Systematic construction of natural deduction systems for many-valued logics

A construction principle for natural deduction systems for arbitrary, finitely-many-valued first order logics is exhibited. These systems are systematically obtained from sequent calculi, which in turn can be automatically extracted from the truth tables of the logics under consideration. Soundness and cut-free completeness of these sequent calculi translate into soundness, completeness, and normal-form theorems for natural deduction systems

مسألة التتميم في منطق الجبر ثلاثي القيمة

درسنا في هذا البحث مسألة التتميم في منطق الجبر ثلاثي القيمة على منطق لوكاسفيتش ثلاثي القيمة بالنسبة لجملة أدوات منطقية تامة حيث تمت دراسة هذه المسألة في منظومة اشتقاق غاسبرغ وتقديم دراسة شاملة تضمنت أهم النظريات و قواعد الاشتقاق التي تم وضعها والتي أدت إلى تقديم برهان مميز لنظرية التتميم في منطق لوكاسفيتش ثلاثي القيمة. We study in this research completeness problem in three valued algebraic logic on Lukasiewicz 3 valued logic for a functionally complete set of connectives where we studied this problem in Wajseberg derivation system and we presented a comprehensive study of the most important theorems and derivation rules which led to a very unique and special proof of completeness theorem in Lukasiewicz 3 valued logic

مسألة التتميم في منطق الجبر ثلاثي القيمة

درسنا في هذا البحث مسألة التتميم في منطق الجبر ثلاثي القيمة على منطق لوكاسفيتش ثلاثي القيمة بالنسبة لجملة أدوات منطقية تامة حيث تمت دراسة هذه المسألة في منظومة اشتقاق غاسبرغ وتقديم دراسة شاملة تضمنت أهم النظريات و قواعد الاشتقاق التي تم وضعها والتي أدت إلى تقديم برهان مميز لنظرية التتميم في منطق لوكاسفيتش ثلاثي القيمة. We study in this research completeness problem in three valued algebraic logic on Lukasiewicz 3 valued logic for a functionally complete set of connectives where we studied this problem in Wajseberg derivation system and we presented a comprehensive study of the most important theorems and derivation rules which led to a very unique and special proof of completeness theorem in Lukasiewicz 3 valued logic

Sequent Calculi for the classical fragment of Bochvar and Halld\'en's Nonsense Logics

In this paper sequent calculi for the classical fragment (that is, the conjunction-disjunction-implication-negation fragment) of the nonsense logics B3, introduced by Bochvar, and H3, introduced by Halld\'en, are presented. These calculi are obtained by restricting in an appropriate way the application of the rules of a sequent calculus for classical propositional logic CPL. The nice symmetry between the provisos in the rules reveal the semantical relationship between these logics. The Soundness and Completeness theorems for both calculi are obtained, as well as the respective Cut elimination theorems.Comment: In Proceedings LSFA 2012, arXiv:1303.713

Neutrality and Many-Valued Logics

In this book, we consider various many-valued logics: standard, linear, hyperbolic, parabolic, non-Archimedean, p-adic, interval, neutrosophic, etc. We survey also results which show the tree different proof-theoretic frameworks for many-valued logics, e.g. frameworks of the following deductive calculi: Hilbert's style, sequent, and hypersequent. We present a general way that allows to construct systematically analytic calculi for a large family of non-Archimedean many-valued logics: hyperrational-valued, hyperreal-valued, and p-adic valued logics characterized by a special format of semantics with an appropriate rejection of Archimedes' axiom. These logics are built as different extensions of standard many-valued logics (namely, Lukasiewicz's, Goedel's, Product, and Post's logics). The informal sense of Archimedes' axiom is that anything can be measured by a ruler. Also logical multiple-validity without Archimedes' axiom consists in that the set of truth values is infinite and it is not well-founded and well-ordered. On the base of non-Archimedean valued logics, we construct non-Archimedean valued interval neutrosophic logic INL by which we can describe neutrality phenomena.Comment: 119 page

The Logic of Internal Rational Agent

In this paper, we introduce a new four-valued logic which may be viewed as a variation on the theme of Kubyshkina and Zaitsev's Logic of Rational Agent \textbf{LRA} \cite{LRA}. We call our logic $\bf LIRA$ (Logic of Internal Rational Agency). In contrast to \textbf{LRA}, it has three designated values instead of one and a different interpretation of truth values, the same as in Zaitsev and Shramko's bi-facial truth logic \cite{ZS}. This logic may be useful in a situation when according to an agent's point of view (i.e. internal point of view) her/his reasoning is rational, while from the external one it might be not the case. One may use \textbf{LIRA}, if one wants to reconstruct an agent's way of thinking, compare it with respect to the real state of affairs, and understand why an agent thought in this or that way. Moreover, we discuss Kubyshkina and Zaitsev's necessity and possibility operators for \textbf{LRA} definable by means of four-valued Kripke-style semantics and show that, due to two negations (as well as their combination) of \textbf{LRA}, two more possibility operators for \textbf{LRA} can be defined. Then we slightly modify all these modalities to be appropriate for $\bf LIRA$. Finally, we formalize all the truth-functional $n$-ary extensions of the negation fragment of $\bf LIRA$ (including $\bf LIRA$ itself) as well as their basic modal extension via linear-type natural deduction systems