Negation of Formal Logic

Question

What domain is U and what is P(x) in order for this statement to be false?

∀x∈U, P(x) ⇒ ∃x∈U, P(x)

I don't think this is possible but I am hoping someone can figure out how to make this statement false.

score 0 · Accepted Answer · answered Sep 30 '13 at 01:33

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I'd imagine that if U is the empty set, that statement is false; there does not exist any element in U, not least one that satisfies P(x). The antecedent is a vacuous truth, but is nevertheless still true.

answered Sep 30 '13 at 01:33

icktoofay

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But if the statement is vacuously true, then it is still true, not false – Brian Huang Sep 30 '13 at 02:14
@Brian: The *antecedent* is a vacuous truth, and is thus true, but the *consequent* is false. Because there is a case where the antecedent is true and the consequent is false, that means that the implication is false. – icktoofay Sep 30 '13 at 03:13
Hmm that's true because the antecedent is only true by default vacuously and not by proof. Thanks! – Brian Huang Sep 30 '13 at 12:21
Yeah an empty set all elements will satisfy all properties but not a single element will. For people in the future for reference: http://en.wikipedia.org/wiki/Empty_set#Properties – Brian Huang Sep 30 '13 at 12:32

Negation of Formal Logic

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