Consequential Implication and the Implicative Conditional

This paper compares two logical conditionals which are strengthenings of the strict conditional and avoid the paradoxes of strict implication. The logics of both may be viewed as extensions of KT, and the two conditionals are interdefinable in KT. The implicative conditional requires that its antecedent and consequent be both contingent. The consequential conditional may be viewed as a weakening of the implicative conditional, insofar as it also admits the case in which the antecedent and the consequent are strictly equivalent (either both necessary or both impossible). The two conditionals share a number of properties, among them Transitivity, Contraposition, Aristotle’s Thesis, Weak Boethius’ Thesis and Aristotle’s Second Thesis. They also share some restricted principles such as Possibilistic Monotonicity, Possibilistic Simplification and Possibilistic Right Weakening. They differ in relation to Identity, which is validated by consequential implication, while the implicative conditional only validates the restricted principle of Possibilistic Identity. The relations between the two conditionals are represented by two Aristotelian cubes of opposition, one involving the contrariety between If A, then B and If A, then ¬B, according to Weak Boethius’ Thesis, and the other the contrariety between If A, then B and If ¬A, then B, according to Aristotle’s Second Thesis. We also explore the relations between the two logical conditionals and natural language conditionals, emphasizing the dependence of the latter on the context, and the need to distinguish natural language conditionals which may be viewed as consequential or implicative, on one side, and concessive and some other types of conditionals, on the other.