Amer Amikhteh; Seyyed Ahmad Mirsanei
Abstract
In this paper, a non-classical axiomatic system was introduced to classify all moods of Aristotelian syllogisms, in addition to the axiom "Every a is an a" and the bilateral rules of obversion of E and O propositions. This system consists of only 2 definitions, 2 axioms, 1 rule of a premise, and moods ...
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In this paper, a non-classical axiomatic system was introduced to classify all moods of Aristotelian syllogisms, in addition to the axiom "Every a is an a" and the bilateral rules of obversion of E and O propositions. This system consists of only 2 definitions, 2 axioms, 1 rule of a premise, and moods of Barbara and Datisi. By adding first-degree propositional negation to this system, we prove that the square of opposition holds without using many of the other rules of classical logic (including double negation elimination).We then show that the Propositional Substructural Logic SLe is the best logic to study Aristotelian Syllogisms. Also, based on the IFLe square of opposition, the rules of conversation and the rules of negation are completely proved in Muzaffar's logic. For this purpose, we used the monadic first-order logic with the same standard deductive apparatus of quantifiers in classical logic, plus the axioms of "some a is an a" and "some not-a is a not-a". Finally, to show that there is no existential commitment to general terms in categorical logic, the Strong Four-Valued Relevant-classical Logic KR4 was used. With the same existential interpretation of the quantifiers and the standard translation of the quarter quantified.
Amer Amikhteh; Lotfollah Nabavi
Abstract
The uninorm logic UL is a fuzzy, substructural and semi-relevant logic. The Gentzen-style system for UL is obtained by removing the contraction rules and weakening from the Gentzen-style system of Godel fuzzy logic. The UL lacks "excluded middle", "positive paradox" and "negative paradox". The truth ...
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The uninorm logic UL is a fuzzy, substructural and semi-relevant logic. The Gentzen-style system for UL is obtained by removing the contraction rules and weakening from the Gentzen-style system of Godel fuzzy logic. The UL lacks "excluded middle", "positive paradox" and "negative paradox". The truth function of uninorm is a relevance weakening of the t-norm function. In this article, we introduce the new logic ULΔ. ULΔ is obtained by adding Δ to UL. ULΔ, an expansion of classical logic, is a normal semilinear modal logic; i.e. it is strongly sound and complete w.r.t. a linearly ordered algebra. And with the theorem of (p→q)∨Δ(q→p) it is distinguished from other standard systems of modal logic. Δφ is intuitively interpreted as "true that φ" or more precisely "classically true that φ". In this paper, we introduce the semi-classical logic ULΔ with four approaches, axiomatizations, hypersequent calculi, algebraic semantics and standard semantics. metatheorems we are considering include Delta deduction, strong soundness, strong standard completeness and definability of classical logic.
