Truthmaker Semantics

Origin. Fine (2010s). Exact verification. States as truthmakers. Hyperintensional. Foundation of exact semantics.

Models. States verify propositions. Exact vs inexact. State space. Fusion. Hyperintensional distinctions.

Formalism.

State space: ⟨S, ⊑, ⊔⟩ where: S: states. ⊑: parthood. ⊔: fusion. States are situations or facts.

Exact verification: s ⊩ A: s exactly verifies A. s is a truthmaker for A. s contains exactly what A requires. No excess.

Inexact verification: s ⊩⁺ A: s inexactly verifies A. s ⊩⁺ A iff s' ⊑ s for some s' with s' ⊩ A. The truthmaker is a part of s, not the other way round — s may contain material irrelevant to A, which is exactly what "inexact" means.

Falsification: s ⊣ A: s exactly falsifies A. Dual to verification. Bilateral semantics.

Clauses: s ⊩ A ∧ B iff s = s₁ ⊔ s₂, s₁ ⊩ A, s₂ ⊩ B. s ⊩ A ∨ B iff s ⊩ A or s ⊩ B. s ⊩ ¬A iff s ⊣ A. Conjunction = fusion.

Hyperintensionality: A ∧ A ≠ A (different truthmakers). Impossible worlds not needed. Finer than possible worlds. Content, not just truth-conditions.

Subject matter: What a proposition is about. Truthmaker states. Relevance logic connection.

Symbols.

Symbol	Unicode	Name	Meaning
⊩	U+22A9	Exact verification	s ⊩ A: s is a truthmaker for A, with nothing to spare
⊣	U+22A3	Exact falsification	s ⊣ A: s is a falsemaker for A
⊔	U+2294	State fusion	Join in the state space; the conjunction clause
⊑	U+2291	Parthood	s' ⊑ s: partial order on states
⊩⁺	—	Inexact verification	Some part of s exactly verifies A
s	—	State	Truthmaker, a possibly partial worldly item
⟨S, ⊑, ⊔⟩	—	State space	Carrier, parthood, fusion
∧	U+2227	Conjunction	Verified by a fusion of verifiers
∨	U+2228	Disjunction	Verified by a verifier of either disjunct
¬	U+00AC	Negation	Verified by what falsifies; requires bilateralism

Metatheory. Exact verification is not closed under logical equivalence, which is the point of the framework. A and A ∧ A can differ in verifiers unless verifier sets are closed under fusion, and absorption fails, since verifiers of A ∧ (A ∨ B) include fusions a ⊔ b that need not verify A. Exact entailment — every verifier of A verifies B — is therefore subclassical. Van Fraassen (1969) showed that the fact-based version of this idea validates exactly Anderson and Belnap's first-degree entailment, and Fine (2016) proved completeness of Angell's logic of analytic containment with respect to exact truthmaker content. Fine (2014) gave a truthmaker semantics for intuitionistic logic with a soundness and completeness proof. Classical equivalences are recovered by imposing closure conditions on verifier sets — closure under fusion, and convexity — so possible-worlds semantics is the degenerate case in which the hyperintensional structure is quotiented away. Negation requires falsification as a primitive: the semantics is bilateral, and ¬ is defined by exchanging the verification and falsification clauses.

Applies to. Relevance. Subject matter. Counterfactuals. Semantics of content.

Limitations. The framework fixes no logic by itself. Different conditions on state spaces — whether fusion is total, whether verifier sets must be closed or convex, whether impossible states are admitted — yield different consequence relations, and completeness results attach to particular systems, intuitionistic logic or analytic containment, not to truthmaker semantics as such. The state space is primitive: nothing internal to the theory settles which states exist, how they fuse, or which are possible, so the ontology carries the explanatory load the semantics was to supply. Because exact verification is not closed under consequence, classical validity is not delivered without imposing the very closure conditions that erase the hyperintensional distinctions motivating the theory. Coverage is uneven — the propositional clauses are settled, while quantification, modality, and the compositional treatment of predication remain contested extensions rather than fixed parts of the framework.