Consequential Implication and the Implicative Conditional

Gilberto Gomes, Claudio Pizzi, Eric Raidl

Abstract:

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.