EXISTENCE, CONSISTENCY. AW OBLIGUE DISCOURSE

Thesis for “IO Degree oi: M. A.
MICHEGAN STATE WEVERSETY

Robert Murray Jones
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EXISTENCE, CONSIQZLECI, AND

OBLIQUE DISCCLnHZ

By

Robert Murray Jones

A THESIS
Submitted to the College of Science and Arts of Michigan
State University of Agriculture and Applied Science
in partial fulfillment of the requirements
for the degree of

MASTER OF ARTS

Department of Philosophy
1957

AN ABSTRACT

This thesis develops a system of logic containing
both modal operators and quantifiers. This system contains
C. I. Lewis' system 84, but it does not contain restrictions
of type theory. In it, certain departures from similar
systems of Ruth Barcan, Rudolf Carnap, and Frederic Fitch
are proposed. One such departure is the inclusion in the
system of a notation for singular existence as this has
been done by Henry Leonard.

The thesis also includes an outline of a
second system which is an attempt to codify G. Freges
notion of the oblique occurrence of a term in a context.
This system is applied to a treatment of the paradoxes
of the theory of types, in order to Justify abandoning
type theoretical restrictions in the first system.

ACI’TN O‘A'LED GEI‘JENT

The author wishes to express grateful thanks to
Professor Henry S. Leonard, Head of the Department of
Philosophy, under whose guidance this thesis was written.
His courses in logic, and especially, his recent paper,
"The Logic of Existence", have been the source of this
thesis.

I wish also to thank Dr. Richard S. Rudner,
Assistant Professor of Philosophy, for his many dis-

cussions of various positions that are maintained here.

CHAPTER

II

III

CONTENTS

INTRODUCTION . . . . . . .t. . . .
1.1 The Logic of Existence . . .
1.2 Existence and Modal Logic . .

1.3 Parmenides and Assertions of
I‘IOH‘EXiStenCe o o o o o o o o

1.4 The "Logic of Possibles". . .

1.5 Direct and Oblique Modes of
Raference . o o o o o o o o 0

1.6 Connotative Logic and Russell'

Paradox . . . . . . . . . . .

DIRECT DISCOURSE AND DENOTATIVE
LOGIC: SYSTEM 1 . . . . . . . . .

2.1 Formation Rules and
Nomenclature, . . . . . . . .

2.2 Definitions . . . . . . . . .
2.3 Postulates . . . . . . . . .
2.4 Transformation Rules . . . .
2.5 Theorems ... . . . . . . . .

OBLIQUE DISCOURSE AND CONNOTATIVE
LOGIC: SYSTEM 2 . . . . . . . . .

APPLICATIONS . . . . . . . . . . .
4.1 Paradoxes of Logic . . . . .
4.2 Identity of Propositions . .

4.3 A Problem in the Theory of
Perception . . . . . . . . .

MISCELLANEOUS COMMENTS AND FURTHER
TOPICS OF INVESTIGATION . . . . .

ll

l4

l4
19
2O
21
28

55
67

67
74

81

87

CHAPTER I
INTRODUCTION

1.1 Professor Henry S. Leonard has recently published
a paper entitled "The Logic of Existence"1 which modifies
the logical system of Principia gathgmgtigaz in order to
deal with questions of existence of which there is no treat-
ment in the latter system.

The alterations in logic proposed by Professor
Leonard consist, in part, in a notation for singular
existence which takes variables as arguments, recognition
of certain laws governing existence, not expressible in
Principia Mathematigg, and the introduction into logic of
terms which do not denote.

The following two systems are based upon Professor
Leonard's paper.
1.2 Throughout his paper, "The Logic of Existence",
Professor Leonard calls attention to the importance of con-
sidering the topic of modal logic and its bearing upon
questions of existence. This emphasis of Professor Leonard's
paper has influenced the following formulations in many ways.

In the first of the following systems, existence

is not, as in Professor Leonard's system, introduced by

 

1. Henry S. Leonard, "The L0 ic of Existence," Philoso hical
Studies, Vol. VII, Number 4, June 1956), pp. 49- 4.

2. Alfred North Whitehead and Bertrand Russell, Principia
Mathematics, The Cambridge University Press, Firs E on

0.

definition. It is rather taken as primitive. The postu—
lates of System I are used to characterize this primitive.

The considerable hearing which modal logic has
upon existence is illustrated in System I in that, only in
a modal system such as System I, can a primitive "existence"
be adequately characterized. Had System I been an entirely
material logic, the resulting system would not have been
awficiently rich in connections with the primitive "existence"
to have specified the interpretation intended for it.

One such connection, between deducibility and
existence, which is called to our attention in "The Logic
of Existence", consists in the invalidity of the following
argument:

Santa Claus lives at the North Pole. (1)

:.Someone lives at the North Pole. (2)
The argument from (1) to (2) is invalid because, in addition
to premise (l), a premise to the effect that Santa Claus
exists is required in order that (2) might be inferred.

Because of this consideration, System I contains

only the formula:

fx.Elx —3 (3y)fy (3)
rather than the stronger:
fx -3 (3y)fy (4)

Systems of modal logic which leave existence out

of account and contain the invalid formula (4), contain untrue

theorems such as:

 

 

~0~ (3x) (fxv~fx) (5)
Difficulties occasioned by results such an (S) in systems of
quantified modal logic, like those of Ruth Barcan Marcus3
and Rudolf Carnap,4 have caused much controversy among
logicians.

Unlike Mr. Leonard's system, System I contains as
a law, the formula: "Elx", and thereby also contains the
restriction that only terms which denote are allowed as
suitable for substitution. For this reason, System I is so
to speak, a logic of denotation.

A material logic which, unlike Mr. Leonard's system,
does not introduce terms that do not denote, does not require
a notation for singular existence. It will be maintained in
what follows, that a modal logic requires consideration of
singular existence even though that logic does not allow of

terms that do not denote.

 

3. Ruth C. Barcan, "A Functional Calculus of the First Order
Based on Strict Implication." :29 Journal 9; Symbolic Logic,
vol. 11 (1946) p. 1.

_ "The Deduction Theorem in a Functional
Calculus of First Order Based on Strict Implication", The
Journal 9; Symbolic Logic, vol. 11 (1946) p. 115.

‘ g;_ , "The Identity of Individuals in a Strict
Functional CalciIfis of Second Order", Thg Journal 9f Symbolic
Logic, vol. 12 (1947) p. 12.

4. Rudolf Carnap "Modalities and uantification”, 1p;
Journal of Symbolic Logic, vol. 11 1946), p. 33.

 

 

Meaning and Necessit , The University of
Chicago Press, I947.

 

 

 

 

 

 

 

 

 

1.3 To a certain extent, System II is based upon a
criticism of "The Logic of Existence." However, before
considering this criticism, it might be well to review
certain traditional difficulties concerning existence.
Parmenides made claims to the effect that every-
thing that we believe in or speak of must exist; or put in
other words, we cannot believe in or speak of a thing that

does not exist. A fair sample of such doctrines is to be

found in Plato's Thg Sophis :5

Stranger. The truth is, my friend that
we are faced with an extremely difficult
question. This "appearing" or "seeming"
without really being and the saying of ‘
something which yet is not true--all these
expressions have always been and still are
deeply involved in perplexity. It is
extremely hard, Theaetetus, to find correct
terms in which one may say or think that
falsehoods have a real existence, without
being caught in a contradiction by the mere
utterance of such words.
Theaetetus. Why?
Stranger. The audacity of the statement
IIes in its implication that "what is not"
has being; for in no other way could a
falsehood come to have being. But my young
friend when we were of your age the great
Parmenides from beginning to end testified
against this, constantly telling us what he
also says in his poem:
'Never shall this be proved-~that
things that are not are; but do
thou, in thy inquiry, hold back thy
thought from this way.‘
So we have the great man's testimony, and the
best way to obtain a confession of the truth

 

5. Plato, Th; Sophist, 236D-237B.

may be to put the statement itself to a mild
degree of torture. So, if it makes no dif-
ference to you, let us begin by studying it

on its own merits.

In System I, a theorem affirms that everything we speak of
exists, or that it is impossible to speak of a thing that
does not exist.

It can be proved that:

(x)E:x ) (x)(an ) Elx) (6)
(where 'xSy' abbreviates 'x speaks of y')
and further that:

(X)E!x ' (7)
and thereby:

(x)(an ) Elx) (8)
Since it can also be obtained that (8) is analytic, it follows
that it is impossible, in the sense of inconsistency, that
anyone speaks of a thing that does not exist.

However, Parmenides' injunction may rather be to
the effect: "Do not speak with terms that do not denote".
Putting the rule in such a terminology of mention rather than
of use changes the rule from a necessarily true, and hence
unbreakable one, to one which is breakable, and in fact is
broken.

Thus for instance, we might ask, is it true that
"Santa Claus wears a red suit"? Or even, is it true that
"Santa Claus does not exist"? Apparently, each of these

sentences is not true, since were they to be true, the term

 

E v... VI. .3!

 

 

"Santa Claus" must denote something having respectively the
properties of wearing a red suit and of not existing.

The point of the second of "Parmenides' rules"
would then seem to be to prevent us from asserting sentences
which must be automatically untrue because they contain terms
which denote nothing.

Sensible though this second rule seems, the
acceptance of it raises a particularly vexing problem of how
an assertion of non-existence can be true, since such an

assertion seemingly must break the rule if it is to be true.

1.4 The last difficulty was left unresolved. However,
before proceeding to a discussion of any of the several ways
of resolving it that will be recognised here, a way of avoid-
ing it will be examined that will not be followed here.

This way of avoiding the difficulty might be called
"the logic of possibles".

This approach will recognise some things that, while
they do not exist, nevertheless are at least "possibles", and
claim that every term denotes something and that terms such
as ”Santa Claus" merely denote possibles rather than actuals.

"Possible logic" will recognise two systems of
quantifiers. Square brackets might be adopted as a notation
for generalization in an inclusive sense over both possibles
and actuals: "[Xfo" to mean: f is true of everything,

possible or actual. The more usual sort of generalization

 

 

 

over everything that exists might be defined:

(x)fx =Df|:x](E£x)fx). A weak form of existential general-
ization might be defined as:L3fox =Df~IxJ~fx. The usual
strong form of existential generalization might be given the
definition: (3x)fx =Df l3x](E:x.fx).

Mr. W. V. Quine has discussed this problem and
criticised the position of "the logic of possibles" outlined
here.6 He makes the well taken point that while "existence"
is a free word and hence there is nothing to stop us from
using it in such a way as to apply to only a special class of
things rather than to everything unrestrictedly, to do so is
nevertheless to take away the word's usual meaning.

However, it is not only the case that the'logic of
possibles" takes away the meaning of the term "existence,"
but it takes away the meaning of quantifiers as well. In fact
the "logic of possibles" reinterprets the whole of logic as
applying to only a restricted class of things, with a new
set of quantifiers and kind of existence for the things that
are left over. In short, an objection to "the logic of
possibles" is that what was meant by parenthetic quantifiers
in the first place is the meaning which the "logic of possibles"
gives to bracket quantifiers, after the parenthetic quantifiers
have been suitably misinterpreted.

 

6. Willard Van Orman Quins, "On‘What There Is", From A
Logical Point of View, Harvard University Press, l9é3.

The logic of possibles achieves its extension of

the usual logic by misconstruing that logic in a narrow way.

1.5 A clue to a resolution of the problem of how there
can be true statements of non-existence is to be found in
one standard means of making such statements--by employing
Russell's theory of definite descriptions.

The sentence "the man who lives at the North Pole
does not exist" can be true without incurring any paradox.
This is so since it can be read as saying that the preperty
of being a man living at the North Pole has either no in-
stances or more than one instance. A feature of the
sentence that is important to note is that though the definite
description occurring in it does not denote anything, it
nevertheless does refer to something; namely the property of
living at the North Pole.

Frege has proposed the term "oblique" to apply to
linguistic expressions which, although they do refer, do not
do so in the usual mode of denotation. The definite des-
cription in the example in question would seem to be
occurring in a way which perhaps could be described by Frege's
"oblique".

In any case, the term "oblique" will be taken in all
of the following discussion to apply to any reference made

by a term which is not a reference by denotation. The term

 

 

"oblique" in its present meaning might be defined as follows:
("ny" abbreviates "x refers to y", "ny abbreviates "x
denotes y", and "xOy" abbreviates "x obliquely refers to y").

x0y =Df ny.-ny (9)
Such oblique usages of terms allow non-denoting terms to be
used in true sentences because such terms may refer in some
oblique way to something of which the sentence says some-
thing true.

So to speak, the last statement of Parmenides' rule
should be reformulated so as to read: "Do not speak using
terms that do not denote and are not used obliquely". Any such
non-oblique or denotative mode of reference will be referred
to hereafter as "direct reference".

As was mentioned above, System I is a logic of
denotation, and therefore also a logic of direct reference.

System II on the other hand, is a logic of con-
notation and of oblique reference. System II is based upon
a mode of reference in which the referents are connotata
rather than, as in System I, denotata. Angle brackets
("<>") are introduced in System II with the meaning that
an expreSSion together with angle brackets enclosing it shall
be taken to name the connotatum of the expression enclosed in
such brackets. Although the usage of angle brackets is
characterized in System II by postulates and rules of trans-
formation, two points concerning the interpretation might be

mentioned here rather than deferred to Chapter IV.

10

First, the relation of connotation is analogous
to that of denotation in that in either mode of reference,

a term has at most one referent, but differs from the latter
in that every referent in the mode of connotation is a
characteristic or prOperty which is, so to speak, a
definitional criterion by which one identifies the associated
denotatum. This is to say, the possession of, or failure

to possess, the connotatum of a given term is a test by which
a thing may be respectively accepted or rejected as the
denotatum of that term. Secondly, sentences as well as

terms may be enclosed in angle brackets, and if the former is
the case, then the indicated connotatum is a definitional
characteristic of a state of affairs.

However, the meaning of the angle bracket notation
will be explained in more detail later. The last point to
be made in the present section is the criticism of "The Logic
of Existence", which was mentioned at the opening of Section
1.3 but was deferred until a consideration could be made of
the difficulties which lead to Parmenides' injunction in
one or another of its forms.

That criticism is that "The Logic of Existence"
allows statements of non-existence to appear in the system
without an explicit notation indicating the mode of reference
in which such statements are to be interpreted. The absence
of such a notation becomes even more serious in the interpre—

tation of propositional logic than in the interpretation of

11

a functional logic such as "The Logic of Existence". It

 

will be maintained later, that many formulas which are valid
laws of propositional logic if interpreted in a direct mode

of reference, are not valid when interpreted in another mode

of reference.

1.6 Every denotative logic, such as System I, allows
the substitution only of terms which denote in any inference
carried out within that logic. As a consequence of this,
any argument which is purportedly carried out within such a
logic, but which contains steps which make substitutions of
terms which do not denote for free variables in formulas
of the logic, is an invalid argument, and since it is not
carried out within the rules of the logic in question, is in
fact not an inference of that system of logic at all. One of
the purposes of developing System II is to show that at least
some of the paradoxes of the theory of types require such
arguments in order that they might be inferred.

In particular, the term "k" defined:

k =Df f<~rf) (11)
would seem not to denote a property. Speaking obliquely,
the property of non-self—application does not exist. But if
such is the case, then the argument which leads to Russell's

paradox is not an inference within either System I or

Principia Mathematics.

 

12

Russell proscribesin general, all reference to
"illegitimate totalities". It will be suggested that the
phrase "illegitimate totality" might be interpreted to mean
"non-existent totality". Parmenides' injunctions then pro-
vide a means for avoiding the paradoxes of the theory of
types which does not depend upon the reason given by Russell
in formulating the theory of types, namely that unrestricted
generalization is not possible.

In fact, since no term banned from use by Parmenides'
injunction denotes something, it follows that the injunction,
while it does restrict the vocabulary of a language to which
it is applied, does not correspondingly place any limitation
upon the range of subject matters which can be discoursed
about within that language. Or in other words, avoidance of
paradox, and unrestricted generalization, are together possible.

Put differently, since there are no non-existents, and
hence no non-existent totalities, we may restrict general-
ization to "legitimate totalities", and also, generalize quite
unrestrictedly to everything.

As a summary of the position on existence outlined
above, the following quotation from a tract which Cornford

suggests was written in approximately 400 B.C., speaks for

itself:7

 

7. Francis Macdonald Cornford, Plato'g Theory of Knowledgg,
Routledge & Kegan Paul Ltd., p. 209.

13

"It seems to me in general that there
is no art that is not, for it is irrational
to think that something which is is not.
For what 'being' have things that are not
which one could look at and say of it that
'it is'? For if it is possible to see
things that are not, as you can see things
that are, I do not understand how one can
regard them as not being, when you can see
them with your eyes and think of them in
your mind that they are..."

14

CHAPTER II
DIRECT DISCOURSE AND DENGTATIVE LOGIC: SYSTEM I

2.1 Formation rules and Nomenclature.
2.11 A purpose of Chapter II is to develop a quanti-
fied modal logic. This quantified modal logic will be
referred to hereafter as System I.
2.12 The primitives of System I are those of the pro-
positional modal logic of C. I. Lewis,1 and in addition, five
primitives peculiar to this system. Two of the latter
primitives are signs of grouping.
2.121 The primitives of Lewis are the curl ('~'), the '
dot ('.'), and the diamond (’0').
2.122 The first of three additional primitives that are
not signs of grouping is the predicate of universal in-
stantiation ('A'). To assert the sentence 'A is true of f',
or, 'Af' is to assert that 'f is a property possessed by
everything'. (The expression 'A' is adopted to suggest a
contraction of 'all'.) The more usual notation '(x)fx' will
later be introduced by a definition involving 'A'.

In Principia Mathematica '(x)fx' is not interpreted
to be synonymous with 'f is true of everything unrestrictedly'.

The theory of types calls for a limitation upon the

 

1. Clarence Irving Lewis and Cooper Harold Langford, Symbolic
Logic, The Century Co., 1932.

 

15

universality of a generalization.2 However in the present
system, the interpretation of 'Af' is intended in the un-
restricted sense.

2.123 The second primitive in addition to those of C. I.
Lewis is the predicate of singular existence ('EE').

2.124 The third primitive is the cap (”N'). The cap is
placed over variables preceding prepositional formulas. The
resulting formulas signify pgpperties. Although the cap is
often taken to signify classes, a double cap (EAT) will be
used for this purpose in the present system. The double cap
is introduced by a definition Which is essentially the
Principia Mathematicg definition of indefinite descriptions.3
2.125 The two signs of grouping of the present system are
the left parenthesis ('('), and the right parenthesis (’)').
C. I. Lewis uses a dot system of grouping in his systems of
propositional modal logic. The use of parentheses rather than
dots is a departure of the present notation from that of C. I.
Lewis. (Other departures, all minor, will be noted in due
course.)

2.13 Two kinds of variables occur in System I. 'p', 'q',

'r', or one of these variables followed by a numerical sub-

script will be employed as p;ogositional variables. 'x',

 

2. A. N. Whitehead and Bertrand Russell, Principia Mathematica
v.1 Chapter II.

3. Ibido, *20001’ V0 1 p0 188.

16

'y', 'z', 'w', 'f', 'g', 'h', or one of these letters fol-
lowed by a numerical subscript, will be used as n93:
propositional yggigplgs.
2.14 'F' and 'F' followed by a numerical subscript will
be referred to as formula variablgg. These will be the only
meta-linguistic variables used. They do not occur in
System I.
2.15 A variable-sequence is defined to be any formula
satisfying all of the following conditions.
(1) The first sign of the formula is a left
parenthesis, and the last sign of the
formula is a right parenthesis;
(2) Every sign of the formula that is
neither the first nor last sign of the
formula, is a non-propositional variable;
(3) At least two signs of the formula are
variable tokens.
2.16 A first formula is defined to be any one of the fol-

lowing expressions.

(1) (~p)
(2) (p.q)
(3) 60p)
(4) E!
(5) A
2.17 A wgll fggmgd formula is defined to be any first

formula, any variable-sequence, or any expression obtainable

17

from first formulas and variable-sequences by means of one

or more successive applications of the following Formation

Rulgg.

(Not all first formulas and not all well formed

formulas are assertable, or propositional.)

FRl. A well formed formula may be formed by sub-
stituting in any well formed formula, F, any
propositional or non-propositional variable or
well formed formula for any occurrence of any
variable which is respectively propositional

or non-propositional, provided that occurrence

is free in F.4

FR2. A well formed formula may be formed by pre-

fixing any series of one or more capped non-

. propositional variables not containing two

variable tokens of the same variable type, to
any well formed formula F, such that F contains
at least one occurrence which is free in F of

each variable in the series being prefixed to F.

 

2.18 F1 is bound in F2 if and only if:
(1) F1 is a variable, and F2 is a well
formed formula; and
(2) There is a formula, F3, such that:
(a) F1 is contained in F3 and F3 is
contained in F2;
4. "Propositional" and "non—propositional" are defined below,

in 2.111; "free in F" is defined below, in 2.19.

 

 

 

18

(b) F3 can be formed by an application
of FR2; and
(c) F3 contains in its initial series
of capped variables, a variable of the
same variable type as F1.
2.19 F1 is free in F2 if and only if:
(1) F1 is a variable and F2 is a well
formed formula; and
(2) F1 is contained in F2 and F1 is not
bound in F2.
2.110 F1 binds F2 if and only if:
(1) F1 and F2 are tokens of the same variable
type; and
(2) There is a formula F3, a formula F4,
and a formula F5; such that:
(a) 'F1 and F2 are contained in F3; and
(b) F4 is a series of capped variables,
and F; a prOpositional formula, such that
F3 can be formed in accordance with FR2,
by prefixing F4 to F5; and
(c) F1 is contained in F4, and F2 is not
bound in F5.
2.111 A pzopositional formula is any well formed formula
the first sign of which is a left parenthesis. A ppp:

propositional formula is any well formed formula the first

19

sign of which is not a left parenthesis.5

2.2 Definitions.
2.21 A well formed non-propositional formula, F, is proper
if it satisfies either of the following conditions.

(1) No occurrences in F of any variable are free
in F, and the substitution of F for 'x' in the formula 'Elx'
yields a true sentence.

(2) There is at least one occurrence of a variable
which is free in F, and the substitution of F for 'x' in the
formula 'Eix', and subsequent existential generalization of
the resulting formula with respect to every free variable
occurring in it yields a true sentence.

2.22 C. 1. Lewis' definitions of the modal and truth
functional connectives are adopted in this system, with the
exception of the definition of strict equivalence. A double
arrow (Kiy') is here used as a sign of strict equivalence.

System I contains the following additional
definitions.

D1. 331‘ =Df ~(A5‘:(~(rx))

D1 defines the predicate of plural existence.

D2 defines the Principia notation for universal

instantiation.

D2. (x)fx =Df Ai(rx)

 

5. For example, the first three first formulas are pro-
positional, and the last two are non-propositional.

20

D3 defines the Epippipip notation for plural existence.
D3. (3x)fx =Df 3:i(rx)

D4 defines identity.

D4. x=y =Df (f)(fx)fY)

D5 defines the definite description.

D5. f(9x)gx =Df C3x)(fx.gx).(x)(y)((gx.gy)) x=y)
Finally, D6 defines class abstraction.
D6. rich) =Df (3h)(fh.(x)(gx a hx)

Definitions 5 and 6 are essentially the Principip
definitions, *14.01, and *20.01.
2.3 Postulates.
2.31 The propositional logic of System I is C. I. Lewis'
System S4. System I therefore contains Lewis' postulates
B1-B4, B6, B7, and Becker's postulate.6
2.32 Four postulates concerning quantification are assumed
in System I.

Pl (x)fx.E!y-3fy

P2 (x)(p-3(E!x) fx))€3(p-3(x)fx)

 

6. These are:

B1. (p.q)“3(q.p)

B2. (p. )-3

B3.

B4. ((p. q). pr)-—3(p.(q. r))

B6. ((p—Bq).(q-3r))—B(p—3r)

B7. (p .-9(p—3q))

Becker's Postulate. ~O~p—3~<>-<>~p

(B5 of Lewis' original postulate set for S4, has
been shown by McKinsey to be reducible to those above. See:
J. C. C. McKinsey, "A Reduction in Number..." , American
Mathematical Society Bulletin, vol. 40 (1947), BT‘EEfiiI

 

 

21

P3 O~ (3x)E!x
P4 A Elx
2.4 Transformation rules.
2.41 Some expressions involve predicates formed with cap-

ped variables in such a way that the capped variable expression
as a whole, so to speak, "reduces to" an expression not in-
volving capped variables. For example, the expression:
(§(Mx)a) (l)
which says of a, that it is an x such that M is a property
of x, reduces to:
(Ma) (2)
which says of a, that it possesses M.

Again, predicates involving more than one capped
variable may also "reduce to" simpler expressions. The
expression:

(99(ny)ab) (3)
reduces to:
(Rab) (4)

The following definition is an attempt to codify
the relation of reducing to. This definition is not itself
a transformation rule, although whenever a first line reduces
to a second line, the second line is deducible from the first.
The corresponding transformation rule--the rule of reduction--
will be defined later.

2.42 F1 reduces to F2 if and only if:

22

There is a variable sequence, F3, containing n%1
variables (n>O) and a well formed formula, F4, consisting
of a sequence of n capped variables followed by a proposi-
tional formula, F5, such that:

(1) F1 can be obtained from F3 by substituting
F4 for the first variable occurrence in F3, and some non-
propositional variable or non-propositional formula for every
other variable occurrence in F3; and

(2) F2 can be obtained from F5 by sub-
stituting for each variable occurrence in F5 which is bound
by a variable in the kth position in the series of capped
variables preceding F5 in F4, the expression substituted for
the k/lst variable occurrence in F3 in the series of sub-
stitutions prescribed in (l).
2.43 If a formula or variable is substituted in another
formula for one or more expressions to yield a resultant
formula, some of the signs in the resultant formula are
obtained by exchange of a substitute for an expression in the
formula upon which substitution was carried out, while other
signs in the resultant formula are simply copied from the
formula Upon which substitution is carried out. Signs of the
former sort will be said to result from exchangg, and those of
the latter sort, to result gppm pppyipg.
2.44 For the sake of a more usual notation for quantifiers,
and other variable binders, definitions 2, 3, 5, and 6 were

given in section 2.22. However, System I recognises only caps

 

stituting
R's vari
Variables
1559 k/lst
stitutign
2.43

fOI‘rila fl
formula ’

Obtained 1

 

 

22

There is a variable sequence, F3, containing n/l
variables (n>O) and a well formed formula, F4, consisting
of a sequence of n capped variables followed by a prOposi-
tional formula, F5, such that:

(1) F1 can be obtained from F3 by substituting
F4 for the first variable occurrence in F3, and some non-
propositional variable or non-propositional formula for every
other variable occurrence in F3; and

(2) F2 can be obtained from F5 by sub-
stituting for each variable occurrence in F5 which is bound
by a variable in the kth position in the series of capped
variables preceding F5 in F4, the expression substituted for
the k/lst variable occurrence in F3 in the series of sub-
stitutions prescribed in (l).
2.43 If a formula or variable is substituted in another
formula for one or more expressions to yield a resultant
formula, some of the signs in the resultant formula are
obtained by exchange of a substitute for an expression in the
formula upon which substitution was carried out, while other
signs in the resultant formula are simply copied from the
formula Upon which substitution is carried out. Signs of the
former sort will be said to result £30m exchange, and those of
the latter sort, to result {ppm pppyipg.
2.44 For the sake of a more usual notation for quantifiers,
and other variable binders, definitions 2, 3, 5, and 6 were

given in section 2.22. However, System I recognises only caps

 

variables
of a seq:

tional f0

F4 for t‘l

[>4-

- i
p.cposit

.‘b 91- Var

we. v;

stitut ing
bzr a variz
ariables
the k/lst
stitutions

2.43

a
1012313 £6
a

fin
‘ “walla, E

 

 

22

There is a variable sequence, F3, containing n/l
variables (n>O) and a well formed formula, F4, consisting
of a sequence of n capped variables followed by a proposi-
tional formula, F5, such that:

(1) F1 can be obtained from F3 by substituting
F4 for the first variable occurrence in F3, and some non-
propositional variable or non—propositional formula for every
other variable occurrence in F3; and

(2) F2 can be obtained from F5 by sub-
stituting for each variable occurrence in F5 which is bound
by a variable in the kth position in the series of capped
variables preceding F5 in F4, the expression substituted for
the k%lst variable occurrence in F3 in the series of sub-
stitutions prescribed in (1).
2.43 If a formula or variable is substituted in another
formula for one or more expressions to yield a resultant
formula, some of the signs in the resultant formula are
obtained by exchange of a substitute for an expression in the
formula upon which substitution was carried out, while other
signs in the resultant formula are simply copied from the
formula upon which substitution is carried out. Signs of the
former sort will be said to result from exchangg, and those of
the latter sort, to result £39m pppyipg.
2.44 For the sake of a more usual notation for quantifiers,
and other variable binders, definitions 2, 3, 5, and 6 were

given in section 2.22. However, System I recognises only caps

23

as "official" binding signs, hence it is assumed that all
defined expressions introduced by D2, D3, D5, and D6, are
eliminated prior to application of all except the last of the
following transformation rules. The last transformation rule
is a rule for introducing such defined expressions subsequent
to an inference not involving them. The Transformation Rules
of System I are the following.

TRl. (Adjunction, abbreviated 'Adj') If F1
and F2 are postulates or inferred lines, then the
line F1.F2 may be inferred.

TR2. (Detachment, abbreviated 'Detach') If
F1 and F1I3F2 are previously obtained lines, then
F2 may be inferred.

TR3. (Exchange) If F1 is an obtained line, con-

taining F2, and F26$F is an obtained line, then

3
F3 may be substituted for F2 in F1 to yield an
inferred line F4; provided that every variable

type is such that, there is a token of that type
which is free in F2 but bound in F1, if and only if,
there is a token of that type which is free in F3
and bound in F4.

The proviso to TR3 is necessary because without it,
an inference of the following sort would be
sanctioned by TR3.

(A9(fxv~fx)) (5)
(rxv~rx)<=—=> (fyv-i‘y) (6)

(A§(fyv-fy)) (7)

24

Line (7), inferred from (5) and (6), is not well
formed. However, this "inference" violates the requirement
that every variable type of which there is an occurrence free
in the substitute, but bound in the line in which the sub-
stitute occurs, must have an occurrence in the substituted
expression which is free in the substituted expression, but
bound in the line in which the substituted expression occurs.

TR4. (Free variable substitution) Any

variable which is propositional or non-propositional

may be substituted for every free occurrence in any
previously obtained line of any variable which is
respectively propositional or non-propositional
provided; every sign in the resulting line which

results from exchange is free in the line as a

whOle.

TR5. (Bound variable substitution) If F1 is a

capped variable in any obtained line, then any

non-propositional variable may be substituted for

F1 and every variable bound by F1 provided; that in

the resulting line, every variable F2 is such that

F2 is bound by the variable substituted for F1 if

and only if F2 results from exchange.

Examples of the need for the proviso to TR5 are the

following.

(Afi(fyv~fx))) (fyv(A§¢~fx))) (9)

(Ay(fyv~fy))) (fyv(AxC~fx))) (9)

25

If its proviso is ignored, TR5 would sanction the inference of
the invalid (9) from the valid (8). The part of the proviso
to TR5 which is violated in going from (8) to (9), is the
requirement that every variable in the inferred line which is
bound by a capped variable resulting from exchange must itself
result from exchange. The further demand of the proviso to
TR5 that all bound variables in the inferred line which result
from exchange must be bound by the capped variable which
results from substitution, is violated in the following invalid
argument.

(Ai(A§(rxy))) ) (fxy) (lo)

(Ay(A9(fyy))) ) (fxy) (11)

(10) is a valid formula of System I, while (11)

is not even well formed.

' TR6. (Formula substitution.) Any propositional
or non-propositional formula, F1, may be sub-
stituted for every free occurrence of any variable
which is respectively propositional or non-
prOpositional in any obtained line, provided; every
variable in the resulting line which is free in an
occurrence of F1 resulting from exchange, is also
free in the resulting line as a whole, and every
non-propositional formula in the resulting line is
proper.

The following argument illustrates the need for a

part of the proviso to TR6.

26

(Ai(p)rx)) 3 (p)(Ai(rx))) (12)

(Af((rx) J (fx)) 3 ((fx) 3 (A§(fx))) (13)

(12) is a valid law of System I, while (13) is not
valid. The proviso to TR6 is violated in that, not every
variable in the resulting line which is free in an occurrence
of the substitute which results from exchange, is free in the
line as a whole. The latter part of the proviso to TR6 is
worded so as to apply only to occurrences of the substitute
in the resulting line which result from exchange. The purpose
of this restriction is to allow some perfectly valid infer-
ences which would be proscribed were this part of the proviso
made to apply to every occurrence of the substitute in the
resulting line.

<4i<p><rx)>) 3 (p3<A9<ry))) (14>

(ia<(ry) 3 (fx))) ) ((ry) D (Aicry>)> (15)

The inferalce from (14) to (15) is a substitution
of '(fy)' for 'p' in (14) to yield (15). Both the inference
itself and the formulas involved are valid. However, because
(15) contains in its consequent an occurrence of the sub-
stituted '(fy)', which contains an occurrence of 'y' which
is free in that substitute but not free in (15) as a whole,
this valid inference would not be allowed by'TR6 if its
proviso were to be so worded as to apply to any occurrence of
the substitute in the resulting line, rather than to only those

occurrences of the substitute which result from exchange.

27

That part of the proviso to TR6 which demands that
every formula in the resulting line be proper, has to do with
the avoidance of the paradoxes of the theory of types, and its
relevance will be discussed later.

TR7. (Reduction, abbreviated 'Reduc') If F1

is any obtained line which contains F2 and F2 reduces

to F3, then a new line may be inferred by sub-
stituting F3 for F2 in F1.

TR8. (Universal Generalization, abbreviated

'U.G.') If F1 is any postulate or theorem, con—

taining at least one occurrence of some variable

which is free in F1, then a new line may be infer-
red by substituting 'A' for the first variable of
some two variable variable—sequence and a formula
which can be formed in accordance with FR2 by fol-
lowing a capped occurrence of the variable in
question by F1, for the second variable of the
variable-sequence; provided that in the resulting
line, every non-propositional well formed formula
is proper.

A ninth transformation rule will often be appealed
'to in the course of proofs. This rule governs exchange in
zaccordance with definitions. Although listed as one of the
irules of the formalized part of System I, TR9 will only on
<3ccasion be involved in formal proofs. Often TR9 will only

lye used to introduce contextually defined terms and expressions

such as the classical notations for quantifiers, which must
be eliminated prior to application of the other transformation
rules. Therefore, TR9 will frequently be somewhat informal in
application, its proper use being often left in part to intuition.
TR9. (Definitional exchange) Any expression may
be exchanged with its definitional equivalent in
any postulate or theorem. Definitional exchanges
may be preceded by one or more substitutions of
variables or formulas for each occurrence of some
free variable in the definition, subject to the
restrictions of quantifier control. Such sub-
stitutions may be followed by one or more reductions,

also prior to exchange.

2.5 Theorems.

2.51 Proof annotations in the derivations to follow will
follow the method of C. 1. Lewis in Symbolic ngig. The
proofs of Section I will be given in full. The proofs of
later sections will be abbreviated in accordance with con—
ventions which will be introduced prior to the use of these
abbreviations. Certain theorems of Lewis' system S4 not proved
in Symbolic Logic will be stated without proof. The theorems
of section 0 are the latter theorems of S4. The theorems of
section 1 are dependent upon the first postulate, those of
section 2 upon the first two postulates, and those of section 3

upon the first three postulates, and those of section 4 upon

29

upon all four postulates, while those of section 5 are
miscellaneous items dependent upon various combinations of

the postulates.

2.52 Section 0.

'1‘.0.1 ~0~p «3 ~O-0~p

T.O.2 ~0~p .3 (q8~<>~p)

13.0.3 (paq) —3 (r-3(p3q))

T. 04 ~<>-<>~(pv-p)

r.0.5 ((pv~p)-3q)-3~<>~q

’i‘.0.6 (paq)—3~°~(p3q)

2.53 The following is a translation of the f3fi¥i?§¥¥gfis

in §2.32) for functional logic in System 1, into standard
notation.
rl (Mahmud) -3(£x))
£2 ((A2(pat(szx))(rx))))awautm
s3 (<>(~(3'.sz))) '
B4 (E11)
2.54 Section 1.
r1.1 ((A1) 3((sz)3(£x)))
(From 14.25 L and L; (AH/p; (13::qu (rad/r)
((((Af).(Elx))-3(fx)) é? ((Af)-3((E§x))(fx)))) (1)
(rrom (1), and 2.1, by Exchange) ED (2)
In more usual notation, Tl.1 might be written:
(x)fx -3(E'.x)fx).

and could have been proven in this form. However, had T1.1

30

been proven in the latter form, some steps in the proof would

not be explicitly sanctioned by the transformation rules.

In particular, the rule for exchange in accordance with def-

inition would have been somewhat informally applied.

T1.2 (((fx).(3!x)) -3(31f))
(From 12.43 L and L: (Af)/P; ((Elxhlfo/Q)
(((Af)-8((E!x))(fx)))-3((~((Elx))(fx)))-3(~(Af)))) (1)
(From Tl.l, and (l) by Detach) “

((~((E'.x))(fx)))-a(~(Af))) (2)
((2), by Formula Substitution; £(~(fx){/f)
H-Hazxnd(~(unx>n -3(~(.a(~(:xnm (:5)
((3), by Reduo) ‘

((~( (sunHuHH a (~(12(~(rx))m (4)

The formula '(£(~(fx))x)' in (3) reduces to '(~(fx))'.

Applying the definition of 'reduce to' (see section 2.42) with

n‘l, and:

F4;
F5;

«£41:an
'(~(fx))'
'(xx)'
vierw
'(~(fx))'

it can be seen that the conditions of the definition are met.

(From 14.12 L and 1.: (sthp; (”inn/q)
((~((Etx))(~(fx))))(=$ ((E'.x).(~(~tk))))) (5)

31

(From (4). and (5), by Exchange)

(((Elx).(~(~(fx))))-‘3(~(a§(~(fx))))) (a)
(From 12.5 L and L: (fx)/p)

((3)6 (~(~(fx)))) (7)
(From (6), and (7), by Exchange)
(Manama(~(A£(~(:x))m (a)
(From (8), and 01, by exchange in accordance with def.)
(((Etx).(fx))-3 (3'.f)) (9)

Although step (9) was obtained by means of exchange in
Accordance with Definition, it is not actually a part of the
unformalized development of System I. only those definitions
which must be eliminated prior to application of the Trans-
formation Bules are incompletely formalized. To have explicitly
given the conditions under which definitions not eliminated
prior to application of'the Transformation.Rules, may be
exchanged for their definitional equivalents, would have
required two rules for definitional exchange; one applicable
only to definitions eliminated prior to such application, and
another, applicable only to definitions not so eliminated.
Because of its inconvenience, such a procedure is not followed
here. However, the defhnition called for in (9) need not be
eliminated prior to application of the Transformation Rules,
and.therefore (9) is at least in principle, capable of being

formalized.

 

(u

32

(From 12.15, L and L: (szx)/p; (fx)/q)

(((Etx).(12))é¢((fx).(E:x))) (10)
(From (9) and (10) by exchange) QED. (11)
2.55 In subsequent proofs, various abbreviations will be

used in order to simplify exposition. rarentheses will be
omitted if grouping is evident from context. Some lines of
proofs, easily supplied by the reader, will also be omitted.
Although the more conventional notation for quantifiers is
no more compact than the primitive notation for quantifiers
introduced above, the former will for the most part replace
the latter. If signs of grouping are omitted, the scape of
a first connective will extend over the scope of a second,
if the first precedes the second in the following list:
'td'. (43', '2', '3', 'v', '.', '0', '~J, "'. Following
the proofs there is a catalOgue at all formulas assumed to
be proper.
2.56 Section 2.
can (p-3(x)fx)-3(x)[p-B(3'.x)fx)3

(Tl.l, To.3)

(p-3(x)fx)afl(p-3(x)fx).(,(x)fx-3(n‘.'.x)fx)3 (1)
((1), 11.6 L&L) (p.3(x)rx)-3[p-a(szx)rx)l (2)

(15.2 par) [p-3(s:x)rx)J-3[stx)[p-a(s:x)rx)34 (a)

t"

((2). (a), 11.5 L&L)
(p—3(X)fx)sBEth3EP-3(E:x)fx)ll (4)

33

((4), U.G., -2) 13;) (5)
12.2 (x)fp-3(s:x)rx)3tam-hum)

(12, 22.1, 11.03 Lei) :10 (1)
12.3 (x)[(s'.x.rx)—3pj(shank—3p;

(12.2, 12.44 1&1.) (x)[~(szx)rx)-3~p.e¢(~(x)£x-a~p) (1)

((1). ~p/p) .151) (2)
"12.4 (x)(p-3fx)-3(p-3(x1fx)

('11) [(x)(P-3fx)..$'.x)J-3 (p-arx) (1)

(15.2151) mammal) (2)

((1), (2), 110.3)
[(x))p-3fx).otxg-B[(p-fo).£fx-3(ntx)fx)li (3)
((31,112 1a..) [(pr-afx).me-atpamxmn (4)
((4). U. 3., 14.25 1&1) I
(X)[(x)(p-3fx)-3(Jix)[p-3(;S'.x)fx)33 (5)
((5). :2) :33 (6)

Theorem 2.4 illustrates both a way in which quanti-
_ fied modal logic differs from quantified material logic, and
also the neel for including existence in modal legic. Jhile
the analogue of 2.4 with material implication replacing strict
implication is true hiconditionaly, the converse of 2.4 is not
true (see comment after T2.7), and in order to obtain a law
analogous to T2.4 the main connective of which is an equi-
valence relation, the antecedent of 2.4 must be weaxened by

introducing 'E!’ as in T2.2.

34

r2.5 (x)(£x-3gx)-3[(x)£x—3(x)gx3

(1’1) (x)(fx—agx).mx-3(fx-ng) (1)
((1). r1, T0.3)
(x)(fx'381).E'.x-3((x)fx.)£'.x—3fx).(fx-3gx) (2)
((2). 11.6 JUL-L, 14.26 Lax.)

(x)(fx-381)-3 Elx)[(x)fx-3(E$x)gx)] (3)
((3), U. G., 22) can ’ (4)

T2.6 ~<)~(x)£x¢a(x)~0~(E'.x)£x)
(T2.2: pv~P/P) (x)(pv~p—3.E'.x)fx) (=Hpv~p-3(x)fx) (l)

(T0.5) (pv~p-QE'.x)fx)¢=}-O~(E1x)fx) (2)
(TO.5) (pv~p-3(x)fx)®~9~(x)fx (3)
((1), (2). (3)) QED (4)
r2.7 (ad-494:1? ~°~(x)£x
(T2.4: pv~P/C1) (x)(pv~p-3fx)-3(pv~p-3(x)fx) (1)
(Tons: tic/a) (DV~p-3fx)9~O-fx (2)
(T0.5: (x)fx/q) (pv~p-3(x)fx)®~0~(x)fx (3)
((1). (2>,'(3)) can (4)

The converse of T2.7 is not valid. an.exception to
the converse of T2.7 can be obtained by substituting 'E%' for
'f' in such a supposed law. The "law" fails because, while it is
necessarily the case that everything exists (see T2.15). it
is not the case that everything necessarily exists, and in
fact, of anything it is contingent that it exists (see T3.1).
Had the converse of T2.4 been valid, than the con-

verse of 12.? would have followed. Hence the invalidity of

35

this converse, as exhibited above, also exhibits the in-
validity of that converse.
12.8 (x)(fx-3gx)-3~O~(x)(fx)gx)
(12.7) (x)~0~(£x)gx)-3 ~0~(x)(rx)gx) (1)
((1), 18.7 L3G.) QED (2)
T2.9 (3100 (E'.x.fx)® 0 (3x)fx
(132.6, 12.11 L&L: ~p/p) ~(x:)~0~(E'.x)fx) (as)

--Q~(x)fx (1)

((1), 33, 12.3 1.51.) QED (2)
T2.10 0(3x)fx-a(3x)0fx

(22.7, 12.43 1.21.) --0~(x)fx-3~(x)~0~fx (1)

((1). 12.3 1.21., 2.3) can (2)

The counter instance given earlier to disprove
the converse of T2.7 also disproves the converse of T2.10.

22.11 0 (x)fx a (1)0 (Elx)fx)

(11.1, T0.6) ~0~[(x)fx-3(E'.x)fx)3 (1)
(18.53 tax.) [E(inn-3(szxarx)l.o(x)£xJ-amaintain2)
((1), (2), 18.61 eat) cam-3mm...) (3)
r2.12 (3x)+(six.rx)-a +5101:
I (12.43 1.84.. 22.11) ~(x)O(E'.x)fx)-t3 ~O(x)ix (1)
((1), 12.3 1&1, 14.01 L833, 11.3) 9311' (2)

The converses of T2.11 and 132.12 can be proved
only as material implications in this system (see 513.5).

However, these converses could have been obtained as strict

so

implications had Lewis' postulate Cll been assumed, or had
PS of the present system been replaced with V~0~Q~C31)E1x'.
For if it is necessarily contingent that something exists,
then as a result of the so called "paradoxical" properties of
strict implication, the theorems in question could be obtainei.
In any case, both.Qntecedent and consequent of T2.ll are
always true,.while both antecedent and consequent of T2.12 are
always false.

T2.13 is an alternative form of r2.6.
12.13 ~Q~(x)fx€=)(x)(E'.X‘3fx)

(T2.6, 18.7 LdL) QED (1)
r2.14 ~()~(x)s'.x

(22.13, 14/1, 12.1 1.2;.) QED (1)
r2.15 ~9~(x)(tx)mx)

(15.2 LdL) ix-3(fXJE%x) (l)

((1), u.c.) (x)fsix-3(rx)s'.x)3 (2)

(192.13, (2)) QED ' (3)

It is on account of T2.l5 that farmenides' claims
that everything which one talks about, thinks about, etc.,
exists, can be affirmed as so, and indeed, as logically
necessary.

Since however, the converse of T2.8 is invalid,
(f)~0~(x)(fx)mtx) does not imply (f)(x)(fxr33%x). In fact
it is not the case that (f)(x)(eramlx). Jhile all pro-
perties are extensionally included in the pr0perty of

being an existent, existence is not in the intension of

57

every propertyut‘nat is, not all pr0perties "imply

existence“. For example, no necessary properties imply

existence.

T2.15

T2.17

(x)(E'.x)fx) “3(x)fx

(21) [(x)(s'.x)ix).s'.x] -3(E'.x)fx)
((1), 14.26 121, 12.5 121, 12.7 12L)
(x) (ELfox) .E'.x-3fx

(14.26 121, (2). U.G., 12) QED

(x)(E‘Lx)fx)<‘=¢(x)fx

(15.2 121) 11—3 (E'.x)fx)
((1), U.G., 12.5) (x)fx-3(x)(atx)fx)
((2), 22.16, 11.03 1&1) can

(1)

(2)
(3)

(1)
(2)
(3)

T2.17 affirms in effect, that a generalization about

everything that exists is a generalization about every-

thing.
"legitimate," or existent, totalities is an unrestrictedly

As a result or r2.17, a generalization about all

universal generalization.

T2.18

T2.l9

(31)fxd=9 (31)(E'.x.fx)

(11.3 131, 11.2) fx.E'.x-3L3x)(Elx.fx)
((1), U.G., r112.13) (3x)fx-an)(E'.x.fx)
(n.2, T2.3) (3x)(E'.x.fx)—-3(3x)fx
((2). (3)) QED

(x)(fx)-E'.x)4==v ~(3x)fx

(132.1?) (x) (-fx }~E'.x)¢=D-~(x)~fx
((1), 12.3 1&1, D.3) QED

(l)
(2)
(3)
(4)

(1)
(2)

38

T2.l9 asserts that, none of a certain mind of thing

exists, is equivalent to, that kind of thing does not have
plural existence.
€22.20 (x)(y)m-3(N)(x)m
(Pl) (4:)(y)fxy.slx (y)fxy
(T1.l) (y)fxy «3(Ety3fxy)
((1). (2), 11.6 1&1, 14.26 111, 12.15 111)
(x)(3)fxy.dty.Eix-3fxy
((3), 14.26 1&1, U.G., 1'2) (x)(y)fxy.h".y—3(x)fxy
((4), 14.26 1&1, U.G., 22) QED
12.21 (x)(r)fxy<=>(y)(x)£xr
(By proof similar to that for T2.20)
(mum -3 (x)(3)fxy
((1), 12.21, 11.03 1&1) can
T2.22 (fox-ng).(x)($x-3hx)-a(x)(fx-3hx)
(21) (x)(fx-—351).E'.x-3(fx-ng)
((1): g/f. h/g, 19.68 121)
(x)(fxagx).E£x.(x)(gx6hx)-3(fx€gx).(gx€hx)
((2), 11.6 121, 14.26 1&1, U.G.. 22) QED
T2.23 (x)(fx)gx).(x)(nghx)—3(x)(fx3hx)
(Proof similar to that for T2.22)

2.57 Section 3.

1.501 9'”ng

(1)
(2)

(3)

(4)
(5)

(l)

(2)

(l)

(2)
(3)

39

(731.2: s:/r, 12.76 121) s:x«:3(3x)E:x
((1), 23, 18.52 1&1) 130

T3.2 Q~(3x)fx
(n.2, EL/f, 12.76 121) "ax-3 (3x)E‘.x
((1), 19.51 L861.) Etx.fx-3 (3mm:
((2). U.G.. 132.3) (3x)fx-3 (3x)E'.x
((3), 18.52 1&1) 110

T3.3 ¢~9(3x)fx
(T3.2) O~(3x)0fx
((1). 12.10) QED

There are many true cases of plural existence.

Since it is also the case that the formula 'p-30p' is valid,

there are also many cases of consistent plural existence.
By T3.3, these cases of consistent plural existence are
also cases of contingently consistent plural existence.
These are results of the view that plural existence is
always contingent, and of some of the above laws governing
commutation of modal operators with quantifiers.

Because of these results, the present system is
inconsistent with postulate 611 of Lewis and Langford
Symbolic gggig. This postulate is one of several specula-
tions of Professor Oskar Becker. These appear in Appendix
II of Symbolic ngig, as alternative assumptions concerning
iterated modalities. These alternative assumptions are the

following:

(1)
(2)

(1)
(2)
(3)
(4)

(1)
(2)

40

010: ~0~p —3 ~9-0~p

Cll: OLD-3‘0”“?
012: p3 ~0~0p

and in addition to these, one further alternative:
013: 00p

Lewis shows that in a system such as the present
one, which assumes postulate’ClO, postulate (:15 has exceptions.
Furthermore, if 012 be added to such a system, (:11 becomes
a theorem. Since the present system is inconsistent with
C11, it is also inconsistent with 012.

Hence in the present system, all of the above
speculations are decidable. 011, 012, and 013 all have

exceptions, while 010 is a postulate.

T3.4 O(x)fx
(T3.2, D3) Q-(x)~fx (1)
(1) Omar-11: (2)
((2). 12.3 121) QED (3)

TEA affirms that all generalizations are con-
sistent. This view is a result of assuming that it is
contingent that something exists and that a generalization
to everything that exists is a generalization to everything
unrestrictedly. That it is contingent that something

exists is equivalent to its being consistent that nothing

exists. But a generalization to everything that exists

would be guaranteed true were it the case that nothing

41

existed, since the antecedent of that generalization
would be always false. hence any generalization to
everything that exists--and therefore any generaliza-
tion--is consistent. Although T3.4 affirms as consistent,
even a universal generalization over an inconsistent
property, as in 'O(x)(fx.~fx)' , nevertheless, System i
does not allow as valid '0[(3x)E'.x.(x)(fx.—-fx)]' . That

is to say, it is true that everything is (say) red and

not red, only provided that nothing exists.
23.5 +(fx)-3~(fx-3;~3'.x)

(18.52 1&1, 23.1, 12.44 1&1) 122 (1)
23.6 (x)02x)o(x)rx

(23.4, 15.2 1&1) ans (1)
133.7 Q(x)fx)(x)Q(E'.x)fx)

(22.11) ,QED (1)
23.8 (x)O(E:x)£x))o(x)2x

(23.6) (x)<)(s:x)rx)3<)(x)(E:x)2x) (1)

((1). 22.17) QED (2)

23.9 on”: '4 (x)o(s:x)rx)

(23.7, 23.8) QED (1)
T5.10 ¢~Gx)fx 5 (x)0~(E'.x.fx)

(23.9) QED (1')
505.11 ~0(x)fx 5 (3x)~¢(E'.x)fx)

(23.9) QED (1)

42

2.58 Section 4.

2.581 For the most part, the theorems of section four
depend upon the postulate 'Etx'. This postulate is so to
Speak, tacit, in the system of Principia Mathematics, since
while Principia mathematics contains the restriction that
only terms which denote are allowed in the system, 34 of
the present system does not appear in Principia mathematics.
The logic of rrincipia.hathematica is contained in the
present system. T4.4 and T4.5 of System I correSpond
respectively to l"10.1 and *10.21 of Principia. However,

the inference of these formulas characteristic of Brincipia

Mathematics depends upon the postulate 'Elx'.

 

2.582 The first three theorems and the fifth are
presented in section four because they are key theorems in
the inference or the postulates of Principia.mathematica.
Unlike the remaining theorems of section four, they do
not depend upon B4, and might have given in section two.
24.1 (x)(p)fx) '3PD(x)fx

(P1) (x)(p)fx).E'.x --3p3fx

((1), 14.26 1&1) (x)(p)fx).p 42:13::

((2), U.G., 21) (x)(P)fx).P —3(x)2x

((3), 14.26 1&1) 222
T4.2 p)(x)fx—3(x)(p)fx)

(11.7 1&1.) p.(p)(x)fx) a (x)fx

((1), T1.1) p.(p)(x)fx) ‘3 :31fo1:

(1)
(2)
(3)
(4)

(1)
(2)

45

((2), 14.26 1&1) p)(x)fx €3£Lx)(p)fx) (3)

((6), U.G.. 22) 253 (4)
T4.5 (x)(p)fx)¢$p)(x)fx

(T4.1, T4.2, 11.03 Ldl) 23 (1)
T4.4 (x)fxlfx

(21, 14.26 1&1, r4) QED (1)
24.5 (x)(p)fx) ) (p)(x)fx)

(24.1, 14.1 1&1) 150 (1)

Theorems such as T4.5 and 22.2 are sometimes
referred to as "confinement” laws. Such confinement laws
can be validly formulated for material connectives without
intrOducing tne predicate 'Ei'. This fact, plus the
validity or 24.4 in a logic allowing only of terms wnicn
denote, makes possible in such a.logic, a quantified material
calculus that does not contain PEl'.
T4.6 (3X)Etx

(T1.2) .;l‘.x.;-S'.x-8(3X)L3'.x (1)

((1), 24) QED (2)

fihile T4.6 is orten taken to so an assumption or

logic, in a system such as £rincipia hathematica such a

 

 

theorem cannot be obtained, since a notation for singular

existence is not available.

2.59 Section 5.
25.1 x‘y ) (fx-ny)

44

(T4.4, f/x) (f)(fx)fy) D (fxlfy)

(D4, (1)) x‘y J [(fx-fo) ) (fx-ny)]

((2), 15.8 1&1, 12.1 1&1) 122
T5.2 x‘y)~0~x‘y

(19.52 1.1%: fx/q, fy/r, Elf/p, 14.1 L811.)

(fx-ny) ) (ELf.fx-3fy)

((1). T5.1) x‘y ) (2:1.2x-ary)

((2). 14.26 1&1, U.G., 18.7 1&4, 24.3)

x‘y 3 (r)~9~(s:23(rx)£y))

(T2.6, (3), D4) 188 H
25.3 x‘y 2 ~Q~x=y

(18.42 1&1, 25.2)

T5.3 is true if modal terms are interpreted
as terms of the object language. If a is identical with
b then everything true of a is also true of b. Hence
modal terms of the object language which apply to a also
apply to b.

On the other hand, if modal terms are construed
as terms of the metalanguage, then those that apply to
'a' may not also apply to 'b'. This is so because 'a'
and 'b' are not identical, and therefore not every pro-
perty of 'a' is a prOperty of 'b'.

For example, the sentence '~Q~(RaveRa)' meaning,
a is red or a is not red, is true; and would be a theorem

of the present system, were the constants 'a' and 'R'

(l)
(2)
(3)

(l)

(2)

(5)
(4)

(l)

45

added to the list of first formulas for System 1.
Similarly, "(Bav~Ra)' is analytic' is also true.

However, if an occunsnce of 'b' is substituted
for one out not for two of the occurrences of 'a' in each
of these quoted sentences, and further a is identical
with b, then the result of substitution on the first of
these sentences is true while the result of substitution
on the second sentence is a sentence which would in many
languages be false.

In particular, in the language of System I, if
'analytic' were defined to mean a sentence which is a
substitution instance of a theorem of System 1, then
"(Ravaflb)' is analytic' would be false.

But '~9~(Rav~Rb)' makes an assertion about the
same thing as does i'~'°"(Rav~Ra)', and moreover makes the
same claim concerning that thing as does the latter.
Hence both of these last two sentences are true.

T5.5 is not valid if definite descriptions of
the nussellian sort are allowed to replace the variables
in T5.3. No assertion of necessity containing a definite
description the scOpe of which is the sentence or formula
to which the sign of necessity is prefixed, is res-
pectivly true or valid. This is because every statement
containing a definite description is analysed by hussell into
a statement of plural existence; and none of these are
analytic. For example, ‘~O~(qx)(fx):(qx)(fx)' which is

equivalent to, L~0~((3x)(x=x.fx).(x)(y)((fx.fy) x=y))' is
not only invalid but contravalid as well.

 

 

 

46
Definite descriptionsare involved with T§.3 in
the usual formulations of certain "paradoxes" such as the
paradox of morning star and evening star and the paradox
of analysis.7 A. F. Smullyan was the first writer to notice
that in systems using Russell's analysis of definite des-
criptions, these "paradoxes" could be traced to certain
fallacies involving scopes of definite descriptions,8 al-
though suggestions of such a solution can be found in the
earlier writings of Alonzo Church.9 W. V. Quins has sug-
gested some of the latest versions of such paradoxes,10

while Frederic B. Fitch has given what is perhaps one of the

latest and most comprehensive analyses of the fallacies

inVOIVEd e 11

In addition to the above reasons, there is at least
one other reason why T5.3 may seem paradoxical. Some
logical writing (for instance Frege's) and perhaps in-
formal discourse, enploys a sense of identity which is
apparently quite different from that of '=' in System
I. By 'a is identical with b' is meant something is

named by 'a' and by 'b'. Under this interpretation, the

 

7. W. V. Quine, "Reference and Kodality", in From 3 Logical

Point 2; View.
"The Problem of Interpreting Modal Logic",

Egg—EEEFEEiffif'symbolio Logic, vol. 12 (1947) p. 43.

8. Arthur Francis Smullyan, Review of "The Problem of Inter-
preting Modal Logic", The Journal 2; Symbolic Logic, vol. 12

(1947) p. 139. .
l: r_____________, "Modality and Description," Thg
Journal pf Symbolic Logic", vol. 13 (1948), p. 31.

 

9. Alonzo Church, The Journal 9: Symbolic ngic, vol. 7
(1942), p. 100.

10. See footnote 7.

11. Frederic Brenton Fitch, "The Problem of Morning Star
and Evening Star", Philosophy pf Science, vol. 16, pp. 137-141.

 

 

 

I l

 

47
term '=' is taken to involve surreptitous mention of
expressions taken as arguments to it. These statements
are also statements of plural existence and so are con-
tingent. In consequence of this, T5.3 again fails to be
valid with '=' so interpreted.

If definite descriptions are not substituted for
variables in T5.5 and the sense of "' is that of D4, and

h49~' is given the interpretation of 'necessarily' rather

than the interpretation of 'is analytic', then T5.3 loses its

paradoxical features. Or rather, T5.5 is a "paradox of
necessity" in the same sense that '~p)(p)q)' is a "paradox
of material implication".
T5.4 ~0~x=x
(12.9 1&1, U.G.) (cumin)
((1), D4) x‘x ’
((2), 25.2) QED
T5.5 31(Qx)(fx)¢$(3x)(fx).(x)(y)[(fx.fy))x‘y]
(D5, 12.11 1&1) '
E'.('(x)(gX)€-9(3x)(;d'.x.gx).(x)(y)[(gx.gl)))x=y3
((1): 1/8; T2.18) QED ‘
If the primitive 'El' takes a definite des-
cription as an argument then T5.5 shows that the resultant
statement is equivalent as a theorem to a condition which

is equivalent to the condition taken in Principia

,Hgthematica to be equivalent by definition to 'E$(Qx)(fx)'.

 

Or, put differently, what is essentially the Principia's

general definition for 'fTQx)(gx)' reduces in System 1 to

(1)
(2)
(5)

(1)
(2)

48

what is essentially the Principia's definition for
'E%(Qx)(fx)' when 'EL' is substituted for 'f'.

This last fact is the justification for using
the same notation fer singular existence in System I
as is used in Brincipia Mathematica.

In T5.5 as in other uses in System I of definite
descriptions, the Principia convention that scOpes are
taken to be the stallest possible when not explicitly
indicated may be followed. It is however, unnecessary
to introduce scope Operators in System I, provided that
the rules of transformation are exactly followed. Perhaps
this can best be made clear by a consideration of the
example used in Principia to justify the introduction of
scOpe Operators.

The example chosen in Principia is, except for
minor notational differences, the following:32

f(Qx)gx ) P
This may be either:

[(31J(fx.gx).(x)(y)((gx.gy))x‘y)J ) p
or:

L31)((fx)p).gx).(x)(y)((gx.gy))x‘y).

But if ~Sl(9x)(gx), then the first of these is true and

the second is false. it would therefore, seem to be

necessary to introduce some such device as the Principia
scOpe Operator in order to distinguish between these two

cases.

 

12. Principia Mathematica, *14, SummarV.

 

49

However, even prior to the introduction of
scOpe Operators, only the first of'the above translations
of the first statement containing the definite des-
cription, can be made in System I.

The second is rather a case of l‘9(fx)p)(’)x)(gx)'.
It might seem plausible that the latter could be inferred
from.'f(Qx)(gx))p' by substituting upon 'p.3p' to Obtain:

2(2x1p)(7x)(gx) ahrxapuvxugx)
then reducing this to obtain:

f(?x)(gX) )p ‘3 Q(fx)p)(’)x)(g1)

But the rule of reduction (and the definition
of 'reduce to') has been so formulated as not to allow
reduction of a sentence consisting of a predicate
followed by a definite description.

'f(7x)(gx))p' can hOwever, be inferred from
'§(rx)p)(?x)(gx)'.

That ambiguity in sentences and formulas con-
taining definite descriptions can be avoided without the
introduction of scope Operators is relevant to the modal
"paradoxes" as they have been treated by Smullyan and
Fitch. Although no explicit rules of reduction appear

in Principia, scOpe Operators appear to have been intro-

 

duced with the intent of distinguishing between sentences

which involve definite descriptions as arguments to

predicates and are obtainable by one or more reductions on

each other.

 

 

.
'
I f
e I
I ~ ' I l V
v u
|
, I
I
p
' I I
C y I ,

 

 

50

However, to allow such reductions prior to the
introduction of scope Operators is to allow'invalid rules
of inference, as the above inference from a true premise
to a false conclusion illustrates. Hence the need for
some such device as the sCOpe Operator.

In order to obtain the modal paradoxes, such
invalid reductions must be allowed in addition to sub-
sequent invalid use of definite descriptions without
their then required scOpe Operators.

T5.6 222(2x)

(12.1 1&1, 14.1 1&1, U.G.) (x)(£x 5 2x) (1)
((1), 24.4) (3g)(x)(fx 2 gr) (2)
((2), T2.18) (33)(E:g.(x)(£x 2 gX)) (3)

((3),.D4) Q.E.D.l

In spite of 15.6, System I is canpatible with
Russell's thesis that classes are fictions. T5.6, as
well as any other statement containing ambiguous des-
criptions, is analysed in such a way that the ambiguous
descriptions occurring in it are syncategorematic. If for
example, the predicate '3' used above is substituted for
'f' in 15.6, the result of substitution is 'Eifile)‘. But
this resultant sentence does not literally assert existence
of a class, it is rather an assertion with a complex

notation for a relation between existence and redness.

51

Similar remarxs concerning scOpes and scope

Operators apply to ambiguous descriptions as were mentioned

immediately preceding $5.6 for definite descriptions. if

System i is formally followed, then no scOpe Operators

need be used, but if reduction is taken to be applicable

to ambiguous descriptions, then SOOpe operators are

necessary.

The following is a cataIOg of formulas assumed

to be prOper in.the course of the above proofs.

Cl.
02.
03.
C4.
05.
06.
07.
08.
09.
010.
011.
012.
013.
014.
015.

A

3'.

aux)

£(~£x)

fiEp-B(Elx)fx)3

9Epa(x)fx)—3 kmxatpa «mum;
9(p-3fx) ’
9[(X)(p-3fX) atazxatpatmxnrxnu
£(fx-agx) I '
§((x)(fx-agX)-33Lx)[(x)fx-3(Elx3gx)3)
§[(x)fx-3(Elx)gx)3 -
§tpv~P-3fX)

£(pv~p-afx)

£(fxlgx)

016.
017.
018.
019.
020.
C21.
022.
023.
024.
025.
026.
027.
028.
029.
030.
031.
032.
C33.
034.
035.
036.
037.
038.

039.
C40.

£~¢~(fx)gx)

9-04 Elx )fx)
QEO~(ELx)fx)J
flomlxmzn
QKNXHX—BOMLIHXH
980(Elx)~fx)3
280~(Elx.fx)3
§L~O~(E$x.fx)3
QEEtx—Sfx]
Qw'maczx)
§[Etx‘3(fx)dix)3
QEfXDEli

9(Elfox1
£[(x)(Eurx)—aa'.x)£xi
QEfx-3(Q(E’.x)fx)x)] -
£(sz)(£(~rx)x)) '
fin-flaunm]
§(E'.x)~£x) '
£(-rx)-E'.x)
9((y)£xy)
magnum

9mm)

QUxHy )rxy.;c'.y 313'.me

QUxHN )fxy.rl'.y-3(£(E'.x)fxy )XU
9[(x)(y )ny ——3 EtyDlexy]

52

041.
042.
043.
044.
045.
046.
047.
C48.
049.
050.
051.
052.
053.
054.
C55.
056.
057.
058.
C59.
060.
061.
062.
C63.
064.

53

y‘UxHnyy -—3 (9(Ety)(x)fxy)y)3

Manx-y)

§[(x)(fx-3gx).(x)(g23hx) -3LB'.x)(fX-3lix)]
£[(x)(fx-3gx).(x)(gfihx) -3 (9(3Lx)(fX—-3hx))x)3
£[(x)(rx3gx).(x)(gx3hx)—3:«3'.x>(£x)hx)3
2[(x)(£x)gx).(x)(gx)hx) .3 (Manxnfxfihxnxn
§(~dix) -
Q[d'.x.fx-3 (3mm;

§(~(£(~fx)x) I

§(Ofx)

Qtprx)

Q[(x)(p)fx).p-3£fl'.x)ij

’x‘uxnpmm a (mum);
QEpNxHx-BB'JMprxU .

Minty) I

9(331’3’)

fEx‘y )~O~(E'.r)( fx)fy ) )J

9E~O~(Eu)(rx)£y)3 I

3tx=y )(?(~Q~(a'.r)(£x)fy)))f)l

¥(£x)£x)

£(~Etx.gx)

£[(y)[(gx.gy})x‘y53

9ng.gy))x=y3 '

9m: 2 fx) '

065.
066.

C67.

§(~(x)(rx 2 gm)

gain)

54

\n
vi

CWAPTTR III

OBLIQUE DISCOURSE AND CONNCTATIVE
LOGIC: SYSTEM II

3.1 The paradoxes of the theory of types were avoided
in System I by the theory of prerequisites together with
the assumption that "Qt~(xx))" and other "paradoxical
predicates" do not denote anything.

While this assumption may be looked upon as to
some degree justified simply by its avoidance of paradox
and by a certain intuitive appeal, nevertheless in the
absence of more conclusive evidence, it seems to have
the rather unsatisfactory appearance of having been intro-
duced ad hoc.

3.2 The purposes of the present section are to suggest
a system of logic within which the above questions may be
more critically investigated and to apply the resulting
system to the investigation Of the particular issue of
‘whether or not "§C~(xx))" denotes something.

3.3 The description of the following logic is not
intended to be complete. The following is an account of
some of the more salient features of a logic suitable to the
above purposes.

3.4 The questions concerning singular existence which
are prerequisites in System I, cannot themselves be inves-

tigated in System I, because these are issues which must be

56

settled prior to an application of System I. This is why
a new system of logic must be developed to investigate
these questions of singular existence.

3.5 Even if a given term does not denote any existent
thing, the term itself exists. This fact suggests a
metalinguistic approach to the investigation of the pre-
requisites to System I. The central question concerning
prerequisites in connection with a term, would be whether
or not the term denoted something. If investigation
revealed that a term "a" denoted something, then the
sentence "Ela" would be true; and if "a" did not denote
anything, then "Ela" would not be true--or false.

This approach to the investigation of prerequi-
sites would be quite in accord with previous suggestions
to the effect that an oblique mode of discourse is necessary
to carry out those investigations.

To discourse in such a way as to mention terms
is to use those terms obliquely since a term in quotes is
not used in order to talk about something denoted by the
term, but rather as a syncategorematic part of a larger
expression consisting of the term in question enclosed in
quotes, which is used to talk about the term itself.

3.6 Though such a metalinguistic approach may seem

promising, it will not be followed here. The drawback of
this approach as far as the present system is concerned

consists in its adoption of semantical terms such as

57

"denotation" as technical terms of the system. The
eXplication of the meaning of such terms is beyond the
SCOpe of the present discussion.

3.7 Although the present system will avoid semantical
terms as formal devices, it nevertheless will be a system
of oblique reference. The mode of reference will be that
of connotation. The meanings of "connotation" and of
"denotation" that are intended, are those which were in-
troduced informally in Chapter 1. Although no extremely
precise explication of these terms will be attempted,

since these terms will be used only to talk about the
system rather than in it, some discussion of the present
usage, in part by way of review of the discussion of
Chapter 1, would seem to be appropriate.

3.8 Both connotation and denotation are modes of ref-
erence. Each mode of reference is analogous to the relation
of naming in that, just as any given term names at most one
thing, so there is at most one thing denoted or connoted

by any term. Of these two modes of reference, denotation
is most similar to the relation of naming, and in fact is

here taken to be synonymous with it.

The single things which are denoted and connoted
by a term will be referred to respectively as that term's
denotatum and connotatum. The connotatum of any term or

of any sentence is always a property.

The connotatum of a term is a characteristic
which is so to speak, a definitional criterion by means
of which one identifies the denotatum of the term. A
candidate for a denotatum of a term may be accepted or
rejected as that term's denotatum, accordingly as it
possesses or fails to possess the connotatum of that term.
3.9 The primary purpose for which System II will be
used will be to investigate questions of singular
existence. A system of connotative logic could investi-
gate the subject of connotative discourse generally, after
the anaIOgy of Russell's general analysis of definite
descriptions in any context in which they might occur, by
means of the contextual definition:

f(9x)gx =Df (3x)(Y)((EYEx=y).fx). (1)

But rather than this, the present investigation
will be concerned only with the analysis of connotative
assertions of existence, after the analogy of Russell's
analysis of this particular context for definite descriptions:

E!(?x)fx =Df Cax)(y)((fy E x=y) (2)

The reason for this restriction is of course,
that the primary purpose for which System II is used here,
is to investigate the prerequisites of System I.
3.10 An expression enclosed in angle brackets such
as "(a)" will be taken to be a name of the connotatum of

the expression enclosed in such brackets. Since the

 

 

IH

HI

5'9

connotatum of an expression is a characteristic possessed
only by a denotatum of the expression, a statement of
singular existence in the connotative mode of interpre-
tation will be expressed by asserting plural existence to
be a property of a terms connotatum:
3: <8) (3)
"Santa Claus does not exist." or, "There is no Santa
Claus." might be written:
~32 (s) (4)
Sentences as well as terms may be enclosed in such
angle brackets and in this case, the connotatum named by
the bracketed expression together with the brackets enclosing
it, is a characteristic satisfied only by a state of
affairs which the enclosed sentence denotes.
In general, a criterion for the truth of a
sentence of the form:
3:<F> (5)
where F is a sentence, will be taken to be whether or not
F denotes a state of affairs.
3.11 Just as denotative logic assumes that every term
used in the logic denotes something, so connotative logic
assumes that every term used in it connotes something.
Rather than being developed as an independent
system, the following logic will be formulated as an

application of System I.

60

Because of the above two points, the following
system cannot be formulated with just one kind of pro-
positional and non-propositional variables. The variables
of System I will be employed in System II with their
previous restriction that only terms which denote may
be involved in any formulas substituted for them. In
addition, in System II, the letters "u", and "v" will be
used as prOpositional variables, while the letters "a",
and "b" will be used as non-propositional variables, each
with the restriction that only terms which connote may be
substituted for them.

3.12 The following is a listing of principles of
System II. This listing is not a list of postulates for
connotative logic, but rather a combination of what might
be both postulates and theorems of connotative logic, in a

deveIOpment which proceeded more rigorously than the present

one.
P1: 33 <(u1)u2)> . 3! <(u23u3)> 3 3: <(u1) u3)>
Principle 1 might be called "The Principle of

Connotative Transitivity".

P2: 31<u> . 31<u3v> )3:<v)

Principle 2 might be called "The Principle of

mcdus Ponens".

P3: 31<u 3~ (326m))
Principle 3 might be called "The Principle of

 

 

61

Double Negation". The converse of Principle 3 is not
valid. Some sentences are such that neither they nor
their negates denote a state of affairs. For example,
neither the sentence "Santa Claus wears a red suit." nor
its negate, "Santa Claus does not wear a red suit" indic-
ates a state of affairs about someone denoted by the term
"Santa Claus".
P4: 33(u.v> E (32(u).31(v>)

Principle 4 will be called "The Principle of
Conjunctive Distribution".
p5: 31(u v v) 3 (3:(u>v3:<v>)

Principle 5 will be called "The Principle of
Di sjunctive Distribution". Unlike the Principle of
Conjunctive Distribution, which is true biconditionally,
the converse of the Principle of Disjunctive Distribu-
tion is not true. Let "RC" abbreviate "Santa Claus wears
a red suit". Let "W" abbreviate "Snow is white.". It is
true that W31<Rc>v 31(W>", because it is true that
"32(W>" and that everything which functions as a term in
the sentence denotes something, and therefore the dis-
junction as a whole is true. However, the sentence
"3:<Rc v W>" is false (and not merely untrue), because
"c" does not denote anything, and therefore the sentence

"Rc v W" cannot denote a state of affairs about something

denoted by "c".

61

Double Negation". The converse of Principle 3 is not
valid. Some sentences are such that neither they nor
their negates denote a state of affairs. For example,
neither the sentence "Santa Claus wears a red suit." nor
its negate, "Santa Claus does not wear a red suit" indic-
ates a state of affairs about someone denoted by the term
"Santa Claus".

P4: 3£<u.v> E (3!<u>.3£<v>)

Principle 4 will be called "The Principle of
Conjunctive Distribution".

P5: 3£<u v v) )(3:(u>v3:<v>)

Principle 5 will be called "The Principle of
Disj'unctive Distribution". Unlike the Principle of
Conjunctive Distribution, which is true biconditionally,
the converse of the Principle of Disjunctive Distribu-
tion is not true. Let "RC" abbreviate "Santa Claus wears
a red suit". Let "W" abbreviate "Snow is white.". It is
true that "3:<Rc>v 3:020", because it is true that
"3!(W>" and that everything which functions as a term in
the sentence denotes something, and therefore the dis-
junction as a whole is true. However, the sentence
"3:<Rc v W)" is false (and not merely untrue), because
"c" does not denote anything, and therefore the sentence

"Re v W" cannot denote a state of affairs about something

denoted by "c".

62

P6

31<u>3 316m u)
Principle 6 is another form of a principle of
double negation.
P7: 31(u)v~3:<u)

Principle 7 is a law of excluded middle.
Principle 7 is actually not a special principle of System
II, but is rather merely a direct application of the usual
law of excluded middle of System I. P7 is mentioned here
mainly in order to allow comparison of it with the invalid
"law of excluded middle": W3!(u>v 316:u>".

P8: (3!<a>.3!<b>)) (31(ab)vfflé~ab>)

Principle 8 is another law of excluded middle
which is very similar to the invalid "law" mentioned
immediately above P8. Through the use of non-propositional
variables, the fact can be expressed that the invalid "law"
above is true under a restricting condition. Principle 8
expresses the fact that so to speak, there are no exceptions
"in nature" to the law of excluded middle; or in other
words, that everything either possesses or fails to possess
any given property.

In addition to stating such particular principles
of connotative logic as those above, certain general meta-
linguistic rules can be formulated which describe how valid

laws of denotative logic can be transformed into valid

laws of connotative logic.

63

Rule 1: Form the negate of any valid formula of the
material propositional calculus. Replace each occurrence

of a denotative propositional variable in the resulting
formula with an occurrence of some connotative proposi-
tional variable. Enclose the resulting formula in angle
brackets, and precede the whole with the expression “~33".
The resulting formula will be a valid formula of connotative
logic.

Rule 2: If F1)F2 is any valid formula of the propositional
calculus, such that every variable an occurrence of which is
in F2, also has at least one occurrence in F1, then form
,3:<F1))3K?éi Replace each occurrence of any denotative
propositional variable in the resulting formula with an
occurrence of some connotative propositional variable. The
resulting formula will be a valid formula of connotative
logic.

The above 8 principles and two rules suffice to
determine a considerable amount of material connotative logic.
3.13 Since System II is being developed within System I,
the methods of inference of System I are also available in
System II.

An additional rule for System II is a rule for
substitution upon free connotative variables, analogous to

the rules of System I for substitution upon free denotative

variables.

64

Two further rules, or perhaps extensions of previous

rules, are a rule of reduction, and a rule for exchange in
accordance with definitions, when the expressions which
are respectively reduced or exchanged, occur within angle
brackets and hence may not denote anything.
3.14 If the above principles and methods of connotative
logic are employed, the formula "~31<k>" may be proved to be
a theorem of System II (where "k" is Russell's predicate) as
follows:

(From P3: kk/u) :3:(kk>)-(3:é-kk>) (1)

((1), and exchange in accordance with the

definition: k =Df {it-13))

310(k)) ~(3:<—i‘(~rf)k>) (2)
(From (2) by reduction) 310(k) )~(3£<—~kk>) (3)
(From P6) ~31<- (kk)>)~ 310(k) (4)
(From (3), and (4)) map 3 ~3£<kk> (5)
(From (5)) ~3i<kk> (6)

(From (6) by exchange in accordance with the
definition of "k")

~3z<14<~rfm> (7)
(From (7) by reduction) ~gfllé-kk> (8)
(From (6), and (8) by adjunction)

~32<kk> ..._3:<. kk> ( 9)

(From P8: k/a; k/b) (31<k>.:31<k>)3(33<kk>v
31<~kk>) (10)

65

(From (10)) 336:) Mazda.) v 326mm (11)
(From (9). and (11)) ~33<k> (12)
3.15 Principle 8 and its use in the above proof sug-

gest that investigation of principles of connotative logic
which involve non-propositional, and possibly, quantified
variables, might prove fruitful.

However, it seems very unlikely that connotative
logic will admit of a calculus of non-propositional var-
iables which will have an importance in connotative logic,
comparable to the importance of the theory of quantifiers
in denotative logic.

One of the difficulties lies in finding an inter-
pretation for bound variables which are enclosed in angle
brackets and bound by capped variables or quantifiers which
are outside the angle brackets. A similar problem of
interpretation arises in all oblique modes of speech.

An expression occurs obliquely in a given sentence
if and only if:

(1) The expression in question purports to denote
something (or is a term), and occurs in the sentence.
and, (2) When taken as asserted, the sentence does not
purport to denote a state of affairs concerning, or to dis—
course concerning, something purportedly denoted by the

expression.

66

A frequent oblique mode of Speech is mention.

The term "water" in the sentence, ""Water" has five letters."
satisfies the two conditions above, and therefore occurs
obliquely in this sentence.

Because of these difficulties of interpretation,
such fundamental methods of inference as Universal and
Existential generalization, are unavailable for connotative
variables. The absence of these methods is in itself a
serious limitation upon connotative logic insofar as it
treats of non-propositional variables.

3.16 As a final attempt to give some explication of the
use of angle brackets in this chapter, it may be helpful to
indicate that §(f)(gf E ~ff) is plausible as a connotatum
of "f(~(ff))". Or, put differently, that:

<f<~ff>> = é(r)(gf s ~ff)

(The present author hopes that'§(f)(gf E ~ff)"

denotes something.)

 

HI

67

CHAPTER IV
APPLICATIONS

4.1: Paradoxes of Logic
4.1 Russell's paradox can be avoided on the grounds
that,

~3:<k), (1)
and (l) was proven as a theorem in the course of develop-
ing System II. In this paradox, 'k' is defined:

k =Df x0~xx). (2)
Paradoxes very similar to Russell's are given in Principia
Mathematigg. These paradoxes involve relations which are
analogous to Russell's monadic predicate. One such
relation is there defined:

R =Df many). (3)
Substitution of 'RRR' for 'p' in 'p E p' yields:

(RRR) 5 (BER). (4)
Substitution on (4) in accordance with (3), and sub-
sequent reduction, yields:

(BBB) E~(RRR). (5)
Although it will not be done here, a proof similar to
that for (1) can be given in System II for:

~3:(R). (6)
By this means, the paradoxical (5) can be avoided in

much the same way as Russell's paradox. Relations with

68

definitions similar to (2) and (3) can be given for any
number of arguments. Each such relation gives rise to

a paradox. All such paradoxes are avoidable by a proof
of non-existence for the relation on which each depends.
Each such proof can be given in System II.

Grelling's paradox is a paradox rather different
from any of these. We might examine how this paradox can
be dealt with in System II.

Grelling's paradox arises from considerations such
as the following. Some terms are applicable to themselves,
others are not. For instance 'short' is applicable to
itself, as is ’word', while 'elephant' is not. Let us
define 'Heterological', (abbreviated 'Het') to mean non-
self-applicable. If 'Het' is heterological, then it is
not heterological; and if 'Het' is not heterological,
it is heterological.

Abbreviating 'is applicable to' by 'App', 'Het'
may be defined as:

Het =Df ie-xAppx). (7)

Grelling's paradox may be obtained as follows.
Taking,

'Het'App'Het' 5 Het'Het', (8)
as a postulate, we can from,
Het'Het‘ 5 Het'Het', (9)

by means of a definitional exchange in accordance with

69

(7), subsequent reduction and use of (8), obtain:
Het'Het' 5-Het'Het’. (10)
(10) is the contradiction of Grelling's paradox.
Another rule in addition to Rules 1 and 2 of
System II, governing transformationofvafim laws of
denotative logic into valid laws of connotative logic,
might be formulated as follows.
Rule 3: If F2 is any valid formula of material logic that
contains no propositional variables, and F1 is a con-
junction of formulas which are substitution instances of
'3:<a)' by replacement of 'a' with a variable, and F1
contains at least one instance of every variable of which
F2 contains an instance; then 1:133:02) is a valid
formula of connotative logic.
If Rule 3 is added to System II, then,
~(3:<aet). 3:<'Het').33QHet'App'Het' 5
Het'Het'>) , (11)
can be proven as follows.
(By Rule 3) (31(a) .3141») )33: {ab 5 ab) (a)
((a): Het/a; 'Het'/b)
(3:<Het).3:('Het'>))3:(Het'Het' 5 Het'Het'> (b)
((b), exchange in accordance with (7), reduction)
(3:(Het>.3:<'net'>))33<Het'31et' 54Het'ipp'
Het'> (C)
(By Rule 2)

7O

3i<aba ca>333Q~aba E.~ca> (d)
((d): 'Het'/a; App/b; Het/c)
3:<'Het'App'Het' 5 Het'Het'>)

32<~'Het'App'Het' 5..Het'_aret') (e)
((c), (e)) (3:<Het).az<'net'>.3:<'net'App'Het'
5 Het'Het'>)) (3!<Het”—T_et' 5 ”"Yet'App'Het'>.

32<~'Het'li.pp'Het' 5 ~Het'Het'>) (f)
Rule 2) _31<(ab E ~bcb).(~bcb E ~abX)3
3:<eb 5 ~ab> (g)

((g): Het/a; 'Het‘/b; App/c)
3:<(Het'Het' 5 ~‘Het'App'Het').(~'Het'App'Het'
5»- Het'Het')>)31<Het'Het' 5 ~Het'IIet'> (h)
((f), (h), by Conjunctive Distribution)
(33(Het).3:<'Het'>.3:<'Het'App'Het'

Het'Het'>) 231(Het'net' 5 ~Het'Het'> (1)
(By Rule 1) A3L§~v(ab E ~ab)> (j)
((3): Het/a; 'Het'/b)

~314~ (Het'Het' 5 ~Het'Het')> (1:)
(From P6 of System II) 3£<Het'I-Iet' 5 enet'netp
333:6 (BIet'Het' 5~ Het'Het')> (l)
((k), (1)) ~31<Het'I-Iet' 5 ~Het'Het'> (m)
((1), (r0) QED (n)

In order that the Grelling paradox be avoided,
it is not necessary to maintain that any particular one

of the factors to the conjunction in (l) is false. Since

III

III

lal

 

7O

3:(aba ca>333<~aba E ~ca> (d)
((d): 'Het'/a; App/b; Het/c)
3:<'Het'App'Het' 5 Het'Het'>)

32<~'Het'App'Het' 5.. Het'Het') (e)
((c), (e)) (3:<Het>.3!<'Het'>.314'Het'App'Het'
5 Bet'Het'>)) (3:<Het'net' 5 ~'U_et'lpp'Het'>.

32<~'Het'App'Het' 5 ~Het'Het'>) (f)
Rule 2) _31<(ab E ~bcb).(~bcb E ~abX))
31<ab 5 ~ab> (g)

((g): Het/a; 'Het'/b; APP/c)

3:<(Het'Het' 5 ~'Het'App'Het') .(~'Het'App'Het'
5~ Het'Het')>)3:<Het'Het' 5 ~Het'Het’> (h)
((f), (h), by Conjunctive Distribution)

(Blast) .3:<'Het 9 .3: ('Het 'App 'Het'

Het'Het'))J;1:<Het'Hetv 5 ~Het'aet'> (1)
(By Rule 1) 43Lé~v(ab E ~ab)> (j)
((3): Het/a; 'Het'/b)

{1:9 (Het'Het' 5 ~Het'Het')> (k)
(From P6 of System II) 3!<Het'Het' 5 ~aet'Het'>
35:9 (Het'Het' 5~ Het'Het')> (l)
((k), (1)) ~33<Het'net' 5 ~Het'Het'> (m)
((1), (m)) QED (n)

In order that the Grelling paradox be avoided,
it is not necessary t0 maintain that any particular one

of the factors to the conjunction in (l) is false. Since

 

Ill

ill

H1

lol

D

71

the derivation of the paradox depends on each of these
factors being true, to Show that their conjunction is
false is sufficient to avoid the paradox.

Another paradox for which interesting results
can be obtained by application of the foregoing logic is
the following.

fa ands~fa appear to Share such consequences as
Eta and (3x)(fxv~fx), but Since these consequences are
contingent, it follows that<O(fa¢~fa), or, that fa and
~fa are not contradictories.

W. V. Quine has made some comments to this paradox
in a review of a discussion of it by Everett J. Nelson.1
Since Mr. Quine's review raises several important questions
concerning both this paradox and other closely allied
paradoxes, the following discussion of the above paradox
will follow this review in the points raised in it.

hr. Quine opens his discussion by making the well
taken point that "the supposed existential consequences
of 'fa' and *‘fa'" should be questioned. However, Mr.
Quine goes on to suggest as evidence for this that the
entity a is not a "constituent of the propositions fa

and ~fa", and as evidence for this in turn:

 

1. Everett J. Nelson "Contradiction and the presupposi-
tion of existence" Mind, n. s. v. 55 (1946), pp. 319- 27.
Review by W. V. Qulne, The Journal of S‘mbolic Log__,

vol. 12, p. 52.

72

... consider the propositions Hesperus is

the Evening Star and HeSperus is the Morning

Star. Being respectively analytic and

synthetic, these propositions are distinct:

yet the supposed constituents Evening Star

and morning Star wherein alone they can

differ are one and the same thing. Clearly

we must View the constituents not as the

Evening Star and the Morning Star (i.e., the

planet named by 'Evening Star' and 'Morning

Star'), but rather as the respective meanings

of 'Evening Star' and 'M rning Star'.

Without here investigating all of the very complex
questions raised by this comment (such as what 'being a
constituent of a proposition' means, under what conditions
propositions are distinct, whether or not the morning star
and the evening star are the only constituents of the
propositions in question wherein they can differ, and
whether named things or meanings are involved in pro-
positions); it will only be argued here that the contention
that HeSperus is the Evening Star and Hesperus is the
Morning Star are respectively analytic and synthetic, is
highly questionable.

Some of the questions involved in the problem of
when statements of identity are analytic and when con-
tingent, were raised during the elaboration of the system
above. In application of these results, it is the case

that, if the phrases 'Evening Star' and 'Horning Star'

are meant to be definite descriptions in the sense of,

say, Russell; then both Hesperus is the Evening Star and

73

Hesperus is the Korning Star are contingent for the
reasons given in the discussion in the system. On the
other hand, if these phrases are meant to be names then
both Hesperus is the Evening Star and Hesperus is the
horning Star are analytic. But in neither case will one
of these be analytic and the other contingent.

In any case, the original point still stands that
fa does not always entail Ela.

Kr. Quine mentions in passing (as they will also
be mentioned here) some further paradoxes such as hat
while 9 is 0, is analytic; 9 is the number of planets, is
contingent.

Hr. Quine next examines certain questions concerning
the contrast between meaning and naming and the conditions
under which sentences of the form 'Elx' can be inferred from
premises of the form 'fx'. These issues lie outside the
scope of the present discussion since they are topics in-
volving metalogic rather than modal connections. Yet in
this discussion Mr. Quine does suggest certain conditions
under which fx may entail Elx. He suggests that logical
functions of E: entail existence, but that other functions
might be regarded as unknowable as to whether or not they
entail existence. The positive part of this comment wauld
certainly seem to be correct, and indeed to constitute

one of the soundest means available for determining if

74

fx entails Elx. On the other hand the negative part of
the suggestion is not always true, Since it can be
proven that analytic properties do not entail existence
(Theorem 3.5).

Finally, Hr. Quine very nearly recognises the
position that is here being proposed when he suggests
that "the usual convenient techniques of logic which
allow proof of '(3x)(fxv~fx)' and inference of '(3x)fx'
from '(x)fx' can be accepted as a semi-logical amalgam,
comprising pure logic plus a true extra-logical premise
to the effect that there is something." The only inac-
curacy in this statement as a description of the present
position is that the true extra-logical premise is not
that 'something exists' but rather the formula ‘Elx', the
result that something exists being a theorem, not a

postulate.

4.2: Identity of States of Affairs.

Truth and falsity are ways of being representative.
A sentence is true or false accordingly as it does or does
not have a denotatum. As mentioned earlier, the denotata
of sentences are here taken to be states of affairs (ahd
the term "proposition" is here taken to be synonymous with
"state of affairs").

Often, in spite of what has just been said, the

terms "true" and "false" are applied to states of affairs

75

(or propositions); rather than sentences. Moreover, it
is clear from an examination of such instances of usage
that the meanings of "true" and "false" are not species
representativeness. Some authors have suggested that
such usages Should be dismissed as mere confusion; on the
grounds that only sentences can be true or false while
propositions or states of affairs simply either exist
or do not exist.
But putting the matter this way actually serves
to bring to light the quite legitimate meaning which "truth"
and "falsity" have in application to states of affairs.
Using "P" for the property of being a proposition,
the terms "true" and "false" might be defined in this
second sense as follows:
Tx =Df Px.E£x (1)
Fx =Df Px.~E!x (2)
Among the consequences of these definitions are the
facts that there are no false states of affairs, but
rather, all states of affairs (or propositions) are true:
~(3X)(PX.FX) (3)
(x)(meTx) (4)
(3) and (4) are consequences of Th. 2.8; the principle
that everything exists.

Furthermore, these results are consonant with the
View stated above thatirue sentences have denotata, while

false sentences do not. If a true sentence is a sentence

le‘A v"i_'_
I
2.

76

that denotes a true state of affairs; then all sentences
that are not true (e.g. false sentences) do not denote a
true state of affairs. But sentences that do not denote
a true state of affairs do not denote any state of
affairs (again because everything exists). Therefore
false sentences do not denote states of affairs.

This View of true sentences as posessing denotata,
while false sentences do not, raises an apparent problem
concerning the connectives of logic.

When we assert a material conditional:

A3B (5)
it would seem natural to interpret such a statement as
asserting that the state of affairs denoted by "A" stands
in a certain relation (material implication) to another
state of affairs denoted by "B". Howeversuch a view cannot
be the case when "A" and "B" are false; since in this case
there are no states of affairs denoted by "A" and "B" to
stand in the relation of material implication (or any
other relation) to one another. Obviously, similar remarks
could be made about other logical connectives as well--what
can the connective 'r" be applying to in the sentence “wA”
if "A" is false?

The answer to this difficulty suggested here is
that the connectives of propositional logic are all re-
lations with twice as many argument places as propositional

variables whichtmey connect. These arguments are properties

ta-“

77

‘5 “his. u‘J -‘ . .1

and individuals, rather than states of affairs. For
instance, rather than writing a material implication as:

p3q (6)
we might more appropriately write this formula as:

(f,x)J(g,y) (7)
construing "D" as denoting a relation of four argument
places rather than two. If this is done, then of course the
above problem about what is being related by the relation
of material implication when sentences which are argu-
ments to "J" are false disappears. What is being related
are properties and perhaps individuals, but in any case
things which exist even if they do not result in a state
of affairs.

Again, similar comments apply to thecmher con—
nectives such as "~". The curl can be construed as a
dyadic relation, and the formula "~p" written as:

~'(f,x) (8)

Such a construal of the propositional connectives
need not involve the abandonment of the use of pro-
positional variables however. In fact, such a construal
need not involve any alterations in the usual prOpositional
logic. Propositional variables can be viewed as, so to
speak, "dummy" variables and the expressions which are
substituted for them as determining the arguments to the

connective in.mxstion though not denoting these arguments.

 

 

78 i f.

In short, the change being suggested is only a change in
interpretation of the usual propositional calculus rather than
a change in the calculus itself.

What this suggested change amounts to is the view
that sentences which occur as arguments to propositional
connectives are not terms (although the same sentence
might very well be a term in other contexts).

However, if this view be a correct one, it has
some significant consequences for the problem of under
what circumstances states of affairs or propositions are
identical with one another.

An important question ini;his regard is: if pro-
positions are such that each is deducible from the other,
then are they always identical with one another?

Suppose we were to attempt to formulate this
question as follows: "Is it the case that:

pééq:):p = q ?" (/)

The Sign of strict equivalence in the antecedent
of (9) however does not relate states of affairs denoted
by substitution instances of "p" and "q" while the Sign
of identity in the consequent of (9) does. For this
reason the problem which has just been eliminated in the
case of propositional connectives is raised again in (9)
in the case of the Sign of identity occuning in it. If
false sentences are substituted for "p" and "q" in (9)

then since these sentences will occur as terms, the

79

VII—mm
V
=.,- q

resulting sentence will be purporting to say something
about their denotata when in fact there are no such
denotata.

Evidently, (9) is not a satisfactory formulation
of the problem. However, the fact that true sentences do
denote states of affairs might suggest that it could be
satisfactorily formulated under the restriction that the
substitution instances of "p" and "q" be true. (Of course
this restriction cannot be formulated merely by asserting
(Q) to be materially implied by the condition that "p.q"
be true:

(p.q)) ®®q=3=p = q) (10)
for (10) involves the same problem as does (9); nanely
that since substitution instances of "p” and "q" occur
as terms in the sentence as a whole, the sentence purports
to denote some fact or state of affairs about things
denoted by those terms which do not denote anything.)

The question might however, be formulated as
follows. Let the term "P" be defined as:

xPy, f : x is a prOposition analysable into y's

having the property f.

Now the restricted form of the problem might be

put:
(fx)¢$(gY). sz,f. wa,g :J: z = w (ll)
This last formulation avoids the difficulties
raised above in that any substitution instance of (ll)

to

will not contain terms that do not denote provided that
only denoting terms are substituted for free variables in
(11).
A perhaps interesting fact; which, though it is
by no means conclusive evidence against (11), might
nevertheless lead one to question it; is that if the
usual logical assumption that any given property and any
given subject of that property determine at most one
state of affairs, is extended to the further assumption
that any given state of affairs and any given property
determine at most one subject of that property; then (11)
is false.
Suppose it to be the case that:

xPz,f. wa,f. x = y :3: z = w (12)
Let "5" name the sun, and "m" name the moon. Since:

3 fl m (13
we can infer from (12) and (13) that:

xPs,f. me,f :3: x f y (14)
Further, let "H" abbreviate "heavy or not heavy". Since
both Hs and Hm are analytic, it follows that:

rise—earn (15)
Yet from (14) it follows that:

xPs,H. me,H :3: x 1 y (16)
It is also true that since H is an analytic property; Hs
and Hm are true, or determine states of affairs, and that

therefore:

v r“

I ‘l’v- 2.5

~
‘a‘—3,

 

,H'

I 'ilr

81

1‘...

(3X)(xPs,H). (3y)(me,H) (17)

But from (15), (16), and (17), the denial of the
principle expressed in formula (11) is deducible.

As was mentioned earlier, this result is not con-
clusive evidence against (11). However the result does
make it seem doubtful that the intuitive acceptability
of (11) is in itself enough to settle the matter.

In any case, the primary purpose of the present
argument is not to examine the question of under what con-
ditions propositions are identical. The purpose of the
present discussion is rather to show that an adequate
treatment of this question will be involved with the

logical problems raised by terms that do not denote.

4.3: A Problem in the Theory of Perception

There is a further problem; in the theory of
perception, which calls for a logic of terms which do not
denote. many authors have discussed this, or related
problems; however, the present account will make primary
reference to a paper on perception by H. A. Prichard.

Prichard asks the question: "What do we see?",
and he concludes that although the naive answer is "bodies"

this is not true because it cannot account for illusions

 

2. H. A. Pritchard, "Perception", in Knowledge and Egrceptigg
Oxford University Press, lQFO, P. 52.

92

and that in fact, in doing what we ordinarily call

"seeing a body" we actually are seeing sensa (as, colors)
and mistaking these for bodies. The purpose of the present
section is to show how formulations of this point, and
others which Prichard makes, require a logic of terms

which do not denote.

To take an example which Prichard himself takes,
suppose that I ask what it is that I see when I have an
experience of seeing the moon as being yellow. If I
describe my experience as 'seeing the moon as being
yellow'; I would appear to be asserting that I am
instantiating a certain triadic relation which might be
symbolized as: "P(x,y,f)", and described in general in
this manner:

P(x,y,f) : x sees (or perceives) y as possessing
the property f.

The instance in question would then be describable as:

"a sees m as possessing the property yellow"
or: "P(a,m,Y)".

But Prichard maintains that the patch of color
which is the sensum of which I am aware when I see the
moon as yellow, is not the moon (nor presumably, the
property of being yellow) and therefore is not included
among the arguments to P in P(a,m,Y). We might therefore
introduce a further relation-~that of sensing-~in order

to have a notation for "a senses the-moon-as-yellow",

83

where the phrase "the—moon-as yellow", names the sensum

in question. However, before doing this, it is as well

to notice immediately a certain inadequacy in formulation
of the phrase "the-moon—as-yellow". Prichard will wish

to maintain that sensa are relative to percipients. The
phrase "the-moon-as-yellow" (say); and thereby the
original sentence amended to: "a senses a-perceiving-the-
moon-as—yellow".

If we adopt the letter "S" as a notation for the
dyadic relation of sensing:

xSy : x scnses y.
then the above sentence might be expressed as:
aS(a—percé1ving—the-moon-as-yellow).
Or alternatively as:
aS(a-perceiving-m—as-Y).
The question naturally arises of how to interpret the
English phrase "a-perceiving-mes-Y".

A suggestion made by several investigators is to
interpret such hyphenated phrases as names for states of
affairs. As the reader may have already guessed, such a
suggestion will be followed here by regarding the phrase
"a—perceiving-m-as-Y" as denoting the state of affairs
that a perceives m as Y. The original sentence "a senses

the moon as yellow" will therefore be taken to beequiva-

lent to "aS(P(a,m,Y))".

:FfirSQIiz-um

84

Although a does not experience either the moon,
or the property yellow in perceiving the moon as yellow;
he nevertheless does experience the state of affairs (or
proposition) that he perceives the moon as yellow. In
fact the state of affairs which a experiences (or senses)
is the patch of yellow color which is his sensum.

The purpose of the present section is not to
argue for this analysis of perceiving and sensing, but
rather to show how this analysis calls for a logic of
non-denoting terms.

Suppose that within this theory, we wished to
give formal expression to what might be regarded as a
plausible law of perception: "Whenever x perceives y as
possessing the property f; then x senses the state of
affairs produced by x's perceiving y as f.". It might
seem to be possible to express this "law" in the following
formula:

(x)(y)(f)(P(x,y,f))xS(P(X,y,f))) (1)
However this "law" is not even an admissably formulated
one in a logical system such as Egipgipig Mathemgticg.
Suppose that a does not perceive b as possessing G. From
(1) the following can be inferred:
P(a,b,G)3aS(P(a,b,G)) (2)

but since the state of affairs that a perceives b as
possessing G does not exist; (2) contains in its conse—

quent the term "P(a,b,G)" which does not denote anything.

85

(Of course the expression"P(a,b,G)" is not a term
occummnce in the antecedent of (2) and therefore this
occurmwpe does not give rise to difficulty.) Ordinarily
a sentence which is a material conditional with a false
antecedent is true. However this principle presupposses
that all of the terms which appear in the material
conditional denote--as does every principle of tradi-
tional logic; and indeed, every sentence in which the
terms appearing in the sentence all occur directly in the
sentence. Since (2) contains such a non-denoting term,
it is unacceptable as it stand; and moreover, the dif-
ficulties of formulation which it raises can be given
no simple solution in any logic which permits only terms
:hich denote. Therefore in spite of the fact that (2) is
a material conditional with a false antecedent, it must
be regarded as either false or meaningless.

Prichard's point that we ordinarily mistake our
sensa for bodies; runs afoul of similar difficulties.

Suppose this point were to be formulated as
follows: "If x perceives y as f and x's perceiving y as
f is a case of ordinary, unreflective, perception; then
x mistakes P(x,y,f) for y." then "P(x,y,f)" occurs as a
term in this sentence. For the same reasons as in the
earlier example, this formulation also will break down in

cases of x not perCEiving y as f.

M __ ,e'

86

Fan? more examples such as these of "principles"
of perception which call for some such logic as will be
proposed in the previous chapter could be found. These
exaMples would for the most part, occur in theories of
perception which take sensa to be states of affairs or
propositions rather than bodies; however theories which
make such proposals are today to be widely found in the
literature on perception. The formulation of such
theories in classical logical systems is not wholly
impossible. However, such formulations would involve
considerable circumlocution at best.

Perhaps enough has been said in this last example
of application, and in the preceding ones; to show that
there are uses of a logic allowing terms which do not

denote over and above whatever intrinsic interest such

a system might have.

87

CHAPTER V
MISCELLANEOUS COMKENTS AND FURTHER
TOPICS OF INVESTIGATION
Sections 5.1 - 5.6 are a series of rather dis-
connected comments on several topics. All of these
comments are intended to illustrate views of their
topics which reflect the position of earlier chapters.
Sections 5:7 - 5.11 briefly indicate some further
problems of this treatment of that earlier position.
5.1 In System I, 'E!’ is a primitive. Two theorems
of System I fix the interpretation of 'E'; in the sense
that they limit the interpretations which can be made of
'E' in such a way as to prevent it from having a great
many unintended interpretations. These are:
‘~04X)B!x (1)
and,
(XX}~E!X (2)
Interpretations of 'E!‘ can easily be found
which will satisfy one of these conditions but not the
other. However, there are few alternative interpretations
which will satisfy both (1) and (2).
For instance, it has often been suggested that
'existence' can be adequately explicated by self identity.
And indeed, self identity does satisfy (1). However, self

identity does not satisfy (2); and therefore, is not

88

adequate as an explication of 'El'.
Again, it is easy to find interpretations which
give conditions that are contingent of everything. But

few of these interpretations will also satisfy (1).

5.2 It has often been maintained that syntactical
criteria of inference are adequate to the purposes of
logic. There would seem to be evidence that this View
is unwise.

One of the conditions necessary to the correct-
ness of an inference isthat no false conclusion can be
inferred from a true premise. However, no sentence con-
taining a subject term that does not denote something is
true, and syntactical criteria alone never suffice to
show whether or not a term denotes. Purely syntatical
criteria of inference that are adequate, may not be as

easily formulated as was at first supposed.

5.3 It is sometimes thought that all uses of modalit-
ies as terms of the object language can be replaced by use!
of the models as terms of the metalanguage. Hore often than
not, the reasons for such a supposition are that for any
assertion of modal connection in the object language

there can always be found a parallel assertion of meta—

linguistic connection.

89

This last claim would indeed, seem to be true.

So that for instance, rather than saying “~0~(fxv~fx)',
we might rather say 'analytic'(fxv~fx)". However the
first claim of the preceding paragraph is much stronger
and more dubious. Moreover, acceptance of the second
claim of the preceding paragraph by no means entails
acceptance of the first.

One of the reasons for this is that while any
statement of modality may be translated into the meta-
1anguage in such a way as to be well formed, often larger
contexts containing statements of modality are not such
that those contained statements of modality can be
translated into the metalanguage in such a way that the
larger context makes sense. For instance if the expres-
sions '9<~‘9~(fxv~fx)' and '(flxa-O ~(fxv~fx)' are translated
by replacing their contained assertions of necessity as in
the preceding paragraph, they result in 'Q analytic'
(fxv~fx)", and 'Gx)analytic'(fxv~fx)". But these last

statements are difficult to allow as well formed.

5.4 Sometimes the law (Ex)fx¢%>(3x)(Elx.fx)' is called
into question on the grounds that there are fictitious
characters in fiction. The argument is often put that
since there are fictitious characters, but no fictitious

characters that exist, the above law fails in this case.

 

 

I

.\|

90

Actually, this last claim that there are ficti-

tious characters, is only a special case of what might
be called truth within a myth. In a sense, it is true
as claimed that in the myth of Oz, the tin woodman and
the cowardly lion exist, and that in reality these things
are not so. But to say this is only to say that given
those things asserted in the story of Oz as premises,
one can deduce that the tin woodman and the cowardly
lion exist. This might be put symbolically by using 'Oz'
as an abbreviation for the conjunction of those things
asserted in the story of Oz, 'Tx" for 'x is a tin
woodman', and 'Cx' for 'x is a cowardly lion'. The last
statements then become:

Cz-BE!(Qx)(Tx) (3)
and,

Oz-3E!(?x)(Cx) (4)
In other words, to be in a myth is to be implied by that
myth. But furthermore, the tin woodman does not exist.
We might therefore assert that in Oz the tin woodman
exists, but he nevertheless does not exist:

(Oz-3E:(lx)(Tx))c~E:(?x)(Tx) (5)
But from (S) we should not infer that there are non-

existent tin woodmen.

‘ 5 The ontological argument for the existence of God

1.,

proceeds by defining the term'God' in such a way that

91

'existence' enters into its definition. The argument
then continues with a line of reasoning to the effect that
with the term so defined, it is an essential feature of
God that he exist. Since anything must possess its
essential features, continues the argument, it must then
be the case that God exists.

This version of the ontological argument might be
put into the notation of System I as follows.

he might introduce the term 'God' by a defini-
tion such as:

God =DF (qx)(E1x.Ox). (6)

Here, '0' is used to abbreviate some further qualifying
condition such as 'is omnibenevolent', or, 'is omnipotent';
over and above existence itself, as in the definition of
'God'. What this further condition means need not be
examined for the purposes of the present remark. The
present comments will be concerned solely with the
logical properties of 'El', rather than with the inter-
pretation of '0'. It may be that 'O' intensionally
contains 'E!’. This is to say, it may be that to assert
'O' of anything, is to imply that 'E:' is applicable to
that thing. In case this is so, (6) contains 'E' in a
way that is superfluous. But again, whether or not this
is so, while it will be relevant to whether or not (6)
is as economical as possible, need not be of concern for

the present point.

92

The argument above infers 'EIGod', presumably
by appeal to some such principle as 'f(flx)(fx)'.

But in System I, this principle is not available,
though,

E!(?x)(fX)-3f(7x)(fx). (7)
is available. However, from (7) and (6) the desired
conclusion does not follow without begging the question,
since we require,

Ez(’)x)(szx.0x) (8)
as a premise in order to conclude 'ElGod' by means of

(6) and (7) 0

5.6.1 Concerning the previously mentioned topic of
truth within a myth, the myth that nothing exists is a
particularly fruitful one. In one sense, there would be
no truths if nothing were to exist. This is true both in
the sense that there would be no true sentences and also
in the sense that there would be no true propositions,
were it the case that nothing exists. These facts are
consequences of
(3X)Elx-€%(3X)fx (9)

by substituting "is a sentence" and "is a proposition"
for "f".

On the other hand, some things are true in an
empty universe, in the sense that there are some things

that are implied by nothings existing. This might be put:

(axli3f)(~43x)E1x-afx) (10)
5.6.2 These facts are more than mere curiosities, for
they indicate (as we see immediately below) that certain
formulas of the quantified material calculus, which are
sometimes called laws of "confinement" and of Which

(3x)(p)£x) ) (p)(3x)fx) (11)
is an example, are such that if the horseshoes appearing
in them are replaced with flowers, the resulting formulas
are not valid laws.

(11) and its converse

(p)(3x)fx) D (EIHPDfx) (12)
are both valid laws of the quantified material calculus
which govern "confinement" of existential quantifiers
over material implication. if the central connective in
(12) is replaced with a flower, the resulting formula

(p)(3x)fx) —3 (aprlfx) (13)
is invalid in System 1. An exception to (15) can be
obtained by substituting 'LBXfo' for 'p' in (13):

((3x)fx)(3x)fx) -—3 (3x)((3x)fx)fx) (14)
But in System i, the antecedent of (14) is analytic, and
the consequent of (14) contingent, and hence (14) itself,
contravalid. on the other hand, the converse of (15) is
a theorem of System i.

"Confinement" of existential quantifiers over flowers

rather than over horseshoes might also be investigated.

94

If the horseshoes in the antecedent and con-
sequent of (11) are replaced with flowers, the result is:
(axnparxi 3 (pecans). (15).
But i15) would seem to have exceptions on the grounds of the
argument concerning the empty universe given at the outset
of 5.6. And of course, if (15) is rejected, then the
formula obtained by replacing the horseshoe in (15) with a
flower, must also be rejected.
rhe converse of (15) can as a strict implication,
be rejected on grounds similar to those which lead to a
rejection of (15). Substituting 'CBx)fx' for 'p‘ in
(p-3L3x)fx) —3 (Epr—fo). (16)
yields
. l(3x)fx-3(3x)fx) ~3 (3x)((3x)fx am). (17)
Which is contravalid in System I on the same grounds as is i
(14).
The converse of (15) is, however, a more difficult
case to decide. it may'be that, as with (11), the con-
verse of (15) is valid as a material, but not as a strict,
implication.
This converse,
(P—3(3x)fx) ) (Bprafx), (18)
can be shown to be equivalent in System i to

(X%Q(P.fx))Q(p.(x)fx). (l9)

 

95 ) ‘I‘ !

(19) may have the rollowing exception, sub-
stituting '(3x)~Rx' for 'p' and 'B' for 'f' in (19),
(where 'Rx' abbreviates 'x is red') yields

(x)0(Gx)~Rx.Rx) )0(Gx)~Rx.(x)Rx) (20)

While it would seem to be so that of everything it is true
that it is consistent both that it be red and that some-
thing fail to be red, it would seem not to be the case
that it is consistent both that something is not red and
that everything is red. But if these last things are so,
then (20) is not true.

These tentative results may be summarised as follows:

(3x)(p)fx))(p)t§x)fx) valid (21)
(p)(3x)fx))(3x)(p)fx) " g (22)
(3x)(p)fx)-3(P)(31)fx) " (25)
(p)(3x)fx)—3(3x)(P)fx) invalid (24)
(3x)(P-3fx))(P-3(3x)fx) " (25)
(p-3(3x)fx))(3x)(p—3fx) " (26)
(3x) (p—3fx)-3(p-3(3x)fx) " (2'7)
(p3(3x)fx)—3(3x)(p-3fx) " (28)

5.6.3 The laws of the above list that are given as valid
can be shown to be theorems of System I. The author is not
aware of a proof for any of the formulas listed as invalid.
There are, however, theorems of System I which are ana-

logues, respectively , of (24)-( 28 ):

96

('p)(3x)£x).ELx-at'aprirx) (29)
Gx)(~(>-E'.x.p-3fx) ) (p-B(’3x)fx) (30)
(P-S(3x)fx).~$~b3'.x) D (BXHP-BfX) (31)
(3X)(~Q~E'.x.(P-3fX))—3(P-3(3x)fx) (32)
(p—3(3x)£x).~<>~ELx—3(3x)(’pgfx) (35)

The only law of this list that is of serious
interest is (29), the other laws being trivializei by
their containing a counter-analytic condition in their
antecedents.

The equivalence laws which are consequences of (50)-
(53) and laws which are similar to their converses, lose
interest for similar reasons. however, the equivalence law:

(P8(BX)fX).(BX)E'.x<—'9 (Hpr‘B‘fx) (34)

is a theorem of System 1, not so trivializei. So to Speak,
the formulas '(pBGx)fx)' , and'(3x)(p—3fx)' , are not equi-
valent because the latter makes a "surplus assumption" of
existence which the former does not make. This "surplus

assumption" is the denial of the myth that nothing exists.

5.7 This and the remaining sections of Chapter 5,
suggest prdblems which will not be investigated in this
thesis, but which might be studied if the topics discussed
here were to be carried further.

The Tarski pardigm of truth does not apply to all

sentences. Some of those to which it does not apply are

   

97

those sentences which have terms which do not denote any-
thing as subjects.

It might be possible to extend the Tarski paradigm
to cover the latter sentences if some such codification
as the following were adopted.

Let ’8' name the sentence 'X'. The following
might then be taken as a criterion of the truth of s:
3!<X> (35)

The Tarski paradigm does not apply to sentences containing

S is true

subject terms which do not denote, because these terms
occur in the direct mode of discourse on one side of the
biconditional equivalence which formulates the test of
truth. However, such terms cannot so occur in (35), and
so such a criterion as this might extend the Tarski test
to sentences containing terms which do not denote as
subjects. I

But this method leads to an ambiguity in the case
of falsity. There are available two alternative ways
of defining falsehood. Letting as before, '8' be a name
of a sentence 'X', falsehood of 8 might be tested by the
criterion:

5 is false 5 ~33<x> (36)

or, on the other hand, by the criterion:
3z<~x> (37)

In order that a sentence be false, (36) so to

S is false

speak, requires that the sentence not say what is so,

98 f I

while (37) requires rather, that the sentence say what
is not so.

If (36) is taken as a criterion of falsity,
sentences with subject terms that do not denote are
false. If (37) is taken as a criterion of falsity,
then such sentences are neither true nor false.

5.8 One of the difficulties of the connotative logic
sketched in Chapter 4 is that the notion of an oblique
usage of a term is not adequately explicated. The
explication of the notion of oblique discourse is another
topic which might be investigated further.

An important feature of this notion is that it,
like the notions of bound and free variable, are ways
in which constituent expressions are related to larger
expressions of which they are parts. For this reason,
it is misleading to Speak as has been done above, of
an expression simply as "oblique" or as "direct", without
including some specification of the context in which the
term is intended to be asserted to be oblique or direct.

The sort ofthing that is meant by saying that a
term is oblique in a sentence of which the term is a
part, is that the sentence does not make an assertion
about something denoted by that term. However, this
definition must wait upon a clarification of the seman—

tical terms that are involved in it before it can be

regarded as an adequate explication. In this respect
the notions of oblique and direct differ from those of
free and bound; the latter being terms which can be
syntactically defined.

Some of the advantages that go with the syntacti-
cal definitions of free and bound can also be enjoyed by
the notions of oblique and direct, provided that a
syntactical criterion of oblique and direct such as the

following is adopted.

A term occurs obliquely in a sentence of System II,

if and only if, there is some pair of angle brackets in
the sentence which enclose the term.

Although this criterion does have much of the
advantageous immediacy of the definitions of free and
bound, it nevertheless also has two serious drawbacks.

First, the definition provides a criterion of
obliquity only for sentences of System II. There are
many cases of obliquity which are, for instance, instances
of expressions which occur within quotes, and which are
not instances of obliquity in System II.

Second, and perhpas more serious, the above
criterion of obliquity provides only an extensional test
of obliquity, not an intensional condition of obliquity.
Because of this last difficulty, the criterion cannot be

taken as a definition of oblique discourse.

1 Fug”:
}‘

100

The first of the above conditions of obliquity would
not seem to share either of these last two disadvantages.
5.9 The formation rules of System II were not
specified. There is however, an important condition that
an adequate set of formation rules for System II should
satisfy. No variable should be bound in such a way as to
occur obliquely in the propositional formula following its
binder. In other words, no bound variable should occur
enclosed in a pair of angle brackets that do not enclose
its binder.

It is a general restriction on all oblique dis-
course, tbat sentences which involve "binding over" oblique
contexts do not make sense. That is, no variable which is
oblique in a context and bound by a binder which is out—
side of that context, occurs in such a way that there is
a sentence containing this binder and context, which makes
sense. i

This difficulty was mentioned previously in
Section 5.3. Ketalanguages which make use of expressions
enclosed in quotes, are examples of languages of oblique
discourse, other than the connotative language of System II.

The logical use of bound variables is such an
effective technique that it is possible that oblique dis-
course vill never be as useful as direct discourse, simply

because this restriction on oblique discourse is so strong.

 

5.10 Various syntactical criteria of analyticity have
been proposed. Fer instance, a criterion due essentially
to Quine, is that a sentence is analytic if and only if,
it is a substitution instance of a law of logic. Some—
times it has been further maintained that these criteria
are adequate as definitions for 'analytic'. Useful though
such criteria may be, they are, however, very likely not
adequate as definitions for 'analytic', because they are
not intensionally equivalent to analyticity.

The following might be considered as an intensional
criterion of analyticity. S is analytic, is equivalent to,
the expression formed by prefixing S with a sign of
logical necessity, is a true sentaice.

5.11 The paradoxes of the theory of types are avoided
in Principia mathematica by proscribing altogether, the
substitution of some expressions, and restricting sub-
stitution of other expressions to special contexts.

The former sort of restriction might be called an
absolute proscription, and the latter, a relative pro-
scription, of substitution.

In System I, much more reliance was put upon
absolute than upon relative prescriptions of substitution,

than is the case in Principia Kathenatica. Yet in both

 

I. This suggestion is due to Mr. Leonard.

 

102

system-, some relative proscriptions of substitution are
necessary. Relative proscriptions are, for instance,
necessary to avoid allowing substitution of non—propositional
expressions for propositional variables. A further

topic of investigation would be that of examining to

what extent absolute restrictions on substitution are

sufficient to avoid paradox, and to what extent relative

restrictions are necessary for this purpose.

 

KL._..
__,—l

103. L;

BIBLIOGRAPHY

Barcan, Ruth C., "A Functional Calculus of the First
Order Based on Strict Implication", The Journal.
9: Symbolic Logic, vol. 11 (1946) p. l.

, "The Deduction Theorem in a Functional
Calculus of First Order Based on Strict Implication",
The Journal 9: Symbolig Logic, vol. 11 (1946) p. 115.

 

, "The Identity of Individuals in a Strict
Functional Calculus of Second Order", The Journal
9: Symbolic Logic, vol. 12 (1947) p. 12.

 

Carnap, Rudolf "Modalities and Quantification" Egg
Jougggi g; Symbolic Logic, vol. 11 (194%) p. 33.

, Meaning and Necessity, The University of
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