Non-classical propositional calculi
Storrs McCall, McCall, Storrs · Oxford University Research Archive (ORA) (University of Oxford) · 1964
There exist well-known varieties of implication, such as strict, intuitionist, three-valued and rigorous, which are non-classical in the sense of being more restrictive than material implication. But there exists also a type of implication, intuitively plausible, which is nonclassical not only in being more restrictive, but in satisfying certain theses which are classically false. These theses are exceedingly venerable, dating back to Aristotle and Boethius, but, despite their plausibility, have been generally rejected by logicians since. It has not been noticed, however, that in Sextus Empiricus reference is made to a species of Stoic implication which fits them perfectly.In this work formal recognition is given to this species of implication, known as connexive implication. It is shown that none of the well-known systems of prepositional logic is connexive, and a new system is accordingly constructed. A proof of consistency is given, and a number of problems posed for further investigation.