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The implicative conditional strengthens the strict conditional with the possibility of the antecedent and of the contradictory of the consequent.<jats:inline-formula><jats:alternatives><jats:tex-math>$${p\\Rightarrow q}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>p<\/mml:mi><mml:mo>\u21d2<\/mml:mo><mml:mi>q<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>is thus defined as<jats:inline-formula><jats:alternatives><jats:tex-math>$${\\lnot } \\Diamond {(p \\wedge \\lnot q) \\wedge } \\Diamond {p \\wedge } \\Diamond {\\lnot q}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mo>\u00ac<\/mml:mo><mml:mo>\u25ca<\/mml:mo><mml:mrow><mml:mo>(<\/mml:mo><mml:mi>p<\/mml:mi><mml:mo>\u2227<\/mml:mo><mml:mo>\u00ac<\/mml:mo><mml:mi>q<\/mml:mi><mml:mo>)<\/mml:mo><mml:mo>\u2227<\/mml:mo><\/mml:mrow><mml:mo>\u25ca<\/mml:mo><mml:mrow><mml:mi>p<\/mml:mi><mml:mo>\u2227<\/mml:mo><\/mml:mrow><mml:mo>\u25ca<\/mml:mo><mml:mrow><mml:mo>\u00ac<\/mml:mo><mml:mi>q<\/mml:mi><\/mml:mrow><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. 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\u2019s theses and weak Boethius\u2019 thesis, as well as some highly entrenched principles of conditionals, such as conjunction of consequents, disjunction of antecedents, modus ponens, cautious monotonicity and cut.<\/jats:p>","DOI":"10.1007\/s10992-023-09715-6","type":"journal-article","created":{"date-parts":[[2023,11,27]],"date-time":"2023-11-27T05:02:21Z","timestamp":1701061341000},"page":"1-47","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["The Implicative Conditional"],"prefix":"10.1007","volume":"53","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6153-4979","authenticated-orcid":false,"given":"Eric","family":"Raidl","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3667-3851","authenticated-orcid":false,"given":"Gilberto","family":"Gomes","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,11,27]]},"reference":[{"key":"9715_CR1","doi-asserted-by":"crossref","unstructured":"Austin, J.L. 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