# not understanding this row of truth table for logical implication

provided we have this truth table where "p->q" means "if p then q":

| p | q | p->q |
| T | T |   T  |
| T | F |   F  |
| F | T |   T  |
| F | F |   T  |


My understanding is that "p->q" means "when there is p, there is q". The second row in the truth table where p is true and q is false would then contradict "p->q" because there is no "q" when "p" is present.

Why then, does the third row of the truth table not contradict "p->q"? If "q" is true when "p" is false, then "p" is not a condition of "q".

I have not taken any logic class so please explain it in laymen's terms.

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 I assume for now that when you say "p is not a condition for q", you mean p is not a necessary condition for q. And so it is not. "If p then q" does not mean p is necessary for q; it means p is sufficient for q. – Michael Hardy Oct 7 '11 at 23:00 Consider the following (totally informal) proposition: if it rains tomorrow, I'll bring my umbrella. However, the fact that I'll bring my umbrella doesn't imply it will rain tomorrow. – Fernando Martin Oct 7 '11 at 23:09

If you don't put any money into the soda-pop machine, and it gives you a bottle of soda anyway, do you have grounds for complaint? Has it violated the principle, "if you put money in, then a soda comes out"? I wouldn't think you have grounds for complaint. If the machine gives a soda to every passerby, then it is still obeying the principle that if one puts money in, one gets a soda out.

Similarly, the only grounds for complaint against $p\to q$ is the situation where $p$ is true, but $q$ is false. This is why the only F entry in the truth table occurs in this row.

If you imagine putting an F on the row to which you refer, the truth table becomes the same as what you would expect for $p\iff q$, but we don't expect that "if p, then q" has the same meaning as "p if and only if q".

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 It does not violate the principle but we cannot be sure that the soda machine works so we cannot be sure that the principle is true. However, it seems that I have misinterpreted the purpose of the truth table. – user701510 Oct 8 '11 at 5:05 @user701510, the truth table is designed to be useful under a quantifier, so that we can define that "the soda machine works" means "every time someone walks past the machine, it is true that if they put money in, a soda comes out". We want to interpret the if-then such that we don't consider the machine to not work simply because some people walk past without buying from it. – Henning Makholm Oct 8 '11 at 17:09

$p\Rightarrow q$ is an assertion that says something about situations where $p$ is true, namely that if we find ourselves in a world where $p$ is true, then $q$ will be true (or otherwise $p\Rightarrow q$ lied to us).

However, if we find ourselves in a woekd where $p$ is false, then it turns out that $p\Rightarrow q$ did not actually promise us anything. Therefore it can't possibly have lied to us -- you could complain about it being irrelevant in that situation, but that doesn't make it false. It has delivered everything it promised, because it turned out that it actually promised nothing.

As an everyday example, it is true that "If John jumps into a lake, then John will get wet". The truth of this is not affected by the fact that there are other ways to get wet. If, on investigating, we discover that John didn't jump in to the lake, but merely stood in the rain and now is wet, that doesn't mean that it is no longer true that people who jump into lakes get wet.

However, one should note that these arguments are ultimately not the reason why $\Rightarrow$ has the truth table it has. The real reason is because that truth table is the definition of $\Rightarrow$. Expressing $p\Rightarrow q$ as "If $p$, then $q$" is not a definition of $\Rightarrow$, but an explanation of how the words "if" and "then" are used by mathematicians, given that one already knows how $\Rightarrow$ works. The intuitive explanations are supposed to convince you (or not) that it is reasonable to use those two English words to speak about logical implication, not that logical implication ought to work that way in the first place.

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