Given 2 variables:
s="it rained" & t="the floor is wet"
we have it that, "if it rained, the floor is wet", i.e. s->t
I read from my lecture notes that the last 2 rows of the truth table is explained by 'the principle of excluded middle' like this:
But isn't principle of excluded middle expressed as "~B v B = T"? How does this law relate to the explanation? Also, if we cannot determine the validity of the implication s->t from s, then how come the statement automatically becomes true? I've been struggling with this for quite a while. Please explain with the use of rain/wet analogy. Thanks.
Best Answer
Well, this isn't exactly right. As it stands, the statement given is the combination of the law of the excluded middle along with the non-contradiction: $\lnot (P \land \lnot P)$, by virtue of how it says "exactly".
Moving on, though, the definition of $P\implies Q$ is $Q \lor \lnot P$. This is referred to as the "material conditional", which can only be violated by $P\land \lnot Q$. Well, law of the excluded middle says that if it isn't false, it's true.
Take some time to wrap your head around this. It doesn't really fit the bill of implication as occurs in natural language, since we would want $P\implies Q$ to be false in certain cases (e.g. when the truth values are "independent" in some sense, but still line up properly), and certain types of conditionals try to reconcile this.