guide to truth predicates.pdf

A Guide to Truth Predicates in the Modern Era

Author(s): Michael Sheard

Source: The Journal of Symbolic Logic, Vol. 59, No. 3 (Sep., 1994), pp. 1032-1054

Published by: Association for Symbolic Logic

Stable URL: http://www.jstor.org/stable/2275927

Accessed: 26/04/2009 07:49

Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at

http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR's Terms and Conditions of Use provides, in part, that unless

you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you

may use content in the JSTOR archive only for your personal, non-commercial use.

Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at

http://www.jstor.org/action/showPublisher?publisherCode=asl .

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed

page of such transmission.

JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the

scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that

promotes the discovery and use of these resources. For more information about JSTOR, please contact support@jstor.org.

Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to The

Journal of Symbolic Logic.

http://www.jstor.org

THE JOURNAL OF SYMBOLIC LOGIC

Volume 59. Number

1994

A GUIDE TO TRUTH PREDICATES IN THE MODERN ERA

MICHAEL

SHEARD

?I. Introduction.A reader coming anew to the recent work on languages which

contain their own truth predicates may be perplexed by the simple question of

where to begin. A first approach to the literaturesuggests a field which is alive and

busy with investigations heading in many differentdirections, but there is much less

indication of how various pieces fit together. There are at least two sources of this

confusion. First, the literature is large and diffuse (as befits a subject which goes

back over 2000 years); Visser'ssurvey [33] aptly describes the literatureas "vast but

scattered, repetitive, and disconnected." Moreover, recent interest in the field has

led to a proliferation of research and publication; it seems that almost any issue of

any philosophical logic journal from the mid-1980s contains some article on the

topic. The second reason, in part a consequence of the first, is that while a typical

article in print usually presents a good internal motivation, with clear reference

to its immediate intellectual antecedents, its place in the broader picture may not

be so easily discerned. The problem can be especially acute in presentations of

axiomatic approaches, because decisions on certain basic questions can lie hidden

in the formal and notational details which abound in any axiomatization.

In fact, though, the recent research on methods for handling self-referential

truth can be seen as a body of work which is very well structured, one in which a few

fundamental decisions suffice to locate any particular approach in its appropriate

place on the landscape. My goal is to describe this structure and in particular

to stress a few critical forks in the road, which will be the recurring metaphor

throughout this paper. I will also pay particular attention to pointing out where

the interesting technical and mathematical questions lie.

The start of the recent era in the study of self-referential truth is easily dated.

Kripke's "An outline of a theory of truth" [20], appearing in 1975, is almost

universallycited as the spark for the explosion of interest in the subject. Certainly

no major intellectual advance ever arises in a vacuum, and any suggestion that

the subject of self-referentialtruth had lain completely dormant prior to Kripke's

paper would be both inaccurate and unfair. Nonetheless, we need only look at

how frequently Kripke's paper is cited in prefatory remarks in the literature, and

especially how many times it is cited at a personal level as having reawakened an

author's interest in the subject, to recognize the catalytic role of that essay in the

recent wave of research and discussion.

Received July 15.1993: revised December 20. 1993.

? 1994 Association for Symbolic Logic

0022-48 12/94/5903-0022/$03.30

1032

3. September

A GUIDE TO TRUTH PREDICATES IN THE MODERN ERA

1033

As should be expected for a subject of this size and inherent interest, the litera-

ture already contains some surveys of various aspects of the topic. Visser's article

in the Handbook of philosophical logic [33] gives a very thorough discussion of

the best-known semantical approaches with both more attention to philosophical

motivation and criticism and far more technical details than will be included

here. The anthology edited by Martin [23] collects several of the most significant

papers from the blossoming of this subject in the mid-1970s and early 1980s,

again mostly (but not exclusively) on the semantical side; these papers make an

excellent introduction for a newcomer to the field. Feferman's papers (e.g., [8],

[9]) provide helpful historical notes, both to the recent work surveyed here and to

more distant antecedents. The interested reader can also look to published reviews

(e.g., [14], [22]) for more comprehensive and specific summaries of the major papers

mentioned here. As much as possible I hope to avoid repeating what is said in all

of those places, although of course some overlap is unavoidable.

While I will aim to be fairly complete concerning the kinds of approaches which

occur and will try to recognize historical primacy for results and constructions, I

will probably be less successful in indicating the intellectual ancestry of particular

presentations in the literature. In keeping with the goal of conveying the structure

of the research effort, I will point to the best known and most influential pre-

sentations of particular approaches, those which have shaped the discourse and

direction of the field: my emphasis is one of taxonomy rather than history. Fuller

historical references can be found in some of the surveys mentioned above.1

I will also not provide philosophical criticism in much depth of the various

approaches described. The vast discussion which has arisen around most of these

approaches is beyond the scope and intent of this survey. Moreover, there are

strong inherent reasons why any approach is liable to provoke an extensive but

ultimately inconclusive round of criticism and rebuttal. As everyone knows, this

is an area in which one simply cannot have all that one might reasonably ask for.

The selection of any particular approach necessarily involves a trade-off, and in the

face of criticism that a given system has some flaw (usually of omission), the most

appropriate response is typically concession, accompanied by an argument that

the advantages outweigh the failings. The resulting colloquy may be informative

or even inspiring, but it can go on forever. Accordingly, I will limit myself to

some fairly straightforward motivational remarks for each approach mentioned

and, dually, to some of the more common criticisms to indicate why each approach

does not simply represent the final word on the issue.

Lest these remarks seem too apologetic, let me suggest that an attitude of

agnosticism toward the philosophical discussion of truth can constitute a fresh

and productive point of view, under which some previously neglected questions

will acquire new importance. That is, the problem of representing truth for a

language within the language itself can be stripped of its philosophical overtones

and phrased purely as a problem of logic or more accurately as a set of related

1 The picture is further complicated because so much of the literature employs eponymous nomen-

clature: the "Kripkeconstruction", a "Gupta sequence", etc. I will avoid this practice, but will indicate

the standard references by which the field is defined.

1034

MICHAEL SHEARD

problems. How close can a model come to containing its own satisfaction relation?

To what extent can a theory, perhaps satisfying specified auxiliary conditions, prove

correct statements about itself? Or essentially mixing the levels of analysis of the

two preceding questions, how fully can a theory characterize the properties of

complete extensions of itself? Despite the fairly neutral and value-free nature

of questions like these, their discussion tends to fall almost automatically into

philosophical terminology. In this spirit, one can enter the discourse on the mer-

its and demerits of systems for truth without either committing to a particular

philosophical perspective or denying the value of the insights provided by such

perspectives.

Notations, conventions, and other preliminaries. The logic employed here will

be classical first-orderpredicate calculus usually; an exception will be occasional

excursions into three-valued logic with an additional truth value u, which might be

interpreted as "unknown" or "undetermined". In any case the classical rules for

functions on {t, f } will hold throughout, so that intuitionistic or other restrictive

systems of logic are not under consideration. Truthwill be treated as a predicate of

sentences. Such a stipulation is already philosophically loaded, but is best suited

to the application of technical machinery.

The underlying domain of discourse must have sufficient richness to manifest

the salient semantic issues, but otherwise the choice of domain admits considerable

flexibility. In order to deal with any worthwhile predicate of sentences, the lan-

guage must have a mechanism for discussing the syntax of formulas; since standard

practice employs arithmetization of syntax, some scheme of arithmetic ought to be

available. Conversely, a setting of arithmetic is quite adequate for highlighting any

of the points at issue. Moreover, there are certain economies of presentation to be

had by working only in arithmetic, such as the adequacy of discussing satisfaction

only for sentences rather than arbitrary formulas.

Accordingly, I will work in a language of arithmetic augmented with a unary

predicate T; the intended meaning of T (x) is "x is the Godel number of a true

sentence in the augmented language". A formula containing no occurrence of the

predicate T is called arithmetical. The specific choice of language for arithmetic

is ultimately irrelevant, but I believe that the smoothest and most convenient

formulation is one with equality as the only predicate and a very large but finite set

of symbols for primitive recursive functions. In particular, I assume a collection

of functions representing all the usual syntactic operations on formulas (relative

to some reasonable Gddel numbering); for example, there is a function neg(x)

such that if x is the Gddel number of the formula X, then neg(x) is the Gddel

number of -,Q. Particularly noteworthy in this regard is the primitive recursive

function sub(x. n), where if x is the Gddel number of a formula with at most one

free variable, then sub(x. n) is the Godel number of the formula resulting from the

relacement of that free variable with the numeral for n. Two useful applications

of the function sub are quantification over the Gbdel numbers of all substitution

instances of a formula and the formulation of axioms and rules of inference with

free parameters within the scope of the predicate T.

I will be intentionally and unapologetically informal about abbreviatingformu-

las of the language. For example, T(-,4) abbreviates T(neg(#O)); an example of

A GUIDE TO TRUTH PREDICATES IN THE MODERN ERA

1035

more substance is the use of (Vn)T(q!(n)) instead of (Vn)T(sub(#0, n)). There

will never be any danger of ambiguity in such informality; the reader,now properly

warned, should have no trouble translating to the more formal expressions. My

own experience from reading in the field is that the resulting benefits in clarity

are tremendous. In fact, in order further to avoid burdening the reader with

clutter, when appropriate I will treat quantifiers as ranging only over Godel num-

bers of sentences, and so for example, write (Vx)(T(x) V T(neg(x)) instead of

(Vx)(x is a sentence -- T(x) V T(neg(x))). This convention avoids an unproduc-

tive discussion on the application of T to numbers which are not Godel numbers

of sentences and allows greatereconomy and concomitant clarity in certain formal

expressions.2

Many presentations in the literature use a second auxiliary predicate F with

the obvious intended meaning of F(x) being "x is the G6del number of a false

sentence". In such cases, the system with F may be regarded as a definitional

extension of the system without F: F(x) +-* T(neg(x)). I will use both T and

F when such use coheres most closely with the literature and streamlines the

presentation. While the presence of F may result in added convenience, I know

of no situation in which it makes a substantive difference.

A few other notational conventions: Whenever a structureM is given as (N. S),

S denotes the extension TM of T in the structure (and thus indication of all

arithmetical apparatus is systematically suppressed). Similarly, if the language

contains both T and F, the notation M = (N, A, B) indicates that A = TM and

B = FM. Since a part of the discussion below involves 3-valued logic, the symbol

l= is reserved for the classical (two-valued) satisfaction relation; since there is no

standard notation, three-valued satisfaction will be indicated either in words with a

statement that a given sentence is satisfied, falsified, or left undetermined by a given

structure, or by reference to truth values {t. f u}. Finally, in prose passages I will

use capitalization to help distinguish the truth predicate from ordinary informal

uses of the word "true": if X is a sentence, "o is True ..." will indicate the

assertability of the sentence T(o), either because it is satisfied by a particular

structure or because it is theorem of a particular formal system.

The questions. Let me begin this section by telling the reader several things which

he or she undoubtedly already knows. I do so in part for the obvious purpose of

specifying the problem to be investigated, but more significantly as background to

illuminate a distinction which I find important for organizing an understanding

of the field as a whole.

The naive desideratum of a truth predicate is summarized in what is known as

Tarski's Convention T: a scheme which asserts for each sentence X of the language,

T(4) <-* Q. This requirement is sometimes described as a materially adequate

condition on a formalization of truth. Of course, as everyone now knows, such a

scheme results in an inconsistent theory: there is a sentence the Liar sentence,

2To make this formal, we could assume that we actually have two systems of Gddel numberings at

work, one for formulas and one for sentences, such that in the latter every natural number represents

a sentence. Lurking in the background would be a primitive recursive function which maps the G6del

number of a formula with no free variables to its Godel number qua sentence.

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika: