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Consider the first-order logic sentence φ ≡ ∃s∃t∃u∀v∀w∀x∀y ψ(s,t, u, v, w, x, y) 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 of size greater than 7.
(D) Every model of φ has a universe of size equal to 7.


Asked On2019-05-29 11:21:55 by:vivekEdu

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Answers

Answer - A.

Quick logic review - 

α:∀x∃yy<x

Is α true for domain of all integers ?, Yes it is true. You pick any number x, I can always give you y that is less than your number x. 

Is α true for domain of  Non Negative integers {0,1,2,3,…} ? No,  it is not true. (You pick any number x) If you pick 0 then I can not give you y which is less than 0.

Definition of Model  - Domain for which my sentence is true. For above sentence α, all integers is model and there can be many other models, like -  real numbers.

(Definition of Co Model - Domain for which my sentence is False.)



Answerd on:2021-12-23 Answerd By:Suresha-N

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