Bruno Whittle (2019) has recently extended Kripke’s semantical theory of truth to languages containing generalized quantifiers. There are reasons for axiomatizing semantical theories, and for regarding Halbach and Horsten’s PKF as a good axiomatization of Kripke’s. PKF is a theory in Partial Logic. The present paper complements Whittle’s by showing how Partial Logic, and then PKF, may be extended to cover binary quantifiers meaning ‘every’, ‘some’, and ‘most’.

The present paper suggests relative truth definability as a tool for comparing conceptual aspects of axiomatic theories of truth and gives an overview of recent developments of axiomatic theories of truth in the light of it. We also show several new proof-theoretic results via relative truth definability including a complete answer to the conjecture raised by Feferman in [13].

We discuss the interplay between the axiomatic and the semantic approach to truth. Often, semantic constructions have guided the development of axiomatic theories and certain axiomatic theories have been claimed to capture a semantic construction. We ask under which conditions an axiomatic theory captures a semantic construction. After discussing some potential criteria, we focus on the criterion of ℕ-categoricity and discuss its usefulness and limits.

Following the finitist’s rejection of the complete totality of the natural numbers, a finitist language allows only propositional connectives and bounded quantifiers in the formula-construction but not unbounded quantifiers. This is opposed to the currently standard framework, a first-order language. We conduct axiomatic studies on the notion of truth in the framework of finitist arithmetic in which at least smash function $\#$ is available. We propose finitist variants of Tarski ramified truth theories up to rank $\omega (...) $, of Kripke–Feferman truth theory and of Friedman–Sheard truth theory, and show that all of these have the same strength as the finitist arithmetic of one higher level along Grzegorczyk hierarchy. On the other hand, we also show that adding Burgess-style groundedness schema, adjusted to the finitist setting, makes Kripke–Feferman truth theory as strong as primitive recursive arithmetic. Meanwhile, we obtain some basic results on finitist theories of (full and hat) inductive definitions and on the second order axiom of hat inductive definitions for positive operators. (shrink)

Following recent developments in the literature on axiomatic theories of truth, we investigate an alternative to the widespread habit of formalizing the syntax of the object-language into the object-language itself. We first argue for the proposed revision, elaborating philosophical evidences in favor of it. Secondly, we present a general framework for axiomatic theories of truth with theories of syntax. Different choices of the object theory O will be considered. Moreover, some strengthenings of these (...) theories will be introduced: we will consider extending the theories by the addition of coding axioms or by extending the schemas of O, if present, to the entire vocabulary of our theory of truth. Finally, we touch on the philosophical consequences that the theories described can have on the debate about the metaphysical status of the truth predicate and on the formalization of our informal metatheoretic reasoning. (shrink)

2. Various philosophical and semantical theories are candidates for axiomatization (but not all, e.g. coherence, pragmatic, fuzzy theories). NB: axiomatizations are not uniquely determined.

§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)

At the centre of the traditional discussion of truth is the question of how truth is defined. Recent research, especially with the development of deflationist accounts of truth, has tended to take truth as an undefined primitive notion governed by axioms, while the liar paradox and cognate paradoxes pose problems for certain seemingly natural axioms for truth. In this book, Volker Halbach examines the most important axiomatizations of truth, explores their properties and shows how (...) the logical results impinge on the philosophical topics related to truth. In particular, he shows that the discussion on topics such as deflationism about truth depends on the solution of the paradoxes. His book is an invaluable survey of the logical background to the philosophical discussion of truth, and will be indispensable reading for any graduate or professional philosopher in theories of truth. (shrink)

We investigate axiomatizations of Kripke's theory of truth based on the Strong Kleene evaluation scheme for treating sentences lacking a truth value. Feferman's axiomatization KF formulated in classical logic is an indirect approach, because it is not sound with respect to Kripke's semantics in the straightforward sense: only the sentences that can be proved to be true in KF are valid in Kripke's partial models. Reinhardt proposed to focus just on the sentences that can be proved to be (...) true in KF and conjectured that the detour through classical logic in KF is dispensable. We refute Reinhardt's Conjecture, and provide a direct axiomatization PKF of Kripke's theory in partial logic. We argue that any natural axiomatization of Kripke's theory in Strong Kleene logic has the same proof-theoretic strength as PKF, namely the strength of the system RA< ωω ramified analysis or a system of Tarskian ramified truth up to ωω. Thus any such axiomatization is much weaker than Feferman's axiomatization KF in classical logic, which is equivalent to the system RA<ε₀ of ramified analysis up to ε₀. (shrink)

The main aim of our paper was to present three formal tools for comparing various axiomatic theories of truth. In Section 2 we aimed at showing that there are indeed many different approaches to defining a set of axioms for the notion of truth. In Section 3 we introduced three different \measures of strength" of axiomatic theories of truth, i.e. three reflexive and transitive relations on the set of axiomatic theories of (...) truth. We have explained the intuition behind each of them. The three relations were called : Fujimoto definability, model-theoretical strength, proof-theoretical strength. Then in the last section we described how they order the truth theories introduced in Section 2. We observed that theories made equivalent by the coarser relation can be strictly ordered by the next one. (shrink)

The work of mathematician and logician Alfred Tarski (1901--1983) marks the transition from substantial to deflationary views about truth.

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)

This paper attempts to address the question what logical strength theories of truth have by considering such questions as: If you take a theory T and add a theory of truth to it, how strong is the resulting theory, as compared to T? It turns out that, in a wide range of cases, we can get some nice answers to this question, but only if we work in a framework that is somewhat different from those usually employed (...) in discussions of axiomatic theories of truth. These results are then used to address a range of philosophical questions connected with truth, such as what Tarski meant by "essential richness" and the so-called conservativeness argument against deflationism. -/- This draft dates from about 2009, with some significant updates having been made around 2011. Around then, however, I decided that the paper was becoming unmanageable and that I was trying to do too many things in it. I have therefore exploded the paper into several pieces, which will be published separately. These include "Disquotationalism and the Compositional Principles", "The Logical Strength of Compositional Principles", "Consistency and the Theory of Truth", and "What Is Essential Richness?" You should probably read those instead, since this draft remains a bit of a mess. Terminology and notation are inconsistent, and some of the proofs aren't quite right. So, caveat lector. I make it public only because it has been cited in a few places now. (shrink)

This is part two of a two-part paper in which we develop an axiomatic theory of the relation of partial ground. The main novelty of the paper is the of use of a binary ground predicate rather than an operator to formalize ground. In this part of the paper, we extend the base theory of the first part of the paper with hierarchically typed truth-predicates and principles about the interaction of partial ground and truth. We show that (...) our theory is a proof-theoretically conservative extension of the ramified theory of positive truth up to. (shrink)

We employ the lens provided by formal truth theory to study axiomatizations of Peano Arithmetic ${\textsf {(PA)}}$. More specifically, let Elementary Arithmetic ${\textsf {(EA)}}$ be the fragment $\mathsf {I}\Delta _0 + \mathsf {Exp}$ of ${\textsf {PA}}$, and let ${\textsf {CT}}^-[{\textsf {EA}}]$ be the extension of ${\textsf {EA}}$ by the commonly studied axioms of compositional truth ${\textsf {CT}}^-$. We investigate both local and global properties of the family of first order theories of the form ${\textsf {CT}}^-[{\textsf {EA}}] +\alpha (...) $, where $\alpha $ is a particular way of expressing “ ${\textsf {PA}}$ is true” (using the truth predicate). Our focus is dominantly on two types of axiomatizations, namely: (1) schematic axiomatizations that are deductively equivalent to ${\textsf {PA}}$ and (2) axiomatizations that are proof-theoretically equivalent to the canonical axiomatization of ${\textsf {PA}}$. (shrink)

In order to be able to express all possible patterns of dependence and independence between variables, we have to replace the traditional first-order logic by independence-friendly (IF) logic. Our natural concept of truth for a quantificational sentence S says that all the Skolem functions for S exist. This conception of truth for a sufficiently rich IF first-order language can be expressed in the same language. In a first-order axiomatic set theory, one can apparently express this same concept (...) in set-theoretical terms, since the existence of functions can be expressed there. Because of Tarski's theorem, this is impossible. Hence there must exist set-theoretical statements, even provable ones, which are said to be true in first-order models of axiomatic set theory but whose Skolem functions do not all exist. Hence there are provable sentences in axiomatic set theory that are false in accordance with our ordinary conceptions of set-theoretical truth. Such counter-intuitive propositions have been known to exist, but they have been blamed on the peculiarities of very large sets. It is argued here that this explanation is not correct and that there are intuitively false theorems not involving very large sets. Hence the provability or unprovability of a set-theoretical statement, e.g. of the continuum hypothesis (CH) in axiomatic set theory is not necessarily relevant to the truth of CH. (shrink)

According to Tarski, a theory of truth for a language L is a theory which logically implies for each sentence S of L a sentence of the form:S is true-in-L if and only if p,where rS1 is replaced by a canonical description of a sentence of L and rp1 is replaced by that sentence if L is contained in the metalanguage or by a translation of S if it is not so contained. Tarski constructed consistent and finitely axiomatized (...) class='Hi'>theories of truth for various formal languages and showed how to explicitly define ‘is true in L’ within these theories. We all agree that Tarski's theories of truth have enormous philosophical significance, but there is much less agreement on precisely what that significance consists in. (shrink)

In this second and last paper of the two part investigation on "Modality and Axiomatic Theories of Truth" we apply a general strategy for constructing modal theories over axiomatic theories of truth to the theory Kripke-Feferman. This general strategy was developed in the first part of our investigation. Applying the strategy to Kripke-Feferman leads to the theory Modal Kripke-Feferman which we discuss from the three perspectives that we had already considered in the first (...) paper, where we discussed the theory Modal Friedman-Sheard. That is, we first show that Modal Kripke-Feferman preserves theoremhood modulo translation with respect to modal operator logic. Second, we develop a modal semantics fitting the newly developed theory. Third, we investigate whether the modal predicate of Modal Kripke-Feferman can be understood along the lines of a proposal of Kripke, namely as a truth predicate modified by a modal operator. (shrink)

We study the relationships between two clusters of axiomatizations of Kripke’s fixed-point models for languages containing a self-applicable truth predicate. The first cluster is represented by what we will call ‘\-like’ theories, originating in recent work by Halbach and Horsten, whose axioms and rules are all valid in fixed-point models; the second by ‘\-like’ theories first introduced by Solomon Feferman, that lose this property but reflect the classicality of the metatheory in which Kripke’s construction is carried out. (...) We show that to any natural system in one cluster—corresponding to natural variations on induction schemata—there is a corresponding system in the other proving the same sentences true, addressing a problem left open by Halbach and Horsten and accomplishing a suitably modified version of the project sketched by Reinhardt aiming at an instrumental reading of classical theories of self-applicable truth. (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)

Part 1 – Formal Development of Theory .................................................................................................4..

What is the simplest and most natural axiomatic replacement for the set-theoretic definition of the minimal fixed point on the Kleene scheme in Kripke’s theory of truth? What is the simplest and most natural set of axioms and rules for truth whose adoption by a subject who had never heard the word "true" before would give that subject an understanding of truth for which the minimal fixed point on the Kleene scheme would be a good model? (...) Several axiomatic systems, old and new, are examined and evaluated as candidate answers to these questions, with results of Harvey Friedman playing a significant role in the examination. (shrink)

History and Philosophy of Logic, Volume 33, Issue 1, Page 99-100, February 2012.

This is part one of a two-part paper, in which we develop an axiomatic theory of the relation of partial ground. The main novelty of the paper is the of use of a binary ground predicate rather than an operator to formalize ground. This allows us to connect theories of partial ground with axiomatic theories of truth. In this part of the paper, we develop an axiomatization of the relation of partial ground over the truths (...) of arithmetic and show that the theory is a proof-theoretically conservative extension of the theory PT of positive truth. We construct models for the theory and draw some conclusions for the semantics of conceptualist ground. (shrink)

This brief, largely expository book—hereafter TT—blends history and philosophy of logic with contemporary mathematical logic. Page 3 says it “is about the relation between formal theories of truth...

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.

Formal theories, as in logic and mathematics, are sets of sentences closed under logical consequence. Philosophical theories, like scientific theories, are often far less formal. There are many axiomatic theories of the truth predicate for certain formal languages; on analogy with these, some philosophers (most notably Paul Horwich) have proposed axiomatic theories of the property of truth. Though in many ways similar to logical theories, axiomatic theories of (...) class='Hi'>truth must be different in several nontrivial ways. I explore what an axiomatic theory of truth would look like. Because Horwich’s is the most prominent, I examine his theory and argue that it fails as a theory of truth. Such a theory is adequate if, given a suitable base theory, every fact about truth is a consequence of the axioms of the theory. I show, using an argument analogous to Gödel’s incompleteness proofs, that no axiomatic theory of truth could ever be adequate. I also argue that a certain class of generalizations cannot be consequences of the theory. (shrink)

Some axiomatic theories of truth and related subsystems of second-order arithmetic are surveyed and shown to be conservative over their respective base theory. In particular, it is shown by purely finitistically means that the theory PA ÷ "there is a satisfaction class" and the theory FS of [2] are conservative over PA.

This book analyses and defends the deflationist claim that there is nothing deep about our notion of truth. According to this view, truth is a 'light' and innocent concept, devoid of any essence which could be revealed by scientific inquiry. Cezary Cieśliński considers this claim in light of recent formal results on axiomatic truth theories, which are crucial for understanding and evaluating the philosophical thesis of the innocence of truth. Providing an up-to-date discussion and (...) original perspectives on this central and controversial issue, his book will be important for those with a background in logic who are interested in formal truth theories and in current philosophical debates about the deflationary conception of truth. (shrink)

This English translation of the author's original work has been thoroughly revised, expanded and updated. The book covers logical systems known as type-free or self-referential . These traditionally arise from any discussion on logical and semantical paradoxes. This particular volume, however, is not concerned with paradoxes but with the investigation of type-free sytems to show that: (i) there are rich theories of self-application, involving both operations and truth which can serve as foundations for property theory and formal semantics; (...) (ii) these theories provide a new outlook on classical topics, such as inductive definitions and predicative mathematics; (iii) they are particularly promising with regard to applications. Research arising from paradoxes has moved progressively closer to the mainstream of mathematical logic and has become much more prominent in the last twenty years. A number of significant developments, techniques and results have been discovered. Academics, students and researchers will find that the book contains a thorough overview of all relevant research in this field. (shrink)

Donald Davidson was one of the most influential philosophers of the last half of the 20th century, especially in the theory of meaning and in the philosophy of mind and action. In this paper, I concentrate on a field-shaping proposal of Davidson’s in the theory of meaning, arguably his most influential, namely, that insight into meaning may be best pursued by a bit of indirection, by showing how appropriate knowledge of a finitely axiomatized truth theory for a language can (...) put one in a position both to interpret the utterance of any sentence of the language and to see how its semantically primitive constituents together with their mode of combination determines its meaning (Davidson 1965, 1967, 1970, 1973a). This project has come to be known as truth-theoretic semantics. My aim in this paper is to render the best account I can of the goals and methods of truth-theoretic semantics, to defend it against some objections, and to identify its limitations. Although I believe that the project I describe conforms to the main idea that Davidson had, my aim is not primarily Davidson exegesis. I want to get on the table an approach to compositional semantics for natural languages, inspired by Davidson, but extended and developed, which I think does about as much along those lines as any theory could. I believe it is Davidson’s project, and I defend this in detail elsewhere (Ludwig 2015; Lepore and Ludwig 2005, 2007a, 2007b, 2011). But I want to develop and defend the project while also exploring its limitations, without getting entangled in exegetical questions. (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.

T-biconditionals have often been regarded as insufficient as axioms for truth. This verdict is based on Tarski’s observation that the typed T-sentences suffer from deductive weakness. As indicated by McGee, the situation might change radically if we consider type-free disquotational theories of truth. However, finding a well-motivated set of untyped T-biconditionals that is consistent and recursively enumerable has proven to be very difficult. Moreover, some authors ) have argued that any solution to the semantic paradoxes necessarily involves (...) ‘inflationary’ means, thus spelling doom to deflationist and minimalist truth theories in particular. The situation is indeed worrisome as formal theories of minimalist truth are missing so far. This makes it very hard to properly evaluate the tenets of minimalism. In the present article, we will show how to find safe instances of the T-schema just by relying on syntactic features of the sentences of our language—in particular, we will explore Quine’s idea of stratification. Based on that, we will introduce some disquotational truth theories that are deductively very strong. (shrink)

The existence of a close connection between results on axiomatic truth and the analysis of truth-theoretic deflationism is nowadays widely recognized. The first attempt to make such link precise can be traced back to the so-called conservativeness argument due to Leon Horsten, Stewart Shapiro and Jeffrey Ketland: by employing standard Gödelian phenomena, they concluded that deflationism is untenable as any adequate theory of truth leads to consequences that were not achievable by the base theory alone. In (...) the paper I highlight, as Shapiro and Ketland, the irreducible nature of truth axioms with respect to their base theories. But, I argue, this does not immediately delineate a notion of truth playing a substantial role in philosophical or scientific explanations. I first offer a refinement of Hartry Field’s reaction to the conservativeness argument by distinguishing between metatheoretic and object-theoretic consequences of the theory of truth and address some possible rejoinders. In the resulting picture, truth is an irreducible tool for metatheoretic ascent. How robust is this characterizaton? I test it by considering: a recent example, due to Leon Horsten, of the alleged explanatory role played by the truth predicate in the derivation of Fitch’s paradox; an essential weakening of theories of truth analyzed in the first part of the paper. (shrink)

As is mentioned in Leigh :845-865, 2015), it is an open problem whether for several axiomatic theories of truth, including Friedman–Sheard theory \ and Kripke–Feferman theory \ :690-716, 1976), there exist cut-elimination arguments that give the upper bounds of their proof-theoretic strengths. In this paper, we give complete cut-elimination results for several well-known axiomatic theories of truth. In particular, we treat the systems \, and \ \\) of Friedman and Sheard’s theories and \.

This book is an examination of the foundations and applications of the program of truth-theoretic semantics for natural languages introduced in 1967 by Donald Davidson in his classic paper “Truth and Meaning.” This is the second of two books on Donald Davidson’s central philosophical project. The first, Donald Davidson: Meaning, Truth, Language and Reality (Oxford: Oxford University Press, 2005), dealt with the basic framework of Davidson’s truth-theoretic approach to providing a meaning theory for a natural language, (...) and then with his development of his general project in the philosophy of language and theory of meaning by way of the analysis of the empirical confirmation of a meaning theory for a speaker, from the stance of the radial interpreter, that is, the interpreter who starts out without any detailed knowledge of the speakers’ meanings or attitudes. This book turns to the development of details of Davidson’s semantic program, specifically, the pursuit of the project of providing a compositional semantic theory for natural languages by making strategic use of an appropriately adapted Tarski-style axiomatic truth theory. Such a theory aims to provide a specification of the truth conditions of any felicitous utterance of a natural language sentence which serves to interpret it, and exhibits in the proof of the theorem that gives the truth conditions the compositional structure of the sentence and how its truth depends on its context of utterance. The book develops the basic framework, applies it to quantifiers, proper names, indexicals, simple and complex demonstratives, quotation, adjectives and adverbs, simple tenses of state and event verbs, temporal adverbs and quantifiers, tense in sentential complements and perfect tenses, opaque contexts, especially indirect discourse and attitude sentences, non-declarative sentences, semantic structure and logical form, and discusses the relation of the framework the metaphysics of truth. (shrink)

The modern notion of the axiomatic method developed as a part of the conceptualization of mathematics starting in the nineteenth century. The basic idea of the method is the capture of a class of structures as the models of an axiomatic system. The mathematical study of such classes of structures is not exhausted by the derivation of theorems from the axioms but includes normally the metatheory of the axiom system. This conception of axiomatization satisfies the crucial requirement that (...) the derivation of theorems from axioms does not produce new information in the usual sense of the term called depth information. It can produce new information in a different sense of information called surface information. It is argued in this paper that the derivation should be based on a model-theoretical relation of logical consequence rather than derivability by means of mechanical (recursive) rules. Likewise completeness must be understood by reference to a model-theoretical consequence relation. A correctly understood notion of axiomatization does not apply to purely logical theories. In the latter the only relevant kind of axiomatization amounts to recursive enumeration of logical truths. First-order “axiomatic” set theories are not genuine axiomatizations. The main reason is that their models are structures of particulars, not of sets. Axiomatization cannot usually be motivated epistemologically, but it is related to the idea of explanation. (shrink)

Proof-theoretic reflection principles have been discussed in proof theory ever since Gödel’s discovery of the incompleteness theorems. But these reflection principles have not received much attention in the philosophical community. The present chapter aims to survey some of the principal meta-mathematical results on the iteration of proof-theoretic reflection principles and investigate these results from a logico-philosophical perspective; we will concentrate on the epistemological significance of these technical results and on the epistemic notions involved in the proofs. In particular, we will (...) focus on the notions of commitment to and acceptance of a theory. Special attention is given to the connection between proof-theoretic reflection and axiomatic truth theories. After distinguishing between different types of proof-theoretic reflection principles, we review some proof-theoretic results concerning extensions of formal theories by (iterated) reflection principles. As basis theories, we concentrate on standard arithmetical and elementary axiomatic truth theories. We then go on to explore the epistemological significance of these results. In this investigation, we aim to show that the epistemic notion of acceptance of (or commitment to) a theory plays a crucial role in the philosophical argumentation for reflection principles and their iteration. (shrink)

The first attempt at a systematic approach to axiomatic theories of truth was undertaken by Friedman and Sheard (Ann Pure Appl Log 33:1–21, 1987). There twelve principles consisting of axioms, axiom schemata and rules of inference, each embodying a reasonable property of truth were isolated for study. Working with a base theory of truth conservative over PA, Friedman and Sheard raised the following questions. Which subsets of the Optional Axioms are consistent over the base theory? (...) What are the proof-theoretic strengths of the consistent theories? The first question was answered completely by Friedman and Sheard; all subsets of the Optional Axioms were classified as either consistent or inconsistent giving rise to nine maximal consistent theories of truth.They also determined the proof-theoretic strength of two subsets of the Optional Axioms. The aim of this paper is to continue the work begun by Friedman and Sheard. We will establish the proof-theoretic strength of all the remaining seven theories and relate their arithmetic part to well-known theories ranging from PA to the theory of ${\Sigma^1_1}$ dependent choice. (shrink)

In the paper we investigate typed axiomatizations of the truth predicate in which the axioms of truth come with a built-in, minimal and self-sufficient machinery to talk about syntactic aspects of an arbitrary base theory. Expanding previous works of the author and building on recent works of Albert Visser and Richard Heck, we give a precise characterization of these systems by investigating the strict relationships occurring between them, arithmetized model constructions in weak arithmetical systems and suitable set existence (...) axioms. The framework considered will give rise to some methodological remarks on the construction of truth theories and provide us with a privileged point of view to analyze the notion of truth arising from compositional principles in a typed setting. (shrink)

This papers deals with the class of axiomatic theories of truth for semantically closed languages, where the theories do not allow for standard models; i.e., those theories cannot be interpreted as referring to the natural number codes of sentences only (for an overview of axiomatic theories of truth in general, see Halbach[6]). We are going to give new proofs for two well-known results in this area, and we also prove a new theorem (...) on the nonstandardness of a certain theory of truth. The results indicate that the proof strategies for all the theorems on the nonstandardness of such theories are "essentially" of the same kind of structure. (shrink)

The first history of postwar mathematics, offering a new interpretation of the rise of abstraction and axiomatics in the twentieth century. Why did abstraction dominate American art, social science, and natural science in the mid-twentieth century? Why, despite opposition, did abstraction and theoretical knowledge flourish across a diverse set of intellectual pursuits during the Cold War? In recovering the centrality of abstraction across a range of modernist projects in the United States, Alma Steingart brings mathematics back into the conversation about (...) midcentury American intellectual thought. The expansion of mathematics in the aftermath of World War II, she demonstrates, was characterized by two opposing tendencies: research in pure mathematics became increasingly abstract and rarified, while research in applied mathematics and mathematical applications grew in prominence as new fields like operations research and game theory brought mathematical knowledge to bear on more domains of knowledge. Both were predicated on the same abstractionist conception of mathematics and were rooted in the same approach: modern axiomatics. For American mathematicians, the humanities and the sciences did not compete with one another, but instead were two complementary sides of the same epistemological commitment. Steingart further reveals how this mathematical epistemology influenced the sciences and humanities, particularly the postwar social sciences. As mathematics changed, so did the meaning of mathematization. Axiomatics focuses on American mathematicians during a transformative time, following a series of controversies among mathematicians about the nature of mathematics as a field of study and as a body of knowledge. The ensuing debates offer a window onto the postwar development of mathematics band Cold War epistemology writ large. As Steingart's history ably demonstrates, mathematics is the social activity in which styles of truth-here, abstraction-become synonymous with ways of knowing. (shrink)

This paper attempts to address the question what logical strength theories of truth have by considering such questions as: If you take a theory T and add a theory of truth to it, how strong is the resulting theory, as compared to T? Once the question has been properly formulated, the answer turns out to be about as elegant as one could want: Adding a theory of truth to a finitely axiomatized theory T is more or (...) less equivalent to a kind of abstract consistency statement. A large part of the interest of the paper lies in the way syntactic theories are 'disentangled' from object theories. (shrink)

This article studies three most basic systems of truth as well as their subsystems over set theory ZF possibly with AC or the axiom of global choice GC, and then correlates them with subsystems of Morse–Kelley class theory MK. The article aims at making an initial step towards the axiomatic study of truth in set theory in connection with class theory. Some new results on the side of class theory, such as conservativity, forcing and some forms of (...) the reflection principle, are also presented. The equivalence results among systems of truth and classes obtained in this article are summarized in Theorem 104 , Theorem 107 and Theorem 108. (shrink)

One of the basic question we can ask about truth in a formal setting is what, if anything, we gain when we have a truth predicate at disposal. For example, does the expressive power of a language change or does the proof strength of a theory increase? Satisfaction classes are often described as complicated model theoretic constructions unable to give useful information toward the notion of truth from a general point of view. Their import is narrowed to (...) a dimension of pure technical utility and curiosity. Here I offer an application of satisfaction classes in order to show that they can have a relevant role in confronting proof theoretical equivalent theories of truth. (shrink)

Axiomatization is a formal method for specifying the content of a theory wherein a set of axioms is given from which the remaining content of the theory can be derived deductively as theorems. The theory is identified with the set of axioms and its deductive consequences, which is known as the closure of the axiom set. The logic used to deduce theorems may be informal, as in the typical axiomatic presentation of Euclidean geometry; semiformal, as in reference to set (...) theory or specified branches of mathematics; or formal, as when the axiomatization consists in augmenting the logical axioms for first‐order predicate calculus by the proper axioms of the theory. Although Euclid distinguished axioms from postulates, today the terms are used interchangeably. The earlier demand that axioms be self‐evident or basic truths gradually gave way to the idea that axioms were just assumptions, and later to the idea that axioms are just designated sentences used to specify the theory. (shrink)

In this article we present a number of axiomatic theories of truth which are conservative extensions of arithmetic. We isolate a set of ten natural principles of truth and prove that every consistent permutation of them forms a theory conservative over Peano arithmetic.