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In the truth table of if P then Q, In the case of P being False and Q being true the whole statement is true. But with examples such as following it wont make sense at all. If you get an A then you go to the movies. P=you get an A Q= you can go to movies If P is not correct and Q is correct then the statement is as follows: If you dont get an A then you can go to the movies. This contradicts the original statement as being true and shouldnt be true. Can you please explain?

Orah
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  • It doesn't really contradict it, you go to the movie if you get A, you don't get A and you don't go to the movies, and the last you don't get A and still go to the movies. Perhaps to better illustrate the point if you fall then you will hurt yourself, if you don't fall you can still hurt yourself. – kingW3 Nov 23 '21 at 17:05
  • The truth table is essentially the definition for the meaning of the implication symbol. Why that definition is more convenient than others is the real question. If you want to express yourself in a form other mathematicians will recognise and understand, the mathematical definition is the one to use. Other truth tables are expressed differently. The fact that the symbol may not match your expectations from natural language or intuition may feel like a problem, but it is not a mathematical problem. – Mark Bennet Nov 23 '21 at 17:12
  • "If P is not correct and Q is correct then the statement is as follows: ... ." No. The original statement "if P then Q" is still, "If you get an A then you can go to movies." What has changed is we now have new information about the truth of parts of that statement. As it happens, these do not contradict the statement. What would contradict the original statement is if you did get the A and then the person who promised to let you see a movie if you got an A broke their promise and refused to let you see a movie. – David K Nov 23 '21 at 17:44
  • Have you ever heard the expression, "damned if you do, damned if you don't"? Let P = "you do" and let Q = "you're damned"; the expression translates to "If P then Q and if not P then Q." Both parts can be true. The result of both parts being true is not a contradiction; it just means that no matter what you did or didn't do, you were going to be damned anyway. – David K Nov 23 '21 at 17:57
  • I like to think of an if-then statement as a promise. And the truth of the statement is whether the promise is broken. – Arthur Nov 23 '21 at 18:08

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It is a common mistake to believe that the negation of a conditional is obtained by negating the hypothesis and maintaining the thesis. In fact, the negation of conditional is obtained by the equivalence $$\neg(p\to q) \iff p\wedge \neg q$$

Then, in your example, the negation of "If you get an A then you can go to the movies" is "You get an A and you can't go to the movies". It's not another conditional.

The sentence "If you don't get an A then you can go to the movies" is just another sentence with same simple propositions, thus don't contradicts.

ecrin
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HoonX
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    I was not trying to negate the statement. What I am asking is that in the case of my example if P is False and Q is True, then the statement can not be True but mathematically should be True because of the truth table. That's confusing. – Orah Nov 23 '21 at 17:24
  • +1 to David K's comment and this answer. @Orah When you created a new statement and claimed that "This contradicts the original statement", you *are* negating the original statement (but doing it wrongly). – ryang Nov 23 '21 at 17:30
  • By contradicting, I meant if the original statement (if you get an A then you can go to movies is true(if P then Q) then the statement (if you dont get an A then you can go to movies) should be true too. But it is not. – Orah Nov 23 '21 at 17:46
  • @Orah What do those statements even mean? Are they universal rules over a set of events -- you will take many tests, and the rule says what happens every time you take a test? Is it a statement about a particular test given in a particular class on Monday, November 29, 2021 and whether you can go to a movie that is showing on the following Saturday evening? – David K Nov 23 '21 at 17:52
  • @Orah if "if you get an A then you can go to movies" is true, then "if you dont get an A then you can go to movies" should be true too. This is also incorrect: say, in fact, and <can't go to the movies>; then notice that the former statement (False implies False) is true, yet the latter statement (True implies False ) is false. P.S. To notify users of messages, insert an @name . Anyhow, here's my response to your main Question. – ryang Nov 23 '21 at 18:06
  • @ryang analyse my example for (False implies True). and . Then it wont make sense. – Orah Nov 23 '21 at 18:34
  • @David K Let's say If you get an A this morning then you can go to movies this afternoon. It is the same argument. If P is false then you didnt get an A and if Q is True, you can go to movies. it still wont make sense. – Orah Nov 23 '21 at 18:40
  • No one is answering my specific question. I know the truth table says for P being False and Q being true then "If P then Q" is True. How does that apply to my specific example. – Orah Nov 23 '21 at 18:56
  • @Orah "If you get an A you can go to the movies." Please explain where in that statement it says what happens if you don't get an A. Or why you can't get a B and then still go to the movies. – David K Nov 23 '21 at 19:54
  • @David K Doesnt that automatically imply that if you dont get an A then you can not go to movies??? Or doesnt it imply that only if you get an A then you can go to movies??? – Orah Nov 23 '21 at 20:08
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    @Orah 1. You are incorrect in insisting that "If you pinch yourself, our planet continues revolving" and "If don't don't pinch yourself, our planet continues revolving" must have different truth values. 2. "IF you get an A you can go to the movies" is $A\Rightarrow M.$ "You can go to the movies ONLY IF you get an A" is $M\Rightarrow A.$ It is accurate & honest for the parent to make the former claim even though they have already (secretly) decided to let the child go to the movies. – ryang Nov 23 '21 at 20:22
  • @ryang can you please apply (False implies Truth) to my example and tell me how that's true? – Orah Nov 23 '21 at 20:29
  • @Orah "Doesnt that automatically imply that if you dont get an A then you can not go to movies??? Or doesnt it imply that only if you get an A then you can go to movies???" Why? Why should either of those statements be implied by "If you get an A you can go to the movies"? Do you think it is bad parenting if the promised reward is delivered regardless of success on the task? Fine, then it is bad parenting. It is still not a contradiction to the plain statement of fact about what would happen if the child got the A. – David K Nov 24 '21 at 01:06