A set of first‑order sentences in a fixed signature (vocabulary) — closed or considered as a generating set for its deductive closure — whose intended models are the structures that satisfy every sentence in the set.

Principle

Mathematical structures are described by quantification over elements (not over sets or relations) in a fixed signature; the theory collects the sentences that are taken as axioms or constraints for those structures within first‑order logic.

Demonstration

The theory of groups is the set of first‑order sentences expressing associativity, identity, and inverses in a language with one binary symbol for multiplication and a constant for the identity; its models are all groups.

Misapplication

Labeling a collection of informal properties as a first‑order theory when some properties require quantification over subsets or functions (second‑order conditions), such as 'every nonempty definable family has a least element' without formal first‑order axiomatization.

Consequence

A proper first‑order theory can be studied via model‑theoretic tools: satisfiability, compactness, Löwenheim‑Skolem phenomena, completions, and completeness/decidability questions are well‑posed and often tractable within first‑order logic.

Reversal

A higher‑order theory permits quantification over relations, functions, or sets and is not confined to first‑order semantics; such theories can express stronger properties but lose certain first‑order meta‑theorems like compactness.

Boundary

Restricts attention to first‑order languages and sentences; excludes infinitary logics (L_{ω1,ω} etc.), second‑order axioms, and semantic frameworks that quantify over classes or categories of structures.

Semantic Tension

The term 'theory' sometimes denotes merely a set of axioms, sometimes the deductive closure; the practical tension is between giving a finite axiomatisation and the full (possibly infinite) first‑order theory determined by intended models.

Synthesis

A first‑order theory is the collection of first‑order statements in a given signature that collectively specify the properties of interest for a class of structures, forming the basis for formal deduction and model‑theoretic analysis.