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connexive implication

Term used in a kind of relevance logic, existing in different versions but similarly motivated and using ideas from Aristotle (384-322 BC) and Boethius (CAD 480-524).

The relevant kind of implication is defined as holding when the antecedent of a conditional proposition is incompatible with the negation of the consequent.

This bans implications of the forms (where P and Q are propositions) 'If not P then P' and 'If P then Q, and if P then not Q', though both of these are valid in classical logic.

Connexive implication has been criticized for leading to the exclusion of logical principles it is in fact implausible to exclude.

Source:
R Routley and H Montgomery, 'On Systems Containing Aristotles' Thesis', Journal of Symbolic Logic (1968); critical, with references to expositions of the idea

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