$$1.\quad p\quad if\quad q\\ \equiv if\quad q\quad then\quad p\\ \equiv q\rightarrow p\\ \\$$$$2.\quad p\quad only\quad if\quad q\\ \equiv if\quad p\quad then\quad q\\ \equiv p\rightarrow q\\ \\$$$$3.\quad p\quad only\quad if\quad q\\ \equiv if\quad q\quad then\quad p\\ \equiv q\rightarrow p\\ \\$$$$4.\quad p\quad iff\quad q\\ \equiv (p\rightarrow q)\wedge (q\rightarrow p)$$
I think #3 is wrong but I'm not sure why.
"$p$, only if $q$" means "if $p$, then $q$." It's infrequently used except as a component of the phrase "if and only if."
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.
A more serious example: If a function is continuous at $x_0$, its limit at $x_0$ exists. However, there exists plenty of discontinuous functions which have limits at their point of discontinuity, so "A function has a limit only if it's continuous" does not hold.
– Steve Pap Jan 08 '14 at 17:23From 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.
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 $Q \subseteq P$.
