Entailment and Bivalence

My purpose in this paper is to argue that the classical notion of entailment is not suitable for non-bivalent logics, to propose an appropriate alternative and to suggest a generalized entailment notion suitable to bivalent and non-bivalent logics alike. In classical two valued logic, one can not infer a false statement from one that is not false, any more than one can infer from a true statement a statement that is not true. In classical logic in fact preserving truth and preserving non-falsity are one and the same thing. They are not the same in non-bivalent logics however and I will argue that the classical notion of entailment that preserves only truth is not strong enough for such a logic. I will show that if we retain the classical notion of entailment in a logic that has three values, true, false and a third value in between, an inconsistency can be derived that can be resolved only by measures that seriously disable the logic. I will show this for a logic designed to allow for semantic presuppositions, then I will show that we get the same result in any three valued logic with the same value ordering. I will finally suggest how the notion of entailment should be generalized so that this problem may be avoided. The strengthened notion of entailment I am proposing is a conservative extension of the classical notion that preserves not only truth but the order of all values in a logic, so that the value of an entailed statement must alway be at least as great as the value of the sequence of statements entailing it. A notion of entailment this strong or stronger will, I believe, be found to be applicable to non-classical logics generally. In the opinion of Dana Scott, no really workable three valued logic has yet been developed. It is hard to disagree with this. A workable three valued logic however could perhaps be developed however, if we had a notion of entailment suitable to non-bivalent logics.     

Modal Logic,Truth, and the Master Modality

In the paper (Braüner, 2001) we gave a minimal condition for the existence of a homophonic theory of truth for a modal or tense logic. In the present paper we generalise this result to arbitrary modal logics and we also show that a modal logic permits the existence of a homophonic theory of truth if and only if it permits the definition of a so-called master modality. Moreover, we explore a connection between the master modality and hybrid logic: We show that if attention is restricted to bidirectional frames, then the expressive power of the master modality is exactly what is needed to translate the bounded fragment of first-order logic into hybrid logic in a truth preserving way. We believe that this throws new light on Arthur Prior's fourth grade tense logic.     

Stability and Paradox in Algorithmic Logic

There is significant interest in type-free systems that allow flexible self-application. Such systems are of interest in property theory, natural language semantics, the theory of truth, theoretical computer science, the theory of classes, and category theory. While there are a variety of proposed type-free systems, there is a particularly natural type-free system that we believe is prototypical: the logic of recursive algorithms. Algorithmic logic is the study of basic statements concerning algorithms and the algorithmic rules of inference between such statements. As shown in [1], the threat of paradoxes, such as the Curry paradox, requires care in implementing rules of inference in this context. As in any type-free logic, some traditional rules will fail. The first part of the paper develops a rich collection of inference rules that do not lead to paradox. The second part identifies traditional rules of logic that are paradoxical in algorithmic logic, and so should be viewed with suspicion in type-free logic generally.     

What Is an Inconsistent Truth Table?

Do truth tables—the ordinary sort that we use in teaching and explaining basic propositional logic—require an assumption of consistency for their construction? In this essay we show that truth tables can be built in a consistency-independent paraconsistent setting, without any appeal to classical logic. This is evidence for a more general claim—that when we write down the orthodox semantic clauses for a logic, whatever logic we presuppose in the background will be the logic that appears in the foreground. Rather than any one logic being privileged, then, on this count partisans across the logical spectrum are in relatively similar dialectical positions.     

Statistics of Intuitionistic versus Classical Logics

For the given logical calculus we investigate the proportion of the number of true formulas of a certain length n to the number of all formulas of such length. We are especially interested in asymptotic behavior of this fraction when n tends to infinity. If the limit exists it is represented by a real number between 0 and 1 which we may call the density of truth for the investigated logic. In this paper we apply this approach to the intuitionistic logic of one variable with implication and negation. The result is obtained by reducing the problem to the same one of Dummett's intermediate linear logic of one variable (see [2]). Actually, this paper shows the exact density of intuitionistic logic and demonstrates that it covers a substantial part (more than 93%) of classical prepositional calculus. Despite using strictly mathematical means to solve all discussed problems, this paper in fact, may have a philosophical impact on understanding how much the phenomenon of truth is sporadic or frequent in random mathematics sentences.     

Labeled Calculi and Finite-Valued Logics

A general class of labeled sequent calculi is investigated, and necessary and sufficient conditions are given for when such a calculus is sound and complete for a finite-valued logic if the labels are interpreted as sets of truth values (sets-as-signs). Furthermore, it is shown that any finite-valued logic can be given an axiomatization by such a labeled calculus using arbitrary "systems of signs," i.e., of sets of truth values, as labels. The number of labels needed is logarithmic in the number of truth values, and it is shown that this bound is tight.     

Hintikka’s Logical Revolution

Hintikka thinks that second-order logic is not pure logic, and because of Gödel’s incompleteness theorems, he suggests that we should liberate ourselves from the mistaken idea that first-order logic is the foundational logic of mathematics. With this background he introduces his independence friendly logic (IFL). In this paper, I argue that approaches taking Hintikka’s IFL as a foundational logic of mathematics face serious challenges. First, the quantifiers in Hintikka’s IFL are not distinguishable from Linström’s general quantifiers, which means that the quantifiers in IFL involve higher order entities. Second, if we take Wright’s interpretation of quantifiers or if we take Hale’s criterion for the identity of concepts, Quine’s thesis that second-order logic is set theory will be rejected. Third, Hintikka’s definition of truth itself cannot be expressed in the extension of language of IFL. Since second-order logic can do what IFL does, the significance of IFL for the foundations of mathematics is weakened.     

Truthier Than Thou: Truth,Supertruth and Probability of Truth

Different formal tools are useful for different purposes. For example, when it comes to modelling degrees of belief, probability theory is a better tool than classical logic; when it comes to modelling the truth of mathematical claims, classical logic is a better tool than probability theory. In this paper I focus on a widely used formal tool and argue that it does not provide a good model of a phenomenon of which many think it does provide a good model: I shall argue that while supervaluationism may provide a model of probability of truth, or of assertability, it cannot provide a good model of truth—supertruth cannot be truth. The core of the argument is that an adequate model of truth must render certain connectives truth‐functional (at least in certain circumstances)—and supervaluationism does not do so (in those circumstances).     

Partiality and Its Dual

This paper explores allowing truth value assignments to be undetermined or "partial" (no truth values) and overdetermined or "inconsistent" (both truth values), thus returning to an investigation of the four-valued semantics that I initiated in the sixties. I examine some natural consequence relations and show how they are related to existing logics, including ukasiewicz's three-valued logic, Kleene's three-valued logic, Anderson and Belnap's (first-degree) relevant entailments, Priest's "Logic of Paradox", and the first-degree fragment of the Dunn-McCall system "R-mingle". None of these systems have nested implications, and I investigate twelve natural extensions containing nested implications, all of which can be viewed as coming from natural variations on Kripke's semantics for intuitionistic logic. Many of these logics exist antecedently in the literature, in particular Nelson's "constructible falsity".     

Gentzen Calculi for the Existence Predicate

We introduce Gentzen calculi for intuitionistic logic extended with an existence predicate. Such a logic was first introduced by Dana Scott, who provided a proof system for it in Hilbert style. We prove that the Gentzen calculus has cut elimination in so far that all cuts can be restricted to very simple ones. Applications of this logic to Skolemization, truth value logics and linear frames are also discussed.     

Acceptable Contradictions: Pragmatics or Semantics? A Reply to Cobreros et al.

Naive speakers find some logical contradictions acceptable, specifically borderline contradictions involving vague predicates such as Joe is and isn’t tall. In a recent paper, Cobreros et al. (J Philos Logic, 2012) suggest a pragmatic account of the acceptability of borderline contradictions. We show, however, that the pragmatic account predicts the wrong truth conditions for some examples with disjunction. As a remedy, we propose a semantic analysis instead. The analysis is close to a variant of fuzzy logic, but conjunction and disjunction are interpreted as intensional operators.     

Dual Intuitionistic Logic and a Variety of Negations: The Logic of Scientific Research

We consider a logic which is semantically dual (in some precise sense of the term) to intuitionistic. This logic can be labeled as “falsification logic”: it embodies the Popperian methodology of scientific discovery. Whereas intuitionistic logic deals with constructive truth and non-constructive falsity, and Nelson's logic takes both truth and falsity as constructive notions, in the falsification logic truth is essentially non-constructive as opposed to falsity that is conceived constructively. We also briefly clarify the relationships of our falsification logic to some other logical systems.     

Taking Degrees of Truth Seriously

This is a contribution to the discussion on the role of truth degrees in manyvalued logics from the perspective of abstract algebraic logic. It starts with some thoughts on the so-called Suszko’s Thesis (that every logic is two-valued) and on the conception of semantics that underlies it, which includes the truth-preserving notion of consequence. The alternative usage of truth values in order to define logics that preserve degrees of truth is presented and discussed. Some recent works studying these in the particular cases of Łukasiewicz’s many-valued logics and of logics associated with varieties of residuated lattices are also presented. Finally the extension of this paradigm to other, more general situations is discussed, highlighting the need for philosophical or applied motivations in the selection of the truth degrees, due both to the interpretation of the idea of truth degree and to some mathematical difficulties.     

Dialogue Games for Many-Valued Logics — an Overview

An overview of different versions and applications of Lorenzen’s dialogue game approach to the foundations of logic, here largely restricted to the realm of manyvalued logics, is presented. Among the reviewed concepts and results are Giles’s characterization of ?ukasiewicz logic and some of its generalizations to other fuzzy logics, including interval based logics, a parallel version of Lorenzen’s game for intuitionistic logic that is adequate for finite- and infinite-valued Gödel logics, and a truth comparison game for infinite-valued Gödel logic.     

On Moderate Pluralism About Truth and Logic

According to moderate truth pluralism, truth is both One and Many. There is a single truth property that applies across all truth-apt domains of discourse, but instances of this property are grounded in different ways. Propositions concerning medium-sized dry goods might be true in virtue of corresponding with reality while propositions pertaining to the law might be true in virtue of cohering with the body of law. Moderate truth pluralists must answer two questions concerning logic: (Q1) Which logic governs inferences concerning each truth-apt domain considered separately? (Q2) Which logic governs inferences that involve several truth-apt domains? This paper has three objectives. The first objective is to present and explain the moderate pluralist’s answers to (Q1) and (Q2). The second objective is to argue that there is a tension between these answers. The answer to (Q1) involves a commitment to a form of logical pluralism. However, reflection on the moderate truth pluralist’s answer to (Q2) shows that they are committed to taking logic to be topic neutrality. This, in turn, forces a commitment to logical monism. It would seem that the moderate truth pluralist cannot have it both ways. The third objective is constructive in nature. I offer an account of what moderate truth pluralists should say about logic and how they might resolve the tension in their view. I suggest that, just like moderate truth pluralists distinguish truth proper and “quasi-truth,” they should endorse a distinction between logic proper and “quasi-logic.” Quasi-truth is truth-like in the sense that instances of quasi-truth ground instances of truth. Quasi-logic is logic-like in the sense that it concerns arguments that are necessarily truth-preserving but are not generally so in a topic neutral way. I suggest that moderate truth pluralists should be monists about truth proper and logic proper but pluralists about quasi-truth and quasi-logic. This allows them to say that logic proper is topic neutral while still accommodating the idea that, for different domains, different arguments may be necessarily truth-preserving.     

On Retaining Classical Truths and Classical Deducibility in Many-Valued and Fuzzy Logics

In this paper, I identify the source of the differences between classical logic and many-valued logics (including fuzzy logics) with respect to the set of valid formulas and the set of inferences sanctioned. In the course of doing so, we find the conditions that are individually necessary and jointly sufficient for any many-valued semantics (again including fuzzy logics) to validate exactly the classically valid formulas, while sanctioning exactly the same set of inferences as classical logic. This in turn shows, contrary to what has sometimes been claimed, that at least one class of infinite-valued semantics is axiomatizable.     

Paradoxes and Failures of Cut

This paper presents and motivates a new philosophical and logical approach to truth and semantic paradox. It begins from an inferentialist, and particularly bilateralist, theory of meaning—one which takes meaning to be constituted by assertibility and deniability conditions—and shows how the usual multiple-conclusion sequent calculus for classical logic can be given an inferentialist motivation, leaving classical model theory as of only derivative importance. The paper then uses this theory of meaning to present and motivate a logical system—ST—that conservatively extends classical logic with a fully transparent truth predicate. This system is shown to allow for classical reasoning over the full (truth-involving) vocabulary, but to be non-transitive. Some special cases where transitivity does hold are outlined. ST is also shown to give rise to a familiar sort of model for non-classical logics: Kripke fixed points on the Strong Kleene valuation scheme. Finally, to give a theory of paradoxical sentences, a distinction is drawn between two varieties of assertion and two varieties of denial. On one variety, paradoxical sentences cannot be either asserted or denied; on the other, they must be both asserted and denied. The target theory is compared favourably to more familiar related systems, and some objections are considered and responded to.     

On Jaśkowski's discussive logics

We expose the main ideas, concepts and results about Jakowski's discussive logic, and apply that logic to the concept of pragmatic truth and to the Dalla Chiara-di Francia view of the foundations of physics.Partially supported by grants from JNICT (Portugal) and FAPESP (Brazil), Philosophy Section. Portions of this paper were concluded while the second author visited the Math-Phys Seminar at the University of Algarve (Portugal).Presented byCecylia Rauszer     

A Normative Pragmatic Perspective on Appealing to Emotions in Argumentation

Is appealing to emotions in argumentation ever legitimate and, if so, what is the best way to analyze and evaluate such appeals? After overviewing a normative pragmatic perspective on appealing to emotions in argumentation, I present answers to these questions from pragma-dialectical, informal logical, and rhetorical perspectives, and note positions shared and supplemented by a normative pragmatic perspective. A normative pragmatic perspective holds that appealing to emotions in argumentation may be relevant and non-manipulative; and that emotional appeals may be analyzed as strategies that create pragmatic reasons and assessed by the standard of formal propriety or reasonability under the circumstances. I illustrate the explanatory power of the perspective by analyzing and evaluating some argumentation from Frederick Douglass’s “What to the Slave is the Fourth of July.” I conclude that a normative pragmatic perspective offers a more complete account of appealing to emotions in argumentation than a pragma-dialectial, informal logical, or rhetorical perspective alone, identifies a range of norms available to arguers, and explains why appealing to emotions may be legitimate in particular cases of argumentation.