Difficulty understanding why $ P \implies Q$ is equivalent to P only if Q.

Question

I have difficulties understanding why $ P \implies Q$ is equivalent to P only if Q. I do understand that in the statement "P only if Q", it means if $ \lnot Q \implies \lnot P$".

Regarding this example, Equivalence of $a \rightarrow b$ and $\lnot a \vee b$

If I win the lottery, then I will give you \$1 billion. This statement has the form $P \implies Q$.

But saying P only if Q, means "I win the lottery only if I give you $1 billion" doesn't sound so right. Is there anything I'm missing here?

Btw I agree that this is confusing. – goblin GONE Oct 05 '14 at 02:15 — goblin GONE, Oct 05 '14 at 02:15
See modus tollens and the rule of transposition. – Oct 05 '14 at 03:00 — , Oct 05 '14 at 03:00
Yep this caused me confusion too – Prince M Nov 30 '19 at 00:59 — Prince M, Nov 30 '19 at 00:59

score 5 · Accepted Answer · answered Oct 05 '14 at 02:00

$P$ only if $Q$ means that when $Q$ is false, then $P$ is false, i.e., ~$Q \implies$ ~$P$.

But the statement of ~$Q \implies$ ~$P$ is equivalent to the statement $P \implies Q$. Specifically, the former is the contrapositive of the latter.

Applying this to the example you gave:

"I win the lottery only if I give you \$1 billion" is equivalent to "if I don't give you \$1 billion, then I won't win the lottery", and the latter statement is the contrapositive to "if I win the lottery, then I will give you \$1 billion."

If you have doubts about why the contrapositive of a statement is an equivalent statement, have you tried making a truth table?

score 4 · Answer 2 · edited Feb 15 '24 at 07:35

"$P$ only if $Q$" means that the only way $P$ can be true is if $Q$ is true.

Now assume "$P$ only if $Q$." Assume also that $P$ is true. Then the only way this can be the case is if $Q$ is true. So $Q$ is true. Hence $P \rightarrow Q$.

Conversely, assume $P \rightarrow Q$. Then from the truth of $P$, we may infer the truth of $Q$. So the only way $P$ can be true is if $Q$ is true. So "$P$ only if $Q$."

score 2 · Answer 3 · answered Jun 04 '16 at 17:50

It really helps me to think about propositions as light bulbs. If a proposition is true, its light bulb is on. If a proposition is false, its light bulb is off. So $p \implies q$ means that whenever $p$ is on, $q$ is on too. If $p$ is off, $q$ could be either on or off. So now let's think about "$p$ only if $q$". If $q$ is off, then $p$ is DEFINITELY off, because if $p$ was on, the $q$ would've been on too. That means that $p$ can only be on when $q$ is on.

user182601 · Answer 4 · 2024-02-15T07:42:01.143

In everyday speech/English, "$P$ only if $Q$" suggests $Q$ causes $P$. Examples:

I'll go only if he isn't going. ("He isn't going" may help to cause "I'll go".)
I'll take on this project only if the company increases my pay. ("The company increases my pay" may help to cause "I'll take on this project".)

In contrast, in logic, "$P$ implies $Q$" or "$P$ only if $Q$" ignores any causality. In particular, there's no suggestion that either $P$ causes $Q$ or $Q$ causes $P$.

So, although "I win the lottery only if I give you \$1 billion" is logically correct, it sounds weird in everyday speech because it suggests that "I give you $1 billion" causes "I win the lottery" (which is of course absurd).

This is perhaps similar to how the statement, "If I'm a trillionaire, then pigs can fly," is logically true (but nonetheless sounds weird to most).

score 0 · Answer 5 · answered Oct 05 '14 at 03:36

I find it easier to think of the signs $$\implies,\impliedby,\iff$$ as inequalities between the truth values of statements. For instance, $$A\implies B$$ is the same as saying $$[A]\leq[B],$$ where $[S]$ denotes the truth value of the statement $S$.

If $[B]=0$, then $[A]\leq[B]\implies[A]=0$, so $A$ is true only (but not necessarily) if $B$ is true.

Difficulty understanding why $ P \implies Q$ is equivalent to P only if Q.

5 Answers5

Linked