How can I negate this conditional statement? [closed]

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The conditional statement is: If today is February 1, then tomorrow is Ground Hog's Day. I need to negate this but I am confused. Would it just be If today is not February 1, then tomorrow is not Ground Hog's Day? I think that is an inverse statement though. Please help me negate this conditional statement.

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2 Answers

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Let$$p: \text{today is February 1}$$ $$q: \text{tomorrow is Ground Hog's Day}$$ Then your statement is, $$p\rightarrow q$$

Informally the negation of "If p then q" is p and not q,

i.e. $\color{green}{\text{today is February 1}}$ and $\color{red}{\text{tomorrow is not Ground Hog's Day}}$.

More formally

$$p\rightarrow q \text{ is equivalent to } (\lnot p\lor q)$$ And

$$\lnot(\lnot p\lor q)\equiv\lnot\lnot p\land\lnot q\equiv p\land\lnot q.$$ by DeMorgan laws.

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The negation would formally be "Today is february the first and tomorrow is not Ground Hog's Day". The reason is that an implication $\phi\rightarrow\psi$ is the same as $\neg\phi\lor\psi$, then negating this using DeMorgan laws you get $\neg(\neg\phi\lor\psi) \equiv \neg\neg\phi\land\neg\psi \equiv \phi\land\neg\psi $

You could also reasoning that the only way an implication is false if the premise is true, yet the consequence is false.

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