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
{t}	—	True	Only true
{f}	—	False	Only false
{t,f}	—	Both	Glut
{}	—	Neither	Gap
⊨	U+22A8	Consequence	Follows from

Metatheory. Sound and complete for FDE. Generalizes classical semantics. Lattice structure on valuations. Basis for Belnap's bilattice. Four-valued matrices equivalent.

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

Limitations. Multiple extensions possible. Implication variants. Philosophy of "both true and false." Set-theoretic foundation needed.