# Why isn't Boethius's Thesis more commonly accepted in mathematics and logic?

Why isn't Boethius's Thesis, that the negation of an implication is another implication where the consequent is negated, a commonly accepted axiom in mathematics and logic? It is an axiom of connexive logic, but are there benefits/obstacles to adopting it more commonly?

• what is "Boethius's Thesis"? Commented Jun 3 at 6:37
• @GeraldEdgar: Boethius’s Thesis is that the negation of an implication is another implication where the consequent is negated. It’s an axiom of connexive logic Commented Jun 3 at 6:53
• @GeraldEdgar, for more information, see this: en.wikipedia.org/wiki/Connexive_logic?wprov=sfti1# Commented Jun 3 at 6:55
• Boethius’ thesis is weaker than equivalence of ¬(p → q) and p → ¬q, Boethius only asserted one direction (p → ¬q) → ¬(p → q) , see Lenzen, Rewriting the History of Connexive Logic. And it is not commonly accepted because the resulting logic is not truth-functional, which makes it harder to use, and because 'causal implication' that motivates it is not relevant in many contexts where logical implication is used. Whether p 'causes' q makes little sense in mathematics, for example. Commented Jun 3 at 13:42
• @GeraldEdgar Because in mathematics we only care that theorems do not admit counterexamples, and this is exactly how material implication is defined - no true antecedent with false consequent. It also helps that this definition is technically easy to handle because it is truth-functional. Commented Jun 3 at 15:29