Let L be a first-order language.

Suppose that we had a general decision algorithm for statements in a first-order language.

Each first-order language is defined by a signature.

To ascribe meaning to all sentences of a first-order language, the following information is needed.

Venn-II is equivalent in expressive power to a first-order monadic language.

Let κ be a finite or infinite cardinal number and M a model in some first-order language.

The set of all consistent theories of a first-order language.

What follows is a formal definition for first-order languages.

A first-order theory is a set of sentences in a first-order formal language .

Structures over a signature, also known as models, provide formal semantics to a signature and the first-order language over it.