TITLE: How can it be that $(P \rightarrow Q) \vee (Q \rightarrow P)$ is a tautology?
QUESTION [7 upvotes]: I consider $(P \rightarrow Q) \vee (Q \rightarrow P)$, that is $(\neg P \vee Q) \vee (\neg Q \vee P)$ and so $\neg P \vee Q \vee \neg Q \vee P$, which is a tautology.
It seems strange to me that, given two arbitrary formulas, necessarily  one implies the other or viceversa. (I am thinking for example of this case: P stands for 'Ann loves animals' and Q 'Ann is a student')
Where is my mistake? Perhaps in the interpretation of $P$ and $Q$?
Any clarification will be much appreciated.

REPLY [7 votes]: The proposition $(P \rightarrow Q) \vee (Q \rightarrow P)$ is always true.  This does not mean that either $P \rightarrow Q$ is always true or $Q \rightarrow P$ is always true.  Isn't that what you said?

REPLY [2 votes]: In your example, it means that either the statement

If Ann loves animals, then Ann is a student

Or the statement

If Ann is a student, then Ann loves animals.

(or both) is correct.
If Ann is, indeed, a student, then the first statement is true if she loves animals or not. If Ann is not a student, then any statement beginning with "If Ann is a student" is correct, so , in conclusion:

If Ann is a student, statement 1 is correct. Statement 2 may or may not be correct (it is correct if Ann indeed loves animals and incorrect if she does not), but that is irrelevant.
If Ann is not a student, then statement 2 is correct. Statement 1 may or may not be correct (it is correct iff Ann does not love animals), but again, that is irrelevant.