A family of semantic frameworks that model vague predicates by assigning graded truth values or degrees of applicability (often on a continuous scale between 0 and 1) so that borderline cases receive intermediate truth‑values rather than binary true/false; degree semantics treats vagueness as genuine gradation rather than mere indeterminacy or epistemic ignorance.

Principle

Vagueness is represented compositionally as gradable predicates whose application yields a degree; logical connectives and quantifiers are interpreted via operations on degrees (e.g., t‑norms, residua), so inferential behavior depends on the chosen algebraic machinery.

Demonstration

Illustrative scenario — Situation: 'Alice is tall' is modeled with a function t(height) returning 0.7 for Alice’s height. Recognition: the predicate applies to Alice to degree 0.7. Action: combining statements uses degree‑operations (e.g., conjunction as minimum), producing intermediate aggregate truth‑values. Consequence: sentences no longer all obey classical bivalence; some classical inferences need reformulation.

Misapplication

Confusing degrees of truth with probabilities about truth (treating 0.7 as the speaker’s credence that the sentence is true); the semantic error is to conflate semantic gradation with epistemic uncertainty rather than treating degrees as semantic values.

Consequence

Degree semantics yields a systematic way to model gradual applicability and supports inferential systems (fuzzy logic) adapted to graded truth; it typically rejects some classical logical laws (e.g., strict excluded middle) or requires reformulated consequence relations, with practical consequences for reasoning systems and formal analyses.

Reversal

Degree semantics is inapt when one wishes to preserve classical bivalence and all standard truth‑functional inferences, or when predicates are best modeled as admitting multiple discrete precisifications rather than continuous gradation.

Boundary

Clearly within: gradable predicates with measurable dimensions (e.g., 'tall', 'warm'). Boundary case: predicates where measurement is possible but mapping to degrees is theory‑laden (choice of scale and anchors contested). Clearly outside: predicates defined by sharp categorical criteria or legal definitions that demand binary classification.

Semantic Tension

Gradation and compositional measurement ↔ Preservation of classical bivalence and truth‑functional logic; degree semantics favors representational fit for gradual phenomena at the cost of modifying classical inference patterns.

Synthesis

Degree semantics relocates vagueness into the semantic values themselves by treating applicability as a measurable continuum; the approach trades classical binary truth for graded representational accuracy and requires corresponding adjustments to logical machinery and consequence relations.