On All Strong Kleene Generalizations of Classical Logic

By using the notions of exact truth (‘true and not false’) and exact falsity (‘false and not true’), one can give 16 distinct definitions of classical consequence. This paper studies the class of relations that results from these definitions in settings that are paracomplete, paraconsistent or both and that are governed by the (extended) Strong Kleene schema. Besides familiar logics such as Strong Kleene logic (K3), the Logic of Paradox (LP) and First Degree Entailment (FDE), the resulting class of all Strong Kleene generalizations of classical logic also contains a host of unfamiliar logics. We first study the members of our class semantically, after which we present a uniform sequent calculus (the SK calculus) that is sound and complete with respect to all of them. Two further sequent calculi (the (Formula presented.) and (Formula presented.) calculus) will be considered, which serve the same purpose and which are obtained by applying general methods (due to Baaz et al.) to construct sequent calculi for many-valued logics. Rules and proofs in the SK calculus are much simpler and shorter than those of the (Formula presented.) and the (Formula presented.) calculus, which is one of the reasons to prefer the SK calculus over the latter two. Besides favourably comparing the SK calculus to both the (Formula presented.) and the (Formula presented.) calculus, we also hint at its philosophical significance

An Epistemic Interpretation of Paraconsistent Weak Kleene Logic

This paper extends Fitting's epistemic interpretation of some Kleene logics, to also account for Paraconsistent Weak Kleene logic. To achieve this goal, a dualization of Fitting's "cut-down" operator is discussed, rendering a "track-down" operator later used to represent the idea that no consistent opinion can arise from a set including an inconsistent opinion. It is shown that, if some reasonable assumptions are made, the truth-functions of Paraconsistent Weak Kleene coincide with certain operations defined in this track-down fashion. Finally, further reflections on conjunction and disjunction in the weak Kleene logics accompany this paper, particularly concerning their relation with containment logics. These considerations motivate a special approach to defining sound and complete Gentzen-style sequent calculi for some of their four-valued generalizations

Strict/Tolerant Logics Built Using Generalized Weak Kleene Logics

This paper continues my work of [9], which showed there was a broad family of many valued logics that have a strict/tolerant counterpart. Here we consider a generalization of weak Kleene three valued logic, instead of the strong version that was background for that earlier work. We explain the intuition behind that generalization, then determine a subclass of strict/tolerant structures in which a generalization of weak Kleene logic produces the same results that the strong Kleene generalization did. This paper provides much background, but is not self-contained. Some results from [9] are called on, and are not reproved here. [9]  Melvin C. Fitting. “A Family of Strict/Tolerant Logics”. In: Journal of Philosophical Logic (2020). Online. Print publication forthcoming

An epistemic interpretation of paraconsistent weak Kleene logic

This paper extends Fitting’s epistemic interpretation of some Kleene logics to also account for Paraconsistent Weak Kleene logic. To achieve this goal, a dualization of Fitting’s “cut-down” operator is discussed, leading to the definition of a “track-down” operator later used to represent the idea that no consistent opinion can arise from a set including an inconsistent opinion. It is shown that, if some reasonable assumptions are made, the truth-functions of Paraconsistent Weak Kleene coincide with certain operations defined in this track-down fashion. Finally, further reflections on conjunction and disjunction in the weak Kleene logics accompany this paper, particularly concerning their relation with containment logics. These considerations motivate a special approach to defining sound and complete Gentzen-style sequent calculi for some of their four-valued generalizations.Fil: Szmuc, Damián Enrique. Universidad de Buenos Aires. Facultad de Filosofía y Letras; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Parque Centenario. Instituto de Investigaciones Filosóficas. - Sociedad Argentina de Análisis Filosófico. Instituto de Investigaciones Filosóficas; Argentin

LP, K3, and FDE as Substructural Logics

Building on recent work, I present sequent systems for the non-classical logics LP, K3, and FDE with two main virtues. First, derivations closely resemble those in standard Gentzen-style systems. Second, the systems can be obtained by reformulating a classical system using nonstandard sequent structure and simply removing certain structural rules (relatives of exchange and contraction). I clarify two senses in which these logics count as “substructural.

Type-free truth

This book is a contribution to the flourishing field of formal and philosophical work on truth and the semantic paradoxes. Our aim is to present several theories of truth, to investigate some of their model-theoretic, recursion-theoretic and proof-theoretic aspects, and to evaluate their philosophical significance. In Part I we first outline some motivations for studying formal theories of truth, fix some terminology, provide some background on Tarski’s and Kripke’s theories of truth, and then discuss the prospects of classical type-free truth. In Chapter 4 we discuss some minimal adequacy conditions on a satisfactory theory of truth based on the function that the truth predicate is intended to fulfil on the deflationist account. We cast doubt on the adequacy of some non-classical theories of truth and argue in favor of classical theories of truth. Part II is devoted to grounded truth. In chapter 5 we introduce a game-theoretic semantics for Kripke’s theory of truth. Strategies in these games can be interpreted as reference-graphs (or dependency-graphs) of the sentences in question. Using that framework, we give a graph-theoretic analysis of the Kripke-paradoxical sentences. In chapter 6 we provide simultaneous axiomatizations of groundedness and truth, and analyze the proof-theoretic strength of the resulting theories. These range from conservative extensions of Peano arithmetic to theories that have the full strength of the impredicative system ID1. Part III investigates the relationship between truth and set-theoretic comprehen- sion. In chapter 7 we canonically associate extensions of the truth predicate with Henkin-models of second-order arithmetic. This relationship will be employed to determine the recursion-theoretic complexity of several theories of grounded truth and to show the consistency of the latter with principles of generalized induction. In chapter 8 it is shown that the sets definable over the standard model of the Tarskian hierarchy are exactly the hyperarithmetical sets. Finally, we try to apply a certain solution to the set-theoretic paradoxes to the case of truth, namely Quine’s idea of stratification. This will yield classical disquotational theories that interpret full second-order arithmetic without set parameters, Z2- (chapter 9). We also indicate a method to recover the parameters. An appendix provides some background on ordinal notations, recursion theory and graph theory

The Power of Naive Truth

While non-classical theories of truth that take truth to be transparent have some obvious advantages over any classical theory that evidently must take it as non-transparent, several authors have recently argued that there's also a big disadvantage of non-classical theories as compared to their “external” classical counterparts: proof-theoretic strength. While conceding the relevance of this, the paper argues that there is a natural way to beef up extant internal theories so as to remove their proof-theoretic disadvantage. It is suggested that the resulting internal theories should seem preferable to their external counterparts