.This paper discusses Aristotle’s thesis and Boethius’ thesis, the most distinctive theorems of connexive logic. Its aim is to show that, although there is something plausible in Aristotle’s thesis and Boethius’ thesis, the intuitions that may be invoked to motivate them are consistent with any account of indicative conditionals that validates a suitably restricted version of them. In particular, these intuitions are consistent with the view that indicative conditionals are adequately formalized as strict conditionals.

This paper investigates the implicative conditional, a connective intended to describe the logical behavior of an empirically defined class of natural language conditionals, also namedimplicative 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.$${p\Rightarrow q}$$p⇒qis thus defined as$${\lnot } \Diamond {(p \wedge \lnot q) \wedge } \Diamond {p \wedge } \Diamond {\lnot q}$$¬◊(p∧¬q)∧◊p∧◊¬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. (shrink)