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Let us refer by MK3 to Kleene’s strong 3-valued matrix. An implicative expansion of MK3 is natural if the conditional function defining it verifies modus ponens, assigns a designated value to a conditional whenever it assigns the same value to its antecedent and its consequent, and, finally, it coincides with the classical conditional function when restricted to the “classical” values \(\mathfrak {t}\) and \(\mathfrak {f}\) . Two are the main results of this paper. (1) It is proven that, from the viewpoint of functional strength, there is only one 3-valued natural implication expansion of MK3 with the variable-sharing property, the logic we dub L3 \({ }^{\text{VSP}}\) . (2) It is shown that L3 \({ }^{\text{VSP}}\) is a significant and strong logic that can be seen from different perspectives, one of them being to consider it an expansion of classical positive propositional logic.

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The Only 3-Valued Logic Which Is a Natural Implication Expansion with the Variable-Sharing Property of Kleene’s Strong Logic

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  • 发表日期：

    2025年

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  • Gemma Robles,
  • José M. Méndez

摘要

Let us refer by MK3 to Kleene’s strong 3-valued matrix. An implicative expansion of MK3 is natural if the conditional function defining it verifies modus ponens, assigns a designated value to a conditional whenever it assigns the same value to its antecedent and its consequent, and, finally, it coincides with the classical conditional function when restricted to the “classical” values \(\mathfrak {t}\) and \(\mathfrak {f}\) . Two are the main results of this paper. (1) It is proven that, from the viewpoint of functional strength, there is only one 3-valued natural implication expansion of MK3 with the variable-sharing property, the logic we dub L3 \({ }^{\text{VSP}}\) . (2) It is shown that L3 \({ }^{\text{VSP}}\) is a significant and strong logic that can be seen from different perspectives, one of them being to consider it an expansion of classical positive propositional logic.