A Remark Concerning Quine’s Paradox about Modality

Citation preview

Mafia"

Eda

5alwwtb$cot+$oeumvz

94‘5‘J9L‘A [P‘6E2“JV"“

{éYfihALI’cI/gas .

IV

IN what

form, reference

(1) (2)

to

El 9

may

‘

9 of

major planets

’

is the notation ‘it is read as

D

for

(3)

El the it

planets hardly

is less a

was

belief

logic and (2) are Both.(l) former on logico—mathematical modal

in standard that’.

and

at

time

one

than in an

major planets

9;

and

believed this belief,

=

9

that

the

number

of

major

though factually false,

was

impossibility. does

Quine‘s original formulation

of the use exactly paradox of however, example but a number others—which, illustrate is evidently the same logical difficulty.And one of Qurne s as the and Modality’ IS in fact the same examples in ‘Reference above except that the premiss (1) is replaced by not

D9>7,

Alonzo

Spanish

George

since

George IV, being this. Moreover,

to

IV believes

that

The

author

version

in Analisis

Scott with

acquainted

it is

Sir Walter

=

Sir

factually true,

Quine’s Paradox

Concerning Filoséfico 2 (1982),

25—32,

About

Modality’,

of

Waverley

(8)

George Walter

IV

believes

Sir Walter

=

By substitutivity of equality it

Scott,

Walter, is unlikely of this same date,

as

4-

to

seems

that

the

Scott,

follow

author

that of

Scott.

Waverley

=

But (8) is known to be false, still as of the same date; and without the factual information it is clearly unreasonable that (8) is a logical consequence suppose of (6) and (7). Or still better we may use the premiss

George

IV believes

which follow

is at least

(10)

George

that

9

=

IV believes

that

Sir

even

to

qt (9)

the number

and

(2)

there

to

seems

w ‘4

of

major planets

=

9.

But

(10) is certainly false, because the discovery of the last two major planets came only after King George’s death. Given the truth of (2), the paradox based on the inference of (3) from (1) differs from that based on the inference of (IO) from (9) only in the replacement of ‘George IV believes that’ by the modal

El ’. It may therefore be argued that there are not two different paradoxes, but only various examples illustrating what must be regarded as a single paradox. And indeed Quine, already in ‘Notes on Existence and Necessity’, makes a close

genuinely

asu -\

9,

probably true. From by substitutivity of equality:

‘

‘A Remark

becomes:

U

operator

Church,

59

7

>

Sir Walter

well

doubt

(9)

this what

(4)

paradoxical conclusion

major planets

(6)

(7)

paradox grounds. substrtutrvtty numberof ‘9’ (I).

of

number

Indeed

of

the

MODALITY

I

9

true, the commonly accepted The latter on astronomical and the grounds of equality, arises if, relying on the principle of we to of (2) in order make use major substltute ‘the in For in this way we of planets’ for the first occurrence what is evrd‘eaLlyfalse: infer from two accepted truths

wruwfim,“ l.

=

necessity

necessary

as

.w

number

(3),

that

number

be

of

ABOUT

inference of (3) from the premisses (1) and (2) has here been chosen as an example in order to exhibit more clearly the very close parallelism between Quine’s paradox and the paradox which Russell2 sought to eliminate by means of his theory of descriptions. For the latter paradox we may use a minor variation of Russell’s example about it true, as of King George IV.3 We may assume some appropriate date, that

sight may seem to be its simplest and most direct paradox about modality may be explained by following example:

the

=

The

Here A

first

at

Quine’s

place

PARADOX

The

CHURCH

ALONZO

in

D the

(5)

QUINE’S MODALITY

ABOUT

PARADOX

that,

so

CONCERNING

A REMARK

QUINE’s

60

ALONZO

association

between

paradoxes

CHURCH

about

belief

QUINE’S and

modality. This paradox that’ and ‘it is

paradoxes

about

arise

not may with ‘believes only in connection that’ but also with any of various necessary other we phrases which may speak of as introducing intensional contexts. Carnap calls it the antinomy 0f the name relation.4 And this terminology is useful as a reminder that, in place of substitutivity of equality, the paradox may be made to depend on the semantical principle that (in Carnap’s words) tftwo expressions name the same entity, then a true sentence remains true when the one is replaced in it by the other. But since—contrary to Carnap—there is no actual antinomy or contradiction, but only such results as (3), (8), (10), which are factually false or unacceptably counterintuitive, the writer prefers to speak of the paradox of the name relation. Once it is seen that Quine’s paradox about modality and the paradox about King George [V and Sir Walter Scott must be seen as instances of the same paradox, it is not surprising that Smullyan is able to resolve Quine’s paradox by means of Russell’s theory of descriptions.5 Moreover, it seems certain in advance that whatever objections may apply to Smullyan’smodal logic with descriptions (e.g. Quine’s objection in ‘Reference and Modality‘ that it requires excessive attention to matters of scope) must equally apply to the logic of belief statements with descriptions which is in Russell’s resolution implicit in ‘On Denoting’ of the paradox about George IV and Sir Walter.“ Besides the complications regarding scope which arise when the paradox about modality is resolved of by Russell’s doctrine descriptions, Quine’s ‘Reference and Modality’ raises also a different objection, which depends on reformulating the paradox in a way that refers only to variables and makes no use either of names (or naming expressions) or of desriptions. The point is that although Russell largely reconstrues names and naming expressions as descriptions, and then eliminates the descriptions by his device of contextual definition,7 of course the use of variables is not

thereby eliminated. Citing Ruth Barcan8 Quine (1]) Indeed

x

=

y

nyElx

this theorem

=

calls

attention

to

the

theorem

y.

follows by

elementary logic alone, independently

of the

PARADOX

ABOUT

exact

meaning or definition of equality, provided only that we have (12) x y 3,”, F(x)DF(y)

MODALlTY

the

sign

‘=’

61 of

identity

or

=

.

and

(13) For

(x)

El

=

x

x.

by substitution

(14)

=

x

And

y

in

(12) D[x

31),.

we

get

x]

=

(11)

then follows by The result is perhaps possibility rather than

(13)

3

is

~F(x)

(14). striking

more

equally

denied.

(16)

if

allow

we

¢

(x) ~x

~ix

Then

from

*

(16)

(>[x

(If

things

two

1":

and

y] Dny

are

The

in

put

of

terms

at: y

Dyx

of that

further

in

(15)

1‘13»le

Murphy’s Law: (18)

when

identity

which

can

hardly

be

x,

by substitution

(17)

y].

For

elementary property

an

And

obtain

we

3),,F(y)

=

and

necessity.

(15)

El[x

(17)

that

i we

yIDyx have

the

¢

y.

following

variant

of

#5 y.

possibly different,

then

they

are

different.)

theorems (11) and (18) are hardly avoidable if modal logic is formulated in such a way that modal are operators prefixed directly to sentences is now (as indeed Quine in usual).9 ‘Reference and Modality’objects to these theorems as compelling of ‘Aristotelian acceptance essentialism"”~which he as

philosophically with

the

idea

suspect,

which DS

and

regards

as

being,

moreover,

incompatible

modal many logicians have held, that the is true if and only if the corresponding unmodalized sentence S is analytic. This second form of Quine‘s paradox about modality, which refers to variables rather than names,” can be paralleled by a paradox about belief statements in place of modality. By substitution in (15) we get: modal

sentence

62

ALONZO

(19) This

For every x :9 x, if be

may

(20)

For

(21) And

and

we

of

in

as

analogous

being

IV no

every

does more

it,

accept

For every x and then x 1": y. this

QUINE’s

every y, if George IV does not believe that IV believes that x ¢ y, then x i y.

George

x,

(16) if

CHURCH

George

thought

every

differs from certain. But conclusion:

x

we

y, if

to

(17)‘ The believe

not

second

that

x

premiss, #

x,

than (18) the

IV believes

that

x

#

y,

otherwise

surprising power of King George’s beliefs to facts about x and y can be explained only on the doubtful assumption that belief ‘to the fulfilment properly applies of conditions by objects’ quite “apart from special ways of control

the actual

specifying‘ the objects.'2 This assumption principle of transparency of belief) is the same

of belief

that

consequences

(let us call it the thing to the notion necessity. But the

essentialism is to the notion of of the former for the ordinary notion of belief to be even more repellent than the consequences for modal notions.

may be thought of essentialism To illustrate this last point, let us that George IV is suppose convinced (and in fact on good evidence) that there is one and only one who wrote Waverley, that there is one and only one who wrote Ivanhoe, and that the two authors are the same. Then let us ask for what objects13 (or individuals) y we have that

(22) The holds

George

IV believes

that

y wrote

ABOUT

63

MODALITY

significant objections in ‘Referenceand in complications about scope which arise connection with the use of descriptions, and the transparency of both belief and necessity which is forced by use of the theory of descriptions to resolve the paradox of the name relation: But be pointed out that finally it must Qumes objections, of the resolution though strong, are no firm refutation 'Russellian of the paradox. There of the may be those who, in the interest resolution of the paradox, are willing to accept both the complications about descriptions and the strange transparent notions of belief and necessity which result. And to them it can only be said that well, it does seem strange. The sort of In

the

summary,

Modality’

are

two:

the

.

than very likely rather have as analogous to

George

PARADOX

Waverley.

essentialism to which the transparency of mucus as belief and necessity leads does not rise above the level of and constants. And the Russellian With the primitive may proceed more confidence because he is able, besides the transparent notions of belief, necessity, and possibility, to express also the more usual non-transparent notions. Namely if B(x, p) is used to mean that x believes that p,15 S(x) that x scottizes, W(x) that x is author of Waverley, G(x) that x is George IV (or that x georgivizes), he may write:16

such variables

(23) (24)

=

.

The

(23) (24)

ordinary

notion of belief seems to require that although (22) when y is specified in a special way, namely as having written Ivanhoe, it may yet fail when the same y is specified in some other special way, for example as scottizing. Our conclusion is that Quine’s objections against the Russellian treatment” of modal logic, according to which modal operators are prefixed to sentences, do have some considerable force. But it is better to present them in a way that exhibits the nearly complete parallelism between the objections against a Russellian modal and those against a Russellian logic logic of belief (or of denying, wishing to know, or the like). For this has the effect of putting the objections in perspective and 0f clarifying both their and strengths their weaknesses.

B( (1x)G(x), (1 x)S(x) (1x)W(x) ). #5 O (1x)S(x) (1x)W(x). convention of minimum to be understood in scope is of course and (24). And notwithstanding (18), it does not follow from

that

(10506)

#5

(1 X)W(X)-

Notes 1.

2.

V. Quine, ‘Notes on and Necessity’, Journal Existence 40 of Philosophy (1943), 113—27 (reprinted in Leonard Linsky (ed), Semantics andthe Philosophy ofLanguage (Urhana, 1952), 77—91)and ‘Reference and Modality in From a Logical Pom! of View (Cambridge, Mass, 1901), D9759 (reprinted in Leonard Linsky (ed.), Reference and Modality (London, 197l), I7734. Bertrand Russell, ‘On Denoting’, Mind 14 (1905), 479793.Reprinted'." Herbert Sellars Feigl and Wilfred (eds), Readingx in Philosophical Analysis C. Marsh (New York, l949), l(l3—15; Robert (ed), Logic and Knowledge, Exmy: [90171950, by Beriarid Russell (London, 1956),fill—So; Irvmg M. Copi and James A. Gould (eds), Contemporary Readings in Logical Theory (New W.

.

64

ALONZO York and

.

and

CHURCH

u.» .

.

.

Russell

has

informal

an

connection with ‘I thought your

.

.

.

belief

treatment statements

The

Scott

yacht

was

matter

of scope

of

descriptions

was

writer

believes that a more Fregean version of modal logic might be in which the modal are operators prefixed not directly to a sentence but to any name of the proposition which the sentence Quine's expresses. misgivings about this in ‘Reference and Modality' can be dispelled only by a detailed development of such a Fregean modal logic, explicitly exhibiting the ‘interplay'which he fears may be wanting; but Quine‘s further objection (in W. V, Quine, Word and Object (Cambridge, Mass., 1960); cf. printing of 1973, p. 198) that there is some ad hoc restriction on quantifying into modal contexts seems to be

preferable,

based

on a misunderstanding.(For historical accuracy it should be Frege himself disbelieved in modal logic.) Cf. also Word and Object, § 41. Or naming expressions. No distinction is intended in this or paper, by Carnap, between names and naming expressions. And the idea is Russellian rather than Fregean, that a name must be an unanalysed primitive and hence normally a single word or a single symbol. The quoted phrases are from Section III of ‘Reference and and are used in order to emphasize that what is here said about beliefModality‘ closely parallels what is said by Quine about modality. We assume that human beings are included among objects, or

added

10. ll.

12.

13.

individuals

in

that

among

(as

PARADOX

it may

be better

ABOUT

MODALITY

65

say in order to allow for type theory and its follow Quine in avoiding the semantical formulation that consists in asking what values of the variable ‘y‘ satisfy the form ‘George IV believes that propositional y wrote this Waverley—although alternative formulation might otherwise be helpful, eg. in bringing out that what is at issue concerns the values of a variable, and not (as in the original paradox of the name relation) the dcnotution of a name or names. 14. We call it Russellian in spite of Russell's own rejection of modality, because it is the standard

.

of the

(and the like), using the particular larger than it is' and ‘George IV wished examples to know the author of Wuverley'; but in his later writings, leading up to and including Principia Muthemau'ca, (A. N. Whitehead and Bertrand Russell, Principia Mathematica, 3 vols. (Cambridge, 191043)), he does not return to this. His formal language is confined to what is needed for the of the foundations Principia account of mathematics, and he therefore never considers a formalized logic of belief in connection with the theory of descriptions. Nevertheless, the complications about scope are already implicit in Russell’s paper of 1905 and are not due to changes by Smullyan. As Russell writes in ‘On Denoting‘:‘The phrase perse has no meaning, because in any proposition in which it occurs the proposition, fully expressed, does not contain the phrase, which has been broken up. Ruth C. Barcan, ‘The of Identity Individuals in a Strict Functional Calculus of Second Order', Journal of Symbolic Logic 12 (1947), 12—15,and her earlier papers. whether

QUINE‘S

London, 1967),

93-105; Ausanis Marras (ed.), lnlenrionality, Mind, Language (Urbana, Chicago, and London, 1972), 362—79. The example is due to ‘On Russell, Denoting‘; the point which it illustrates, to Gottlob Frege, ‘Uber Sinn und Bedeutung', Zeilschrifl ftir Philasophie und Philasophische Kririk 100 (1892), 25—50. (Reprinted in Gunther Panzig (ed), Funku'on, Begn'flr,Bedeu/ung: ["an logische Sludien2 (Gottingen, 1966), 4065. English translation in Feigl‘Sellars, pp. 85—102,reprinted in Cord-Gould, pp. 75—92,and in Marras, pp. 337-61.) Rudolf Carnap, Meaning and Necessity2(Chicago, 1956). Arthur F. Smullyan, Review of Quine‘s ‘The Problem of Interpreting Modal Logic‘ (Journal of Symbolic Logic 12 (1947), 43-8; reprinted in pp. 267—73)in Journal ofSymbo/ic Logic 12 (1947), 139-41 and Copi—Gould, ‘Modalityand Description', Journal ofSymbolr'c Logic l3 (1948), 3|~7 (reprinted in Linsky, Reference and Modality, pp. 35—43).

terminology).

And

to

we

treatment appropriate to Russell's explicit and implicit especially propositions as values of the propositional variables, and semantics, the sort of transparency in belief contexts, modal contexts, and the like that is required by the theory of as resolution of the paradox of the name descriptions relation. To avoid antinomies such as the Epimenides, it may be necessary either to distinguish different orders of propositional variables (by adopting ramified type theory) or to distinguish different orders of belief by writing B', 82, B3, etc. We ignore this here as not being immediately relevant to what is being said.

16. These e.g.

examples illustrate that Russell's unitary propositional is unitary‘, is defined as (3x). F(y) x, y y

‘F

E

considerable appear

as

extent

senses

of

as

surrogates

names.

belief appears as a relation between 3(2) and W0?)

functions

must

For as

for

the

entities

functions, where

:

which

may in

serve

to

a

Frege's theory

example in (23) King George’s non-transparent the propositional functions S and W (or would write), and these propositional

between Russell

be unitary if King the same belief

George's belief is not mistaken. And in a by George IV would appear as a relation the senses which ‘Sir Walter belong to the names Scott‘ and ‘the author of Waverley‘.It would be of interest to look into the question how far the Fregean theory can be reproduced within the Russellian by the Fregean senses with propositional functions (in Russell's sense, identifying to which propositional according functions are intensional entities). Indeed, the Russelhan theory might have to be rather drastically mutilated to obtain the Fregean fragment. But it remains true that any significant partial success in representing one within the other would throw theory light on the relationship of the two theories. Fregean theory

between

rap in :‘r‘ r.-

.uv

ca