# 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.**

| Symbol | Unicode | Name | Meaning |
|--------|---------|------|---------|
| T | — | True only | Affirmed |
| F | — | False only | Denied |
| B | — | Both | Glut |
| N | — | Neither | Gap |
| ≤_t | — | Truth order | Falsity to truth |
| ≤_k | — | Knowledge order | Gap to glut |
| ⊗ | U+2297 | Consensus | Meet in ≤_k |
| ⊕ | U+2295 | Gullibility | Join in ≤_k |
| ⊨ | U+22A8 | Entails | Designated-preservation |

**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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