Difference between "only if" and "if and only if"

Solution 1:

"$p$, only if $q$" means "if $p$, then $q$." It's infrequently used except as a component of the phrase "if and only if."

Solution 2:

Yes, the third statement is incorrect. It should start "$q$ only if $p$".

The "only" makes all the difference! "$p$ if $q$" means that whenever $q$ is true, $p$ is necessarily true as well. This is the same as "if $q$ then $p$" or "$q\rightarrow p$.

On the other hand, "$p$ only if $q$" means that, unless $q$ is true, $p$ cannot be true; equivalently "if $q$ is false, then $p$ must be false as well, which is written $\neg q \rightarrow \neg p$, which is the same (by contrapositive) as $p \rightarrow q$.

Thus, we have demonstrated that "$p$ if $q$" and "$p$ only if $q$" are logical converses of each other.

Solution 3:

From a just published book by Jan von Plato, Elements of Logical Reasoning (Cambridge UP, 2013), pag 11:

The two sentences if A, then B and B if A seem to express the same thing. Natural language seems to have a host of ways of expressing a conditional sentence that is written $A \rightarrow B$ in the logical notation. Consider the following list :

From A, B follows; A is a sufficient condition for B; A entails B; A implies B; B provided thet A; B is a necessary condition for A; A only if B.

The last two require some thought. The equivalence of $A$ and $B$, $A \leftrightarrow B$ in logical notation, can be read as A if and only if B, also A is a necessary and sufficient condition for B. Sufficiency of a condition as well as the 'if' direction being clear, the remaining direction is the opposite one. So A only if B means $A \rightarrow B$ and so does B is a necessary condition for A.

It sound a bit strange to say that B is a necessary condition for A means $A \rightarrow B$. When one thinks of conditions as in $A \rightarrow B$, usually $A$ would be a cause of $B$ in some sense or other, and causes must precede their effects. A necessary condition is instead something that necessary follows, therefore not a condition in the causal sense.

Solution 4:

A nice way to think about this is with sets.

If:

Consider outcome sets $P \subseteq Q$, then if an outcome in $P$ occurs, that outcome will also be in $Q$ i.e. $Q$ if $P$.

Only If:

Now consider $Q \subseteq P$, in this case the outcome is in $Q$ only if it is also in $P$ (note here it is not correct to say the outcome is in $Q$ if it is in $P$).

Hence $Q$ if $P$ is like saying $P \subseteq Q$ and $Q$ only if $P$ is like saying $Q \subseteq P$.