The Problem:
Consider the first-order logic sentence
$φ≡∃s∃t∃u∀v∀w∀x∀y\psi(s,t,u,v,w,x,y)$ 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 φ 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 $\varphi$ with universe of size less than or equal to $3$
- There exists no model of $\varphi$ with universe of size less than or equal to $3$
- There exists no model of $\varphi$ with universe size of greater than $7$
- Every model of $\varphi$ has a universe of size equal to $7$
My attempt:
There exist at least one s,t, and u in some domain. It is also given that there exist a model with 7 elements. i.e there is at least one instance of $v,w,x$ and $y$ as well, together making $7$ elements with s,t,u. So any other model of $\varphi$ must have at least these $7$ elements as well. Any model cannot have more than seven elements because there are only seven given. i.e every model of $\varphi$ will have exactly seven elements. So Option $4$ seems to be the right one.
Being just a beginner I wish to develop intuition to solve these problems. Also I want to know your thought process and how you solve this problem. Thanks.