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.
On
Remember the definition of implication.
$P \rightarrow Q$ means $\lnot P \lor Q$ In words, either $P$ is false or $Q$ is true.
$Q\rightarrow P$ means $\lnot Q \lor P$ In words, either $Q$ is false or $P$ is true.
Putting it all together gives us $$(P\rightarrow Q) \lor (Q\rightarrow P) \equiv \lnot P \lor Q \lor \lnot Q \lor P \equiv P\lor \lnot P \lor Q \lor \lnot Q$$
The above cannot logically be false. Either $P$ is true or $\lnot P $ is true, not to mention we also have $Q \lor \lnot Q$. You can verify the tautology by using a truth table.
Recall that material implication tells us nothing about the relevance of $P$ with respect to $Q$, nor that of $Q$ with respect to $P$. That is, material implication does not necessarily signify any causal relation between one proposition and another.
This is what may seem very counter-intuitive because in natural language, we think of one event implying another event as being causal, or otherwise being based on some relevant relationship between one statement and another. In logic, we are not to take $P\rightarrow Q$ as saying anything more than "either $P$ is false, or $Q$ is true."
On
The kicker is to remember that a true statement is implied by any statement, and that a false statement implies every statement. (This is one upshot of the fact that $A\to B\equiv\neg A\vee B.$ The other is that a true statement does not imply a false statement.) Hence, if P is true or P is false, then your compound statement is true.
To clarify that last sentence, let's consider your particular example statements, in light of the possibilities. Does Ann love animals? If so, then $Q\to P$ holds; if not, then $P\to Q$ holds. Regardless, one of the two implications is true, and so $(P\to Q)\vee(Q\to P)$ is necessarily true.
On
Sometimes logical implication is not that intuitive.
For your $P$ and $Q$ we have the following:
Either Ann loves animals, or she doesn't. If she does, then $Q\to P$ is true.
If she doesn't, then $P\to Q$ is true.
Either way one of the components in the disjunction is true.
On
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:
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?