Paraconsistent logic

About: Paraconsistent logic is a research topic. Over the lifetime, 1610 publications have been published within this topic receiving 28842 citations.

On the formulae-as-types correspondence for classical logic

Abstract: The Curry–Howard correspondence states the equivalence between the constructions implicit in intuitionistic logic and those described in the simplytyped lambda-calculus. It is an insight of great importance in theoretical computer science, and is fundamental in modern approaches to constructive type theory. The possibility of a similar formulae-as-types correspondence for classical logic looks to be a seminal development in this area, but whilst promising results have been achieved, there does not appear to be much agreement of what is at stake in claiming that such a correspondence exists. Consequently much work in this area suffers from several weaknesses; in particular the status of the new rules needed to describe the distinctively classical inferences is unclear. We show how to situate the formulae-as-types correspondence within the proof-theoretic account of logical semantics arising from the work of Michael Dummett andDag Prawitz, and demonstrate that the admissibility of Prawitz’s inversion principle, which we argue should be strengthened, is essential to the good behaviour of intuitionistic logic. By regarding the rules which determine the deductive strength of classical logic as structural rules, as opposed to the logical rules associated with specific logical connectives, we extend Prawitz’s inversion principle to classical propositional logic, formulated in a theory of Parigot’s lambda-mu calculus with eta expansions. We then provide a classical analogue of a subsystem of Martin-Lof’s type theory corresponding to Peano Arithmetic and show its soundness, appealing to an extension of Tait’s reducibility method. Our treatment is the first treatment of induction in classical arithmetic that truly falls under the aegis of the formulae-as-types correspondence, as it is the first that is consistent with the intensional reading of propositional equality.

Logic Colloquium 2006: The inevitability of logical strength: Strict reverse mathematics

Abstract: An extreme kind of logic skeptic claims that "the present formal systems used for the foundations of mathematics are artificially strong, thereby causing unnecessary headaches such as the Gödel incompleteness phenomena". The skeptic continues by claiming that "logician's systems always contain overly general assertions, and/or assertions about overly general notions, that are not used in any significant way in normal mathematics. For example, induction for all statements, or even all statements of certain restricted forms, is far too general mathematicians only use induction for natural statements that actually arise. If logicians would tailor their formal systems to conform to the naturalness of normal mathematics, then various logical difficulties would disappear, and the story of the foundations of mathematics would look radically different than it does today. In particular, it should be possible to give a convincing model of actual mathematical practice that can be proved to be free of contradiction using methods that lie within what Hilbert had in mind in connection with his program”. Here we present some specific results in the direction of refuting this point of view, and introduce the Strict Reverse Mathematics (SRM) program.

Reductions of higher-order logic

Abstract: A higher-order logic is considered which contains variables of all finite and transfinite types, together with a more restricted logic in which the types of variables are all in a sense describable ordinals. It is shown that several important problems connected with the restricted higherorder logic, pertaining, for instance, to the determination of spectra and the characterization of logical truth, are reducible to the corresponding problems for second-order logic; the principal results of this sort extend earlier results of Zykov and Hintikka. It is obtained as a corollary that the set of logical truths of second-order logic does not appear in any reasonable extension of the Kleene hierarchy.

Paraconsistent query answering systems

Abstract: Classical logic predicts that everything (thus nothing useful at all) follows from inconsistency. A paraconsistent logic is a logic where inconsistency does not lead to such an explosion, and since in practice consistency is difficult to achieve there are many potential applications of paraconsistent logics in query answering systems. We compare the paraconsistent and the non-monotonic solutions to the problem of contradictions. We propose a many-valued paraconsistent logic based on a simple notion of indeterminacy. In particular we describe the semantics of the logic using key equalities for the logical operators. We relate our approach to works on bilattices. We also discuss and provide formalizations of two case studies, notably the well-known example involving penguins and a more interesting example in the domain of medicine.

If A, Then B: How the World Discovered Logic

Abstract: PrefaceIntroduction: What Is Logic?1 The Dawn of Logic2 Aristotle: Greatest of the Greek Logicians3 Aristotle's System: The Logic of Classification4 Chrysippus and the Stoics: A World of Interlocking Structures5 Logic Versus Antilogic: The Laws of Contradiction and Excluded Middle6 Logical Fanatics7 Will the Future Resemble the Past? Inductive Logic and Scientific Method8 Rhetorical Frauds and Sophistical Ploys: Ten Classic Tricks9 Symbolic Logic and the Digital Future10 Faith and the Limits of Logic: The Last Unanswered QuestionAppendix: Further FallaciesNotesBibliographyIndex