# Thread: Why is "P only if Q" another way of representing "Q if P"

1. ## Why is "P only if Q" another way of representing "Q if P"

They are very similar in form and it confuses me why it makes sense to reverse P and Q while using "only if".

Similarly, what's the reason for using "P is sufficient for Q" and "Q is necessary for P". Sufficient in what way? Necessary in what sense?

2. ## Re: Why is "P only if Q" another way of representing "Q if P"

Originally Posted by Elusive1324
They are very similar in form and it confuses me why it makes sense to reverse P and Q while using "only if".
Similarly, what's the reason for using "P is sufficient for Q" and "Q is necessary for P". Sufficient in what way? Necessary in what sense?
1) If $\displaystyle 2^2=4$ then $\displaystyle 3^2=9$.
2) if $\displaystyle 2^2=4$ then $\displaystyle 1^2=2$.
3) if $\displaystyle 2^2=3$ then $\displaystyle 1^2=1$.
4 if $\displaystyle 2^2=3$ then $\displaystyle 1^2=2$.

Of those four statements only one is false, all the others are true. Can you see which?

Both "P is sufficient for Q" and "Q is necessary for P" are equivalent to "P implies Q"
The truth of P implies the truth of Q.
Which of those four fails to satisfy that definition? WHY?

3. ## Re: Why is "P only if Q" another way of representing "Q if P"

Originally Posted by Plato
1) If $\displaystyle 2^2=4$ then $\displaystyle 3^2=9$.
2) if $\displaystyle 2^2=4$ then $\displaystyle 1^2=2$.
3) if $\displaystyle 2^2=3$ then $\displaystyle 1^2=1$.
4 if $\displaystyle 2^2=3$ then $\displaystyle 1^2=2$.

Of those four statements only one is false, all the others are true. Can you see which?

Both "P is sufficient for Q" and "Q is necessary for P" are equivalent to "P implies Q"
The truth of P implies the truth of Q.
Which of those four fails to satisfy that definition? WHY?
The truth of the implication depends on the truth value of the conclusion when the hypothesis is true. When the hypothesis is false, the implication is always true (vacuously true -- no way to check). That's how an implication is defined. By definition of implication, 2 is false. The implication is defined such that "anything follows from a false premise". Thus, if it is known that the implication is true and the hypothesis is true, it must follow that the conclusion is true. Also if the implication is true and the hypothesis is false, nothing is said about the conclusion.

Is my understanding of implication sufficient to answer my question? If not, can you be more precise with your definition: "the truth of P implies the truth of Q"? Do you mean "whether Q is true or false depends on the truth of P (that is if P is true then Q is true. If P is false then Q is false)?

4. ## Re: Why is "P only if Q" another way of representing "Q if P"

Originally Posted by Elusive1324
Is my understanding of implication sufficient to answer my question?

Originally Posted by Elusive1324
{C]an you be more precise with your definition: "the truth of P implies the truth of Q"?
Well, it is certainly not my definition. It has been in use for centuries. The operative word is implies. The truth of P implies the truth of Q; Q follows P; if P is true then Q is true.

Originally Posted by Elusive1324
Do you mean "whether Q is true or false depends on the truth of P (that is if P is true then Q is true. If P is false then Q is false)?
Absolutely NOT.

Let's agree that this is an empirically true statement: "If it rains then the grass is wet".
I wake today and see that my yard is thoroughly wet. May I conclude that it has rained. No indeed, the sprinkler came on at 3am to water the grass. So P, it rained, is false but Q, the grass is wet, is true. Does that make the statement "If it rains then the grass is wet" incorrect? Not at all! It is still a fact.

What would make it not a fact? Well suppose you saw a really hard rain, but the grass around you is still bone dry. That falsifies the statement.

5. ## Re: Why is "P only if Q" another way of representing "Q if P"

Well, it is certainly not my definition. It has been in use for centuries. The operative word is implies. The truth of P implies the truth of Q; Q follows P; if P is true then Q is true.

Absolutely NOT.

Let's agree that this is an empirically true statement: "If it rains then the grass is wet".
I wake today and see that my yard is thoroughly wet. May I conclude that it has rained. No indeed, the sprinkler came on at 3am to water the grass. So P, it rained, is false but Q, the grass is wet, is true. Does that make the statement "If it rains then the grass is wet" incorrect? Not at all! It is still a fact.

What would make it not a fact? Well suppose you saw a really hard rain, but the grass around you is still bone dry. That falsifies the statement.
I would like to examine why it is that this example would constitute a counterexample.

The unstated assumption is that the conditional is true and has not yet been disproved. If the conditional is true, then if the hypothesis is true, the conditional (with the imply operator) is defined such that it follows that the conclusion must be true. However suppose we find an example where the hypothesis is true and the conclusion false under the same unstated assumption. There must be a contradiction. By our assumption, if the hypothesis is true, the conclusion must be true. The conclusion is false. Therefore, our assumption is wrong and that the conditional is actually false.

What I would like to emphasize is that I think we define "implication" so that a conditional needs to be proved false, otherwise it is true ONLY because it has not been proven false.

Is my explanation the basis for why a counterexample proves a conditional statement false? (And also what it means for a statement to be true i.e., a conditional that has not yet been proven false?