This chapter contains sections titled: References.

We prove that: • if there is a model of I∆₀ + ¬ exp with cofinal Σ₁-definable elements and a Σ₁ truth definition for Σ₁ sentences, then I∆₀ + ¬ exp +¬BΣ₁ is consistent, • there is a model of I∆₀ Ω₁ + ¬ exp with cofinal Σ₁-definable elements, both a Σ₂ and a ∏₂ truth definition for Σ₁ sentences, and for each n > 2, a Σ n truth definition for Σ n sentences. The latter result (...) is obtained by constructing a model with a recursive truth-preserving translation of Σ₁ sentences into boolean combinations of $\exists \sum {\begin{array}{*{20}{c}} h \\ 0 \\ \end{array} } $ sentences. We also present an old but previously unpublished proof of the consistency of I∆₀ + ¬ exp + ¬BΣ₁ under the assumption that the size parameter in Lessan's ∆₀ universal formula is optimal. We then discuss a possible reason why proving the consistency of I∆₀ + ¬ exp + ¬BΣ₁ unconditionally has turned out to be so difficult. (shrink)

§1. The mission of axiomatic set theory. What is set theory needed for in the foundations of mathematics? Why cannot we transact whatever foundational business we have to transact in terms of our ordinary logic without resorting to set theory? There are many possible answers, but most of them are likely to be variations of the same theme. The core area of ordinary logic is by a fairly common consent the received first-order logic. Why cannot it take care of itself? (...) What is it that it cannot do? A large part of every answer is probably that first-order logic cannot handle its own model theory and other metatheory. For instance, a first-order language does not allow the codification of the most important semantical concept, viz. the notion of truth, for that language in that language itself, as shown already in Tarski. In view of such negative results it is generally thought that one of the most important missions of set theory is to provide the wherewithal for a model theory of logic. For instance Gregory H. Moore asserts in his encyclopedia article “Logic and set theory” thatSet theory influenced logic, both through its semantics, by expanding the possible models of various theories and by the formal definition of a model; and through its syntax, by allowing for logical languages in which formulas can be infinite in length or in which the number of symbols is uncountable. (shrink)

The paper discusses the notion of finite model truth definitions (or FM-truth definitions), introduced by M. Mostowski as a finite model analogue of Tarski's classical notion of truth definition. We compare FM-truth definitions with Vardi's concept of the combined complexity of logics, noting an important difference: the difficulty of defining FM-truth for a logic ᵍ does not depend on the syntax of L, as long as it is decidable. It follows that for (...) a natural ᵍ there exist FM-truth definitions whose evaluation is much easier than the combined complexiy of ᵍ would suggest. We apply the general theory to give a complexity-theoretical characterization of the logics for which the $\Sigma_{m}^{d}$ classes (prenex classes of higher order logics) define FM-truth. For any $d \geq 2.m \geq 1$ we construct a family $\{[\Sigma_{m}^{d}]^{\leg\kappa}\}\kappa\in\omega$ of syntactically defined fragments of $\Sigma_{m}^{d}$ which satisfy this characterization. We also use the $[\Sigma_{m}^{d}]\leg\kappa$ classes to give a refinement of known results on the complexity classes captured by $\Sigma_{m}^{d}$ . We close with a few simple corollaries, one of which gives a sufficient condition for the existence, given a vocabulary $\sigma$ , of a fixed number $\kappa$ such that model checking for all first order sentences over $\sigma$ can be done in deterministic time $n^{\kappa}$. (shrink)

Tarski's model-theoretic truth definition of the 1950s differs from his 1930s truth definition by allowing the language to have a set of parameters that are interpreted by means of structures. The paper traces how the model-theoretic theorems that Tarski and others were proving in the period between these two truth definitions became increasingly difficult to fit into the framework of the earlier truth definition, making the later one more or less inevitable. The paper also maintains (...) that neither recursiveness nor satisfaction are essential features of the truth definition . The recursive form was a ‘practical’ step towards the explicit definition that was Tarski's target. There are model-theoretic languages for which satisfaction is provably too crude a notion for expressing those properties of formulas that are needed for determining the truth of sentences. (shrink)

Tarski's manner of defining truth is generally considered highly significant. About why, there is less consensus. I argue first, that in his truth-definitions Tarski was trying to solve a set of philosophical problems; second, that he solved them successfully; third, that all of these that are simply problems about defining truth are as well or better solved by a simpler account of truth. But one of his crucial problems remains: to give an account of validity, (...) one requires an account not just of truth but of truth under varying interpretations. Tarski's account has the merit of generalizing to this, to model theory and to abstract algebra. (shrink)

In this paper we show that first-order languages extended with partially ordered connectives and partially ordered quantifiers define, under a certain interpretation, their own truth-predicate. The interpretation in question is in terms of games of imperfect information. This result is compared with those of Kripke and Feferman.

We study the structure of the partial order induced by the definability relation on definitions of truth for the language of arithmetic. Formally, a definition of truth is any sentence \(\alpha \) which extends a weak arithmetical theory (which we take to be \({{\,\mathrm{I\Delta _{0}+\exp }\,}}\) ) such that for some formula \(\Theta \) and any arithmetical sentence \(\varphi \), \(\Theta (\ulcorner \varphi \urcorner )\equiv \varphi \) is provable in \(\alpha \). We say that a sentence \(\beta (...) \) is definable in a sentence \(\alpha \), if there exists an unrelativized translation from the language of \(\beta \) to the language of \(\alpha \) which is identity on the arithmetical symbols and such that the translation of \(\beta \) is provable in \(\alpha \). Our main result is that the structure consisting of truth definitions which are conservative over the basic arithmetical theory forms a countable universal distributive lattice. Additionally, we generalize the result of Pakhomov and Visser showing that the set of (Gödel codes of) definitions of truth is not \(\Sigma _2\) -definable in the standard model of arithmetic. We conclude by remarking that no \(\Sigma _2\) -sentence, satisfying certain further natural conditions, can be a definition of truth for the language of arithmetic. (shrink)

This chapter contains sections titled: Truth The Liar Tarskian Truth Definition Discussion Conclusion.

We introduce a notion of semantical closure for theories by formalizing Nepeivoda notion of truth. [10]. Tarski theorem on truth definitions is discussed in the light of Kleene's three valued logic (here treated with a formal reinterpretation of logical constants). Connections with Definability Theory are also established.

Criticism of Soames' understanding of Tarskian theories of truth.

Alfred Tarski's work on truth has become a touchstone for a great deal of philosophical work on truth. A good grasp of it is critical for understanding the contemporary literature on truth and semantics. In this paper, I present a fresh interpretation of Tarski's view, one which aims to draw it out more fully in areas of philosophical interest.

This paper shows that the logic known as Information-friendly logic (IF-logic) introduced by Jaakko Hintikka and Gabriel Sandu defines its own truth-predicate. The result is interesting given that IF logic is a much stronger logic than ordinary first-order logic and has also a well behaved notion of negation which, on its first-order subfragment, behaves like classical, contradictory negation.

According to an influential view, asserting a proposition involves undertaking some “commitment” to the truth of that proposition. But accounts of what it is for someone to be committed to the truth of a proposition are often vague or imprecise, and are rarely put to work to define assertion. This paper aims to fill this gap. It offers a precise characterisation of assertoric commitment, and shows how it can be applied to define assertion. On the proposed view, acquiring (...) commitment is not sufficient for asserting. To assert, commitment must be acquired by explicitly presenting a proposition as true. (shrink)

In “Truth – A Traditional Debate Reviewed”, Crispin Wright proposed an inductive definition of “coherence truth” for arithmetic relative to an arithmetic base theory B. Wright’s definition is in fact a notational variant of the usual Tarskian inductive definition, except for the basis clause for atomic sentences. This paper provides a model-theoretic characterization of the resulting sets of sentences "cohering" with a given base theory B. These sets are denoted WB. Roughly, if B satisfies a certain minimal condition, (...) then WB is the Th, where M is the canonical model of the set At of atomic sentences provable in B. The paper also shows that the disquotational T-scheme is provable from Wright’s inductive definition just in case the base theory B is sound and complete for arithmetic atomic sentences. (shrink)

The age-old question “What is truth?” is not an unambiguous one. There are at least two different meanings. In one sense, it is a semantic question about the meaning of the word “truth” and/or a metaphysical question about the nature of the property of truth, that is, how truth can be defined in terms of other notions, if it is definable at all. In another sense, it is an epistemological question about the criterion or test of (...) truth, that is, how we can recognize or accept (empirical) propositions as true. Ayer, in Language, Truth and Logic (LTL) and some other articles, tried to convince us that when we attempt to elucidate a theory of truth, we are only engaging in the epistemological project of finding the criterion of truth. This anti-semantical and anti-metaphysical view is based on Ayer’s (radical) deflationism about truth. In this chapter, I will argue that since Ayer did not accept that the criterion of truth is intimately related to the concept of truth, he can be seen as a deflationist about the nature of truth and as an advocate of a certain (Schlickian) kind of correspondence conception about the criterion of truth. His position would not be tenable if he had maintained what some of his contemporaries accepted, namely that there is a close connection between the two truth-theoretical tasks: the definitional and the criterial one. (shrink)

In this article we study the systems KF and VF of truth over set theory as well as related systems and compare them with the corresponding systems over arithmetic.

Are the small and indolent cancers found in abundance in cancer screening normal variations, risk factors, or disease? Naturalists in philosophy of medicine turn to pathophysiological findings to decide such questions objectively. To understand the role of pathophysiological findings in disease definition, we must understand how they mislead in diagnostic reasoning. Participants on all sides of the definition of disease debate attempt to secure objectivity via reductionism. These reductivist routes to objectivity are inconsistent with the Bayesian nature of clinical reasoning; (...) when they appeal to the sciences, they are inconsistent with what philosophy of biology tells us about its natural kinds. Proposals that we narrow the scope of our claims in the disease definition debates are useful, but paradigms can still distort our reasoning in particular cases, even when we are self-conscious about their status. (shrink)

This article investigates the connection and dependence between the definiteness of the totalities involved in mathematical structures and the determinateness of statements about that structure. From a logical perspective, we investigate whether logical principles expressing the definiteness of totalities license the use of classical logic. From a philosophical perspective, this article provides a reconstruction of Solomon Feferman’s claim that the definiteness of the natural number conception implies the determinateness of arithmetical statements and therefore justifies the adoption of classical logic for (...) arithmetical theories. (shrink)

Kant claims that the nominal definition of truth is: “Truth is the agreement of cognition with its object”. In this paper, I analyse the relevant features of Kant's theory of definition in order to explain the meaning of that claim and its consequences for the vexed question of whether Kant endorses or rejects a correspondence theory of truth. I conclude that Kant's claim implies neither that he holds, nor that he rejects, a correspondence theory of truth. (...) Kant's claim is not a generic way of setting aside a correspondence definition of truth, or of considering it uninformative. Being the nominal definition of truth, the formula “truth is the agreement of cognition with its object” illustrates the meaning of the predicate “is true” and people's ordinary conception of truth. True judgements correspond to the objects they are about. However, there could be more to the property of truth than correspondence. (shrink)

Since the publication of Hartry Field’s influential paper “Tarski’s Theory of Truth” there has been an ongoing discussion about the philosophical import of Tarski’s definition. Most of the arguments have aimed to play down that import, starting with that of Field himself. He interpreted Tarski as trying to provide a physicalistic reduction of semantic concepts like truth, and concluded that Tarski had partially failed. Robert Stalnaker and Scott Soames claimed then that Field should have obtained a stronger conclusion, (...) namely that Tarski’s failure in his allegedly intended physicalistic reduction was total. From another front, Hilary Putnam argued for an even more sweeping thesis: “[a]s a philosophical account of truth, Tarski’s theory fails as badly as it is possible for an account to fail.” Scott Soames followed suit also on this count, endorsing Putnam’s argument in section iii of the aforementioned paper; and John Etchemendy used a very similar argument to contend that “a Tarskian definition of truth [...] cannot possibly illuminate the semantic properties of the object language.” The paper argues that these contentions are based on several misunderstandings about the nature of a definition like Tarski’s. Regarding the Putnam-Soames-Etchemendy argument, we must apply a distinction Frege found natural to draw—and is natural to draw anyway—between what he called “constructive” definitions, definitions properly so called, which are mere stipulations, and what he described as “analytic” definitions, definitions which aim to make explicit the meaning of a term already in use. Regarding the Field-Soames-Stalnaker criticism, the paper contends that the true aims of a Tarskian truth definition are perfectly well fulfilled without Field’s amendments. (shrink)

This is a collection of eighteen solicited papers on the topics of the title: circularity, definition, and truth. The papers are loosely connected in subject matter, but present a great variety of issues, theories, and approaches. Amongst the many subjects discussed are: the revision theory of truth and applications of revision rules, partiality and fixed point constructions, substitutional quantification, fuzzy logic, negation, belief revision, context dependence, hierarchies, Tarski on truth, deflationism, correspondence theories of truth, and normative (...) aspects of truth. The Liar paradox figures prominently in this collection, but is not alone. Other familiar paradoxes are also discussed, including paradoxes of definability, the Sorites paradox, and the Surprise Exam paradox. So-called paradoxes of rationality, such as the Prisoner’s Dilemma, receive attention as well. (shrink)

In this paper we consider truth as a vague predicate and inquire into the relation between truth and definite truth. We use some tools from modal logic to clarify this distinction, as done in McGee . Finally, we consider the question whether some of the results given by McGee can be transferred to the case in which the underlying logic is stronger than first-order logic. The result will be seen to be negative.

The author defends a combination of Strawson’s account of definite descriptions as devices of singular reference par excellence with the Russellian truth-evaluation of utterances of sentences with descriptions. The complex Russellian proposition is, according to the author’s view, introduced by such utterances into communication as a by-product of the instrumental side of an attempt to make a singular statement. This, precisely like the instrumental aspects of similar attempts exploiting names or demonstratives has to be reflected by analysis but should (...) not be confused with the communicative function of utterances. The success of all these attempts depends on the fulfilment of empirical conditions of various types, given by semantics of descriptions, names or demonstratives . But their communicative function does not consist in claiming that these conditions are fulfilled.The author agrees with Strawson that the first two conjuncts of the complex Russellian proposition are introduced into communication as presuppositions: but argues in favour of defining presupposition in normative, rather the intentional terms. (shrink)