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<p>This paper develops a logic that invalidates <i>Ex Contradictione Quodlibet</i> but delivers a package of other results that are unusual for a paraconsistent logic. In particular, it validates Disjunctive Syllogism. But it also is structurally classical, meaning that its consequence relation is reflexive, monotonic, and transitive. As a trade-off it will invalidate the Law of Addition which allows us to infer a disjunction from one of its disjuncts. The logic will also allow for certain logical theorems such as instances of the Law of the Excluded Middle and the Law of Non-Contradiction. This logic is captured with both a natural proof system and a simple modal model theory, showing the relevant completeness theorem. I discuss adding a conditional operator and briefly explore routes for responding to worries about some less intuitive results of the logic such as the logical invalidity of modus tollens.</p>

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A Simple Paraconsistent Logic without Addition

  • 期刊论文

  • 发表日期：

    2026年1月5日

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  • Philipp Mayr

摘要

This paper develops a logic that invalidates Ex Contradictione Quodlibet but delivers a package of other results that are unusual for a paraconsistent logic. In particular, it validates Disjunctive Syllogism. But it also is structurally classical, meaning that its consequence relation is reflexive, monotonic, and transitive. As a trade-off it will invalidate the Law of Addition which allows us to infer a disjunction from one of its disjuncts. The logic will also allow for certain logical theorems such as instances of the Law of the Excluded Middle and the Law of Non-Contradiction. This logic is captured with both a natural proof system and a simple modal model theory, showing the relevant completeness theorem. I discuss adding a conditional operator and briefly explore routes for responding to worries about some less intuitive results of the logic such as the logical invalidity of modus tollens.