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Consider the first-order logic sentence ? ≡ ∃?∃?∃?∀?∀?∀?∀? ?(?, ?, ?, ?, ?, ?, ?) where ?(?, ?, ?, ?, ?, ?, ?) is a quantifier-free first-order logic formula using only predicate symbols, and possibly equality, but no function symbols. Suppose ? has a model with a universe containing 7 elements.

Consider the first-order logic sentence ? ≡ ∃?∃?∃?∀?∀?∀?∀? ?(?, ?, ?, ?, ?, ?, ?) where ?(?, ?, ?, ?, ?, ?, ?) is a quantifier-free first-order logic formula using only predicate symbols, and possibly equality, but no function symbols. Suppose ? has a model with a universe containing 7 elements. Correct Answer There exists at least one model of 𝜑 with universe of size less than or equal to 3.

∀ 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 1 is necessarily TRUE.

Important Points:

∀ → for all

∃ → there exists

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