Question

What is the universal set of values over which x is quantified?

Answer 1

When you see a quantification like ‘∀ϕx:ψx∀ϕx:ψx’, this is shorthand for ‘∀x:ϕx→ψx∀x:ϕx→ψx’. Since ‘x∈∅x∈∅’ is false for all ‘xx’, the antecedent of ‘x∈∅→ϕxx∈∅→ϕx’ will always be false, thus the entire conditional statement is always true.

Edit: In the case of existential quantification, the statement ‘∃ϕx:ψx∃ϕx:ψx’ is shorthand for ‘∃x:ϕx∧ψx∃x:ϕx∧ψx’, so in this case, the conjunction ‘x∈∅∧ϕxx∈∅∧ϕx’ will always be false.