First-Degree Entailment

Origin. Anderson, Belnap, and Dunn developed FDE as the first-degree fragment of relevance logic (Dunn 1976; Belnap 1977, "A useful four-valued logic" and "How a computer should think"). Four values—true, false, both, neither—let a reasoner handle information that is incomplete or contradictory without explosion. It is the propositional core shared by Belnap's and Belnap–Dunn's four-valued systems.

Models. Four truth values for partial and inconsistent information: true only (T), false only (F), both (B), neither (N). A source may tell us a sentence is true, false, both, or neither. The values form a bilattice under two orderings—truth and information—the motivating structure for Belnap's database reasoner.

Formalism.

Four values:

• T: told true, not told false
• F: told false, not told true
• B: told both (glut)
• N: told neither (gap)

Bilattice (two orderings): Truth order ≤_t: F <_t N,B <_t T (N, B incomparable). Knowledge order ≤_k: N <_k T,F <_k B (T, F incomparable). Each ordering is a lattice; together, the four-element bilattice FOUR.

Truth-order connectives: ∧ = meet in ≤_t, ∨ = join in ≤_t, ¬ swaps T↔F and fixes B, N.

| ¬ | | | ∧ | T | F | B | N | | ∨ | T | F | B | N | |---|---| |---|---|---|---|---| |---|---|---|---|---| | T | F | | T | T | F | B | N | | T | T | T | T | T | | F | T | | F | F | F | F | F | | F | T | F | B | N | | B | B | | B | B | F | B | N | | B | T | B | B | B | | N | N | | N | N | F | N | N | | N | T | N | B | N |

Knowledge-order operations: Consensus ⊗ = meet in ≤_k (what two sources agree on). Gullibility ⊕ = join in ≤_k (all information pooled).

Entailment: φ ⊨_FDE ψ iff every valuation with v(φ) designated has v(ψ) designated. Designated = {T, B} ("at least true").

No explosion, no excluded middle: p ∧ ¬p ⊭ q (paraconsistent); ⊭ p ∨ ¬p (paracomplete).

Relevance: FDE is the first-degree (no nested →) fragment of relevance logics E and R. Adding a genuine → requires more structure.

Catuṣkoṭi: Nāgārjuna's four koṭis—is, is not, both, neither—map to FDE's {T}, {F}, {T,F}, {}. Priest reads the third koṭi as a glut, giving a paraconsistent formalization. The Buddhist use is soteriological, not a semantics of designated values.

Symbols.

Metatheory. Decidable (finite-valued). Paraconsistent and paracomplete. Self-dual under T↔F, B↔N. Algebraic semantics: De Morgan lattices; with both orderings, the four-element bilattice FOUR. Sound and complete first-degree calculus. Classical logic embeds by restricting to {T, F}. Infinite-valued bilattices extend it.

Applies to. Databases with null and conflicting values. Logic programming (well-founded and stable semantics). Belief revision from conflicting sources. Paraconsistent reasoning. Relevance-logic foundations. Multi-source information integration.

Limitations. No implication connective (first-degree only). Extending to a full logic loses some properties. Choice of designated values matters. Limited proof-theoretic development beyond first degree. Philosophical debate over "both true and false."

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