Truth Values and Connectives in Some Non-Classical Logics
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Abstract:
The question as to whether the propositional logic of Heyting, which was a formalization of Brouwer's intuitionistic logic, is finitely many valued or not, was open for a while (the question was asked by Hahn). Kurt Gödel (1932) introduced an infinite decreasing chain of intermediate logics, which are known nowadays as Gödel logics, for showing that the intuitionistic logic is not finitely (many) valued. Now we know that the propositional intuitionistic logic is infinitely many valued (with a countably many logical values). In this paper we provide another proof for this result of Gödel, from the perspective of Kripke model theory. Švejadr and Bendova (2000) proved that in Gödel fuzzy logic the conjunction and implication are not definable by the rest of the propositional connectives (while disjunction is definable by conjunction and implication). In this paper, we show that disjunction is not definable by implication and negation in Gödel fuzzy logic; two proofs, one by Kripke models and one by fuzzy semantics, are provided for this new theorem.
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Journal title
volume 5 issue 19
pages 31- 36
publication date 2019-08-01
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