Paraconsistent logic

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

Behavioral algebraization of da Costa's C-systems

Abstract: It is well-known that da Costa's C-systems of paraconsistent logic do not admit a Blok-Pigozzi algebraization. Still, an algebraic flavored semantics for them has been proposed in the literature, namely using the class of so-called da Costa algebras. However, the precise connection between these semantic structures and the C-systems was never established at the light of the theory of algebraizable logics. In this paper we propose to study the C-systems from an algebraic point of view, and to fill in this gap by using the tools and techniques of the newly developed behavioral approach to abstract algebraic logic. As a by-product of the approach, we also rediscover the bivaluation semantics of the logics.

Toward the Reconstruction of the Early History of Paraconsistent Logic: The Prerequisites of N.A.Vasiliev's Imaginary Logic

Abstract: On the basis of recently unknown archive and manuscript documents the article reveals the informal and heuristic prerequisites of N.A.Vasiliev's imaginary logic. N.A.Vasiliev's imaginary logic (1908 - 1914) became precursor of paraconsistent logic. His path to imaginary logic was steep and toilsome. The starting point of his way to be found in youth animations and "vague sensations" of the future scholar related to the radically new treatment of contradiction and approach to logic. At present moment we are able to reconstruct the very early history of N.A.Vasiliev's imaginary logic and its formal or informal prerequisites and heuristic prompts. It is possible due to the the discovering of scholars personal archive and unknown before early 1990-s his works and letters. Some Soviet scientists worked diligently but unsuccesfully to find the archives of the Vasiliev's family (for instance, prominent algebraists A.I.Mal'tsev and V.V.Morozov). I too engaged in this search, and was fortunate to find two of his logical manuscripts and the "remains" of archive (diary, letters, photographs, books with Vasiliev's annotations, etc.) . This material enabled me to write a scientific biography of Vasiliev and study his way to imaginary logic [1] and in collaboration with V.A.Smirnov to publish for the first time in Soviet era the works of N.A.Vasiliev itself [2]. Later some new documents and Vasiliev's unknown papers were found by me. A close look at Vasiliev's life and work shows us that he is not only the founder of original non-classical logical theories, a forerunner of paraconsistent logic but a thinker with very wide interests - philosopher, ethician, psychologist, historian, poet and even skilled interpreter. All components of Vasiliev intellectual activity are bounded. What vague, uncertain and barely formulated analogies fed Vasiliev's pioneer work? To my mind they can be specified due to new findings: 1) C.S.Peirce's logic of relatives, which Vasiliev learned when he was only seventeen ; 2) the symbolist poetry that paid a great deal of attention to the subject of ''another world''; 3) the special psychological standpoint and approach used for the critical assessment and analysis of Aristotelian logic; 4) Charles Darwin's ideas on the evolution of life; 5) the analogy with non-Euclidian geometry construction and method. Soon (according to historical scale) after Kant wrote his Preface to "CRITIQUE OF PURE REASON" in Aristotelian logic the powerful movement emerged, resulted in eventual drastic changes in logic. Assessing this movement Vasiliev names its following landmarks: Hegel's dialectical logic, Mill's inductive logic and his critical approach towards Aristotelian syllogistic, Sigwart's critique of the classical doctrine of modal judgements and, at last, the development of mathematical logic by Boole, Schroder, Poretsky, Peano, Frege and Russell.

LK, LJ, Dual Intuitionistic Logic, and Quantum Logic

Abstract: In this paper, we study the relationship among classical logic, intuitionistic logic, and quantum logic (orthologic and orthomodular logic). These logics are related in an interesting way and are not far apart from each other, as is widely believed. The results in this paper show how they are related with each other through a dual intuitionistic logic (a kind of paraconsistent logic). Our study is completely syntactical.