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:
| ∧ | T | F | U | ∨ | T | F | U | ¬ | |||
|---|---|---|---|---|---|---|---|---|---|---|---|
| T | T | F | U | T | T | T | U | T | F | ||
| F | F | F | U | F | T | F | U | F | T | ||
| U | U | U | U | U | U | U | U | U | U |
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.
| Symbol | Unicode | Name | Meaning |
|---|---|---|---|
| T | — | True | Definitely true |
| F | — | False | Definitely false |
| U | — | Undefined | Meaningless |
| ∧ | U+2227 | And | Internal (weak) conjunction |
| ∨ | U+2228 | Or | Internal (weak) disjunction |
| ¬ | U+00AC | Not | Internal negation |
| ∘ | U+2218 | Assertion | External "meaningful-true" |
| ∼ | U+223C | External negation | "Is false" |
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: Kleene algebras with an infectious element. Paracomplete: A ∨ ¬A can fail. Bochvar's B₃ = weak Kleene internal connectives + external assertion and negation.
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 very strong: F ∧ U = U is controversial. No internal tautologies. Limited inference without the external layer. Strong Kleene often preferred. The external operators are non-truth-functional over the internal ones.
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