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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?

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    $\begingroup$ what is "Boethius's Thesis"? $\endgroup$ Commented Jun 3 at 6:37
  • $\begingroup$ @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 $\endgroup$
    – AUTIST INC
    Commented Jun 3 at 6:53
  • $\begingroup$ @GeraldEdgar, for more information, see this: en.wikipedia.org/wiki/Connexive_logic?wprov=sfti1# $\endgroup$
    – AUTIST INC
    Commented Jun 3 at 6:55
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    $\begingroup$ 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. $\endgroup$
    – Conifold
    Commented Jun 3 at 13:42
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    $\begingroup$ @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. $\endgroup$
    – Conifold
    Commented Jun 3 at 15:29

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