Problem Statement:
Consider the first order logic sentence:
$$\mathbf \phi \equiv \exists s\exists t\exists u \forall v \forall w \forall x \forall y\; \psi(s,t,u,v,w,x,y)$$
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 $\phi$ has a model with a universe containing 7 elements.
Which of the following statements is necessarily true?
(A) There exists at least one model of $\phi$ with a universe of size $\leq3$.
(B) There exists no model of $\phi$ with a universe size $\leq3$.
(C) There exists no model of $\phi$ with a universe size $\gt7$.
(D) Every model of $\phi$ has a universe of size $=7$.
I wanted to know the approach to solve this problem, as from the I do understand that:
A FOL is defined using $\phi$ which contains only predicate symbols and equality and $\psi(s,t,u,v,w,x,y)$ is defined over a domain of discourse containing only 7 elements, but I am not able to understand how to deduce anything from the given information.
The explanation given for the solution is:
P.S There is one doubt which I have when checking whether the given propositional/predicate logic is valid or not. As per the definition of the validity of a logic:
"A propositional formula is called valid when it evaluates to T no matter what truth values are assigned to the individual propositional variables."
Does that mean that if a formula is valid then it's a tautology?
E.g: $S = p \implies q \; and \;T = \lnot q \implies \lnot p$ then $S \iff T$ is always true, means tautology.
The key fact is that since $\psi$ has no quantifiers, its meaning does not depend on the domain.
Now, notice that the statement asserts the existence of three elements of the domain (which may be equal to one another) that satisfy the universal formula. What if you restrict your seven element model to only consist of these three (or possibly one or two) objects?
Edit More details.
As Ross Millikan has noted, answers B,C,D can be ruled out by taking $\psi$ to be something always true, e.g. $s=s.$ In this case, any (non-empty) structure is a model of $\phi,$ in particular, there is a model of every size, which conflicts with B-D. The "solution" you screenshotted says that the three existential quantifiers means there are at least three elements, but this is wrong since $s,t,$ and $u$ could all be the same element. (And this is the only statement in the solution that even makes sense to me... the rest seems like complete nonsense.)
A is correct. Let $M_7$ be the seven-element model that exists by assumption. Then there are an $s,t,u\in M_7$ such that for all $v,w,x,y\in M_7,$ $\psi(s,t,u,v,w,x,y)$ holds in $M_7$. Choose any such $s,t,u$ and let $M_3$ have domain $\{s,t,u\}.$ As explained before, it is possible that $s,t$ and $u$ are not distinct, so $M_3$ has between $1$ and $3$ elements.
$M_3$ can be considered as a substructure of $M_7$ with respect to the language consisting of all the symbols that occur in $\phi.$ There are no function symbols (and I presume this means no constant symbols since they also say "only" predicate symbols), so there is no issue with $M_3$ being closed under the functions. So all you have to do to get symbol interpretations for $M_3$ is just restrict the interpretation of the relation symbols for $M_7$ to the subset $\{s,t,u\}.$
Now we can argue that $\phi$ is satisfied in $M_3$ and thus $M_3$ is a model of $\phi$ of size $\le 3.$ This means we need to show there are an $s,t,u\in M_3$ such that for all $v,w,x,y\in M_3,$ $\psi(s,t,u,v,w,x,y)$ holds in $M_3.$ We simply take $s,$ $t$ and $u$ to be the very same elements we chose from the domain of $M_7$ to comprise the domain of $M_3.$
We must show that $\forall v,w,x,y(\psi(s,t,u,w,x,y))$ still holds true in $M_3$. Well, we know that it is true in $M_7,$ i.e. for any $v,w,x,y\in M_7,$ $\psi(s,t,u,w,x,y)$ holds in $M_7.$ Since the domain of $M_3$ is a subset of $M_3,$ this implies that for all $v,w,x,y \in M_3,$ $\psi(s,t,u,w,x,y)$ holds in $M_7.$
So we just need to show that the fact that $\psi(s,t,u,w,x,y)$ holds in $M_7$ means it holds in $M_3$ as well. This follows from the fact that $\psi$ has no quantifiers. Thus it is a propositional function of atomic sentences of the form $x=y$ or $R(x,y,\ldots)$ for predicate symbols $R.$ But, since all the predicate symbols' interpretations are just the restrictions of their interpretations in $M_7,$ the atomic sentences hold in $M_3$ if and only if they hold in $M_7.$ Thus any propositional function of these atomic sentences holds in $M_3$ if and only if it holds in $M_7.$ And we're done.