# Is the following statement true, false, or can't be determined? Why?

"If it snows and we don't have school, then (x^3)<0 implies x is negative" (Assume x is a real number).

• Any idea ? Why "argumentation" ? An argument needs at least one premise and a conclusion; thus, more than one statement. – Mauro ALLEGRANZA Feb 19 '19 at 16:03
• You need to be more specific. Do you mean is the statement true in reality or is the statement true in Mathematical logic. You should not believe every claim with the format "If . . . Then" format will be true on reality. As far as this being a homework question as written is an issue because all you did was ask the question without any work. You should explain why you are unsure, confused, or why you are lost with the question. Do you know truth tables at this point? A truthtable for the conditional connector would answer the question in Mathematical logic. – Logikal Feb 19 '19 at 16:07

## 2 Answers

A statement of the form "If X then Y" where Y is true, is always true in classical logic. If the consequent of a conditional is true, then it matters neither what the antecedent is, nor whether there's any actual connection between them.

In this case, your "Y" expresses a mathematical truth, so we can take it as being a proposition that is always and necessarily true (although from a mathematical point of view, not a logical one). Therefore, your particular "If X then Y" is always true no matter what your X is. For instance "If pigs can fly then x^3 < 0 implies x < 0" is likewise true.

It's worth nothing that there are other systems of logic that do try to explicitly capture relevance and/or causal relationships. Your statement might not be evaluated as true in one of those systems.

• You might want to add that x^3 is negative iff x is negative (considering real numbers; there is no ordering for complex numbers), so that Y is definitely true. – David Thornley Feb 19 '19 at 17:27
• One might add as a corollary that in a relevance logic this sentence may come out as false because there is no shared proposition or term between the antecedent and consequent. But as you say, in classical logic, and in most others that I can think of, it is true. – Bumble Feb 19 '19 at 19:18
• @Bumble, David: Thanks for the suggestions, I have incorporated them. – Chris Sunami supports Monica Feb 19 '19 at 19:21
• I would write 'antecedent' instead of 'premise' in the first paragraph. Premises are parts of arguments, not of conditional statements. Arguments and conditional statements are different things in logic, though they are often confused. – Eliran Feb 19 '19 at 19:56

this could be simplified as :

I like pineaple therefore sky is blue

This is a false statement, the 2 propositions can be true, but they are connected with a non-sequitur falsely implying there is a connection.

• But the statement in the OP's question is not of the form "P, therefore Q", but of the form "if P, then Q". – Mauro ALLEGRANZA Feb 19 '19 at 19:15
• @MauroALLEGRANZA , "if" [] "then" [] implies a strong logical cause/consequence , just like "therefore", so much so that programming languages use if/then. en.wikipedia.org/wiki/… – Manu de Hanoi Feb 20 '19 at 1:19