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Why is F $\implies$ T taken as true? Why is this the "convention"?

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So is this operation defined. –  Harold May 13 '13 at 8:21
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Otherwise "$\implies$" would be the same thing as "$\iff$". –  Marc van Leeuwen May 13 '13 at 8:25
    
You can prove, using some logic axiomatization, that for $x=False$ $x\Rightarrow \overline x$ –  Harold May 13 '13 at 8:27
    
This might help to understand the pragmatic point, which might by taken before the interpretation of the arrow $P \Longrightarrow Q$ as "$P$ implies $Q$": You consider function of two statements ($Q,P$) which both have two possible values ("true" and "false") and a composed statement (like $P \Longrightarrow Q$) also has one of two possible values. This is why there are $2^4$ truth functions and all must have some name. "It's defined that way" means the arrow is one of the functions for which $( false \Longrightarrow true ) = true$. –  NikolajK May 13 '13 at 8:46

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up vote 10 down vote accepted

It's a convention in classical logic. Not all logical systems make this convention.

To see why this convention makes sense, consider a statement of the form $P\implies Q$, which I make, swearing that I'm telling the truth. When will you be able to sue me for lying? Well, if $P$ is true, but $Q$ is false then certainly you can sue me. But if $P$ is not true and $Q$ is, and you try to sue me, the judge will wonder what the hell you want. I only said that if $P$ is true then $Q$ is true. So, if $P$ is false, I did not commit to anything and thus your case is dismissed (i.e., I was telling the truth).

The point is that the implication is material implication, not that of causal implication. You may disagree with that interpretation and develop a logic where it is interpreted differently (for instance, in paraconsistent logic a statement of the form $F\implies T$ is dismissed as meaningless, not having any truth value (simply saying, don't bother about such nonsense as pondering what results from a false assumption)). Still, using classical logic in mathematics is very powerful, and so we make this convention.

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