What is a universal sentence?

Is it just a sentence that starts with a universal quantifier? If so, isn't every sentence A equivalent to the sentence

$$\forall x A$$

Where x does not appear in A?

• It's not necessarily equivalent, if it's possible for a model to be empty. – Daniel Schepler Jun 29 '17 at 22:25

A universal sentence is a sentence of the form $$\forall x_1\forall x_2\cdots\forall x_n A$$ where $A$ is a quantifier-free formula.

A first-order sentence $\sigma$ is logically equivalent to a universal sentence if and only if every substructure of a model of $\sigma$ is a model of $\sigma;$ see this question

I am not sure how you draw that conclusion, but as long as you are not dealing with free logics (where the domain can be empty, and hence every universal is automatically true, regardless of the truth of $A$), the equivalence does indeed hold.

Most logics assume the assumption of Existential Import, which is the assumption that any domain we consider is non-empty, and with that Assumption $\forall x A$ is equivalent to just $A$. Indeed, we typically agree to the following equivalence principle:

Null Quantification

where $A$ is a formula that does not contain $x$ as a free variable:

$\forall x A \Leftrightarrow A$

$\exists x A \Leftrightarrow A$