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What is the exact meaning of "only" in "only if"?

It is not clear to me whether "x only if y" means that

  • y is the only factor that guarantees x and denies the possibility of any other thing guaranteeing x,

or whether it merely means that

  • y by itself has the power to guarantee x and still leaves open the possibility of factors other than y being sufficient conditions for x.

So far, it seems that if y by itself can guarantee x, this removes the possibility of any other thing being a necessary condition of x, as this contradicts the fact that y by itself serves as a necessary condition for x.

ryang
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Princess Mia
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    Consider a (stern) parent saying: "You will get ice cream only if you clean up your room". – ahulpke Feb 11 '23 at 19:21
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    $A$ if $B$ means "If $B$ then $A.$" $$$$ $A$ only if $B$ means "If $A$ then $B.$" – Adam Rubinson Feb 11 '23 at 19:23
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    The statement "x only if y" does not mean that y guarantees x (alone or otherwise). To express that y guarantees x, you can say "x if y" or "if y then x". – Andreas Blass Feb 11 '23 at 19:37
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    "x only if y" means that it can not be the case that x holds but y doesn't. – Natalie Clarius Feb 11 '23 at 20:56
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    Basically, it is only a way of saying... We have that “P if Q” means “if Q, then P". And this reads "no Q without P". On the other hand, “P only if Q” is the other case: without "only", we cannot have a difference between the two phrases. It means "if P, then Q", and it reads "no P without Q". – Mauro ALLEGRANZA Feb 12 '23 at 08:38
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    Single words do not have an "exact meaning", each must be read in context. So you should be asking "what is the exact meaning of 'only if' ?" – ancient mathematician Jun 11 '23 at 06:35

2 Answers2

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There are $3$ scenarios to consider. I will address each one by one to hopefully clarify the terminology being used here. First note that the symbol ($\implies$) can be understood to mean the word "implies" when reading through my answer below.

"$P$ is true if $Q$ is true"

This can be rewritten as $Q \implies P$. In other words, if we know that $Q$ is true, then we know that $P$ must be true.

Note$_1$: This doesn't dismiss the possibility that $P$ could be true even when $Q$ is not true. All we know from this statement is that if $Q$ is true, then $P$ must be true - it doesn't tell us anything about whether or not $P$ can or can't be true when $Q$ does not hold.

"$P$ is true only if $Q$ is true"

This can be rewritten as $P \implies Q$. In other words, if we know that $P$ is true, then we know that $Q$ must be true.

Note$_2$: This doesn't dismiss the possibility that $Q$ could be true even when $P$ is not true. All we know from this statement is that if $P$ is true, then $Q$ must be true - it doesn't tell us anything about whether or not $Q$ can or can't be true when $P$ does not hold.

"$P$ is true if and only if $Q$ is true"

This can be rewritten as $P \iff Q$. In other words, if we know that $P$ is true, then we know that $Q$ must be true. And if we know that $Q$ is true, then we know that $P$ must be true. This means that they are equivalent since if one of them is true, then the other must also be true (and if one of them is false, then the other must also be false).

Note$_3$: This is equivalent to $(Q \implies P)$ and $(P \implies Q)$. Unlike the above two examples, this one gives us information about $P$ and $Q$ regardless of whether or not one is known to be true or false. All we need to know is the truth value of either $P$ or $Q$ and this immediately tells us the truth value of the other.

FD_bfa
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What is the exact meaning of "only" in "only if"?

  1. In logic and mathematics, the conventional reading of ‘B only if A’ is that for B to be true, A (or equivalent) is a necessary condition. As such, we can mechanistically decode the phrase ‘only if’ as is a sufficient condition for, that is, implies.

  2. However, in everyday English, it is not unnatural to read ‘B only if A’ to mean that B is true just if A is true (and not if A is false), that is, B is true if and only if A is true. (In this reading, ‘only if’ is a stronger form of and communicates more than ‘if’.)

    In everyday English, while the mathematics reading above is reasonable too, considering the pragmatics of social interaction, the utterance ‘B only if A’ frequently gets taken to mean ‘B if and only if A’ anyway!

So far, it seems that if y by itself can guarantee x, this removes the possibility of any other thing being a necessary condition of x, as this contradicts the fact that y by itself serves as a necessary condition for x.

You're incoherently conflating two key ideas. y guaranteeing x (y⇒x; y being sufficient for x) and y being necessary for x (y⇐x; x⇒y) are independent notions.

It is not clear to me whether "x only if y" means that y is the only factor that guarantees x, and denies the possibility of any other thing guaranteeing x,

I don't think this particular reading is common; after all, if y is sufficient for x, then surely any condition equivalent to y is also sufficient for x. Oh, but if you mean to say, "y, or equivalent, is the only condition that guarantees x", then this reading corresponds to the second paragraph above (so isn't the correct reading in mathematics).

or whether it merely means that y by itself has the power to guarantee x, and still leaves open the possibility of factors other than y being sufficient conditions for x.

No, ‘x only if y’ is never equivalent to ‘x if y’.

ryang
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