same (Rd3) Deontic necessitation: System OM-OM'I Axioms: i) for OM (A1)-(A3) Same (AN4) I- P 3 Op (AN5) I- OIpvq] 5 [Ovaq] From A we may inferr 0A1. 271 ii) for OM' Axioms for OM plus (AN6) (AN6) (— 00p 3 Op iii) for OM" Axioms for OM' plus (AN7) (AN7) [— O~Op3~0p Rules (for OM-OM"): (R1)-(R2) Same (RN3) Extensionality: From r A :3 B.' we may infer VGA 3 03". (RN4) Necessitation: From A we may infer r ~<>~A-I . B. DEFINITIONS System vW (D2.1) '06“ = ''~ ~ " Df p a (D2.2) 'F ‘ _—.- "~ 1 a Df Pa (D2.3) rI " = " " 01 Df PU & P~O’ Systems 0T*-055* (Ddl) rPA1 = 'U» " Df O~A (DdZ) rFA": '~ ' Df PA (Dd3) ’lA" = rPA & P~A‘ Df Systems OM-OM" (DNl) 'DA‘ = I'~ ~ I Df 0 A (DNZ rA-aB‘: .- 1 ) Df [MA :13] (DN3) s = ' .... Df .13.”. 3* °~§' r (DN4) PA" :Df 'OEA & ~p,]1 FA.‘ 2Df r~PA1 I” (DN5 ) (DN7) r1A1 =Df'PA ea P~A1 Systems CMo, T*-CMo,ss* r *1 (D17.l) ORA = Rrequires that A*. Df I' *‘I (Dl7.2) PRA = l"~o,,~A""' Df D17. " *‘— " ( 3) FRA _. ~P,A*" Df I” 'I (D17.4) I,A* = "P,A* isil‘.->,,~.z>.""I Df 272 273 Systems CMl,T*-CMI,55* (D25. 3) l-(!,,)A’M :Df R requires that A* . (D24. 1) "(,,/,,)A"“=Df"~(l,,)~A*1 (D24. 2) "(X, )A*"= f Pq./R )A* " D (1324.3) reludes“=Df"(J..)A"‘ & (J.)~A*' (0T*1) (OT*2) (0T*3) (OT*4) (OT*S) (0T*6) (0T*?) (OT*8) (0T*9) (0T*10) (0T*11) (0T*12) (OT*13) (OT*14) (OT*15) (OT*16) (OT*17) (0T*18) (0T*19) (OT*ZO) (0T*21) c. THEOREMS (0T*-055*) l- Op '=' ~P~p I-Op E P~p l-PP =—=’~Fp |—Pp§~O~p |—Fp '-_=~Pp |—Fp E O~p !-0~pE"Pp Ill 2 O "U r-P~P |-Ip:-:.= Pp& P~p l—OPDPp F0~p3~0p t~PP3P~p I-Fp3~0p tFpD~F~p I-F~pDPp l-O(p V~p) f-F(p&~p) r-~(Op & 0~p) t'PPV P~p )-~(Op&Fp) 274 (OT*22) (0T*23) (OT*24) (OT*25) (OT*26) (OT*27) (0T*28) (OT*29) (0T*30) (0T*31) (OT*32) (OT*33) (OT*34) (0T*35) (0T*36) (0T*37) (0T*38) (0T*39) (OT*40) (0T*41) (0T*42) (0T*43) (OT*44) (0T*45) 275 I-0(p &~p)30q r-OPDOIPVq) I-Pp3P(p Vq) l—FPDNP & q) l-F(qu)3Fp |—O(p &q) s. Op8zOq t-P(pV q) '-—=- PpVPq I-F(qu) 5. Fp & Fq }-0(qu) 3- va Pq l—P(p&q) 3. P198: Pq |-Op v Oq. 30(pvq) l-Fp30(p3q)l l-OqDO(qu) t0p&0(p=>q)-30q I-Pp&0(p3q)-=>Pq lr-Fq 8: O(qu)-3Fp f—(Fq&Fr) & O(p:(qu)).DFp I-~(0(qu)&(Fp&Fq)) I—Op & 0((p&q)Dr). DO(qu) I-FP & F91-3F(p&q) |-Fp&0q- DI-*“(p&q) I-Fp8r1p- 3F(p&q) I’FP V 0p V Ip |-Ip D I~p (0T*46) (0T*47) (0T*48) (0T*49) (OT*SO) (OT*Sl) (0T*52) (0T*53) (0T*54) (0T*55) (0T*56) . (054*57) (084*58) (054*5 9) (034*60) (084*6 l) (084*62) (085*63) 276 I—oop: Op l—OFpD Fp t—FPp D Fp t—FPNp DOp I-O(FP D~p) l-O(pDPp) I—OpDOPp l—FpDFOp |—PFpDP~p |——POp DPp )- Pp 3 PPp |-O~p3 OO~p I—O~p'='OO~p l—PPps Pp I-OOP 5 Op |—O~ Op D~Op l—O~Op-=-:~Op BIB LIOGRAPHY I) A General Bibliography of Deontic Logic and Imperative Logic This bibliography includes most of the important works in deontic logic and imperative logic published before 1968. However, only those publications appearing in English are listed except in some cases we give non-English titles for completeness. The starred items are explicitly mentioned or referred to in this dissertation. Allen, L. E. (1960) "Deontic Logic", Modern Uses 2f Logic i_n Law, vol. 60, pp. 13-27. Anderson, A.R. *(1956) The Formal Analysis if. Normative Systems, A Tech- nical Report to the Office of NavalResearch, Yale University. (1958a) "The Logic of Norms", Logigue_e_t Analyse, vol. 1, pp. 84-91. (1958b) "A Reduction of Deontic Logic to Alethic Modal Logic", Mind, vol. 67, pp. 100-103. (1959) "On the Logic of ’Cornrnitment' ", Phil. Studies, vol. 10, pp. 23~27. (1962) "Reply to Mr. Rescher", Phil. Studies, vol. 13, pp.6-8. (1967) "Some Nasty Problems in the Formal Logic of Ethics", Nous, vol. 1, pp. 345-360. Anderson,A.R. and O. K. Moore (1957) "The Formal Analysis of Normative Concepts", The American Sociological Review, vol. 22, pp. 9-17. Apostel, L. (1960) "Game Theory and the Interpretation of Deontic Logic", Logiqueit Analyse, vol. 3, pp. 70-90. Bar-Hillel, Y. (1966) "Imperative Inference", Analysis, vol. 26, pp. 79-82. 277 278 Beardsley, E. L. (1 944) "Imperative Sentences in Relation to Indicatives", The Phil. Review, vol. 53, pp, 175-185, Berg, J. (1960) "A Note on Deontic Logic", Mind, vol. 69, pp. 566-567. . Bergstrfim, L. (1962) Imperatives and Ethics, Filosofiska studier utgivna av Filosofiska Institutionen vid Stockholms Universitet, Stockholm. Beth, E.W. (1946-47) "Discussion", Synthese, vol. 5, pp. 94-95. Bohnert, H. G. *(1945) "The Semiotic Status of Commands", Phil. 9_f_ Science, vol. 12, pp. 302-315. C as tafieda, H. N. (1955) "A Note on Imperative Logic", Phil. Studies, vol. 6, pp. 1-4. (1957) "On the Logic of Norms", Methodos, vol. 9, pp. 209-215. (1958) "Imperatives and Deontic Logic", Analysis, vol. 19, pp. 42-48. (1959) "The Logic of Obligation", Phil. Studies, vol. 10, pp. l7-22. (1960a) "Obligation and Modal Logic", Logigueit Analyse, vol. 3, pp. 40-48. *(l960b) "Imperative Reasonings", Phil. and Phenom. Research, vol. 21, pp. 21-49. (1964) "Correction to "The Logic of Obligation" (A Reply)", Phil. Studies, vol. 15, pp. 25-28. (1966) "A Note on Deontic Logic (A Rejoinder)", The Journal £31111” vol. 63, pp. 231-234. (1967a) "Actions, Imperatives, and Obligations", Aristotelian Societ ’2797, pp. 25-48. (1967b) "Indicators and Quasi-Indicators", Am. Phil. Quart, vol. 4, pp. 85-100. (1968) "Acts, the Logic of Obligation, and Deontic Calculi", Phil. Studies, vol. 19, pp. 13-26. Castafieda, H. N. and G. Nakhnikian (1962) See Nakhnikian. 279 Chisholm, R. M. *(1963a) *(l963b) (1964) Cohen, J. (1951) Cresswell, M. (1967) Davidson, D. (1967) Dawson, E. E. (1959) Downing, P.B. (1961) ESper sen, J. *(1967) Fenstad, J. E. (1959a) (1959b) Fisher, M. (1961a) *(1961b) (1962a) (1962b) (1962c) (1965) "Contrary-to-duty Imperatives and Deontic Logic", Analysis, vol. 24, pp. 33-36. "Supererogation and Offence", Ratio, vol. 5, pp. 1-14. "The Ethics of Requirement", Am. Phil. Quart. , vol. 1, pp. 147-153. "Three-Valued Ethics", Philosophy, vol. 26, pp.208-227. J. "Some Further Semantics for Deontic Logic", Logigue gtAnalyse, vol. 10, pp. 179-191. "The Logical Form of Action Sentences", In Rescher (1967a), pp. 81-95. "A Model for Deontic Logic", Analysis, vol. 19, pp. 73-78. "Opposite Conditionals and Deontic Logic", Mind, vol. 70, pp. 491-502. "The Logic of Imperative 5", Danish Yearbook of Phil. , vol. 4, pp. 57-112. "Notes on Normative Logic", Avhandlinger utgitt fl Det Norske VidenskaRs-Akademi i Oslo, 0810. "Notes on the Application of Formal Methods in the Soft Sciences", Inquiry, vol. 2, pp. 34-64. "A Logical Theory of Commanding", Logique _e_t_ Analyse, vol. 4, pp. 154-169. "A Three-Valued Calculus for Deontic Logic", Theoria, vol. 27, pp. 107-118. "On a so-called Paradox of Obligation", The Journal of Phil. , vol. 59, pp. 23-26. 7Strong and Weak Negation of Imperatives", Theoria, vol. 28, pp. 196-200. "A System of Deontic-Alethic Modal Logic", Mind, vol. 71, pp. 231-236. "A Contradiction in Deontic Logic? ", Analysis, vol. 25, pp. 12-13. 280 Fitch, F.B. (1963) "A Logical Analysis of Some Value Concepts", The Journalo_fSJ/m. Logic, vol. 28, pp. 135-142. _— *(1966) "Natural Deduction Rules for Obligation", Am. Phil. Quart., vol. 3, pp. 27-38. Geach, P.T. (1958) "Imperative and Deontic Logic", Analysis, vol. 18, pp. 49-56. *(1963) "Imperative Inference", Analysis, vol. 23 (Suppl. ), pp. 37-42. (1966) "Dr. Kenny on Practical Inference", Analysis, vol. 26, pp. 76-79. Gibbons, P.C. (1960) "Imperatives and Indicatives", Australasian Journal of Phil., vol. 38. pp. 107-119 and 207-219. Goble, L. F. *(1966) "The Iteration of Deontic Modalities", Logigue _e_t Analyse, V01. 9, pp. 197-209. Grelling, K. *(1939) "Zur Logik der Sollsatze", Unity o_f Science Forum, pp. 44-47. Hanson, W. H. *(1965) "Semantics for Deontic Logic", Logigue gt Analyse, vol. 8, pp. 177-190. (1966) "A Logic of Commands", Logigue _e_t Analyse, vol. 9, pp. 329-318. Hare, R. M. (1949) "Imperative Sentences", Mind, vol. 58, pp. 21-39. (1961) The Larguaig 2f. Morals, Oxford. (1967) "Some Alleged Differences between Imperatives and Indicatives", Mind, vol. 76, pp. 309-326. Hintikka, J. J. K. *(1957) "Quantifiers in Deontic Logic", Societas Scientiarum Fennica, Commontationes humanarum letterarum, Helsingfors, vol. 23, p. 1023. Hofstadter, A. and J. C. C. McKinsey *(1939) "On the Logic of Imperative", Phil. ngcience, vol. 6, pp. 446-457. Holmes, R. L. (1967) Jarvis, J. (1962) Jérgensen, J. *(1937) (1938) Keene, G. B . *(1966) Kenny, A. J. *(1966) 281 "Negation and the Logic of Deontic Assertions", Inquiry, vol. 10, pp. 89-95. "Practical Reasoning", Phil. Quart., vol. 12, pp. 316-328. "Imperatives and Logic", Erkenntnis, vol. 7, pp. 288-296. "Imperatives og Logik" ("Imperative and Logic"), Theoria, vol. 4, pp. 183-190. "Can Commands Have Logical Consequences? ", Am. Phil. Quart, vol. 3, pp. 57-63. "Practical Inference", Analysis, vol. 26, pp. 65-75. Lemmon, E. J. (1960) (1962) (1965) McKinsey, J. (1939) See Nowell-Smith. "Moral Dilemmas", Phil. Review, vol. 71, pp. 139-158. "Deontic Logic and the Logic of Imperatives", Logigue§_tAnalyse, vol. 8, pp. 39-71. C. C. See Hofstadter. McLaughlin, R. N. (1955) Mally, E. *(1926) "Further Problems of Derived Obligation", Mind, vol. 64, pp. 400-402. Grundgesetze des Sollens: Elemente der Logic des Willens, Graz. Marcus, R. B. *(1966) Menger, K. *(1939) "Iterated Deontic Modalities", Mind, vol. 75, pp. 580- 582. "A Logic of the Doubtful. On Optative and Imperative Logic", Reports 5153 Mathematical Colloquium, Univ. of Notre Dame, pp. 53-64. 282 Meredith, D. (1956) "A Correction to von Wright's Decision Procedure for the Deontic System P", Mind, vol. 65, pp. 548-550. Moore, O. K. (1957) See Anderson. Nakhnikian, G. and H. N. Castafieda (eds.) (1962) Morality and the Language 9_f Conduct, Detroit. Nowell-Smith, P. H. and E. J. Lemrnon (1960) "Escapism: The Logical Basis of Ethics", Mind, vol. 69, pp. 289-300. Nozick, R. and R. Routley (1962) "Escaping the Good Samaritan Paradox", Mind, vol. 71, pp. 377-382. Peters, A. F. (1949) "R. M. Hare on Imperative Sentences: A Criticism", Mind, vol. 58, pp. 535-540. Prior, A. N. *(1954) "The Paradoxes of Derived Obligation", Mind, v01. 63, pp. 64-65. *(1956) "A Note on the Logic of Obligation", Revue PhilosoPhique d_e_Louvain, vol. 54, pp. 86-87. (1957) Time and Modalitl, Oxford. (1958) "Escapism: The Logical Basis of Ethics", in Melden, A. 1. (Ed.) Essays i_nMoral PhilOSOphy, Seattle, pp. 135-146. *(1962) Formal Logic, Second ed. , Oxford. (1964) "The Done Thing", Mind, vol. 73, pp. 441-442. (1967) "Logic, Deontic", The Encyclgpedia (iPhilosophy (ed.) P. Edwards, New York, pp. 509-513. Rand, R. (1962) "The Logic of Demand-Sentence", Smthese, vol. 14, pp. 237-254. Rescher, N. *(1958) ‘ "An Axiom System for Deontic Logic", Phil. Studies, vol. 9, pp. 24-30 (A Corrigenda on p. 64) *(1962) "Conditional Permission in Deontic Logic", Phil. Studies, vol. 13, pp. 1-6. 283 (1964) "Can One Infer Conunands from Commands? ", Analysis, vol. 24, pp. 176-179. (1966a) Logic 2f Commands, New York. (1966b) "Recent Trends and Developments in Logic", Logigue gtAnalyse, vol. 9, pp. 269-279. *(l967a) _T_l_l§ Logic o_f Decision and Action (ed. ), Pittsburgh. (1967b) "Aspects of Action", in Rescher (1967a), pp. 215-219. Rickman, H. P. (1963) "Escapism: The Logical Basis of Ethics", Mind, vol. 72, pp. 273-274. Robison, J. (1964) "Who, What, Where, and When: A Note on Deontic Logic", Phil. Studies, vol. 15, pp. 89-92. Routley, R. (1962) See Nozick. Ross, A. *(1941) "Imperative and Logic", Theoria, vol. 7, pp. 53-72. *(1944) "Imperatives and Logic", Phil. _ofScience, vol. 11, pp. 30-46, (Reprint of (1941)). (1964) "On Moral Reasoning", Danish Yearbook _of Phil. , vol. 1, pp. 120-132. *(1966) Directives and Norms, London Sellars, W. (1956) "Imperatives, Intentions and the Logic of 'Ought' ", 9 Methodos, vol. 8, pp. 227-268. Sluya, H. D. (1963) "Some Remarks on Deontics", Theoria, vol. 29, pp. 70-78. Smiley, T. J. (1963a) "Relative Necessity", The Journal 91 Sym. Logic, vol. 28, pp. 113-134. (1963b) "The Logical Basis of Ethics", Acta Philosophica Fennica, vol. 16, pp. 237-246. Sosa, E. (1965) "Actions and Their Results", Logigue _e_t_ Analyse, vol. 8, pp. 111-125. (1966a) "The Logic of Imperatives", Theoria, vol. 32, pp. 224-235. (1966b) (1966c) (1967) Stenius, E. (1963) 284 ”On Practical Inference and the Logic Imperatives", Theoria, vol. 32, pp. 211-223. "Imperative and Referential Opacity", Analysis, vol. 27, pp. 49-52. "The Semantics of Imperatives", Am. Phil. Quart. , vol. 4, pp. 57-64. "The Principles of a Logic of Normative Systems", Acta PhilOSOphica Fennica, vol. 16, pp. 247-260. Williams, S. A. O. *(1963) Wright, G. H. *(l951a) *(1951b) (1952) (1955) *(1956) (1957) (1963a) *(1963b) (1963c) *(1964) *(1965a) *(1965b) *(1966) (1967a) (1967b) Aqvist, L. (1962a) (1962b) "Imperative Inference", Analysis, vol. 23 (Suppl.), pp. 30-36. von "Deontic Logic", Mind, vol. 60, pp. 1-15. All Essay 2'. Modal Logic, Amsterdam. "On the Logic of Some Aciological and Epistemological Concepts", Ajatus, vol. 17, pp. 213-234. "Om s. k. praktiska Slutledningar", Tidsskriftfgr Rettsvitenskap, vol. 68, pp. 465-495. "A Note on Deontic Logic and Derived Obligation", Mind, vol. 65, pp. 507-509. Logical Studies, New York The Logic 9_f_ Preference, Edinburgh. Norm and Action, London. The Varieties o_f Goodness, London. "A New System of Deontic. Logic", The Danish Yearbook £3121. , vol. 1, pp. 173-182. "A Correction to a New System of Deontic Logic", The Danish Yearbook _o_f£_h1_1., vol. 2, pp. 103-107. " "And Next" ", Acta Philosophica Fennica, vol. 18, pp. 293-304. " "And Then" ", Societas Scientiarum Fennica, Commentationes Physico-Mathematicae, vol. 32, pp. 1-11. "Deontic Logic", Am. Phil. Quart., vol. 4, pp. 136-143. "The Logic of Action--A Sketch", in Rescher (1967a), pp. 121-139. "Interpretations of Deontic Logic", Filosofiska Studier tillégglade Konrad Marc-Wgau den 4 April 1962, Uppsala, pp. 15-23. "A Binary Primitive in Deontic Logic", Logique _e_t Analyse, vol. 5, pp. 90-97. 285 (1963a) "A Note on Commitment", Phil. Studies, vol. 14, pp. 22-25. _ *(1963b) "Postulate Sets and Decision Procedures for Some Systems of Deontic Logic", Theoria, vol. 29. pp 154-175. >.'<(l963c) "Deontic Logic Based on a Logic of 'Better' ", Acta PhilOSOphica Fennica, vol. 16, pp. 285-290. (1964a) "On Dawson-models for Deontic Logic", Logigue_e_t Analyse, vol. 7, pp. 14-21. (1964b) "Interpretations of Deontic Logic", Mind, vol. 73, pp. 246-253. *(1965) "Choice-offering and Alternative-presenting Disjunc- tive Commands", Analysis, vol. 25, pp. 185-187. *(1966) ""Next" and "Ought"", Logiguee_tAnalyse, vol. 34, pp. 231-251. *(1967) "Good Samaritans, Contrary-to-Duty Imperatives, and Epistemic Obligations", Nofis, vol. 1, no. 4, pp. 361-379. II) Other References Church, A. (1956) Introduction to Mathematical Logic, vol. I, Princeton. Feys, R. (1937-38) "Les logiques nouvelles des modalités", Revue Neo- scholastiqueglgPhilosophie, vol. 40, pp. 517-553, and vol. 41, pp. 217-252. (1955) "EXpression modale du 'devoir-étre' ", Journal 2f Sym. Logic, vol. 20, pp. 91-92. ( 1956) "Reply to An. n Prior, 'A Note on the Logic of Obligation' Revue PhilosophiquedELouvain, vol. 54, pp. 88-89. (1965) Modal Logics, Louvain and Paris. Halldén, S. (1957) The Logic _<_)_f_ Better, Copenhagen. Hintikka, J. "The Modes of Modality", Acta Philosophica Fennica, vol. 16, pp. 65-81. Kripke, s. A. (1963) "Semantical Analysis of Modal Logic (1), Normal Modal Propositional Calculi", Zeitschrift 5&1: Mathe- matische Logik und Grundlagen der Mathematik, vol. 9, pp. 67-96. 286 Leonard, H. S. (1957) (1959a) (1959b) (1961) (1967) Lukas iewicz, (1930) Massey, G. J. (1969) Quine, W. V. (1950) Sobocinski, B . (1953) Tar ski, A. (1930) Principles_9_f Right Reason, New York. "Interrogatives, Imperatives, Truth, Falsity and Lies", Phil. ngcience, vol. 26, pp. 172-186. "Authorship and Purpose", £1111: o_fScience, vol. 26, pp. 277-294. "A Reply to Professor Wheatley", Phil. 9f vol. 28, pp. 55-64. PrincipleS_o_f Reasoning, New York. Science, J. and A. Tarski "Unter suchungen iiber den Aussagenkalkiil", Sprawozdania 5 posiedzefi Towarzystwa Naukowejg War szawskiegg, vol. 23, no. 1-3 (in Comptes rendus des Seances £1313 Société des Sciences gt deS lettres _d_e Varsovie, Classe III) pp. 30-50. If Under standingSyrnbolic Logic, New York. 0. Methong Logic, New York. (Rev. ed. , 1959) "Note on a Modal System of Feys-von Wright", 92352212 Compugrlg Sistems. vol. 1. pp. 171-178. See Lukasiewicz.