There are several ways for a formula to be “the contrary” of a thesis. When there is a negation we could call a formula an “antithesis” if it is the negation of a thesis—or its negation is a thesis—both properties being equivalent when the negation is classical.
When there is no negation or when the connective called “negation” is very different from classical negation, we are forced to look for a different notion.
Definition 1. In a propositional calculus, a substitution antithesis—or, for short, an s-antithesis—is a formula no instance of which is a thesis.
There is another kindred notion, but only for calculi for which a deducibility has been defined:
Definition 2. A deduction antithesis—for short, a d-antithesis—is a formula from which every formula is deducible.
Both notions have been used in the study of the deducibility of certain systems (see below, §4).
In the classical propositional calculus, the s-antitheses are simply the formulas which are negations of theses (and whose negations are theses).
In S5 and in all the weaker modal propositional calculi, there are s-antitheses which are not negations of theses (and whose negations are not theses); typical examples are ¬(Mp ⊃ Lp) and ¬(p ⊃ Lp) where p is a propositional variable, and ⊃, ¬, M and L are respectively material implication, negation, possibility and necessity (see ).