# Conditional Statements only if.

Conditional Statements "only if."

For some reason, be it some bad habit or something else, I can not understand why the statement "p only if q" would translate into p implies q. For instance, I have the statement "Samir will attend the party only if Kanti will be there." The way I interpret this is, "It is true that Samir will attend the party only if it is true that Kanti will be at the party;" which, in my mind, becomes "If Kanti will be at the party, then Samir will be there."

Can someone convince me of the right way?

Student100
Gold Member
If Samir is at the party, then it also implies that Kanti will be there. Since you know Samir will only attend if Kanti attends.

So if you have p, then it implies q.

It doesn't work the same the other way around, having q doesn't always imply you also have p unless the statement is true both ways. So Kanti could attend the party, but Samir decides to sleep in instead.

So you could have q without p. Although the example above isn't a very good one because you explicitly said Samir will attend if Kanti is there, which implies that the statement works both ways. However, p only if q, doesn't give any information if q always guarantees the existence of p, so the the safer bet is to assume q can exist independently of p, but p requires the existence of q.

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Just write a truth table. P only if Q will mean that P will never be true whenever Q is false.

P: T
Q: T
P only if Q: T

P: T
Q: F
P only if Q: F

P: F
Q: T
P only if Q: T

P: F
Q: F
P only if Q: T

Thus, P only if Q is false when P is true and Q is false, and true otherwise. Thus it is equivalent to P => Q

The way I interpret this is, "It is true that Samir will attend the party only if it is true that Kanti will be at the party;" which, in my mind, becomes "If Kanti will be at the party, then Samir will be there."

But this is explicitly wrong.

The first sentence is false whenever Samir attends but Kanti does not, because we said that Samir will only be there when Kanti is. The first statement is S => K

The second sentence still allows for Samir to attend without Kanti. This statement is K=> S

PeroK
Homework Helper
Gold Member
2020 Award
For some reason, be it some bad habit or something else, I can not understand why the statement "p only if q" would translate into p implies q. For instance, I have the statement "Samir will attend the party only if Kanti will be there." The way I interpret this is, "It is true that Samir will attend the party only if it is true that Kanti will be at the party;" which, in my mind, becomes "If Kanti will be at the party, then Samir will be there."

Can someone convince me of the right way?

"p only if q" means "you can't have p without q" means "if you have p you must have q" means "p implies q"

"p is going to the party only if q is going to the party" means "p doesn't go without q" means "if p is going, then q must be going too" means "p is going to the party implies q is going to the party"

(In this case, p might not go, even if q goes, but he won't go without him! So, you don't have q implies p)

No, everyone, I am familiar with the definition of a conditional statement (that is, when it is true and when it is false); and I am well-acquainted with truth tables. I am having an issue with interpreting what "p only if q" translates to. Read my original post, that it exactly what I said.

P only if q: q is a necessary condition for p. Therefore p implies q.

Blahdeblah, you, just as everyone else, have only spewed facts. I am asking for insight as to why "p only if q" translates into "p implies q." I understand what a necessary condition is, but why is q the necessary condition?

You misinterpreted your own example. Samir will be present only if Kanti is present. This allows Kanti to turn up at the party beforehand without Samir. In which case your last assertion is incorrect.

I think it is the difference between "only if" and "if and only if".

Three cases:
If Manti is at the party then I will be there
- doesn't exclude the possibility that I will be there if Manti is not
Only if Manti is at the party. Then I will be there
- doesn't exclude the possibility that I might not be there if Manti is
If and only if Manti is at the party then I will be there
- if Manti is there then I will be too. If Manti is not there, then neither will I.

I hope this is clear enough.

Student100
Gold Member
P can only exist if q exists. That's all the statement is trying to tell you.

Did you read my post? I'm not sure how you can put it another way.

Blahdeblah, you, just as everyone else, have only spewed facts. I am asking for insight as to why "p only if q" translates into "p implies q." I understand what a necessary condition is, but why is q the necessary condition?

P implies Q
Let P
Then Q

Thus, Q occurs any time P does. P does not occur without Q. Thus P only if Q.

What else is there? I don't understand what you're asking. What do you mean by "translate?" They are two ways of saying the same thing because the truth tables match, that's all that matters. Seeing that they have the same logical outcomes means they are the same, period. Asking how they "translate" doesn't mean anything.