How do ∀x [P(x) ∨ Q(x)] and ∀x [P(x)] ∨ ∀x [Q(x)] differ in their meaning?

Question

How do ∀x [P(x) ∨ Q(x)] and ∀x [P(x)] ∨ ∀x [Q(x)] differ in their meaning?

I thought it was because when ∀x is defined twice there is a possibility of a different x value for both P(x) and Q(x) but then I noticed it was for all so this would not be the case as it would use every x value.

Could you have two different sets of x?

Yes, every `∀` creates its own identifier scope. The first `x` is a different variable than the second `x`. You can (should) rename one of them. — Bergi, Oct 28 '15 at 01:41

score 1 · Answer 1 · answered Oct 28 '15 at 01:18

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The first means:

For every thing condition P is true or condition Q is true (or both).

The second means:

Condition P is true for all things or condition Q is true for all things.

answered Oct 28 '15 at 01:18

Thilo

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score 0 · Accepted Answer · answered Oct 28 '15 at 01:18

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Suppose P(X) means "X is even" and Q(X) means "X is odd" and assume you're taking about integers. Is the first formula true? How about the second? Does that explain why they're not the same?

answered Oct 28 '15 at 01:18

templatetypedef

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How do ∀x [P(x) ∨ Q(x)] and ∀x [P(x)] ∨ ∀x [Q(x)] differ in their meaning?

2 Answers2