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The conditional operator, $\phi \implies \psi$, is True for the values $TT, FT, FF$ and false for $TF$. I can easily understand why it's true for $TT$ and false for $TF$, but why is it for $FT$ and $FF$? As far as I can see, we can't come to any conclusion in those cases:

If $\phi$ is false but $\psi$ is true, to me that doesn't imply anything about the truth or falsity of $\phi \implies \psi$. Couldn't it be that when $\phi$ is true then $\psi$ is false? Why does this situation imply that $\psi$ follows from $\phi$? e.g.

$\phi : n < 4$, $\psi : n > 2$. If $\phi$ is false then clearly $\psi$ is true, but why does that imply that the truth of $\psi$ follows from the truth of $\phi$? If $\phi$ is true then either $\psi$ is true or $\psi$ is false, depending on the situation.

Similarly, $\phi: n < 4$, $\psi : n < 2$. If $\phi$ is false then $\psi$ is false but if $\phi$ is true then $\psi$ is not necessarily true.

Where did I go wrong with my reasoning?

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  • $\begingroup$ You didn't go wrong. This has long been a matter of dispute. See for example the Logic section of the Wikipedia article on Clarence Irving Lewis. He wrote a book almost a century ago, expressing the same concerns. $\endgroup$ – MikeC Feb 18 '14 at 2:32
  • $\begingroup$ Yea, if I were the one to decide I would just say it's not defined on this values, but the lecture said the point of the conditional was to have an allegory to implication that was defined in all cases, and I guess just saying it's true makes more sense than saying it's false. $\endgroup$ – MCT Feb 18 '14 at 2:33
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The point behind the truth table is that you want it to behave in the following way: It is not the case that $\phi$ is true but $\psi$ is false. So you set it up to reflect that.

The reason it is DEFINED for other truth values as you mention above is this: Suppose you want to verify the truth of the statement If $x$ is even, then $x^{2}$ is even. Now suppose that $x=3$. Then $x^{2}=9$. If you didn't define the truth values in the way above, you have that the statement is false (since you would have defined $FF$ is $F$). But this doesn't reflect what we want (intuitively the statement should only be wrong if we can find $x$ even but $x^{2}$ false). Similar examples can be thought of for the other case: Think about "If $x>0$, then $x^{2}>0$".

This does create some weird situations sometimes: If $x\neq{x}$, then dolphins are sharks is a true statement. However this doesn't affect things much when you do math: You use modus ponens to prove things (i.e. if $p\implies{q}$ and $p$, then $q$). Here while If $x\neq{x}$, then dolphins are sharks is a true statement, you don't have $x\neq{x}$ to conclude that dolphins are sharks, so it doesn't really hurt anything.

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I think that this explanation of S.C.Kleene, Mathematical Logic (1967) [pag.10 - footnote 12] is the best "short" elucidation of it :

The ordinary usage certainly requires that "If $A$ , then $B$" to be true when $A$ and $B$ are both true, and to be false when $A$ is false but $B$ is false. So only our choice for $T$ in the third and fourth lines can be questioned. But if we changed $T$ to $F$ in both these lines, we would simply get a synonym for $\land$; in the third line only, for $\lnot$ . If we changed $T$ to $F$ in the fourth line only, we would loose the useful property of our implication that "If $A$ , then $B$" and "If $\lnot B$ , then $\lnot A$" are true under the same circumstances [...].

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"Follows from" is often not the best interpretation of the conditional operator. ϕ⟹ψ can probably be better understood to mean that ψ is at least as true as ϕ, or no less true than ϕ. If we know that ϕ = F, then whether ψ = T or ψ = F doesn't matter. Either way, ψ is at least as true as ϕ, and not less. The content or meaning of ϕ and ψ don't matter; we are only comparing truth values.

How we establish that ϕ⟹ψ in the first place is another question entirely. There are at least three ways to do this. For one, we could claim ϕ⟹ψ simply from the knowledge that ϕ is false or ψ is true. But while that is perfectly true, it's also totally useless. We can't use that fact to deduce anything else meaningful. For a second, We could simply assume it is true and examine the consequences. That's a common intermediate step in more involved proofs. For a third, we could assume ϕ and through some chain of reasoning deduce ψ. Then, we can say that ψ follows from ϕ. But ϕ⟹ψ alone says nothing about which of these three ways we used, so "follows from" is not the best way to read it.

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