scholarly journals Paradoxes of truth-in-context-X

2021 ◽  
Author(s):  
Christopher Gauker
Keyword(s):  
Truth Predicate

AbstractWe may suppose that the truth predicate that we utilize in our semantic metalanguage is a two-place predicate relating sentences to contexts, the truth-in-context-X predicate. Seeming paradoxes pertaining to the truth-in-context-X predicate can be blocked by placing restrictions on the structure of contexts. While contexts must specify a domain of contexts, and what a context constant denotes relative to a context must be a context in the context domain of that context, no context may belong to its own context domain. A generalization of that restriction appears to block all of the paradoxes of truth-in-context-X. This restriction entails that, in a certain sense, we cannot talk about the context we are in. This result will be defended, up to a point, on broadly ontological grounds. It will also be conjectured that our semantic metalanguage can be regarded as semantically closed.

scholarly journals NOTES ON BOUNDED INDUCTION FOR THE COMPOSITIONAL TRUTH PREDICATE

2017 ◽  
Vol 10 (3) ◽  
pp. 455-480 ◽  
Author(s):  
BARTOSZ WCISŁO ◽  
MATEUSZ ŁEŁYK
Keyword(s):  
Reflection Principle ◽  
Peano Arithmetic ◽  
Truth Predicate ◽  
Base Theory ◽  
Modified Theory ◽  
Induction Scheme

AbstractWe prove that the theory of the extensional compositional truth predicate for the language of arithmetic with Δ0-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.

Tarski

Truth ◽  
2011 ◽  
Author(s):  
Alexis G. Burgess ◽  
John P. Burgess
Keyword(s):  
Mathematical Logic ◽  
Model Theory ◽  
The State ◽  
Truth Predicate ◽  
Object Language ◽  
French And English ◽  
Direct Definition ◽  
Definition Of ◽  
Basic Features ◽  
Self Reference

This chapter offers a simplified account of the most basic features of Alfred Tarski's model theory. Tarski foresaw important applications for a notion of truth in mathematics, but also saw that mathematicians were suspicious of that notion, and rightly so given the state of understanding of it circa 1930. In a series of papers in Polish, German, French, and English from the 1930s onward, Tarski attempted to rehabilitate the notion for use in mathematics, and his efforts had by the 1950s resulted in the creation of a branch of mathematical logic known as model theory. The chapter first considers Tarski's notion of truth, which he calls “semantic” truth, before discussing his views on object language and metalanguage, recursive versus direct definition of the truth predicate, and self-reference.

Soundness, Epistemology, and Frege’s Project

2020 ◽  
pp. 115-169
Author(s):  
Joan Weiner
Keyword(s):  
Natural Language ◽  
Formal System ◽  
Truth Predicate

Insofar as the use of natural language to introduce, discuss, and argue about features of a formal system is metatheoretic, there can be no doubt: Frege has a metatheory. But what kind of metatheory? Although the model theoretic semantics with which we are familiar today is a post-Fregean development, most believe that Frege offers a proto-soundness proof for his logic that intrinsically exploits a truth predicate and metalinguistic variables. In this chapter it is argued that he neither uses, nor has any need to use, a truth predicate or metalinguistic variables in justifications of his basic laws and rules. The purpose of the discussions that are typically understood as constituting Frege’s metatheory is, rather, elucidatory. And once we see what the aim of these particular elucidations is, we can explain Frege’s otherwise puzzling eschewal of the truth predicate in his discussions of the justification of the laws and rules of inference.

A theory of properties

1987 ◽  
Vol 52 (2) ◽  
pp. 455-472 ◽  
Author(s):  
Ray Turner
Keyword(s):  
Recent Work ◽  
Set Theory ◽  
Semantic Theory ◽  
Truth Predicate ◽  
Minimal Logic ◽  
History Of ◽  
Concept Of Truth ◽  
Almost All ◽  
The Liar ◽  
Self Reference

Frege's attempts to formulate a theory of properties to serve as a foundation for logic, mathematics and semantics all dissolved under the weight of the logicial paradoxes. The language of Frege's theory permitted the representation of the property which holds of everything which does not hold of itself. Minimal logic, plus Frege's principle of abstraction, leads immediately to a contradiction. The subsequent history of foundational studies was dominated by attempts to formulate theories of properties and sets which would not succumb to the Russell argument. Among such are Russell's simple theory of types and the development of various iterative conceptions of set. All of these theories ban, in one way or another, the self-reference responsible for the paradoxes; in this sense they are all “typed” theories. The semantical paradoxes, involving the concept of truth, induced similar nightmares among philosophers and logicians involved in semantic theory. The early work of Tarski demonstrated that no language that contained enough formal machinery to respresent the various versions of the Liar could contain a truth-predicate satisfying all the Tarski biconditionals. However, recent work in both disciplines has led to a re-evaluation of the limitations imposed by the paradoxes.In the foundations of set theory, the work of Gilmore [1974], Feferman [1975], [1979], [1984], and Aczel [1980] has clearly demonstrated that elegant and useful type-free theories of classes are feasible. Work on the semantic paradoxes was given new life by Kripke's contribution (Kripke [1975]). This inspired the recent work of Gupta [1982] and Herzberger [1982]. These papers demonstrate that much room is available for the development of theories of truth which meet almost all of Tarski's desiderata.

Disquotationalism, Minimalism, and the Finite Minimal Theory

2004 ◽  
Vol 34 (1) ◽  
pp. 61-86 ◽  
Author(s):  
Jay Newhard
Keyword(s):  
Prima Facie ◽  
Truth Predicate ◽  
Minimal Theory ◽  
Paul Horwich ◽  
Minimalist Theory

Recently, Paul Horwich has developed the minimalist theory of truth, according to which the truth predicate does not express a Substantive property, though it may be used as a grammatical expedient. Minimalism shares these Claims with Quine's disquotationalism; it differs from disquotationalism primarily in holding that truth-bearers are propositions, rather than sentences. Despite potential ontological worries, allowing that propositions bear truth gives Horwich a prima facie response to several important objections to disquotationalism. In section I of this paper, disquotationalism is given a careful exegesis, in which seven known objections are traced to the disquotational Schema, and two new objections are raised. A version of disquotationalism which avoids two of the seven known objections is recommended. In section II, an examination of minimalism shows that it faces eight of the nine objections facing disquotationalism, plus a new objection.

Construction of Truth Predicates: Approximation Versus Revision

1998 ◽  
Vol 4 (4) ◽  
pp. 399-417 ◽  
Author(s):  
Juan Barba
Keyword(s):  
Liar Paradox ◽  
Predicate Symbol ◽  
Main Idea ◽  
Truth Predicate ◽  
Order Language ◽  
Fixed Model ◽  
Contextual Dependence ◽  
The Subject ◽  
Concept Of Truth ◽  
The Liar

§1. Introduction. The problem raised by the liar paradox has long been an intriguing challenge for all those interested in the concept of truth. Many “solutions” have been proposed to solve or avoid the paradox, either prescribing some linguistical restriction, or giving up the classical true-false bivalence or assuming some kind of contextual dependence of truth, among other possibilities. We shall not discuss these different approaches to the subject in this paper, but we shall concentrate on a kind of formal construction which was originated by Kripke's paper “Outline of a theory of truth” [11] and which, in different forms, reappears in later papers by various authors.The main idea can be presented as follows: assume a first order language ℒ containing, among other unspecified symbols, a predicate symbol T intended to represent the truth predicate for ℒ. Assume, also, a fixed model M = 〈D, I〉 (the base model)where D contains all sentences of ℒ and I interprets all non-logical symbols of ℒ except T in the usual way. In general, D might contain many objects other than sentences of ℒ but as that would raise the problem of the meaning of sentences in which T is applied to one of these objects, we shall assume that this is not the case.

scholarly journals MODALITY AND AXIOMATIC THEORIES OF TRUTH II: KRIPKE-FEFERMAN

2014 ◽  
Vol 7 (2) ◽  
pp. 299-318 ◽  
Author(s):  
JOHANNES STERN
Keyword(s):  
Truth Predicate ◽  
General Strategy ◽  
Modal Operator ◽  
Axiomatic Theories Of Truth ◽  
Theories Of Truth ◽  
Modal Semantics

AbstractIn 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.

SUPER LIARS

2010 ◽  
Vol 3 (3) ◽  
pp. 374-414 ◽  
Author(s):  
PHILIPPE SCHLENKER
Keyword(s):  
Fixed Points ◽  
Detailed Analysis ◽  
Expressive Power ◽  
Truth Predicate ◽  
Truth Values ◽  
Kripke’S Theory Of Truth ◽  
Domain Restrictions ◽  
The Liar ◽  
Self Reference ◽  
Linguistic Means

Kripke’s theory of truth offered a trivalent semantics for a language which, like English, contains a truth predicate and means of self-reference; but it did so by severely restricting the expressive power of the logic. In Kripke’s analysis, the Liar (e.g., This very sentence is not true) receives the indeterminate truth value, but this fact cannot be expressed in the language; by contrast, it is straightforward to say in English that the Liar is something other than true. Kripke’s theory also fails to handle the Strengthened Liar, which can be expressed in English as: This very sentence is something other than true. We develop a theory which seeks to overcome these difficulties, and is based on a detailed analysis of some of the linguistic means by which the Strengthened Liar can be expressed in English. In particular, we propose to take literally the quantificational form of the negative expression something other than true. Like other quantifiers, it may have different implicit domain restrictions, which give rise to a variety of negations of different strengths (e.g., something other than true among the values {0, 1}, or among {0, 1, 2}, etc). This analysis naturally leads to a logic with as many truth values as there are ordinals—a conclusion reached independently by Cook (2008a). We develop the theory within a generalization of the Strong Kleene Logic, augmented with negations that each have a nonmonotonic semantics. We show that fixed points can be constructed for our logic, and that it enjoys a limited form of ‘expressive completeness’. Finally, we discuss the relation between our theory and various alternatives, including one in which the word true (rather than negation) is semantically ambiguous, and gives rise to a hierarchy of truth predicates of increasing strength.

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