Q28

MCQ2 marks, −0.67

Consider the first-order logic sentence

\varphi \equiv \exists \: s \: \exists \: t \: \exists \: u \: \forall \: v \: \forall \: w \forall \: x \: \forall \: y \: \psi(s, t, u, v, w, x, y)

where $\psi(s, t, u, v, w, x, y, )$ is a quantifier-free first-order logic formula using only predicate symbols, and possibly equality, but no function symbols. Suppose $\varphi$ 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 $\varphi$ with universe of size less than or equal to $3$

(B)

There exists no model of $\varphi$ with universe of size less than or equal to $3$

(C)

There exists no model of $\varphi$ with universe size of greater than $7$

(D)

Every model of $\varphi$ has a universe of size equal to $7$

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