Connexive logic
Connexive logic is a class of non-classical logics designed to exclude the paradoxes of material implication. The characteristic that separates connexive logic from other non-classical logics is its acceptance of Aristotle's thesis, i.e. the formula,
as a logical truth. Aristotle's thesis asserts that no statement follows from its own denial. Stronger connexive logics also accept Boethius' thesis,
which states that if a statement implies one thing, it does not imply its opposite.
Relevance logic is another logical theory that tries to avoid the paradoxes of material implication.
Failure in classical logic
Adding Aristotle's thesis to classical logic leads to a contradiction, and therefore any logic emploing it needs to giv up one or more principles of classical logic. If is true, then the thesis comes out as false, because it is logically equivalent to.| 1 | ||
| 2 | material implication | |
| 3 | double negation elimination | |
| 3 | idempotency of disjunction |
History
Connexive logic is arguably one of the oldest approaches to logic. Aristotle's thesis is named after Aristotle because he uses this principle in a passage in the Prior Analytics.
It is impossible that the same thing should be necessitated by the being and the not-being of the same thing. I mean, for example, that it is impossible that B should necessarily be great if A is white, and that B should necessarily be great if A is not white. For if B is not great A cannot be white. But if, when A is not white, it is necessary that B should be great, it necessarily results that if B is not great, B itself is great. But this is impossible. An. Pr. ii 4.57b3.
The sense of this passage is to perform a reductio ad absurdum proof on the claim that two formulas, and, can be true simultaneously. The proof is:
| 1 | hypothesis | |
| 2 | hypothesis | |
| 3 | ~B | hypothesis |
| 4 | ~A | 1, 3, modus tollens |
| 5 | B | 2, 4, modus ponens |
| 6 | 3, 5, conditional proof |
Aristotle then declares the last line to be impossible, completing the reductio. But if it is impossible, its denial,, is a logical truth.
Aristotelian syllogisms appear to be based on connexive principles. For example, the contrariety of A and E statements, "All S are P," and "No S are P," follows by a reductio ad absurdum argument similar to the one given by Aristotle.
Later logicians, notably Chrysippus, are also thought to have endorsed connexive principles. By 100 BCE logicians had divided into four or five distinct schools concerning the correct understanding of conditional statements. Sextus Empiricus described one school as follows:
And those who introduce the notion of connexion say that a conditional is sound when the contradictory of its consequent is incompatible with its antecedent.
The term "connexivism" is derived from this passage.
It is believed that Sextus was here describing the school of Chrysippus. That this school accepted Aristotle's thesis seems clear because the definition of the conditional,
- := ~, where ° indicates compatibility,
The medieval philosopher Boethius also accepted connexive principles. In De Syllogismo Hypothetico, he argues that from, "If A, then if B then C," and "If B then not-C," we may infer "not-A," by modus tollens. However, this follows only if the two statements, "If B then C," and "If B then not-C," are considered incompatible.
Since Aristotelian logic was the standard logic studied until the 19th century, it could reasonably be claimed that connexive logic was the accepted school of thought among logicians for most of Western history. However, in the 19th century Boolean syllogisms, and a propositional logic based on truth functions, became the standard. Since then, relatively few logicians have subscribed to connexivism. These few include Everett J. Nelson and P. F. Strawson.