Structure (mathematical Logic) - Structures and First-order Logic

Structures and First-order Logic

See also: Model theory#First-order logic and Model theory#Axiomatizability, elimination of quantifiers, and model-completeness

Structures are sometimes referred to as "first-order structures". This is misleading, as nothing in their definition ties them to any specific logic, and in fact they are suitable as semantic objects both for very restricted fragments of first-order logic such as that used in universal algebra, and for second-order logic. In connection with first-order logic and model theory, structures are often called models, even when the question "models of what?" has no obvious answer.