A Paraconsistent Higher Order Logic

Classical logic predicts that everything (thus nothing useful at all) follows from inconsistency. A paraconsistent logic is a logic where an 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 knowledge-based systems, logical semantics of natural language, etc. Higher order logics have the advantages of being expressive and with several automated theorem provers available. Also the type system can be helpful. We present a concise description of a paraconsistent higher order logic with countable infinite indeterminacy, where each basic formula can get its own indeterminate truth value (or as we prefer: truth code). The meaning of the logical operators is new and rather different from traditional many-valued logics as well as from logics based on bilattices. The adequacy of the logic is examined by a case study in the domain of medicine. Thus we try to build a bridge between the HOL and MVL communities. A sequent calculus is proposed based on recent work by Muskens.Comment: Originally in the proceedings of PCL 2002, editors Hendrik Decker, Joergen Villadsen, Toshiharu Waragai (http://floc02.diku.dk/PCL/). Correcte

The enduring scandal of deduction: Is propositional logic really uninformative?

“The original publication is available at www.springerlink.com”. Copyright Springer DOI: 10.1007/s11229-008-9409-4Deductive inference is usually regarded as being “tautological” or “analytical”: the information conveyed by the conclusion is contained in the information conveyed by the premises. This idea, however, clashes with the undecidability of firstorder logic and with the (likely) intractability of Boolean logic. In this article, we address the problem both from the semantic and the proof-theoretical point of view and propose a hierarchy of propositional logics that are all tractable (i.e. decidable in polynomial time), although by means of growing computational resources, and converge towards classical propositional logic. The underlying claim is that this hierarchy can be used to represent increasing levels of “depth” or “informativeness” of Boolean reasoning. Special attention is paid to the most basic logic in this hierarchy, the pure “intelim logic”, which satisfies all the requirements of a natural deduction system (allowing both introduction and elimination rules for each logical operator) while admitting of a feasible (quadratic) decision procedure. We argue that this logic is “analytic” in a particularly strict sense, in that it rules out any use of “virtual information”, which is chiefly responsible for the combinatorial explosion of standard classical systems. As a result, analyticity and tractability are reconciled and growing degrees of computational complexity are associated with the depth at which the use of virtual information is allowed.Peer reviewe

Craig Interpolation in Displayable Logics

Abstract. We give a general proof-theoretic method for establishing Craig interpolation for displayable logics, based upon an analysis of the individual proof rules of their display calculi. Using this uniform method, we establish interpolation for a spectrum of display calculi differing in their structural rules, including those for multiplicative linear logic, multiplicative additive linear logic and ordinary classical logic. Our analysis at the level of proof rules also provides new insights into the reasons why interpolation fails, or seems likely to fail, in many substructural logics. Specifically, we identify contraction as being particularly problematic for interpolation except in special circumstances.

A dialogue game to offer an agreement to disagree

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