Pablo Valencia-Equy
University of Santiago de Compostela
The paper develops a non-Boolean exact truthmaker semantics for First Degree Entailment (FDE). It begins with the truthmaking principle: every true proposition is made true by some state, and, in a bilateral framework, every false proposition is made false by some state. Truthmakers and falsitymakers may be exact or inexact, depending on whether they are fully relevant to the truth or falsity they ground.
The main motivation is that standard FDE semantics, although designed to model informational states in computation (Belnap, 1977), is not sufficiently fine-grained for this task: formulas that are FDE-equivalent may still encode different informational patterns—e.g., (φ∨ψ)∧(φ∨λ) and φ∨(ψ∧λ). An exact truthmaker semantics for FDE is therefore introduced as a hyperintensional framework that distinguishes these formulas by assigning them different exact truthmakers. In this respect, it recovers FDE’s original motivation.
Accordingly, the paper’s main formal contribution is an exactification theorem for FDE’s standard semantics. It shows that, for any formula A, an FDE valuation assigns A the value t,f,b, or i if and only if it extends, respectively, only an exact verifier of A, only an exact falsifier of A, both an exact verifier and an exact falsifier of A, or neither. The proof relies on two compatibility principles: exact verifiers are incompatible with falsifiers, and states containing no exact truthmaker for a proposition are compatible with its exact falsitymakers.
The paper then defines an exact truthmaker consequence relation: Γ⊨A if and only if every state that extends exact verifiers for all formulas B in Γ also extends an exact verifier for A. I prove that this relation coincides with the standard FDE consequence, thereby establishing the promised exactification result.
Finally, the paper shows that other logics can be recovered by imposing ontological restrictions on the space of states, rather than merely introducing semantic constraints on truth-values. Excluding impossible states yields K3; requiring states to be maximally compatible yields LP; and combining both restrictions yields classical logic. The exactification of FDE therefore suggests that differences among logics reflect dissimilarities in the underlying structure of reality.
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SOPhiA 2026 