Logical Semantics & Many-Valued Systems

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Abstractions about how formal logics build formulas and assign them truth, covering syntactic units (atomic sentences, propositional formulas), semantic relations (logical consequence, tautology, equisatisfiability), quantifier-alternation definability hierarchies, and many-valued and fuzzy systems such as four-valued and Gödel–Dummett logic.

11 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Atomic Sentence — A closed formula formed by an atomic base clause, not by a connective or quantifier, in a specified formal logic.
• Equisatisfiability — The logical relation in which two formulas share the same existence-of-a-model status—both satisfiable or both unsatisfiable—without requiring the same models, enabling compact satisfiability-preserving transformations such as Tseitin encoding and Skolemization.
• Exportation (logic) — Convert an implication from a conjunction, (P ∧ Q) → R, into nested implications, P → (Q → R), under formal conjunction and implication rules.
• Finite-Valued Logic — A logic is characterized by a finite logical matrix: finitely many semantic values, designated values defining consequence, and truth functions interpreting its connectives.
• Four-Valued Logic — A finite-valued logical system whose semantics distinguishes exactly four values, with the meaning and connective tables fixed by a declared logic rather than by cardinality alone.
• Fuzzy rule — Represent a graded IF-THEN relation whose linguistic antecedents and consequent are fuzzy sets, then compute a consequent degree or fuzzy output through declared connective, implication, aggregation, and output conventions.
• Gödel–Dummett Logic — The prelinear family of intermediate and many-valued logics characterized by linearly ordered Heyting semantics, with conjunction and disjunction interpreted by minimum and maximum and implication by the Gödel residuum.
• Logical Consequence — A logic-relative relation licensing a formula as following from premises under a specified semantic or derivational criterion.
• Propositional formula — Type of logical formula in propositional logic.
• Quantifier-Alternation Hierarchy — Stratify definable properties by alternating existential and universal quantifier blocks over a declared base class, keeping the quantified objects and equivalence criterion explicit.
• Tautology (Logic) — A formula that evaluates as true under every admissible valuation in a specified logic, with propositional tautology distinguished from broader first-order logical validity and from merely repetitive language.