Negating a statement with quantifiers and conditionals

[yxgao]

What is the negation of the statement "For each s in R, there exists an r in R such that if f(r) >0, then g(s) >0."

The answer is "There exists an s in R such that for each r in R, f(r) >0 and g(s) <0."

What is the general method to find the negation of any logical statement?

Thanks!

 

[Doc Al]

Originally posted by yxgao
What is the general method to find the negation of any logical statement?

While I can't give you a general method, you may find it useful to review the concept of contradictory statements from Boolean logic:

All S is P is contradictory to Some S is not P

No S is P is contradictory to Some S is P

A statement and its contradictory cannot both be true (or both be false). Thus if "All S is P" is not true, then "Some S is not P" must be true. Of course, this only applies to statements that can be put in standard categorical form.

 

[Hurkyl]

Basically, you just want to distribute the negation. Use the laws

$$\neg \forall x: P(x) = \exists x: \neg P(x)$$
$$\neg \exists x: P(x) = \forall x: \neg P(x)$$
$$\neg(x \wedge y) = \neg x \vee \neg y$$
$$\neg(x \vee y) = \neg x \wedge \neg y$$
$$\neg(x \Rightarrow y) = x \wedge \neg y$$
$$\neg(\neg x) = x$$