In mathematical logic

Mathematical logic

Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

, an algebraic theory is one that uses axioms stated entirely in terms of equations between terms with free variables. Inequalities and quantifiers are specifically disallowed. Sentential logic is the subset of first-order logic

First-order logic

First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...

 involving only algebraic sentences.

Saying that a theory is algebraic is a stronger condition than saying it is elementary

Elementary theory

In mathematical logic, an elementary theory is one that involves axioms using only finitary first-order logic, without reference to set theory or using any axioms which have consistency strength equal to set theory....

.