How do you determine if a formula is satisfiable in Predicate Logic?

Question

For example:

$ (\forall x)(P(x) \rightarrow Q(x))$

Are you suppose to invent your own Interpretation (domain, and giving the meaning to the predicates), and make it satisfiable under that Interpretation?

Attempt:

Interpretation I:

• Domain: $ \mathbb{N} $
• P(x) - "x is a natural number"
• Q(x) - "x is an integer"

I is a model of $ (\forall x)(P(x) \rightarrow Q(x)) $

Therefore, $ (\forall x)(P(x) \rightarrow Q(x)) $ is satisfiable.

Answer 1

From

Yes. More precisely, you have to construct your interpretation such that the formula becomes satisfied under it. - @lemontree

It is clear how a formula can be satisfied.