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)

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)

In this investigation we explore a general strategy for constructing modal theories where the modal notion is conceived as a predicate. The idea of this strategy is to develop modal theories over axiomatic theories of truth. In this first paper of our two part investigation we develop the general strategy and then apply it to the axiomatic theory of truth Friedman-Sheard. We thereby obtain the theory Modal Friedman-Sheard. The theory Modal Friedman-Sheard is then discussed from three different perspectives. First, we (...) show that Modal Friedman-Sheard preserves theoremhood modulo translation with respect to modal operator logic. Second, we turn to semantic aspects and develop a modal semantics for the newly developed theory. Third, we investigate whether the modal predicate of Modal Friedman-Sheard can be understood along the lines of a proposal of Kripke, namely as a truth predicate modified by a modal operator. (shrink)

This paper outlines an account of numbers based on the numerical equivalence schema, which consists of all sentences of the form ‘#x.Fx=n if and only if ∃nx Fx’, where # is the number-of operator and ∃n is defined in standard Russellian fashion. In the first part of the paper, I point out some analogies between the NES and the T-schema for truth. In light of these analogies, I formulate a minimalist account of numbers, based on the NES, which strongly parallels (...) the minimalist account of truth. One may be tempted to develop the minimalist account in a fictionalist direction, according to which arithmetic is useful but untrue, if taken at face value. In the second part, I argue that this suggestion is not as attractive as it may first appear. The NES suffers from a similar problem to the T-schema: it is deductively weak and does not enable the derivation of any non-trivial generalizations. In the third part of the paper, I explore some strategies to deal with the generalization problem, again drawing inspiration from the literature on truth. In closing this paper, I briefly compare the minimalist to some other accounts of numbers. (shrink)

Horwich proposes a solution to the liar paradox that relies on a particular notion of grounding—one that, unlike Kripke’s notion of grounding, does not invoke any “Tarski-style compositional principles”. In this short note, we will formalize Horwich’s construction and argue that his solution to the liar paradox does not justify certain generalizations about truth that he endorses. We argue that this situation is not resolved even if one appeals to the \-rule. In the final section, we briefly discuss how Horwich (...) might respond to the situation. (shrink)

Truth-theoretic deflationism holds that truth is simple, and yet that it can fulfil many useful logico-linguistic roles. Deflationism focuses on axioms for truth: there is no reduction of the notion of truth to more fundamental ones such as sets or higher-order quantifiers. In this paper I argue that the fundamental properties of reasonable, primitive truth predicates are at odds with the core tenets of classical truth-theoretic deflationism that I call fix, express, and quantify. Truth may be regarded as a broadly (...) logical, sui generis notion. However, this has little to do with the original aims of the deflationary theory of truth. (shrink)

This paper investigates a set of issues connected with the so-called conservativeness argument against deflationism. Although I do not defend that argument, I think the discussion of it has raised some interesting questions about whether what I call “compositional principles,” such as “a conjunction is true iff its conjuncts are true,” have substantial content or are in some sense logically trivial. The paper presents a series of results that purport to show that the compositional principles for a first-order language, taken (...) together, have substantial logical strength, amounting to a kind of abstract consistency statement. (shrink)

A common objection to hierarchical approaches to truth is that they fragment the concept of truth. This paper defends hierarchical approaches in general against the objection of fragmentation. It argues that the fragmentation required is familiar and unprob-lematic, via a comparison with mathematical proof. Furthermore, it offers an explanation of the source and nature of the fragmentation of truth. Fragmentation arises because the concept exhibits a kind of failure of closure under reflection. This paper offers a more precise characterization of (...) the reflection involved, first in the setting of formal theories of truth, and then in a more general setting. (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)

The notion of implicit commitment has played a prominent role in recent works in logic and philosophy of mathematics. Although implicit commitment is often associated with highly technical studies, it remains an elusive notion. In particular, it is often claimed that the acceptance of a mathematical theory implicitly commits one to the acceptance of a Uniform Reflection Principle for it. However, philosophers agree that a satisfactory analysis of the transition from a theory to its reflection principle is still lacking. We (...) provide an axiomatization of the minimal commitments implicit in the acceptance of a mathematical theory. The theory entails that the Uniform Reflection Principle is part of one’s implicit commitments, and sheds light on why this is so. We argue that the theory has significant epistemological consequences in that it explains how justified belief in the axioms of a theory can be preserved to the corresponding reflection principle. The theory also improves on recent analyses of implicit commitment based on truth or epistemic notions. (shrink)

Disquotational theories of truth are often criticised for being too weak to prove interesting generalisations about truth. In this paper we will propose a certain formal theory to serve as a framework for a solution of the generalisation problem. In contrast with Horwich’s original proposal, our framework will eschew psychological notions altogether, replacing them with the epistemic notion of believability. The aim will be to explain why someone who accepts a given disquotational truth theory Th, should also accept various generalisations (...) not provable in Th. The strategy will consist of the development of an axiomatic theory of believability, one permitting us to show how to derive the believability of generalisations from basic axioms that characterise the believability predicate, together with the information that Th is a theory of truth that we accept. (shrink)

Answering a question formulated by Halbach (2009), I show that a disquotational truth theory, which takes as axioms all positive substitutions of the sentential T-schema, together with all instances of induction in the language with the truth predicate, is conservative over its syntactical base.

Conservativeness has been proposed as an important requirement for deflationary truth theories. This in turn gave rise to the so-called ‘conservativeness argument’ against deflationism: a theory of truth which is conservative over its base theory S cannot be adequate, because it cannot prove that all theorems of S are true. In this paper we show that the problems confronting the deflationist are in fact more basic: even the observation that logic is true is beyond his reach. This seems to conflict (...) with the deflationary characterization of the role of the truth predicate in proving generalizations. However, in the final section we propose a way out for the deflationist — a solution that permits him to accept a strong theory, having important truth-theoretical generalizations as its theorems. (shrink)

Definitional and axiomatic theories of truth -- Objects of truth -- Tarski -- Truth and set theory -- Technical preliminaries -- Comparing axiomatic theories of truth -- Disquotation -- Classical compositional truth -- Hierarchies -- Typed and type-free theories of truth -- Reasons against typing -- Axioms and rules -- Axioms for type-free truth -- Classical symmetric truth -- Kripke-Feferman -- Axiomatizing Kripke's theory in partial logic -- Grounded truth -- Alternative evaluation schemata -- Disquotation -- Classical logic -- Deflationism (...) -- Reflection -- Ontological reduction -- Applying theories of truth. (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 notion of implicit commitment has played a prominent role in recent works in logic and philosophy of mathematics. Although implicit commitment is often associated with highly technical studies, it remains so far an elusive notion. In particular, it is often claimed that the acceptance of a mathematical theory implicitly commits one to the acceptance of a Uniform Reflection Principle for it. However, philosophers agree that a satisfactory analysis of the transition from a theory to its reflection principle is still (...) lacking. We provide an axiomatization of the minimal commitments implicit in the acceptance of a mathematical theory. The theory entails that the Uniform Reflection Principle is part of one's implicit commitments, and sheds light on the reason why this is so. We argue that the theory has interesting epistemological consequences in that it explains how justified belief in the axioms of a theory can be preserved to the corresponding reflection principle. The theory also improves on recent proposals for the analysis of implicit commitment based on truth or epistemic notions. (shrink)