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?

Answer 1

$P$ 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?

Answer 2

"$P$ 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 $Q$."

Answer 3

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.

Answer 4

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).

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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).

Answer 5

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.