# In symbolic logic why (p->q) and (p->~q) don't contradict each other?

I am a beginner in symbolic logic and still mostly using intution to solve the problems. After spending some time trying to figure out why (p -> q) is true if p is false I've realized that this also means (p -> q) & ( p -> ~q) is not a contradiction.

How come statements like "if it rains the road will be wet" and "if it rains the road will not be wet" don't contradict each other?

When we write the truth table we can see that the both statements are true when p is false. So according to this both of those intuitively contradictory statements are true when it doesn't rain. Am I missing something or is symbolic logic supposed to depict something other than everyday usage of logic?

• No, they do not. The first one is False only when p is T and q is F, while the second one is F only when both are F. Both are "compatible" with the fact that it doers not rain. Commented Jan 27, 2021 at 7:03
• @MauroALLEGRANZA Can we say that in symbolic logic "true" does not mean it is actually "true" it just means it can not be disproven? it seems `p->q being true` does not make a statement about the laws of universe, it only says that the current situation does not disprove it? It is the only way I can make sense of it. If we can't disprove it we call it true(for whatever reason) If it doesn't rain we can not disprove any statement about what happens when it rains, so we say all of them are true. Commented Jan 27, 2021 at 7:25
• NO; in classical logic True means True. Commented Jan 27, 2021 at 7:26
• As per infinitely many similar posts on this site, the symbolic translation of "if p, then q" does not mean: "p is the cause of q" and neither "q follows logically from p". Commented Jan 27, 2021 at 7:39
• The answer is because in deductive reasoning the term contradiction means something specific. In slang or street variations people use the word contradiction different which is what you are doing now. You don't show you understand the relationships between propositions. Aristotle came up with the SQUARE OF OPPOSITION which indicates relationships between propositions. Mathematical logic does not use the original square of Opposition but a modern version where only contradiction exist. So you won't learn about the original proposition relationships in Mathematical logic. Commented Jan 27, 2021 at 17:01