Weak Kleene Logic

Origin. Kleene's weak three-valued logic (1952) coincides with Bochvar's B₃ (1938): a third value that infects every connective, modelling meaninglessness or nonsense. Contrast Strong Kleene, where some operations ignore the undefined value.

Models. Nonsense propagation. Three values: T (true), F (false), U (undefined/meaningless). Any U input yields U output. "The present king of France is bald and 2+2=4" is undefined; the meaningless part infects the whole. Bochvar reads U as meaningless, so the internal connectives are the infectious ones and a separate external layer recovers classical assertion.

Formalism.

Truth values: {T, F, U} — true, false, undefined/meaningless.

Internal (infectious) tables:

Key property: Any U in any position → U. Contrast Strong Kleene: F ∧ U = F, T ∨ U = T.

No internal tautologies: Even A ∨ ¬A is U when A is U. The infectious value blocks every internal validity.

External assertion (Bochvar): ∘A ("A is true and meaningful"): ∘A = T iff A = T, else F. Classical output; escapes U-propagation.

External negation: ∼A := ∘(¬A) ("A is false"): ∼A = T iff A = F, else F. The external connectives are bivalent; practical reasoning runs through them.

Internal vs external: Internal connectives infectious (Bochvar's B₃ core). External connectives classical. Liar and other paradoxes take value U—meaningless, not contradictory.

Symbols.

Metatheory. Decidable (three-valued). U is neither designated nor anti-designated, so pure weak Kleene has no tautologies; the external layer restores classical assertion. Algebraically: involutive bisemilattices, obtained as Płonka sums of Boolean algebras (Bonzio, Gil-Férez, Paoli, Peruzzi) — not De Morgan or Kleene algebras, since the infectious element breaks absorption. Paracomplete: A ∨ ¬A can fail. Bochvar's B₃ = weak Kleene internal connectives + external assertion and negation. What the infectious value does to consequence is exactly a variable-inclusion constraint on classical consequence, and the constraint falls on opposite sides depending on which value is designated. With D = {T} the conclusion must be forced classical, so for classically satisfiable Γ, Γ ⊨_WK φ iff Γ ⊨ φ classically and every variable of φ occurs in Γ — which is why nothing is valid, the empty premise set containing no variables; a classically unsatisfiable Γ is never designated and so entails everything. With D = {T, U} the logic is paraconsistent rather than paracomplete, and Γ ⊨_PWK φ iff some subset of Γ whose variables all occur in φ classically entails φ; there the classical tautologies survive and it is the premise side that is constrained. Halldén (1949) reached the same tables independently as a logic of nonsense.

Applies to. Meaninglessness and category mistakes. Presupposition failure. Undefined values in computation (nulls, type errors, division by zero). Semantic paradoxes read as meaningless.

Limitations. Infection is absolute, so F ∧ U = U: a conjunction with a false conjunct is meaningless rather than false, and the tables cannot model short-circuit evaluation, which is the main computational reason for wanting a third value. Absorption fails — A ∧ (A ∨ B) is U when A is T and B is U — so the algebras are not lattices and none of the De Morgan or Kleene machinery applies. With no internal validities there is nothing to axiomatize as theorems, and the deduction theorem fails for the definable material conditional, since A ⊨ A while ¬A ∨ A takes U when A does. The variable-inclusion characterization shows the logic only subtracts from classical consequence and never adds. The external operators are not definable from the internal ones — they are not infectious, so no composition yields them — and once present they are bivalent, so all substantive inference runs through a classical layer the three-valued tables do not touch. Nothing distinguishes a category mistake from a divergent computation.

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