A formula with no free variables — a closed well‑formed formula — that asserts a proposition about the domain of discourse and can be assigned a truth value relative to an interpretation or structure.

Principle

Close all free variables (by quantification or substitution) so that the expression denotes a complete proposition whose truth or falsity is decidable relative to a model, enabling its use as an axiom, theorem, or query.

Demonstration

In arithmetic, the Peano statement ∀x ∃y (y = x + 1) is a sentence: it contains quantifiers binding its variables and, given the standard interpretation of numbers, has a truth value.

Misapplication

Calling an open formula like P(x) a sentence and attempting to evaluate its truth without specifying x or quantifying over it, which conflates parameterized expressions with propositional claims.

Consequence

Sentences are the units to which semantics (truth, falsity, satisfaction) apply; they can be theorems of a formal system, asserted in a theory, or tested against models to probe consistency and completeness.

Reversal

An open formula with free variables — instead of asserting a truth about the domain, it defines a property or relation parametrized by values; the reversal converts a sentence into a schema or predicate.

Boundary

A sentence is syntactic and requires an interpretation for a truth value; provability in a formal system and truth in an interpretation are distinct notions, and sentences with undefined interpretation domains fall outside its scope.

Semantic Tension

Tension exists between sentence as syntactic closure and sentence as semantically evaluated truth: a closed formula may be syntactically a sentence but undecidable or independent relative to a theory, creating pragmatic ambiguity.

Synthesis

A sentence is a closed formula that expresses a complete proposition about a domain; by eliminating free variables and supplying an interpretation it becomes the object of semantic truth evaluation and formal reasoning.