The p implies q statement is often described in various ways including:
(1) if p then q (i.e. whenever p is true, q is true)
(2) p only if q (i.e. whenever q is false, p is false)
I see the truth table for (1) as
p | q | if p then q
-------------------
T | T | T
T | F |
F | T |
F | F |
I see the truth table for (2) as
p | q | p only if q
-------------------
T | T |
T | F | F
F | T |
F | F | T
How are the two statements the same? What is wrong with my understanding?
Addenda
1). There are some excellent excellent answers/suggestions here but what really worked for me was the following tip:
I think the intuitive way to think of this is if something is
contradicted then it is false but if nothing can be contradicted it is
by default true.
2). I have now learned that a conditional statement that is true by virtue of the fact that its hypothesis is false (i.e. true by default) is called vacuously true.
Best Answer
"If P then Q" means that whenever P is true, Q is true as you have observed. Thus, if both are true, it follows that the statement as a whole is true. Now, what if P is true, but Q is false? What does this imply about the statement as a whole? In this case, it is not true that "if P is true, Q is true". Hence, the statement as a whole is false.
Now, consider the cases where P is false. Does it matter whether Q is true or false? Again, we are evaluating the truthfulness of the statement that "If P is true, then Q is also." Since P is false, the statement no longer has a bearing on the factuality of the statement we are assessing. By convention, we thus assert that the statement is still true. (At the very least, it doesn't contradict the original statement. This is a complex topic; you would do well to look up some related questions regarding this. See this one for instance.)
Now, using the advice above, try to fill in the truth table for the other one. You will find that they are the same; therefore, the two are logically equivalent.