# What is the statement “If not p then q” called?

Let's say I have a statement: if p then q.
The converse would be: if q then p.
The inverse would be: if not p then not q.
The contraposition would be: if not q then not p.
What would you call the following? if not p then q.
Thanks.

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negation of converse? – hjpotter92 Oct 3 '13 at 3:00
I don't think it has a name. – Trevor Wilson Oct 3 '13 at 3:31
@hjpotter92 can you show it to us how it's called negation of converse? – b16db0 Oct 3 '13 at 4:07
@Rustyn One reaason I could think of why I would not call it "p or q" is when the statement "p implies q" has a name, e.g. the "p or q" of principle x – b16db0 Oct 3 '13 at 4:20

There's no name for it, because there's no real connection between them. The inverse and converse exist because they still assert a direct correlation between p and q; it's just that, as opposed to the regular and contrapositive forms, the condition for failure is reversed.

On the other hand, "if not p then q" is a completely different assertion.

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I'm not sure what you mean by "correlation" here. For any definition I can think of, if $p$ and $q$ are correlated then so are $\neg p$ and $q$. – Trevor Wilson Oct 3 '13 at 3:29
There is no reason why $\lnot p$ and $q$ must be correlated if $p$ and $q$ are correlated. All four of the related forms (regular, contrapositive, converse, and inverse) state that $p$ and $q$ should be true simultaneously under some specific condition. There is no inherent correlation between $\lnot p$ and $q$, because $\lnot p$ does not imply, or even suggest, anything about $q$ when $p$ implies $q$. – Glen O Oct 3 '13 at 3:50
@GlenO can you cite a specific condition for example? – b16db0 Oct 3 '13 at 4:22
This is what I'm saying: in the truth tables for the regular, contrapositive, converse, and inverse, if both p and q are true, then the statement is true. Similarly, if both p and q are false, then the statement is true. On the other hand, with "if not p then q", if both p and q are true, the statement is false. – Glen O Oct 3 '13 at 4:25
So, "if it is hot, then I'm sweating" is the regular statement. A reasonable statement. "If I'm not sweating, then it is not hot" is also a reasonable statement, the contrapositive. "If it is not hot, then I'm not sweating" is a statement with less accuracy, as it is possible for me to be sweating when it's not hot, and similarly for "if I'm sweating, then it is hot" - but they're mostly consistent with the original observation that heat and sweating are related. But "if it is not hot, then I'm sweating" is a completely different statement. – Glen O Oct 3 '13 at 4:27

$$p \Rightarrow q \equiv \lnot p \lor q$$

$$\lnot p \Rightarrow q \equiv p\lor q$$ It is logically equivalent to "$p$ or $q$"

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That's not really a name for it so much as a statement that happens to be equivalent to it in classical logic. I don't think the equivalence holds intuitionistically, so it probably wouldn't be good to call the one a "name" for the other. – Trevor Wilson Oct 3 '13 at 3:31
@TrevorWilson can you expound what you mean by intuitionistically? – b16db0 Oct 3 '13 at 4:16
@bimboxX I mean that, assuming $\neg p \implies q$, it seems like one might need to use the law of excluded middle ($p \vee \neg p$) to conclude $p \vee q$. – Trevor Wilson Oct 3 '13 at 4:32
Ah, so it's like you have to "cross a bridge" to get to "p or q". But the two statements are still equal, right? I'm just thinking if there's a name for "p or q" in relation to "not p implies q" then I could gladly use it. – b16db0 Oct 3 '13 at 4:46