a problem in justifying ONLY IF truth table

Question

EDITED:I saw the link above but it does't answer my question.I have a clear understanding of "P ONLY IF Q", I know that it equals "IF P THEN Q" .. but I see that there is a difference in the case where Q is True and P is False... in "IF P THEN Q" I know that it's true .. but in the "P only if Q" it seems like t violates the "if Q" part because it means when having Q is True then P is true - by understanding the "if" part in the "if Q"-.

I am facing a problem in understanding one of the cases in the "P only if Q" truth table-which I didn't find anyone asking about it- that is: when Q is True and P is False ..I am trying to justify how this can be a true condition, I understand that the "only" part means P must be False When Q is False but the "if" part means -as we know from implication- when the "if" part-Q- is True, the other proposition -P- should be true- but this is not the case-

the following is an example for more illustration: P is:"we'll go swimming" Q is:"weather is sunny" then P ONLY IF Q: "we'll go swimming only if the weather is sunny" in the case when Q is True but P is False: "the weather is sunny but we'll not go swimming" here the violation of "IF" part is clearer

Answer 1

Consider: "You are a bachelor only if you are male"

That makes sense, right? You can be a bachelor only if you are male: if you are not a male, then you cannot be a bachelor. So, being male is a necessary condition for being a bachelor, which is why we use 'only if'

However, does it follows from this that if you are male, then you are automatically a bachelor? Of course not! Plenty of males are not bacheler. Thus, being male is not a sufficient condition for being a bachelor. Hence, from 'P only if Q' you cannot infer that it must be the case that 'if Q then P'. Indeed, when all you have is that 'P only if Q', then Q can be true and P false: you can be a bachelor only if you are male is true, and it can also be true that one is a male but not a bachelor.

Answer 2

Consider the example:

$P$ is:"we'll go swimming" and $Q$ is:"weather is sunny". Then "$P$ ONLY IF $Q$" will be:

"we'll go swimming only if the weather is sunny".

The statement states that the only case when we'll go swimming is when weather is sunny.

It does not exclude the possibility that we refuse to go swimming in a sunny day.

In terms of "truth condition", thus means that the case "$\text {F only if T}$" is not in contradicition with the truth-table for "$\text { if P, then Q}$".

Answer 3

EDITED:I saw the link above but it does't answer my question.I have a clear understanding of "P ONLY IF Q", I know that it equals "IF P THEN Q"

Yes, read it as: "$P$ can be true only when $Q$ is true".

The claim "$P$ only if $Q$" is not a guarantee that $P$ will be true when $Q$ is.

The claim is a promise that we will not have $P$ true while $Q$ is false.

Thus in the truth table, this promise holds for all rows except the one where $P$ is true and $Q$ false.   Thus it is "If $P$, then $Q$".

$$\boxed{\begin{array}{c:c|c:c} P & Q & P\text{ only if }Q&\text{if }P\text{, then }Q\\\hline \top&\top & \top&\top\\\color{red}\top&\color{red}\bot&\color{red}\bot&\color{red}\bot\\\bot&\top&\top&\top\\\bot&\bot&\top&\top \end{array}}$$