[Logic] [Metaphysics] - Anil Gupta and Nuel Belnap - The Revisionist Theory of Truth.pdf - sikula

Preface

This book developsa way of viewing the concept of truth (and several

other concepts) that we havecometo believe, after much hesitation, to be

fruitful and correct. The view can be stated quite simply, though its exact

meaningcan be madeplain only during the courseof the book: Truth is a

circular concep. We shall argue that this viewpoint enablesone to make

senseof the perplexing behavior of truth (e.g., its behavior in the Liar

paradox), and to explain much of its ordinary unproblematic behavior.

The material principles that are widely recognizedas governing truth

(the Tarski biconditionals) suggeststrongly that truth is circular. (Seesec-

tion IV of chapter 4.) The difficulty in arguing for the thesisis createdby

widely acceptedformal strictures which dictate that circular conceptsdo

not exist and that circular definitions are illegitimate. Against thesestric-

tures we shall put forth and arguefor the following claims:

. Generaltheories of definitions are possible within which circular def-

initions - and, more generally, systems of mutually interdependent

definitions- make logical and semanticsens.

. In the context of certain sorts of logical and philosophical inquiries,

theseformal strictures ought to be abandoned.

The theory of definitions, we believe, is the proper framework for the

construction of a theory of truth . Indeed, the theory of truth is an immedi-

ate corollary of the theory of definitions.

We hope to showin this book that the viewpoint just sketchedis attrac-

tive and plausible. We do not claim, however, to be setting down the

theoriesof definitions and truth in their final form. Part of the reasonfor

this is persona. Despite the many yearswe have spenton the subjec, our

study remainsincomplete. But part of the reasonis intrinsic to the subjec.

There are several conflicting desiderata one can impose on theories of

definitions and truth , and it is not yet completely clear how theseconflicts

are bestresolved.

Howto ReadtheBook

This book may be read in two ways: either straight through or by first

reading chapters 1, 4, and 7 and then reading the remaining chaptersin

order. Chapters 1, 4, and 7 explain the main philosophical ideaswe wish

to put forward in an accessibleway. The remaining chapters are more

Preface

technical. Chapters2 and 3 presentsomebasicfacts about the fixed-point

approach to the theory of truth and develop our argument against it .

Chapters 5 and 6 contain a formal developmentof our approach. (Chap-

ters 2, 3, 5, and 6 contain severalopen problems, which may be found via

the index entry " open problems.")

A citation of the form Author 198+ or Author 199+ refers to an un-

published paper listed in the bibliography. For example, Thomason 198+

refersto Richmond Thomason's paper "Paradoxesof intentionality ?" and

indicatesthat we receivedit in the 1980.

Prerequisites

Different parts of this book have different logical prerequisite. Chapters

1, 4, and 7 presupposevery little logical background and can be under-

stood by anyone who has had a course in first-order logic. Chapters 2

and 3 presupposesomeknowledge of set theory (in particular, of Zorn 's

lemma). Thesechaptersdo not presupposeany knowledge of arithmetiz-

ation (except in a few skippable passage) or of the theory of ordinals.

Chapters5 and 6 presupposea rudimentary acquaintancewith the theory

of ordinals, and threesectionsof chapter 6 (6C- 6E) presupposefamiliarity

with arithmetization. All the technical chapters(2, 3, 5, and 6) assumethat

the reader has pencil and paper handy and is willing to work out the

detailsof an exampleand to supply the missingproof of a straightforward

theorem.

6 Truth, Categoricalnes, and Necessity

In this chapter we apply the theory of definitions to the concepts of truth ,

categoricalness , and necessity.

6A Truth I : The Main Lemma

Kripke sketches in the following well -known passage an attractive picture

of the concept of truth :

We wish to capture an intuition of somewhat the following kind . Suppose we are

explaining the word 'true ' to someone who does not yet understand it . We may say

that we are entitled to assert (or deny ) of any sentence that it is true precisely under

the circumstances when we can assert (or deny ) the sentence itself . Our interlocutor

then can understand what it means, say, to attribute truth to (6) ('snow is white ')

but he will still be puzzled about attributions of truth to sentencescontaining the

word ' true ' itself . . . .

Nevertheless , with more thought the notion of truth as applied even to various

sentences themselves containing the word 'true ' can gradually become clear . Sup-

pose we consider

the sentence

(7) Some sentence printed in the New York Daily News , October 7, 1971, is true .

. . . [ I ] f (7) is unclear , so still is

(8) (7) is true .

However , our subject , if he is willing to assert 'snow is white ' , will according to the

rules be willing to assert '(6) is true ' . But suppose that among the assertions printed

in the New York Daily News , October 7, 1971, is (6) itself . Since our subject is

willing to assert '(6) is true ', and also to assert '(6) is printed in the New York Daily

News, October 7, 1971', he will deduce (7) by existential generalization . Once he is

willing to assert (7), he will also be willing to assert (8). In this manner , the subject

will eventually be able to attribute truth to more and more statements involving

the notion of truth itself . There is no reason to suppose that all statements will

become decided in this way , but most will . (Kripke 1975, pp . 65- 66)

Kripke goes on to show that the statements so decided can plausibly be

identified with those that have a truth value in the least fixed point (the

grounded statements ) of the Strong Kleene scheme.

One part of Kripke 's picture can hardly be doubted : that the grounded

statements are unproblematic . It is not so plausible to maintain , however ,

that only these statements are unproblematic . The instructions given by

Kripke to his imaginary subject leave out something important : They do

not specify how the subject is to work with the concept of truth in hypotheti -

cal contexts . Suppose that the subject does not know that

200

Chapter6

Gillian owns a lamp that once belonged to Carnap (P),

but that in the course of some reasoning he is led to hypothesize that P .

The instructions given by Kripke do not entitle the subject to deduce,

under the hypothesis P, that " P" is true . If the subject can assert that P,

then the instructions entitle him to assert that " P" is true . If the subject can

deny that P, then again the instructions entitle him to deny that " P" is true .

But if the subject merely supposesthat P, they do not entitle him to con-

cl ude that " p " is true .

This points to one important difference between the intuitive picture

of truth we wish to put forward and Kripke 's. The instructions we would

give to the subject consist simply of the Tarski biconditionals (understood

definitionally ). No restrictions would be placed on their use; in all con-

texts , categorical or hypothetical , their use would be uniform . What sepa-

rates the situation under consideration from countless others in which the

meaning of a word is explained is that the (partial ) definitions we would

give our subject are circular . This circularity does not make the definitions

illegitimate or wrong or senseles . One does not have to eliminate the

circularity of truth (say, via a hierarchy of truth predicates or by invoca -

tion of new semantic values) to make sense of the biconditionals . The

biconditionals make sense on their own . Indeed , the roots of both the

ordinary nonpathological behavior of the concept of truth and its perplex -

ing pathological behavior lie in their circularity .

The picture we wish to put forward retains the legitimacy of Kripke 's

procedure . It affirms that a sentence is nonpathological if it receives a

truth value by this procedure (seeLemma 6B.5). But it denies the converse.

Indeed , as we shall see below , there are conditions under which the revi -

sion process dictates that truth has a classical signification while Kripke 's

procedure does not (Examples 6B.6, 6B.7).

Let us apply the theories of definitions we constructed in chapter 5 to

the concept of truth . We begin with some notational matters . Let 2

( = ( L , M , t ) ; M = ( D , I ) be an interpreted classical language (seepp . 44-

45), and let 2 + be obtained by adding a new one-place predicate T to 9! .

We suppose that the set S of the sentences of 2 + is included in the domain

D and that 2 has a quotation name 'A ' for each sentence A of L + (so

1('A ') = A ). We interpret T as a predicate defined via the partial definitions

T('A ') = DfA .

Truth , Categoricalnes, and Necessity

201

Thesebiconditionals yield the rule of revision tM (seeDefinition 2B.8) as

the signification of T.l Revision sequencesof length On for 'tM will be

called tM-sequences

for various normal monotonic three- and four-valued scheme.

Of particular importance are the schemesJl and K (introduced in chapter

2) and the supervaluation scheme(J of van Fraassen(1968). Truth and

falsity are evaluatedin the supervaluation schemeasfollows.

6A.l Definition (Supervaluation scheme0-) Let ~ be a three-valued

model of a language L . A sentenceA of . P is true (false) in ~ by the

supervaluationscheme0' iff A is true (false) in all classicalmodelsJ!{' ~ JIt

(seep. 39). (I'Mshall be thejump yieldedby the supervaluation schemein a

ground model M .

that have the value

t (f ) in the least fixed point of PMwill be said to be p-groundedtrue (false)

in M ; p-groundedsentencesare thosethat are either p-grounded true or p-

grounded false. We shall understand"p-intrinsic truth," " PM-paradoxical,"

etc. in a parallel way.2

that truth behaveslike an ordinary class-

cal concept under certain conditions- conditions that can roughly be

characterizedas those in which there is no vicious referencein the lan-

guage. This can be shown on the basisof the Main Lemma, which we will

state after two preliminary definitions.

6A.2 Definition (Degree) The degreeof terms and formulas is defined

recursively:

(i) The degreeof atomic terms (i.e., variablesand nonquotational names)

and of -L is O.

1We are assuming, as usual~that only sentencesare true. If fl7 hasnamesof all the objectsin

its domain~then the revision rule is weakly expressiblein it .

2A sentenceA is a PM-intrinsic truth iff it is true in the largest intrinsic fixed point of PM;

A is PM-paradoxical iff it is neither true nor falsein eachfixed point of PM.

of M ).

For purposesof comparison, we shall also be interestedin the revision

sequences

(alternatively, 't'-sequences

Note that 0' is normal and monotonic, but not truth functional. Given a

schemep, the revision sequencesof length On for PMwill be called PM-

sequences

(alternatively, p-sequencesof M ). Sentences

An important feature of the revision theory, and one that prompted our

interest in it , is its consequence

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