Free Logic

Origin. Lambert introduced free logic (1960). Logic free of existential assumptions. Terms may denote nothing (non-denoting terms). Addresses: "The present king of France is bald." Empty domains allowed. Important for definite descriptions and fiction.

Models. Logic without existence assumptions. Classical: every term denotes, domain non-empty. Free logic: terms may fail to denote. "Pegasus flies" doesn't presuppose Pegasus exists. Inner/outer domain: existing vs possible objects. E! predicate: "exists."

Formalism.

Existence predicate: E!t — "t exists" (t denotes something in domain)

Positive free logic: Some atomic formulas with non-denoting terms are true — at least the identities. t = t holds for every term, denoting or not; "Pegasus is a winged horse" can be true.

Negative free logic: Every atomic formula containing a non-denoting term is false. P(t) → E!t is a theorem (predication implies existence); t = t fails for non-denoting t.

Neutral free logic: Atomic formulas with non-denoting terms are neither true nor false — a truth-value gap. Partial interpretation function; requires a three-valued or supervaluational treatment of the compounds.

Modified quantifier rules: ∀x.φ(x) → φ(t) only if E!t ∀x.φ(x) → (E!t → φ(t)) always valid

Universal instantiation: From ∀x.φ(x) and E!t, infer φ(t).

Definite descriptions: ιx.φ(x) — "the x such that φ" May fail to denote if no unique φ-satisfier. Russell: contextual definition. Free logic: allows non-denoting ι-terms.

Symbols.

Symbol	Unicode	Name	Meaning
E!	—	Exists	Existence predicate
ι	U+03B9	Iota	Definite description
∀	U+2200	Universal	For all existing
∃	U+2203	Existential	There exists
↓	U+2193	Defined	Term denotes
↑	U+2191	Undefined	Term fails to denote

Metatheory. Free logic is first-order. Completeness holds for various semantics. Decidability same as classical first-order (undecidable in general). Conservative extension: restricting to E!-true terms recovers classical logic. Multiple free logics: positive, negative, neutral. Identity: t = t holds for all terms in positive free logic, fails for non-denoting t in negative free logic.

Applies to. Philosophy (fictional entities, abstract objects). Definite descriptions (Russell, Strawson, Frege). Database theory (null values). Partial functions in CS. Modal logic (possibilia). Meinongian objects. Scientific theories with theoretical terms.

Limitations. Multiple incompatible versions. No consensus on "correct" free logic. Metatheory less developed than classical. Tool support limited. Interaction with other logics complex. Empty domain adds complications. Non-denoting detection not always decidable.