Using Semantic Arguments in Proving Validity

Question

I have to show that $\forall x (\phi \to \psi) \to (\forall x \phi \to \forall x \psi)$ is valid using semantic arguments. However, what does "semantic arguments" mean? Do I use rules of inference?

Answer 1

When having a logical formula, there're usually two methods to prove validity.

A syntactic one where we use rules of inference only by looking at the syntactic structure of the formula. For instance if we have $P$ and $P \rightarrow Q$ then we can syntactically produce Q with a rule called modus ponens $\frac{P\quad P\rightarrow Q}{Q}$.

A semantic one which consists of interpreting/evaluating the meaning of the formula with respect to a subjective conception of meaning. For instance if we have the propositional formula $\phi = P \land Q$ its interpreation will be $[\![ A \land B ]\!]_\sigma = AND([\![ A ]\!]_\sigma, [\![ B ]\!]_\sigma$) where $\sigma$ associates a truth value in $\{0, 1\}$ to each variables and $AND : \{0, 1\} \times \{0, 1\} \longrightarrow \{0, 1\}$ is a function reproducing the truth table of $\land$. To be valid is to be evaluated to the truth value $1$ for any interpretation.

In first-order logic, we have to extend the method to models to interpret entities in a particular universe. When we write $\forall x. P(x)$, the $x$ doesn't refer to a truth value but to an entity (e.g a number). Then we seek an universe making the formula true.

The theorem of soundness and completude relates these two methods by saying that they are equivalents.

Answer 2

By proving that a formula is valid by semantic arguments one usually means to prove that it is logically valid, that is that it is true in every possible interpretation.

So what you are asked is to prove that the formula $$\forall x (\varphi \to \psi) \to (\forall x \varphi \to \forall x \psi)$$ is true in every interpretation.

In order to do that you have to use the basic definition of truth for a first-order formula.

I suggest you try to continue from here and in case you find any difficulty fell free to ask here for some hints.

I hope this helps.