Logically Proper Definite Descriptions

8/11/2019 Logically Proper Definite Descriptions

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Logically proper definite descriptions*

An Essay in Honor of RUTHMARCUS

By

KAREL

LAMBERT**

Abstract

This essay notes a striking parallel between the original Hilbert-Bernays treatment of d ef-

inite descriptions and Russells theory of logically proper nam es. The forma l language for the

original theory is laid out and the imp lications of a theory of vis a vis the statements that qual-

ify as predications in a logically proper definite descriptionssense of the word predication

different from the espoused by F rege, Russell and M einong.

I Introduction

In

The Philosophy

of

Logical Atomism,

Russell declared that a [logically

proper] name can just name a particular, or, if it does not, it is not a [logically

proper] name at all, it is a noise. This is the case because the particulars so

named are the meanings of those [logically proper] names. In

Principia Math-

ematica

Russell said, in effect, that the expression n exists, where n is a

[logically proper] name would be insignificant, but would not be wheren n

is replaced by its counterpart definite description.* Elsewhere Russell ex-

Two acknowledgements are in order. First, much of what is contained in the formal

part of

this

article is

the

result of joint work done with Paul Schweizer during an increasing-

ly rare period in which he w as not in the rarified air of som e peak in the Himalayas, or the like.

Whether he would subscribe to the philosophical use to w hich it has been put here, I have not

been able to scertain. Second, I

am

pleased to be invited to contribute a piece in honor of Ruth

Marcus, colleague and friend As the inven tor

of

the theory of d irect reference, definite descrip-

tions have always been of considerable interest to her. So I hope she will find the animal

that follows deserving enough to be included in w hat Russell called ou r philosophical zoo.

**University f California, Irvine and University of Salzburg

B. Russell, Logic and Know ledge ed. R. Marsh), Allen and Unwin, London, 1966,

A. Whitehead B. Russell, Principiu Mathematics: Vol 1,

2nd

Edition, Cambridge,

p. 187.

University Press, 1910, pp. 174-175.

Dialectica

Vol. 53, No 14 1999)


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272 Karel Lam bert

plained, if a is a name, it

must

nam e ~ o m e t h i n g .~ence, in contrast to a

sentence of the form The so and so exists, it would be impossible for a state-

ment of the fonn a exists to be false.

So

here the word insignificant appar-

ently does not mean without sense but rather som ething like is trivial.

Many grammatically proper names -Vulcan, the name of the putative

planet, for instance could not be logically proper names in the Russellian

sense.

Failing to have a referent at all, they could not name particulars, and so,

when spoken, would only

be

noises. Of course, Russell himself would not have

regarded the gram matically proper nam e Vulcan, when spoken, as m erely a

noise. He would have believed it to be a meaningful expression falling in

the

category of truncated definite descriptions, and the statem ent Vulcan exists

nontrivially false.

An insufficiently stressed feature in the discussion whether g ramm atically

proper names such as Vulcanor DeGaulle

are

truncated definite descriptions

is that their assimilation to the category of definite descriptions depends on

ones of definite descriptions. One reason the proviso is important is that, con-

tra expectation perhaps, the theory of definite descriptions need not be Russel-

lian. It might be what nowadays is called applied) negative

free

definite

description theory. This kind of theory agrees with Russell vis

u

vis the truth

values assigned to statements containing definite descriptions along with

the

necessity of scope distinctions), but rejects the peculiarly Russellian view that

definite descriptions are not singular terms.5 In fact, this way treating definite

descriptions is attractive to some who distinguish between names a s rigid des-

ignators and definite descriptions as nonrigid des ignators, and treat Vulcan

for example) as a disguised definite description.Soone could assimilate many,

B. Russell, Introduction to Mathem atical Philosophy, George Allen and Unw in, LTD,

London, 1919, pp. 178-179.

B. Russell,

Logic and Knowledge

ed. R. Marsh), Allen and Unw in, London, 1966,

p. 243.

A

singular term is an expression purporting to refer to exactly one thing. As every

graduate student knows, Russell urged, in his famous

1905

essay, On Den oting, that expres-

sions of the formthe

so

and

so

are not really refemng kinds of expressions, but rather func-

tion in discourse more like the logical particles all and some. Negative free definite descrip-

tion theory NFDT)was invented by R. Schock. See his Logics

Without Existence

Assumptions, Almqv ist and Wiksell, Stockholm, 19 68, for reference to his pioneering work of

1963). L ater other versions of NFD T were developed by R onald Scales Attribution and exis-

tence, University of M ichigan Microfilms, 1 969) and Tyler Burge Singular Terms and Truth,

No[FB]s, VIII, 1974, pp. 309-325.). In these theories, expressions such asvulcan and the

planet causing perturbations in the orbit of Mercury are singular terms, and statements such as

The planet causing perturbations in the o rbit

of

Mercury exists and The planet causing per-

turbations in the orbit of M ercury rotates are predications, but false ones. The conception of

Fortsetzung siehe S. 73


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Logically proper defi nite descriptions 273

if not all, grammatically proper names to definite descriptions without thereby

subscribing to the Russenian view that the statements, Vulcan exists, Vulcan

rotates, DeGaulle exists and DeGaulle pontificates are not predications.

Another reason the proviso is important is that certain theories of definite

descriptions are not acceptable candidates for such assimilation at all. For in-

stance, the

original

Hilbert-Bernays theory of definite descriptions is inade-

quate because word strings like Vulcan exists would paradoxically, upon

natural paraphrase, be without sense. If paraphrased as The planet causing

perturbations in the orbit of mercury exists, the locution in question

turns

out

not false, trivial or otherwise, but without sense because the basis of the con-

stituent definite description is not provably unique; as a matter of fact, it is log-

ically ungrammatical. And presumably the same would be true of statements

containing DeGaulle or its counterpart definite description The man who

wanted to die in his own arms.

It is clear, then, that those who subscribe to the view that (at least) irref-

erential grammatically proper names are truncated definite descriptions must

be prepared to defend the philosophical superiority of the Russellian (or

asymptotic Russellian) logics of definite descriptions over their adversaries,

the Hilbert-Bernays treatment, for instance. Yet one need not look

far

for philo-

sophical defenders of the superiority of a Hilbert-Bernays like logic of defi-

nite descriptions for the purposes of mathematics or even for the purposes of

capturing natural language reasoning. An example

of

the first sort is Abraham

Robinson, and an example of the second

sort

is Soren Stenlund.6 Be that as it

Fortsetzung von S.272

predication in negative free logics is different from the conception shared by F rege, Meinong

and Russell. The latter threesome held a statement to be a predication just in case it joins a

general term (or n-adic predicate) to n singular terms such that its truth value true (or false)

depends on its general term being true (or false) of the n-tuple of n-objects specified by the n

singular terms. This cannot be

the

conception of predication in negative free logics because,

for instance, in the simplest case, a predication can be false even when there is no object spec-

ified by the singular term for

the

general term to

be

true or false) of. Nevertheless, exponents

of the different conceptions often agree on cases. For example, Meinong, Frege in his scien-

tific mood) and negative free logicians would ag ree that The planet causing perturbations in

the orbit of Mercury rotates is a predication; only Ru ssell would h old otherwise.

A. Robinson, Constrained Denotation in

Selected Papers:

Vol 2 eds. J. Keisler, et.

Al.), Yale University Press, New Haven, 1979; and

S.

Stenlund, The Logic

of

Existence and

Description, Filosofiska Studier, Uppsala, 197

3.

Consider, also, the following quotation by

G.

Kneebone Mathematical Logic and the Foundations of Mathematics, Van Nostrand,

London, 1963, p. 93) speaking of

the

motivation for the Hibert-Bernays approach. He writes:

In comm on speech we are not troubled by sentences containing descriptions which refer

either to no object at all

or

to a multiplicity of objects, never use a phrase

of

the form

the individual with the property P except when speaking loosely or idiomatically

unless we believe that it refers to a unique individual; and this sugg ests that in the for-

mal system we m ight make the introduction of a description symbol... onditional on the

prior derivation of [an] associated [formula]


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274

Karel Lambert

may, here the purpose is to examine some consequences of the original

Hilbert-Bernays theory, namely, that when properly formalized and ap-plied

it yields a view about definite descriptions that, in analogy with Russells dec-

larations above about [logically proper] nam es, simply craves to be called

the

theory

of

logically proper definite descriptions.

2.

A

Formalization

of

the Hilbert-Bemays Theory

of

Definite Descriptions

The theory to be formalized here is the original theory of H ilbert-Bernays

HB), the informal theory in the first edition of their famous

Grundlagen der

Mathematik.

It is neither the Frege like theory of the second edition nor the

presentation in most neo-H ilbert-Bernays treatments in w hich expressions *of

the form the

so

and so are treated as logically grammatical even when soand

so is not provably ~ n i q u e . ~

The distinctive feature of the original theory is a syntactical strategy de-

signed to ensure that definite descriptions have well-defined referents. For a

definite description to coun t as logically grammatical, the basis of the definite

description

so

and so in the so and

so

must be

provably

unique. This re-

quirement preserves the classical notion that all singular terms, including defi-

nite descriptions , have referents.

Two features of the current formalization of

HB

need to be emphasized.

First, the apparent circularity inherent in H B that the set of logically gram-

matical expressions of the language depends on what is provable, but w hat is

provable depends on the set of logically grammatical expressions can be cir-

cumvented. The language of H B can be defined in stages, beginning with a

definite description free base. Definite descriptions of a given language level

can then be introduced on the basis of what is provable in the language of the

previous level. All finite depths of em bedding of definite descrip tions are ob-

tained by defining HB as the union of the results of the procedure just outlined

over the finite ordinals. Second, the distinctive feature of the original HB,

namely, that the formation rules allow the introduction of a definite descrip-

tion just in case the basis of the defin ite descrip tion is provably unique, has the

consequence that though the ensuing form ation rules provide an inductive

definition of the language of HB, the set of expressions is not decidable; since

the underlying first order logic is undecidable, the syntax of H B will a lso be

undecidable.

D. Hilbert

nd

P. Bernays, Grundlagen der Mathematik, Vol.

1

Springer Verlag,

Berlin,

1968,

esp.

pp.

392-401.


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Logically proper definite descriptions

275

yntax of L,,

1.Logical constants:

- , 3 v,

3, i

2.

Variables: x,yz,.

.

with o r without subscripts

3.

Individuals constants: a,b,c, with or withou t subscripts

4. Predicates:

P,Q,R,

with o r without subscripts

Symbols from

1 -4.

Are used autonom ously.

The definite description free base language IA is a standard first order lan-

guage. That is,

1.The singular)

terms

of Lo

are

all and only individuals constants,

and

2.

the

statements

of

Lo

are defined as follows:

a) If P is an n-ary predicate constan t, and a l,.

..,

a,, are individuals

b) a

=

b is a statement;

c) If A, B are statements,

so

is - A, A B), A V B), and A 3 B);

d) If A a/x)

is

a statement, so are

v

x A and 3 x A.

constants, P a, , .

,

an ) s a statement;

The language

L,,+],

containing definite descriptions formed on the basis of the

language

L,,,

is defined as follows:

3.

Th e singular) terms of L,,+l

are

defined as follows:

a) If t

is

a singular) term of L,,, then t is a singular) term of

L,,+,;

b) If A is a statement of L,,, and a is a constant occurring in A, then

if

it

is

provable that

3

x

V y

A y/a)

=

y

=

x)), ixA x /a) is a sin-

gular) term of

L

4.

The statements of L,+] are defined as in

2.

with

Ln+ ,

eplacing Lo,

t

replacing by herein.

t,, replacing a l,. .,a,, and

s

replacing a in a = b and t

The language L=

UL,,,

for

ri E a

A statement A is provable in L,, if A is a cons equenc e of the axioms of L,, by

Detachment M odus Ponens), and an xiom of

L,,

is any tauto-

logy or instance in L n of the following schem ata:


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276

Karel

Lambert

5

V

x

(A B) V x A V x B)

6 A 3 V x A

1 V

x A x A ( a / x )

8 .a = a

9 . a = b b ( A ~ A ( a / / b ) )

10. V x A (x/a) provided A is an axiom and a is a (singular) term in A

11. A (ixA(x/a) provided A is a statement of L,+, and ixA(x/a) is a (sin-

gular) term of L n

The Hilbert-Bernays infortnal treatment in the first edition of the Grundlagen

is not hierarchical (as in the formulation above), and indeed, if not rec-

onstructed, is potentially circular. In place of

11

they have an inference rule,

the i-rule, such that any instance of 11. is derivable given the provability of the

appropriate uniqueness condition. But how can this rule be used to define the

syntax of their object language? To what language do the premises of the in-

ference rule belong if the formation rules of the language being defined ap-

peal to that inference rule? Evidently the premises of the i-rule are tacitly re-

stricted to a sub-language like

L,+l,

a restriction made explicit above, and,

hence, an appropriate version of the i-rule is derivable in the preceding treat-

ment.

Semantics

of

L B

A

model

for

L,,

is a pair .Ds a non-empty set, and I

=

UI,, where Ii(i

n) is an interpretation function such that

(la) I,(a)

E

D for any (singular) term a of Lo

(2a) I,(P") is a set of n-tuples of elements of D for any n-adic predicate

(3a) I, maps the (singular) terms of Loonto D.

of Lo;

I, induces a valuationfunction VM(o) efined on the set of statements of

Lo

as

follows:

(la(o)) VM(,)

a

= b) = T(rue) if and only if Io(a) is the same as Io(b) for

any (singular) terms a, b of

Lo:

otherwise Vm(o)(a

=

b)

=

F(a1se);


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217

(lb(o)) VM(o, Pn(al ..., an)

=

T if and only if E

Io(Pn),where al,

..,

an are (singular) terms of

Lo:

otherwise

VM(o, Pn(a ... an)=

F;

(2a(o)) VMc0,(-A)

=

T if and only if VM(o,(A)

= F,

and otherwise it

F =;

(2b(o)) V,,,,(A B) =T if and only if VM(o,(A)

=

T and VM(o,(B)

=

T,

(2c(o)) V,,,,(A V B) = T if and only if VM(o,(A)= t or VM(o,(B)

=

T, and

and otherwise it = F;

otherwise it

=

F;

(2d(o)) V,(,,(A B) = T if and only if V,(,,(A) =

F

or VM(o,(B)

=

T, and

otherwise it

=F:

(2e(o)) VMo (A (B) = T if and only if VM(o,(A)= VM(o,(B),and other-

wise it

= F:

3)

V,(,,('d x A) = T if and only if V,(,,(A(a/x)) for every (singular)

term of Lo, and otherwise it =

F.

To accommodate definite descriptions in L,,, first the interpretation function

In+l

s defined as follows:

(la(n+l)) In+l(t)

=

I,(t) provided t is a (singular) term of Li (i I );

(lb(n+ll)) In+l(ixA(x/a))= d E D such that for some (singular) term b

of Lo, Io(b)

=

d, and V,(,,,(A(b/a))=T, where ixA(x/a) is a (singular) term

OfLn+1;8

In

nduces a valution function VM(,,)defined on the statements of Li (i n)

as follows:

(la(n)) V,(,,(s

=

t)

=

T if and only if In(s) is the

same

as In(t), and other-

wise it = F;

(lb(n))V,~,,)(Pn(al,..,a,)=T if and only if Io(Pn),and

otherwise it

=

F;

This clause ensures that each term

of

the

form

ixA x/a) is assigned the unique element

d

of

D that satisfies the basis A x/a).

Its

adequacy is guaranteed by con dition 3a) in the defi-

nition of a model.


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278

Karel Lambert

The cases (2a(n))-(2e(n)) for the connectives are similar to the cases of

(1

a 0))

(

e(o)) above;

(3n) VMMcn)(d

A )

=

T if and only if V,(, A(a/x))

=

T

for every (singular)

term a of Lo, and otherwise it

= F.

3. Logically Proper De3nite Descriptions

Let a be ones favorite logically proper name perhaps the demonstra-

tive this, to take a word Russell himself once favored. Consider now the

predicate same as this. It is provable in L HB that there is exactly one thing

the same as this.

So

the definite description ix(x

=

this) is a singular term of

LHB, nd, indeed, picks out the same element of

D

of which the basis of that

definite description is provably unique he object of

D

assigned to this. If

arithmetic is added to

LHB,

a taken to be 4nd b taken to be S he same

holds for the expression ix(x is a positive whole number between nd

6 ,

except that the referent in this case would be 5 Moreover,

if

one wishes to

Quine-ize singular terms by adding special predicates such as thisizes to the

language, with appropriate axioms governing these special predicates, the

expression ix(x thisizes) would also qualify as a singular term in the extended

language in question.

Consider now statements of the forms, respectively, t exists and a does

not exist, where t is any of the definite descriptions mentioned in the previ-

ous paragraph. Let exists be understood as an abbreviation for x(x

=... ).

Statements of the first form would be true, and trivially true at that. Similarly

statements of the second form would be trivially false. Moreover, one can say

precisely what it would mean to say that the definite descriptions making up

the subjects of these statements would be mere noises, were they not to def-

initely describe an individual in D; they would be logical gibberish. It seems

appropriate, therefore, to call such expressions

logically proper definite

descriptions

at least as measured by the characteristics associated with log-

ically proper names enunciated in the opening paragraph of this essay.

This gives rise to the Russell-like questions: Which coroquial definite

descriptions are logically proper definite descriptions? How are statements

containing logically improper definite descriptions to be analyzed? The

answer to the first question is straightforward enough; no colloquial expres-

sion of the form the

so

and so, where so and so is not provably unique can

be a logically proper definite description. In particular, neither the planet

causing the perturbations in the orbit of Mercury, nor any of its relatives, can

be logically proper definite descriptions. Presumably also the man who


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279

wanted to die in his own

arms

is not a logically proper definite description.

But suffixing exists to this pair of expressions does not yield a pair

of

non-

sensical statements. What one gets is, respectively, a nontrivial falsehood,

French refuter notwithstanding, and a no ntrivial truth. H ow to analy ze them?

Exactly as Q uine does in Word and Object. To illustrate, the statement The

planet causing perturbations in the orbit of M ercury would b e paraphrased as

There is exactly on e thing that

is

a planet c ausing perturbations in the orbit

of Mercury, a false statement that does not contain a phrase of the form

the

so

and

so.

I conjecture that any truth o r falsehood containing a phrase of the

form the

so

and

so,

where so and

so

is not provably unique, can be treated a

la Quine, given the requisite sensitivity to scope considerations in the case of

non-atomic statements. Of course, the Russell program about the elimination

of grammatically proper names via logically improper definite descriptions

proceeds as usual. Vulcan may be taken

as

a truncated version of the planet

causing perturbations in the orbit of M ercury, DeG aulle as a truncated ver-

sion of the man w ho wanted to die in his own arms, and the resulting state-

ments containing these descriptive phrases paraphrased into the purely quan-

tificational fragment of

L,,

a la Quine.

Turning to the pair of colloquial statements The num ber between

4

nd 6

exists and The king of France in 1999exists, it might be thought that exists

means different things in these two cases because their paraphrases into

L,,

are quite different. Th e former, containing a logically proper definite descrip-

tion, gets paraphrased as 3x x

=

the number between 4 nd 6), but the latter

gets paraphrased as3 z V y) y is a king of France in 1999= y

=

z .

However,

this conclusion

is

incorrect. exists is a silent partner in the latter paraphrase

because that paraphrase is logically equivalent to 3z Vy) y is a king of

France in 1999=

y =

z)

3

x x

=

z)). W hat is true and important) is that

only the first of conoquial statements is a predication because only the first

definite description is logically proper, and hen ce qualifies a s a singular term.

So

only the first of this pair of statements gets evaluated by appealing in part)

to the object which the constituent definite description refers

to.

In virtue of their similarity of properties, the question a rises whether, con-

tra Russell, logically proper nam es m ight not be eliminated in favor of logi-

cally proper definite de sc r i p ti ~ n s. ~his would have the advantage of giving

The attempt here to produce a theory of logica lly proper definite descriptions deliber-

ately avoids the ep istemological overtones in Russels account of logically proper names, an

account that appeals

to

entities directly apprehended sense data for the Russell

of

The

Philosophy

ofLogicuZAtornisrn).t is more in the spirit

of

Priors attempts to get at the logical

essence of Russells notion, what Prior called Ru ssellian names. Se e Objects of Thought

Fortsetzung siehe S.280


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280

Karel Lambert

formal substance to Russells view that logically proper names that failed to

name would be insignificant because one could then pin the insignificance on

a concrete violation of the rules of logical grammar.

There is, however, a serious problem confronting the current proposal to

eliminate logically proper names. The problem is that since what is provable

depends on what the axioms and/or rules of inference are in a given theory,

which definite descriptions will turn out to be logically perfect becomes a rel-

ative matter. For instance, in some set theories the expressionix(Vy)(ye x =

y = y) will be logically perfect and in others not because in some set theories

the basis of this definite description is not provably unique. And the same

would be true of definite descriptions in opposed empirical theories. But the

relativity outlined seems quite alien to Russells conception of a logically

proper name. Moreover, if a, b, c... Of L are taken to be the formal coun-

terparts of logically proper names, the very characterization of ixA(x/a)as a

singular term requires, on pain of circularity, that there be some (unanalyzed)

singular terms, hence some irreducible logically proper names.

The provability requirement for a definite description to qualify as a sin-

gular term, and thence as logically proper, is very strong. Following Carnap,l0

one might think of replacing it in clause 3(b) above by the weaker requirement

that the uniqueness of the basis of the putative description simply be true. This

might now allow the president of the

U.S.

in

1999

to count as a (singular)

term, but this proposal to weaken the condition determining when an expres-

sion of the form ixA(x/a) is to count as a (singular) term faces a possible for-

mal obstacle. Provability is a syntactical property, but truth is a semantical

property. Since the true statements of the formal language require an inde-

pendent characterization of statement, it is not clear that a non-circular for-

malization of

L

is possible. Perhaps a levels of language approach of the

sort above can be achieved, but that is a beyond the narrow purposes of this

essay.

The undecidability of

Lm vis a vis

what constitutes a statement is seen by

many as a serious disadvantage, given provability (or truth) of the unique-

ness of the basis of a definite description as a requirement for (singular) term-

hood. Here is the way Carnap puts the matter in

eaning A n d Necessity.

Fortsetzung von

S. 79

eds.

P.

Geach and

A

Kenny), Clarendon Press, Oxford, 1971, esp. Chapter

10.

For a brief

examination of Priors

efforts,

see K. Lambert, Russellian Nam es: Notes on a

theory

of Arthur

Prior

in

Logic

nd

Reality,

ed.

i.

Copeland) Clarendon Press, Oxford, 1996, pp.

41

1-419)

OR.

Carnap,

Meaning

And

Necessity

Univ. of Chicago Press, Chicago,

1947,

pp. 33-34.


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28

1

... this method has a serious disadvantage, although of chiefly a theo-

retical nature: the rules of, Formation become indefinite, i.e. there is no

general procedure for determining whether any given expression of the

form

[

e.g., PixQx] is a [statement] of the system (no matter whether

true or false or provable or not).

In addition, he says:

For systems containing factual [statements], the disadvantage would be

still greater, because here the question of whether a given expression is

a statement or not would, in general, depend on the contingency of

facts.

In the first quotation, it is important to note Carnaps emphasis on the

chiefly theoretical nature of the disadvantage to the original

HB

theory.

Now it may be annoying not to know whether the statement The explanation

of the results in Lamberts Experiment on latent inference learning by the

Expectancy Theory shows the transitive character of e~pectation~eally is a

predication, and hence whether certain inferential procedures are applicable to

it (for instance, substitutivity of identity). But philosophers need not be com-

pletely disabled by that consideration

vis

a vis computation of its truth or fal-

sity. The example expression clearly is a sensible statement and its truth or

falsity may be ascertained by paraphrasing it into

L,

a la Quine. If it were to

turn out to be a predication after all, this would not change its truth-value, only

the method (or methods) of computing that truth-value.

Turning to the second quotation, whether a statement is to be regarded as a

predication often depends on how the facts

tur

out. For instance, suppose it

were to

tur

out that the name, Baxter, picked out by U.S. citizens Smith and

his wife for their planned, but not yet conceived first boy, should fail to refer

(because, for example, Smiths wife subsequently miscarried). Then the truth

value of Baxter (that is, the planned first born son of Smith and his wife) is

an American could not be ascertained by seeing whether the predicate is an

American is true (or false) of the individual picked out by the singular term the

first born son of Smith andhis wife, there being no such individual.

So,

though

a false statement, it would not be a predication. The tie between singular term,

reference and predication characteristic of Russells theory of logically proper

names is reflected in the current theory of logically proper definite descriptions.

Ibid., p. 34.

l

Ibid., p. 34.

l 3

K. Lambert,

A

study

of

latent in ference, Canadian Journal

of

Psychology, 14, 1960,

pp. 45-50.


12/12

282

Karel

Lambert

If a definite description is a singular term it must refer, and (at least) any atomic

statement in which it occurs must be a predication. On the other hand, deter-

mination of whether a definite description is a singular term, a logically signif-

icant unit, hence whether statements in which it is contained are predications,

may be indefinite for lack of proof of the uniqueness of the basis of that definite

description (or, less strictly, because the facts are not

all

in).

This talk of predication shows a particular value of HB qua theory of logi-

cally proper definite descriptions. On the classical view of predication, state-

ments containing definite descriptions in subject position were uniformly

taken by Russell to be non-predications because definite descriptions could

never be singular terms, but Frege (in his scientific mood) and Meinong re-

garded all such statements

s

predications. These seemed to be the, only ways

of regarding such statements given the principle of compositionality, the prin-

ciple that the value of a complex is a function of the values of its logically

grammatical parts. Russell preserved the principle by denying that expressions

of the form the so and so are ever logically grammatical parts of their host

statements, but Frege and Meinong held such expressions to be logically sig-

nificant parts of their host statements and thus had to have values, invented or

otherwise. HB qua theory of logically proper definite descriptions represents

an intermediate view because i-expressionscannot fail to have referents, and

hence statements containing them will obey the principle of compositionality.

However, in contrast to Frege and Meinong, some such expressions in collo-

quial discourse will not

be

singular terms, yet compositionality is preserved

by treating the statements in which they occur as shorthand for existential

statements essentially in the manner of Russell.

To

sum up, if an expression

of

the form the

so

and so is to be regarded as

a singular term a la

HB,

then it will have the status of something like a logi-

cally proper name. To

be

sure this means that there is a certain indefiniteness

about which expressions of ordinary discourse of the form the so and

so

so

qualify. However,

this

should cause no more discomfiture than the familiar

indefiniteness surrounding the question which statements of colloquial dis-

course qualify s (or will turn out to be) predications, in the classical sense of

predication.

Dialectica

Vol. 53,No /4 1999)

Logically Proper Definite Descriptions

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