##  [Sentence](/sentence-0) 

 Definition

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

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

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

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

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

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

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

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

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.