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 | F | | B | T | B | B | T | | N | N | | N | N | F | F | N | | N | T | N | T | 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. Dunn (1976) gave the equivalent relational semantics in which a valuation simply relates each formula to truth and to falsity with no constraint either way, so the four values are the four subsets of {true, false}; the two clauses for truth-relatedness and falsity-relatedness are computed independently and monotonically, which is why the four-valued and relational presentations agree. Routley and Routley (1972) gave a second, two-valued semantics using a star involution on worlds, and the two approaches — the American and Australian plans — validate the same first-degree inferences. Algebraically FDE is the logic of De Morgan lattices: A ⊨ B holds iff the inequality A ≤ B holds in every De Morgan lattice, the four-element one being characteristic. Decidability is immediate from the finite matrix, and the consequence problem lies in coNP. The degeneracies of K3 and LP combine: the valuation sending every atom to N makes every formula N, so there are no valid formulas, and the valuation sending every atom to B makes every formula B, so there are no unsatisfiable ones. Every FDE-valid inference is both K3- and LP-valid, since the glut-free and gap-free valuations are special cases, but the inclusion is strict — p ∧ ¬p ⊨ q ∨ ¬q holds vacuously in K3 and trivially in LP yet fails in FDE at p = B, q = N.

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. FDE has no valid formulas and no unsatisfiable ones, so it can be presented neither as a set of theorems nor by any refutation method that works by driving premises to absurdity; it exists only as a relation between formulas. There is no conditional in the language, and the material ¬A ∨ B is neither detachable nor a deduction-theorem partner, so entailment cannot be expressed inside the object language and nesting is exactly what "first-degree" excludes. Recovering a genuine arrow means the Routley–Meyer ternary-relation semantics of R and E, and Urquhart (1984) proved those systems undecidable, so the decidability here belongs to the fragment and not to the logic it is a fragment of. Disjunctive syllogism, modus ponens, excluded middle and explosion all fail together. The values record what one has been told but not by whom or how often, so pooling is idempotent and a single contradictory source is indistinguishable from two conflicting ones. The bilattice operations ⊗ and ⊕ are not connectives of FDE; adding them changes the logic.

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