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Discussion

GATE preparation for Computer Science & Information Technology
(25)

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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.

Which one of the following statements is necessarily true?

 
There exists at least one model of with universe of size less than or equal to 3.
There exists no model of with universe of size less than or equal to 3.
There exists no model of with universe of size greater than 7.
Every model of has a universe of size equal to 7.

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Bhanu Pratap Singh Rathore

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.