Abstract
This paper investigates the implicative conditional, a connective intended to describe the logical behavior of an empirically defined class of natural language conditionals, also named implicative conditionals, which excludes concessive and some other conditionals. The implicative conditional strengthens the strict conditional with the possibility of the antecedent and of the contradictory of the consequent. \({p\Rightarrow q}\) is thus defined as \({\lnot } \Diamond {(p \wedge \lnot q) \wedge } \Diamond {p \wedge } \Diamond {\lnot q}\). We explore the logical properties of this conditional in a reflexive normal Kripke semantics, provide an axiomatic system and prove it to be sound and complete for our semantics. The implicative conditional validates transitivity and contraposition, which we take to be integral parts of reasoning and communication. But it only validates restricted versions of strengthening the antecedent, right weakening, simplification, and rational monotonicity. Apparent counterexamples to some of these properties are explained as due to contextual factors. Finally, the implicative conditional avoids the paradoxes of material and strict implication, and validates some connexive principles such as Aristotle’s theses and weak Boethius’ thesis, as well as some highly entrenched principles of conditionals, such as conjunction of consequents, disjunction of antecedents, modus ponens, cautious monotonicity and cut.
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Acknowledgements
We are grateful to two anonymous reviewers, whose detailed comments led to substantial improvements in our paper.
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Open Access funding enabled and organized by Projekt DEAL. Eric Raidl’s work was funded by the Deutsche Forschungsgemeinschaft (EXC number 2064/1, project no. 390727645) and the Baden-Württemberg Foundation (program ‘Verantwortliche Künstliche Intelligenz’).
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G.G. proposed the idea of the implicative conditional and wrote Sections 1–3. E.R. developed the logical system and wrote Sections 4–7 and Appendices A–B. All sections and the article as a whole were enriched by the authors’ joint work and continuous exchange.
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Raidl, E., Gomes, G. The Implicative Conditional. J Philos Logic 53, 1–47 (2024). https://doi.org/10.1007/s10992-023-09715-6
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DOI: https://doi.org/10.1007/s10992-023-09715-6