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I understand that "P iff Q" means that "if P is true then Q is true" and also "if Q is true then P is true".

However, I don't understand why can't we just say "P only if Q" to mean the same thing? How does "only if" mean differently than "if only if"?

Claire
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    Hello, it's a convention between mathematicians, but it is not used by the rest of people. It would be better to say "is equivalent to", but habits don't change easily. – Mateo_13 Oct 14 '22 at 03:45
  • "only if" would imply that $P$ is only possible when $Q$ is true, but it wouldn't necessitate that $P$ is true when $Q$ is true, i.e. you could have $Q$ and $¬P$. – Set Oct 14 '22 at 03:45
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    John Conway argued for using "just if" to mean "if and only if" in a more concise way, but I don't think most people would understand that if you started using it. – Misha Lavrov Oct 14 '22 at 03:49
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    You will see the show only if you have a ticket. Does that also mean that you will see the show if you have a ticket? No. You might have a ticket and skip the show, or you might be refused entrance for some other reason, or the show might be canceled. – MJD Oct 14 '22 at 03:54
  • More bout John Conway's brilliant thoughts can be found here. – Kurt G. Oct 14 '22 at 06:29

2 Answers2

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P only if Q is another way of saying if P then Q, but does not imply if Q then P. P if Q is another way of saying if Q then P. If you think about them you should be able to make truth tables that show they are synonyms. Iff includes both of these and if you want it you need to say it.

Ross Millikan
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In the following paragraphs $P$ and $Q$ denote statements.

I personally agree with you that "only if" is a biconditional statement that includes the "if" case, as reflected by the fact that linguistically we can infer that "only if" is stronger than "if" and hence the use of the word "if" in "if and only if" is redundant. That is to say, in my opinion, "if $P$ then $Q$" is in math symbols $P \implies Q$; and "only if $P$ then $Q$" is in math symbols $P\iff Q$.

However, it is an unfortunate convention at this point that mathematicians take "if $P$ then $Q$" to be the good old implication $P\implies Q$, and "only if $P$ then $Q$" to be the converse implication $P \impliedby Q$. That is why most mathematicians understand "if and only if" to be the conjunction of both the conditional statement and its converse. One may argue this convention came to be because if we go by the linguistically obvious way, namely if we take "only if $P$ then $Q$" to mean $P\iff Q$ as you do, then there would be no convenient way to refer to the converse implication. Why would we need a convenient way to refer to the converse implication? Why not simply restate it as an implication? To understand the answer look at the following statements:

  1. If I need help, I will scream
  2. Only if I need help, I will scream.
  3. Only if I need help, I may scream.

Forget about the math convention for a moment. Let's just think about these statements logically:

In statement 1, multiple things can cause my scream, one of which is my need for help. Another could be my pain, or just fun if I am on a roller coaster. Suppose you see me screaming. You can't be sure which of the reasons actually caused my scream. i.e. you can't be sure whether I need help or not. My scream is necessary for me to need help, but it's not sufficient to conclude I need help. In math symbols statement 1 is $$\text{Ardy needs help}\implies\text{Ardy screams}$$

In statement 2 we know that the only thing in the world that can cause me to scream is my need for help. Nothing else can result into my scream -- I am some tough guy that no amount of pain can make him scream. Suppose you see me screaming. In this tough guy scenario you can be certain that I need help; because my need for help is the only thing that can cause me to scream. In a sense my need for help and me screaming are equivalent. That is my screaming is not only necessary for me to need help, but it's also sufficient. In math symbols, in my opinion, statement 2 is $$\text{Ardy needs help}\iff\text{Ardy screams}$$

In statement 3 it can happen that sometimes I need help but I don't scream. Suppose you see me screaming. Then you can conclude I need help, because despite the fact that sometimes I am in need of help and I don't scream, the only thing that may cause me to scream is my need for help. So my scream is sufficient to conclude I need help, but it's not necessary. In math symbols, in my opinion, statement 3 is $$\text{Ardy needs help}\impliedby\text{Ardy screams}$$

As you can see statements 2 and 3 sound very similar to each other. So it is not too hard to see why the wrong convention has prevailed among mathematicians i.e. why most mathematicians take "only if" to mean the converse implication and not the biconditional statement that you and I understand it to be.

Ardy
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  • Your explanation is the reason why I forego writing logical statements in english altogether and just go straight to using $\implies$. There's just too many ways to get a sentence with double meanings in english, it's like the meme that asked what 8/2(2) equals. – Dark Rebellion Oct 14 '22 at 17:05
  • You’re confusing P only if Q with Q only if P. Only if P, Q means Q->P, and P only if Q means P->Q. – PW_246 Nov 16 '22 at 18:40