Partial Logic

Origin. Kleene's partial recursive functions and three-valued tables (1938, 1952); Blamey's survey "Partial logic" (Handbook of Philosophical Logic, 1986); Langholm, Partiality, Truth and Persistence (1988); Muskens (1995) for the type-theoretic version. Partial functions. Undefined terms. Gaps. Foundation of partiality semantics.

Models. Partial interpretations. Undefined values. Three-valued or more. Gap propagation or absorption.

Formalism.

Partiality sources: Partial functions: f(a) undefined. Empty descriptions: ιx.φ undefined. Presupposition failure. Non-terminating computation.

Denotation: [[t]] ∈ D ∪ {↑}. ↑: undefined. Terms may lack denotation.

Atomic formulas: P(t₁,...,tₙ) undefined if some tᵢ undefined. Undefinedness propagates. Gap at atomic level.

Logic options: Strong Kleene: gaps propagate. Weak Kleene: gaps dominate. Supervaluations: classical on all completions. Blamey: transvaluation.

Kleene approach: Third value: neither true nor false. Connectives as in strong Kleene. Gaps preserved compositionally.

Supervaluation approach: φ supertrue if true in all completions. φ superfalse if false in all completions. Otherwise gap. Classical validities preserved.

Existence predicate: E!(t): t is defined. E!(t) ↔ t = t (sometimes). Explicit definedness.

Scott's approach: Information ordering. ⊥ ⊑ true, ⊥ ⊑ false. Monotonicity. Domain theory connection.

Symbols.

Metatheory. Partiality. Gaps. Completion semantics. Information ordering.

Applies to. Partial functions. Computation. Presupposition. Non-termination.

Limitations. Multiple approaches. Compositionality tensions. Classical reasoning lost. Complexity.

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