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.

This chapter contains sections titled: References.

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.

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)

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 article aims to fill this gap. It offers a precise characterisation of assertoric commitment, and applies it 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)

§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)

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

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)

Keith Donnellan’s distinction between the attributive and referential uses of definite descriptions has spurred debates regarding the truth conditions of the utterance “the F is G” with definite descriptions used referentially. In this article, I present a semantic account of referential descriptions, grounded in the contextual factors of the utterance, including the speaker’s intention and presupposition as well as the interlocutor’s recognition of them. This account is called the IPR-semantic account, according to which the speaker’s intention (I), (...) presupposition (P), and the interlocutor's recognition (R) jointly determine whether “the F” in an utterance “the F is G” is used referentially or attributively, and the meaning of “the F” is determined by whether it is used referentially or attributively. Moreover, I argue that the meaning of the referential description “the F” is the intended object e, embodied with a property H that has prompted the speaker to presuppose that e is F and to intend to use “the F” to refer to e, as well as the interlocutor to recognize the presupposition and intention. According to the IPR-semantic account, the utterance “the F is G” with “the F” used referentially expresses a singular proposition, namely, that e is G, and it is true if and only if the intended object e is G. Additionally, I argue that the IPR-semantic account not only surpasses some alternative semantic accounts but also outperforms Kripke’s pragmatic account. (shrink)

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)

Of his numerous investigations ... Tarski was most proud of two: his work on truth and his design of an algorithm in 1930 to decide the truth or falsity of any sentence of the elementary theory of the high school Euclidean geometry. [...] His mathematical treatment of the semantics of languages and the concept of truth has had revolutionary consequences for mathematics, linguistics, and philosophy, and Tarski is widely thought of as the man who "defined truth". (...) The seeming simplicity of his famous example that the sentence "Snow is white" is true just in case snow is white belies the depth and complexity of the consequences which can be drawn from the possibility of giving a general treatment of the concept of truth in formal mathematical languages in a rigorous mathematical way. (J.W. Addison). (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)

For the sake of simplicity, we adopt the same logical frame as Tarski's in his Wahrheitsbegriff (Wb). There, Tarski is mainly interested in the possibility of explicitely defining truth for an object-language, he does not pay much attention to recursive definitions of truth. We say why. However, recursive definitions have advantages of their own. In particular, we prove the positive theorem: if L is of finite order ≥ 4, then a recursive definition is possible for L in a (...) metalanguage of the same order as L. We indicate how this result could be used for a solution of a generalized version of Frege's paradox. (shrink)

In this paper Rudolf Carnap's definition of 'analytic truth' based upon a meaning postulate At, for theoretical predicates of a given scientific theory is subjected to critique. It is argued that this definition is both too exclusive and too inclusive. Assuming that the preceding is correct, At is subjected to further scrutiny to determine how to interpret it and whether, and under what conditions, it need even be true. It is argued that a given At need not be true (...) as a sentence in a scientific theory and that it must be regarded as a confusedly presented rule for the same. It is then argued that even as a semantical rule for the theory, At cannot provide a guarantee of truth for some meaning postulate similar to At in the theory itself. It is thus argued that Carnap's definition of 'analytic truth' for theoretical predicates is unsatisfactory. (shrink)

This original and enticing book provides a fresh, unifying perspective on many old and new logico-philosophical conundrums. Its basic thesis is that many concepts central in ordinary and philosophical discourse are inherently circular and thus cannot be fully understood as long as one remains within the confines of a standard theory of definitions. As an alternative, the authors develop a revision theory of definitions, which allows definitions to be circular without this giving rise to contradiction (but, at worst, to “vacuous” (...) uses of definienda). The theory is applied with varying levels of detail to a circular analysis of concepts as diverse as truth, predication, necessity, physical object, etc. The focus is on truth, and hope is expressed that a deeper understanding of the Liar and related paradoxes has been provided: “We have tried to show that once the circularity of truth is recognized, a great deal of its behavior begins to make sense. In particular, from this viewpoint, the existence of the paradoxes seems as natural as the existence of the eclipses” (p. 142). We think that this hope is fully justified, although some problems remain that future research in this field should take into account. (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)

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)

Tarski's definition of truth has rehabilitated the application of the word "true" to sentences of formalized languages. But a correspondence theory according to which a sentence is true if, And only if, It is related in the peculiar way of correspondence to the facts, Is incompatible with tarski's definition. Actually no theory of truth, Which claims to make proper assertions about sentences when calling them true, Is compatible with tarski's definition. Hence they all have to find their own (...) solution to the problem of the semantical antinomies and have to explain what it means to say that sentences are true. (shrink)

What is the nature of law? The question that St. Thomas Aquinas answers in Summa Theologica I-II continues to be a crucial question in contemporary philosophy of law. Various scholars of jurisprudence attempt to identify the necessary features of law. Yet they struggle with the question, what kind of necessity is involved? Is it conceptual necessity? Metaphysical necessity? In this paper, I explore an alternative way of distinguishing different kinds of necessity that is found in Aquinas’ Commentary on Aristotle’s Physics. (...) I argue that the three kinds of necessity simpliciter, from the formal, material, and efficient cause, and hypothetical necessity, from the final cause, are relevant for understanding how Aquinas’ definition of ‘law’—an ordinance of reason, for the common good, made by the one who has care for the community, and promulgated—is a necessary truth. This historically interesting approach offers insights for contemporary jurisprudence. (shrink)

A hallmark of correspondence theories of truth is the principle that sentences are made true by some truth-makers. A well-known objection to treating Tarski’s definition of truth as a correspondence theory has been put forward by Donald Davidson. He argued that Tarski’s approach does not relate sentences to any entities (like facts) to which true sentences might correspond. From the historical viewpoint, it is interesting to observe that Tarski’s philosophical teacher Tadeusz Kotarbinski advocated an ontological doctrine of (...) reism which accepted only concrete individuals and rejected all such abstract entities as facts, states of affairs, properties, and sets. Kotarbinski’s physicalism influenced Tarski who also avoided concepts like “fact” and “property” in his theory of truth, but—unlike Kotarbinski—he used freely set-theoretical terminology. In his mature work in model theory in the 1950s, Tarski used systematically the notion of a relational system (i.e., a domain of objects with designated elements, subsets, and relations). Wilfrid Hodges has argued that the notions of “structure” and “truth in a structure” appeared in Tarski’s work only in 1950. In my view, one can find the main ingredients of the model-theoretic account of truth already in the 1930s. These considerations suggest, against Davidson, that Tarski’s definition presupposes that material truth is always related to some kind of truth-maker. Further, facts as truth-makers can be reconstructed by employing the resources of model theory. (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)

This article argues that two new definitions are needed for what constitutes a truth commission. The first new definition that is needed is a different backward-looking definition that is used reflectively to contrast, compare and research past and present truth commissions. It is argued that the variety of definitions that exist about what constitutes a Truth Commission have a number of problems, and that a better definition is needed to categorise past mechanisms, make comparisons and improve comparative (...) research. The second goal of the article is to investigate why it is necessary to create a non-prescriptive explanatory forward-looking definition focusing on the compositional elements of a TC. This is because definitions are also useful to assist those who wish to establish TC on what the compositional elements need to be, to ensure that these institutions are less likely to fail in their endeavours. This is important, as TCs have been extensively criticised for not achieving the goals that they are meant to. To determine why a new backward-looking definition is needed, which is broader and more inclusive than existing definitions, some of the current TC definitions are deconstructed. The various elements are re-examined and reconstituted to ensure a more holistic and relevant definition for what a TC is. As far as a forward-looking definition is concerned, the article examines what the compositional elements ought to be to make them more likely to be contextually applicable, to have more of a positive effect and have more of a constructive role in the places where they are created. Issues focused on that need greater delineation include the idea that TCs—besides focusing on the past—also, in some cases, investigate on-going human rights abuse. It is contended that TCs must be created with the support of the people in the country and be reflective of their needs. It is maintained that a TC should be a temporary body and that independence, legitimacy and sufficient resources are crucial factors in determining whether a TC will be successful. Also, considered is why having an independent process to choose commissioners, who are diverse and representative of that society is critical to the outcome of the process. (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)

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 paper proposes substitutional definitions of logical truth and consequence in terms of relative interpretations that are extensionally equivalent to the model-theoretic definitions for any relational first-order language. Our philosophical motivation to consider substitutional definitions is based on the hope to simplify the meta-theory of logical consequence. We discuss to what extent our definitions can contribute to that.

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.

This chapter provides a detailed explanation of Tarski's definition of truth for formalized languages. It begins by indicating how he conceived the problem, how his criterion of adequacy guarantees that any definition satisfying it introduces a predicate that applies to all and only object‐language truths, and how he approached the technical problem of formulating a definition that would allow him to derive what he regarded as a “partial definition” of truth for each sentence of the object language. Next, (...) the formal techniques employed in his inductive definitions are explained, along with the method of turning those definitions into explicit definitions, and the way in which his definitions can be shown to be materially adequate. The explication concludes with a discussion of the relationship between truth and proof in the language of arithmetic, and the outlines of Tarski's theorem of the arithmetic indefinability of arithmetical truth. (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)