Dunn Semantics

Origin. J. Michael Dunn (1966, 1976). Generalized semantics via sets of values. Valuations assign {t}, {f}, {t,f}, or {} to formulas. Relational semantics for relevance logic. Foundation for Belnap's four-valued logic.

Models. Formulas get sets of truth values. v(A) ⊆ {t, f}. Four possibilities: {t}, {f}, {t,f}, {}. Generalizes to arbitrary truth value sets. Clauses for connectives via set operations.

Formalism.

Valuation: v: Form → P({t, f}) v(A) = {t}: A is true only v(A) = {f}: A is false only v(A) = {t,f}: A is both (inconsistent) v(A) = {}: A is neither (gap)

Negation: t ∈ v(¬A) iff f ∈ v(A) f ∈ v(¬A) iff t ∈ v(A)

Conjunction: t ∈ v(A ∧ B) iff t ∈ v(A) and t ∈ v(B) f ∈ v(A ∧ B) iff f ∈ v(A) or f ∈ v(B)

Disjunction: t ∈ v(A ∨ B) iff t ∈ v(A) or t ∈ v(B) f ∈ v(A ∨ B) iff f ∈ v(A) and f ∈ v(B)

Consequence: A ⊨ B iff for all v: t ∈ v(A) implies t ∈ v(B)

Intensional operators: For relevant implication and fusion: Ternary relation semantics for →. R(a,b,c): if true at a,b then true at c.

FDE (First-Degree Entailment): Fragment without conditionals. Semantics: all four-valued valuations.

Symbols.

Symbol	Unicode	Name	Meaning
¬	U+00AC	Negation	De Morgan negation, defined by swapping t and f
→	U+2192	Rightwards arrow	Relevant implication; not definable from the four values
∈	U+2208	Element of	Membership of a truth value in v(A)
∧	U+2227	Conjunction	Extensional meet
∨	U+2228	Disjunction	Extensional join
⊨	U+22A8	Consequence	Follows from: t ∈ v(A) implies t ∈ v(B)
⊭	U+22AD	Does not entail	Failure of disjunctive syllogism and of explosion
⊆	U+2286	Subset	v(A) ⊆ {t, f}
{t}	—	True	Only true
{f}	—	False	Only false
{t,f}	—	Both	Glut; designated
{}	—	Neither	Gap; undesignated
v	—	Valuation	v: Form → P({t, f})
P	—	Power set	The four subsets of {t, f}
R	—	Ternary relation	Routley–Meyer accessibility for →
FDE	—	First-degree entailment	The conditional-free fragment
LP	—	Logic of Paradox	FDE plus the no-gap constraint
K3	—	Strong Kleene	FDE plus the no-glut constraint
FOUR	—	Belnap's bilattice	Two orders on the four values: truth and information

Metatheory. Dunn's relational semantics is sound and complete for first-degree entailment, the first-degree fragment of Anderson and Belnap's relevance logics E and R, and it is equivalent to the four-element De Morgan matrix with {t} and {t,f} designated — the relational and the matrix presentations pick out the same consequence relation. Belnap (1977) reorganized the four values as a bilattice, with a truth order and an information order, and read the extra values as what a database has been told rather than as how the world is. Two constraints generate the neighbouring logics: forbidding gaps gives Priest's LP, forbidding gluts gives strong Kleene K3, and FDE is exactly their intersection as consequence relations. The systems separate sharply on validities: LP has precisely the classical tautologies while failing modus ponens, K3 has none at all, and FDE has none either, since the all-gaps valuation leaves every formula undesignated. The Routley–Routley star (1972) gives an equivalent two-valued treatment of the same negation.

Applies to. Relevance logic semantics. Paraconsistent reasoning. Truth value gaps and gluts. Philosophical logic. Database semantics.

Limitations. FDE has no valid formulas at all: the valuation sending every atom to {} leaves every formula undesignated, so this is a consequence relation with empty theoremhood, and there is no deduction theorem because the first-degree fragment contains no detachable conditional. Paraconsistency costs disjunctive syllogism — A ∨ B, ¬A ⊭ B — and the failure is unconditional, not confined to inconsistent premise sets. The four values fix ¬, ∧, and ∨ but leave → undetermined; a relevant implication must be supplied by the Routley–Meyer ternary relation, and the frame conditions separating E, R, and T are invisible at this level, so Dunn semantics selects no relevance logic on its own. The connectives are functionally incomplete over FOUR, since only operations monotone in the information order are definable. And adding a Boolean negation restores explosion, so the paraconsistency belongs to the De Morgan negation, not to the framework.