Consider the first-order logic sentence
where ψ(s, t, u, v, w, x, y) is a quantifier-free first-order logic formula using only predicate symbols, and possibly equality, but no function symbols. Suppose φ has a model with a universe containing 7 elements.
Which one of the following statements is necessarily true?
(A) There exists at least one model of φ with universe of size less than or equal to 3
(B) There exists no model of φ with universe of size less than or equal to 3
(C) There exists no model of φ with universe size of greater than 7
(D) Every model of φ has a universe of size equal to 7
Explanation: Let’s interpret the problem this way :
∀ are always True and ∃ are always False for empty sets.
So there exists at least one model with universe of size 3 (or less than).
Therefore, option (A) is necessarily TRUE.
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