A declarative formula or sentence that has a definite truth value (true or false) under a specified interpretation or model; in logic a proposition expresses a propositionally evaluable claim rather than a question, command, or expression with an undetermined truth value.

Principle

A proposition must be a well-formed formula with no free variables under the given interpretation (or else be closed by quantifiers) so that its truth value is determined by the semantics of the logical system and the interpretation of non-logical symbols.

Demonstration

The arithmetic statement '2 + 2 = 4' is a proposition in the standard model of arithmetic because under that interpretation it is true; by contrast, 'x > 0' without a valuation for x is not a proposition until x is given a value or quantified.

Misapplication

Treating open formulas with free variables (like P(x)) as propositions in deductive moves or truth-value assignments without specifying valuations or quantifiers, which leads to category errors and invalid inferences.

Consequence

Propositions serve as the basic units of assertion and inference: they are the inputs and outputs of proofs, the items combined by logical connectives and rules of inference, and the entities to which truth conditions are assigned.

Reversal

Non-declarative linguistic acts (questions, commands, expressions of desire) or syntactic fragments with indeterminate truth-value are the opposite of propositions and require different semantic or pragmatic treatment.

Boundary

Includes any closed formula or sentence with a truth value within a formal interpretation; excludes open formulas without valuation, performative utterances, and expressions whose truth value is undefined or context-dependent without additional specification.

Semantic Tension

Tension exists between the syntactic view (closed formula as proposition) and the pragmatic/epistemic view (assertions tied to speaker meaning and context); also between propositions as abstract truth-bearers and sentences as linguistic tokens.

Synthesis

A proposition is a truth-bearing declarative formula: ensure closure or valuation so its truth is determinate, then use it as the atomic unit of logical reasoning, distinct from imperatives, questions, or underspecified predicates.