Substantivism is a general philosophical methodology advocating a substantive approach to philosophical theorizing. In this article, I present an overview of this methodology with a special emphasis on the field of truth. I begin with a framework for understanding what is at stake in the substantivist–deflationist debate and describe the substantivist critique of deflationism. I then proceed to discuss contemporary substantivism as a positive methodology, present examples of recent substantivist theories of truth, delineate several principles of philosophical substantivism, and connect (...) it to contemporary thought about the nature and methods of philosophy. Due to limitations of space, I am unable to discuss all the forms contemporary substantivism has taken. But I try to give a clear sense of the central principles, challenges, and promise of this methodology. (shrink)

The aim of this paper is to show that it’s not a good idea to have a theory of truth that is consistent but ω-inconsistent. In order to bring out this point, it is useful to consider a particular case: Yablo’s Paradox. In theories of truth without standard models, the introduction of the truth-predicate to a first order theory does not maintain the standard ontology. Firstly, I exhibit some conceptual problems that follow from so introducing it. Secondly, I show that (...) in second order theories with standard semantics the same procedure yields a theory that doesn’t have models. So, while having an ω- inconsistent theory is a bad thing, having an unsatisfiable theory of truth is actually worse. This casts doubts on whether the predicate in question is, after all, a truthpredicate for that language. Finally, I present some alternatives to prove an inconsistency adding plausible principles to certain theories of truth. (shrink)

Proof-theoretic reflection principles are schemas which attempt to express the soundness of arithmetical theories within their own language, e.g., ${\mathtt{{Prov}_{\mathsf {PA}} \rightarrow \varphi }}$ can be understood to assert that any statement provable in Peano arithmetic is true. It has been repeatedly suggested that justification for such principles follows directly from acceptance of an arithmetical theory $\mathsf {T}$ or indirectly in virtue of their derivability in certain truth-theoretic extensions thereof. This paper challenges this consensus by exploring relationships between reflection principles (...) and principles of mathematical and transfinite induction as well as the status of the latter with respect to various foundational characterizations of number theory. (shrink)

In this dissertation, I argue that we should be pluralists about truth and in turn, eliminativists about the property Truth. Traditional deflationists were right to suspect that there is no such property as Truth. Yet there is a plurality of pluralities of properties which enjoy defining features that Truth would have, were it to exist. So although, in this sense, truth is plural, Truth is non-existent. The resulting account of truth is indebted to deflationism as the provenance of the suspicion (...) that Truth doesn't exist. But it would be hasty to simply classify the account as deflationary. Each of the 'truth-like' properties that it recognizes is highly substantive--that is, complex and explanatorily potent. So we should deflate Truth by recognizing that it doesn't exist, but we should also recognize that one of the most vital tasks in truth theory is to discover the essences of the many truth-like properties. My aim here is to do precisely this. (shrink)

Che cos'è la verità? A questa domanda le teorie deflazioniste rispondono in modo sorprendente: niente, o quasi. Secondo il deflazionismo la verità, come proprietà, semplicemente non esiste o è priva di qualsiasi sostanza. In questo contributo presenterò tale posizione offrendo un breve resoconto critico dell'evoluzione della proposta e una disamina delle sue tesi centrali.

The paper draws attention to an important, but apparently neglected distinction relating to axiomatic theories of truth, viz. the distinction between weakly and strongly truth-compositional theories of truth. The paper argues that the distinction might be helpful in classifying weak axiomatic theories of truth and examines some of them with respect to it.

Is the restricted, consistent, version of the T-scheme sufficient for an ‘implicit definition’ of truth? In a sense, the answer is yes (Haack 1978 , Quine 1953 ). Section 4 of Ketland 1999 mentions this but gives a result saying that the T-scheme does not implicitly define truth in the stronger sense relevant for Beth’s Definability Theorem. This insinuates that the T-scheme fares worse than the compositional truth theory as an implicit definition. However, the insinuation is mistaken. For, as Bays (...) rightly points out, the result given extends to the compositional truth theory also. So, as regards implicit definability, both kinds of truth theory are equivalent. Some further discussion of this topic is mentioned (Gupta 2008 , Ketland 2003 , McGee 1991 ), all in agreement with Bays’s analysis. (shrink)

The proof-theoretic results on axiomatic theories oftruth obtained by different authors in recent years are surveyed.In particular, the theories of truth are related to subsystems ofsecond-order analysis. On the basis of these results, thesuitability of axiomatic theories of truth for ontologicalreduction is evaluated.

A new approach to truth is offered which dispenses with the truth predicate, and replaces it with a special kind of quantifier which simultaneously binds variables in sentential and nominal positions. The resulting theory of truth for a (first-order) language is shown to be able to handle blind truth ascriptions, and is shown to be compatible with a characterization of the semantic and syntactic principles governing that language. Comparisons with other approaches to truth are drawn. An axiomatization of AU-quantifiers and (...) a model theory for them is given, and an appendix contains a completeness proof. (shrink)

What requirements must deflationary formal theories of truth satisfy? This chapter argues against the widely accepted view that compositional and Tarskian theories of truth are substantial or otherwise unacceptable to deflationists. First, two purposes that a formal truth theory can serve are distinguished: one descriptive, the other logical (i.e., to characterise the correctness of inferences involving ‘true’). The chapter argues that the most compelling arguments for the incompatibility of compositional and Tarskian theories concern descriptive theories only. -/- Second, two requirements (...) that any deflationist truth theory intended to serve a logical purpose must satisfy are put forward. These requirements, it is argued, suggest that (i) many well-known compositional and Tarskian theories are acceptable from a deflationist standpoint (including CT); (ii) certain other popular theories of truth (including KF and FS) are not similarly acceptable; (iii) there are no conclusive reasons to impose a conservativeness requirement on deflationary theories of truth. (shrink)

Many philosophers believe that a deflationist theory of truth must conservatively extend any base theory to which it is added. But when applied to arithmetic, it's argued, the imposition of a conservativeness requirement leads to a serious objection to deflationism: for the Gödel sentence for Peano Arithmetic is not a theorem of PA, but becomes one when PA is extended by adding plausible principles governing truth. This paper argues that no such objection succeeds. The issue turns on how we understand (...) the notion of logical consequence implicit in any conservativeness requirement, and whether we possess a categorical conception of the natural numbers. I offer a disjunctive response: if we possess a categorical conception of arithmetic, then deflationists have principled reason to accept a rich notion of logical consequence according to which the Gödel sentence follows from PA. But if we do not, then the reasons for requiring the derivation of the Gödel sentence lapse, and deflationists are free to accept a conservativeness requirement stated proof-theoretically. Either way, deflationism is in the clear. (shrink)

Tarski’s Convention T is often taken to claim that it is both sufficient and necessary for adequacy in a definition of truth that it imply instances of the T-schema where the embedded sentence translates the mentioned sentence. However, arguments against the necessity claim have recently appeared, and, furthermore, the necessity claim is actually not required for the indefinability results for which Tarski is justly famous; indeed, Tarski’s own presentation of the results in the later Undecidable Theories makes no mention of (...) an assumption to the effect that the definition of truth implies the biconditionals. This raises a question: was Tarski in fact committed to the necessity claim in the important papers of the 1930s and 40s? I argue that he was not. The discussion of this apparently esoteric interpretive issue in fact gets to the heart of many important questions about truth, and in the final sections of the paper I discuss the importance of the T-biconditionals in the theory of meaning and the relation of deflationary and inflationary theories of truth to the semantic paradoxes. (shrink)

In this paper, we investigate the phenomenon ofspeed-upin the context of theories of truth. We focus on axiomatic theories of truth extending Peano arithmetic. We are particularly interested on whether conservative extensions of PA have speed-up and on how this relates to a deflationist account. We show that disquotational theories have no significant speed-up, in contrast to some compositional theories, and we briefly assess the philosophical implications of these results.

In recent decades deflationary theories of truth have been challenged with a technical argument based on the notion of conservativeness. In this paper, I shall stress that conservative extensions of theories and expandability of their models are not equivalent notions. Then, I shall argue that the deflationary thesis of the unsubstantiality of truth is better understood as leveraging on the stronger notion of expandability of models. Once expandability is involved in the argument, some notable consequences follow: the strategy proposed by (...) Hartry Field in response to the conservativeness objection is strongly undermined and even simpler proposals (such as theories based on standard t-sentences) are shown to be quite problematical. (shrink)

I clarify how the requirement of conservative extension features in the thinking of various deflationists, and how this relates to another litmus claim, that the truth-predicate stands for a real, substantial property. I discuss how the deflationist can accommodate the result, to which Cieslinski draws attention, that non-conservativeness attends even the generalization that all logical theorems in the language of arithmetic are true. Finally I provide a four-fold categorization of various forms of deflationism, by reference to the two claims of (...) conservativeness and substantiality. This helps to clarify the various possible positions in the deflationism debate. (shrink)

In this article, the prospects of deflationism about the concept of truth are investigated. A new version of deflationism, called inferential deflationism, is articulated and defended. It is argued that it avoids the pitfalls of earlier deflationist views such as Horwich’s minimalist theory of truth and Field’s version of deflationism.

In 1999, Jeffrey Ketland published a paper which posed a series of technical problems for deflationary theories of truth. Ketland argued that deflationism is incompatible with standard mathematical formalizations of truth, and he claimed that alternate deflationary formalizations are unable to explain some central uses of the truth predicate in mathematics. He also used Beth’s definability theorem to argue that, contrary to deflationists’ claims, the T-schema cannot provide an ‘implicit definition’ of truth. In this article, I want to challenge this (...) final argument. Whatever other faults deflationism may have, the T-schema does provide an implicit definition of the truth predicate. Or so, at any rate, I shall argue. (shrink)

Truth predicates are widely believed to be capable of serving a certain logical or quasi-logical function. There is little consensus, however, on the exact nature of this function. We offer a series of formal results in support of the thesis that disquotational truth is a device to simulate higher-order resources in a first-order setting. More specifically, we show that any theory formulated in a higher-order language can be naturally and conservatively interpreted in a first-order theory with a disquotational truth or (...) truth-of predicate. In the first part of the paper we focus on the relation between truth and full impredicative sentential quantification. The second part is devoted to the relation between truth-of and full impredicative predicate quantification. (shrink)

Reflection principles are of central interest in the development of axiomatic theories. Whereas they are independent statements they appear to have a specific epistemological status. Our trust in those principles is as warranted as our trust in the axioms of the system itself. This paper is an attempt in clarifying this special epistemic status. We provide a motivation for the adoption of uniform reflection principles by their analogy to a form of the constructive \(\omega \) -rule. Additionally, we analyse the (...) role of informal arithmetic and the conception of natural numbers as an inductive structure, also with regard to extra conceptual resources such as a primitive truth predicate. (shrink)

This paper is an exposition of some recent results concerning various notions of strength and weakness of the concept of truth, both published or not. We try to systematically present these notions and their relationship to the current research on truth. We discuss the concept of the Tarski boundary between weak and strong theories of truth and we give an overview of non-conservativity results for the extensions of the basic compositional truth theory. Additionally, we present a natural strong theory of (...) truth which admits a number of apparently unrelated axiomatisations. Finally, we discuss other possible explications of the notion of ‘strength’ of axiomatic theories of truth. (shrink)

We prove that the theory of the extensional compositional truth predicate for the language of arithmetic with \Delta 0 -induction scheme for the truth predicate and the full arithmetical induction scheme is not conservative over Peano Arithmetic. In addition, we show that a slightly modified theory of truth actually proves the global reflection principle over the base theory.

There are two perspectives from which formal theories can be viewed. On the one hand, one can take a theory to be about some privileged models. On the other hand, one can take all models of a theory to be on a par. In contrast with what is usually done in philosophical debates, we adopt the latter viewpoint. Suppose that from this perspective we want to add an adequate truth predicate to a background theory. Then on the one hand the (...) truth theory ought to be semantically conservative over the background theory. At the same time, it is generally recognised that the central function of a truth predicate is an expressive one. A truth predicate ought to allow us to express propositions that we could not express before. In this article we argue that there are indeed natural truth theories which satisfy both the demand of semantical conservativeness and the demand of adequately extending the expressive power of our language. (shrink)

A common objection against deflationist theories of truth is that they cannot do justice to the correspondence intuition, i.e. the intuition that there is an explanatory relationship between, for instance, the truth of ‘Snow is white’ and snow's being white. We scrutinize two attempts to meet this objection and argue that both fail. We then propose a new response to the objection which, first, sheds doubt on the correctness of the correspondence intuition and, second, seeks to explain how we may (...) nonetheless have come to have that intuition. (shrink)

According to the implicit commitment thesis, once accepting a mathematical formal system S, one is implicitly committed to additional resources not immediately available in S. Traditionally, this thesis has been understood as entailing that, in accepting S, we are bound to accept reflection principles for S and therefore claims in the language of S that are not derivable in S itself. It has recently become clear, however, that such reading of the implicit commitment thesis cannot be compatible with well-established positions (...) in the foundations of mathematics which consider a specific theory S as self-justifying and doubt the legitimacy of any principle that is not derivable in S: examples are Tait’s finitism and the role played in it by Primitive Recursive Arithmetic, Isaacson’s thesis and Peano Arithmetic, Nelson’s ultrafinitism and sub-exponential arithmetical systems. This casts doubts on the very adequacy of the implicit commitment thesis for arithmetical theories. In the paper we show that such foundational standpoints are nonetheless compatible with the implicit commitment thesis. We also show that they can even be compatible with genuine soundness extensions of S with suitable form of reflection. The analysis we propose is as follows: when accepting a system S, we are bound to accept a fixed set of principles extending S and expressing minimal soundness requirements for S, such as the fact that the non-logical axioms of S are true. We call this invariant component the semantic core of implicit commitment. But there is also a variable component of implicit commitment that crucially depends on the justification given for our acceptance of S in which, for instance, may or may not appear reflection principles for S. We claim that the proposed framework regulates in a natural and uniform way our acceptance of different arithmetical theories. (shrink)

Recent work on formal theories of truth has revived an approach, due originally to Tarski, on which syntax and truth theories are sharply distinguished—‘disentangled’—from mathematical base theories. In this paper, we defend a novel philosophical constraint on disentangled theories. We argue that these theories must be epistemically stable: they must possess an intrinsic motivation justifying no strictly stronger theory. In a disentangled setting, even if the base and the syntax theory are individually stable, they may be jointly unstable. We contend (...) that this flaw afflicts many proposals discussed in the literature; we defend a new, stable disentangled theory, double second-order arithmetic. (shrink)

It has been argued in the literature that the deflationists’ thesis about the dispensability of truth as an explanatory notion forces them to adopt a conservative theory of truth. I suggest that the deflationists’ claim that the notion of truth is akin to a logical notion should be taken more seriously. This claim casts some doubts on the adequacy of the conservativity requirement, while it also calls for further investigation to assess its philosophical plausibility.

We show that a typed compositional theory of positive truth with internal induction for total formulae (denoted by PT tot ) is not semantically conservative over Peano arithmetic. In addition, we observe that the class of models of PA expandable to models of PT tot contains every recursively saturated model of arithmetic. Our results point to a gap in the philosophical project of describing the use of the truth predicate in model-theoretic contexts.

Deflationism about truth describes truth as a logical notion. In the present paper, I explore the implication of the alleged logicality of truth from the perspective of axiomatic theories of truth, and argue that the deflationist doctrine of the logicality of truth gives rise to two types of self-undermining arguments against deflationism, which I call the conservativeness argument from logicality and the topic-neutrality argument.

Certain translational T-schemes of the form True(“f”) « f(f), where f(f) can be almost any translation you like of f, will be a conservative extension of Peano arithmetic. I have an inkling that this means something philosophically, but I don’t understand my own inkling.

Certain translational T-schemes of the form True « f, where f can be almost any translation you like of f, will be a conservative extension of Peano arithmetic. I have an inkling that this means something philosophically, but I don’t understand my own inkling.

According to the received view, formalism – interpreted as the thesis that mathematical truth does not outrun the consequences of our maximal mathematical theory – has been refuted by Goedel's theorem. In support of this claim, proponents of the received view usually invoke an informal argument for the truth of the Goedel sentence, an argument which is supposed to reconstruct our reasoning in seeing its truth. Against this, Field has argued in a series of papers that the principles involved in (...) this argument – when applied to our maximal mathematical theory – are unsound. This paper defends the received view by showing that there is a way of seeing the truth of the Goedel sentence which is immune to Field's strategy. (shrink)

A common objection against deflationist theories of truth is that they cannot do justice to the correspondence intuition, i.e. the intuition that there is an explanatory relationship between, for instance, the truth of ‘Snow is white’ and snow's being white. We scrutinize two attempts to meet this objection and argue that both fail. We then propose a new response to the objection which, first, sheds doubt on the correctness of the correspondence intuition and, second, seeks to explain how we may (...) nonetheless have come to have that intuition. (shrink)

Even though disquotationalism is not correct as it is usually formulated, a deep insight lies behind it. Specifically, it can be argued that, modulo implicit commitment to reflection principles, all there is to the notion of truth is given by a simple, natural collection of truth-biconditionals.

Opposing Quine, I defend the view that some of the statements we accept are immune to empirical revision. My examples include instances of Schema and abbreviative definitions. I argue that it serves important cognitive purposes to hold statements of these kinds immune to revision, and that it is epistemically permissible for us to do so. At the end, I briefly consider the question of whether the rationale for these claims might be extended to show that additional statements are unrevisable.

Nous clarifions et critiquons un argument influent opposé au déflationnisme par [Shapiro 1998b] et [Ketland 1999], fondé sur la non-conservativité des extensions des théories formalisées par des principes aléthiques universellement admis. À cette interprétation anti-déflationniste des phénomènes de non-conservativité, nous en opposons une autre, compatible à la fois avec les faits logiques et les thèses déflationnistes.

Nous clarifions et critiquons un argument influent opposé au déflationnisme par [Shapiro 1998b] et [Ketland 1999], fondé sur la non-conservativité des extensions des théories formalisées par des principes aléthiques universellement admis. À cette interprétation anti-déflationniste des phénomènes de non-conservativité, nous en opposons une autre, compatible à la fois avec les faits logiques et les thèses déflationnistes.

Nous clarifions et critiquons un argument influent opposé au déflationnisme par [Shapiro 1998b] et [Ketland 1999], fondé sur la non-conservativité des extensions des théories formalisées par des principes aléthiques universellement admis. À cette interprétation anti-déflationniste des phénomènes de non-conservativité, nous en opposons une autre, compatible à la fois avec les faits logiques et les thèses déflationnistes.

This paper is a follow-up to [4], in which a mistake in [6] was corrected. We give a strenghtening of the main result on the semantical nonconservativity of the theory of PT−with internal induction for total formulae${$, denoted by PT−in [9]). We show that if to PT−the axiom of internal induction forallarithmetical formulae is added, then this theory is semantically stronger than${\rm{P}}{{\rm{T}}^ - } + {\rm{INT}}\left$. In particular the latter is not relatively truth definable in the former. Last but not (...) least, we provide an axiomatic theory of truth which meets the requirements put forward by Fischer and Horsten in [9]. The truth theory we define is based on Weak Kleene Logic instead of the Strong one. (shrink)

It is argued that the blind ascriptive role for the word true, its use, that is, in conjunction with descriptions of classes of sentences or with proper names of sentences (but not quote-names), is one which applies indiscriminately to sentences regardless of whether these are in languages we speak, can understand, or can translate into sentences that we do speak (and understand). Formal analogues of the ordinary word true as they arise in Tarskis seminal work, and in others, cannot replicate (...) this essential role of the ordinary word true. (shrink)