The basics of logic

Question

"$B$ if $A$" means "$A \to B$".
"$B$ only if $A$" means only "$B \to A$".
In 2, in the operator we only added "only". We didn't delete "If". So why the operation caused by "If" ($A \to B$) does not happen anymore?

This is just an unfortunate consequence of the terminology we use. We call $\implies$ ‘if’, and $\impliedby$ ‘only if’. There is no ‘only’ relation, mathematicians just chose confusing names for the operations haha. We can also use the less ambiguous ‘is necessary’ and ‘is sufficient’. Then “$B$ is necessary for $A$” is the sentence $A \implies B$, and “$B$ is sufficient for $A$” is the sentence $B \implies A$. — Lemmon, Jun 11 '23 at 01:12
You are reading 'only if' as 'if, and just if'; in other words, you are reading 'only if' as conveying extra information on top of 'if'. While this reading is neither outlandish nor idiosyncratic, another way is to read 'only if' as conveying necessity; this second reading is the conventional one in all of mathematics. — ryang, Jun 11 '23 at 06:45
If I look for alternative and unconventional reading of "B only if A" as a non-native English user, using line 1 I may (mis)read that as "A -> B only". In the example from Bram28, most students pass all courses, but if you do all the homework ($H$), you can pass this course only ($P$ only). In the example from a comment in a linked question, usually you get full dinner, but today if you clean up your room, you will get ice cream only. — peterwhy, Jun 11 '23 at 14:56
I came across this question in the "reopen" review queue. After some thought, I voted to keep this question closed, as no context was given. It would have been helpful if one of the original reopen voters had explained their action, as I could then have understood why they'd voted and so may have voted differently. — user1729, Jun 13 '23 at 09:55

score 2 · Answer 1 · answered Jun 11 '23 at 02:15

The intuition for ‘B if A’ meaning $A \implies B$ is just that we’re re-arranging the sentence ‘if A, then B’.

For ‘B only if A’ the intuition is a bit trickier to grasp. Basically, it’s saying that the only way B can be the case is if A is also the case. So, we write $B \implies A$ to capture ‘B only if A’. For example, if Alice says, “I’ll go to the party only if Jamal goes,” then if Alice actually goes to the party, we can be sure Jamal also does.

The brute force way to remember it is that $B \iff A$ means ‘B if and only if A’. So, if the intuition for ‘B if A’ makes sense, you just have to remember what ‘B only if A’ means.

Nice tip in the final paragraph. By the way, 'B only if A' is still intuitive when rearranged as 'only if A, then B'; in fact, as long as the comma is not dropped, the rearrangement makes crystal clear the correct reading, in mathematics, of 'B only if A'. — ryang, Jun 11 '23 at 13:26

score 2 · Answer 2 · answered Jun 11 '23 at 12:43

Don't think of 'only if' as a combination of an 'only' operation and an 'if' operation. Rather, it is one operation.

So why use two words? Part of this is how language historically evolved, but I guess that the 'if' is signalling a condition of some kind. However, there is an important distinction between sufficient conditions and necessary conditions. If $P$ is a sufficient condition for $Q$, we symbolize that as $P \to Q$. But is $P$ is a necessary condition for $Q$, then that becomes $Q \to P$. And that is exactly what is going on when we say '$Q$ only if $P$': this is our way of expressing that $P$ is a necessary condition for $Q$

Necessary conditions are always tricky. Because we use the word 'condition', the immediate intuition is put it in the antecedent (the 'if') part of a conditional. This is true for sufficient conditions, but not for necessary conditions. Consider:

'You can pass the course ($P$) only if you do all the homework ($H$)'

Our intuition might be to symbolize this as $H \to P$ ... after all, you first do all the homework, and only later you pass the course. However, this is not correct logically/truth-functionally: Just because someone does all the homework does not make it true that they pass the course, because maybe they also need to do well on the quizzes and final. However, what we can say is that if we ever encounter someone who passed this particular course, then we can conclude that they must have done all the homework. And we can symbolize the given statement as $P \to H$ ... which is rather unintuitive.

Finally, by contraposition, this last statement is equivalent to $\neg H \to \neg P$. And maybe you like that better: at least the condition (doing the homework) is now again back in the antecedent, where we like it to be. However, because it is a necessary condition, we had to negate it: if you do not do the homework, then you will not pass the course either.

The basics of logic

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