Paradoxes of Material Implication (1997)

Material implication has its advantages, but it does sometimes lead to errors. That said, there are ways to reduce the problem.<p>One error is that sometimes people use material implication inside of for-all, and forget that if the antecedent is always false than the entire expression is always true. I specifically created a quantifier called allsome that counters that problem: <a href="https:&#x2F;&#x2F;dwheeler.com&#x2F;essays&#x2F;allsome.html" rel="nofollow">https:&#x2F;&#x2F;dwheeler.com&#x2F;essays&#x2F;allsome.html</a><p>Another problem is using material implication inside there is quantifier. That is almost always a mistake, as usually and is meant instead. However, that is pretty easy to detect automatically. Both Why3 and SPARK already attack that.

There&#x27;s another &quot;solution&quot; to this paradox: if we assert something we are guided by a set of &quot;conversational principles&quot;. For example, asserting &quot;X implies Y&quot; if we <i>know</i> that X is false is inappropriate. If X is false, &quot;not-X&quot; would be the appropriate assertion.<p>According to this theory, there&#x27;s nothing wrong with the truth-functional meaning of &quot;X implies Y&quot;. We just need to take into account what is implied by asserting &quot;X implies Y&quot;, rather than e.g. &quot;not-X&quot;, or &quot;X and Y&quot;.<p>Same with disjunction: &quot;X or Y&quot; is true if we know that X is true. However, if we assert &quot;X or Y&quot;, it is implied that we&#x27;re not certain that X is true, otherwise we would have used &quot;X&quot;, which is the simplest way to convey what that fact.<p>This is known as Grice&#x27;s Pragmatic Defence of Truth-Functionality.

<i>Entailment: The Logic of Relevance and Necessity</i> suggested in the article as further reading on relevance logics is on the expensive side. If you can&#x27;t get hold of it, the Standford Encyclopedia of Philosophy goes on at some length into the subject of relevance (or relevant) logics:<p><a href="https:&#x2F;&#x2F;plato.stanford.edu&#x2F;entries&#x2F;logic-relevance&#x2F;" rel="nofollow">https:&#x2F;&#x2F;plato.stanford.edu&#x2F;entries&#x2F;logic-relevance&#x2F;</a><p>And also the related subject of necessity and sufficiency:<p><a href="https:&#x2F;&#x2F;plato.stanford.edu&#x2F;entries&#x2F;necessary-sufficient&#x2F;" rel="nofollow">https:&#x2F;&#x2F;plato.stanford.edu&#x2F;entries&#x2F;necessary-sufficient&#x2F;</a><p>Btw, all these are issues with material implication in propositional logics. I&#x27;m not sure, but I think, in first-order (and, I guess, higher order) logics you can determine the relevance of premises to conclusions more easily, thanks to quantifiers.<p>For example, if I say that ∀x,y P(x) → Q(y), or even P(x) → P(y), it&#x27;s easy to see that the premises are irrelevant to the conclusions (and if not, I can always add that x ≠ y and, in higher order logics, that P ≠ Q).<p>But, I don&#x27;t know, there may be something I&#x27;m missing. Am I wrong about this?

This was one of my headaches while studying any type of logic - material implication properties just sound weird and don&#x27;t model what I call implication in my brain. IMO the only case when we can say anything about two unrelated propositions in general, is when T -&gt; F yields F; the rest should be either undefined or defined based on their relationship&#x2F;context (something like what relevance logic tries to do).

Classical logic is optimized for the verification that no contradiction is made and convenient mathematical manipulation: Boolean algebras are very nice. Unfortunately intuition goes out the window.<p>On the other hand I found intuitionistic implication more... intuitive :)