Connexive Logic - History

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, (A → B) and (~A → B), can be true simultaneously. The proof is,

1. (A → B) hypothesis
2. (~A → B) hypothesis
3. (~B → ~A) 1, Transposition
4. (~B → B) 2, 3, Hypothetical Syllogism

Aristotle then declares step 4 to be impossible, completing the reductio. But if step 4 is impossible, it must be because Aristotle accepts its denial, ~(~B → B), as a logical truth.

Aristotelian syllogisms (as opposed to Boolean 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 C.E. logicians had divided into four or five distinct schools concerning the correct understanding of conditional ("if...then...") 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 (as translated by Kneale and Kneale).

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,

• (p → q) =df ~(p ° ~q) - where ° indicates compatibility,

requires that Aristotle's Thesis be a logical truth, provided we assume that every statement is compatible with itself - which seems fairly fundamental to the concept of 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. (Of course, logicians were not necessarily aware of belonging to the connexivist school.) 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 E. J. Nelson and P. F. Strawson.