
Paradoxes of Material Implication (1997) - jpelecanos
https://legacy.earlham.edu/~peters/courses/log/mat-imp.htm
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dwheeler
Material implication has its advantages, but it does sometimes lead to errors.
That said, there are ways to reduce the problem.

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:
[https://dwheeler.com/essays/allsome.html](https://dwheeler.com/essays/allsome.html)

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.

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tr352
There's another "solution" to this paradox: if we assert something we are
guided by a set of "conversational principles". For example, asserting "X
implies Y" if we _know_ that X is false is inappropriate. If X is false,
"not-X" would be the appropriate assertion.

According to this theory, there's nothing wrong with the truth-functional
meaning of "X implies Y". We just need to take into account what is implied by
asserting "X implies Y", rather than e.g. "not-X", or "X and Y".

Same with disjunction: "X or Y" is true if we know that X is true. However, if
we assert "X or Y", it is implied that we're not certain that X is true,
otherwise we would have used "X", which is the simplest way to convey what
that fact.

This is known as Grice's Pragmatic Defence of Truth-Functionality.

~~~
YeGoblynQueenne
Thanks, I didn't know about that. Here's the relevant article on the Stanford
encyclopedia of Philosophy, which your comment prompted me to read; others
might also find it useful:

[https://plato.stanford.edu/entries/conditionals/#GriPraDefTr...](https://plato.stanford.edu/entries/conditionals/#GriPraDefTru)

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YeGoblynQueenne
_Entailment: The Logic of Relevance and Necessity_ suggested in the article as
further reading on relevance logics is on the expensive side. If you can't get
hold of it, the Standford Encyclopedia of Philosophy goes on at some length
into the subject of relevance (or relevant) logics:

[https://plato.stanford.edu/entries/logic-
relevance/](https://plato.stanford.edu/entries/logic-relevance/)

And also the related subject of necessity and sufficiency:

[https://plato.stanford.edu/entries/necessary-
sufficient/](https://plato.stanford.edu/entries/necessary-sufficient/)

Btw, all these are issues with material implication in propositional logics.
I'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.

For example, if I say that ∀x,y P(x) → Q(y), or even P(x) → P(y), it'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).

But, I don't know, there may be something I'm missing. Am I wrong about this?

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bitL
This was one of my headaches while studying any type of logic - material
implication properties just sound weird and don'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 -> F yields F; the rest should be
either undefined or defined based on their relationship/context (something
like what relevance logic tries to do).

~~~
pulisse
> one of my headaches while studying any type of logic

Non-classical logics don't typically feature a material conditional.

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danharaj
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.

On the other hand I found intuitionistic implication more... intuitive :)

