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.

Metatheory. Free logic is a proper sublogic of classical first-order logic in the same language: every free validity is classically valid, since classical structures are just the free structures in which every term denotes, while ∀x.φ → φ(t), φ(t) → ∃x.φ and ∃x(x = t) are all classically valid and free-logically not. Classical logic is recovered by adding the axiom that every term denotes, together with non-emptiness of the domain. Completeness theorems are available for each of the main variants — Leblanc and Thomason (1968) proved them for the presupposition-free systems with inner and outer domains — and the inner/outer-domain semantics also shows that positive free logic adds no expressive power, since relativizing each quantifier to the existence predicate translates it faithfully into classical two-sorted logic. Undecidability is therefore inherited unchanged; free logic decides nothing that classical logic does not. Lambert's Law, ∀y(y = ιx.φ(x) ↔ ∀x(φ(x) ↔ x = y)), is the standard axiom governing descriptions, and unlike Russell's contextual definition it treats ι-terms as genuine terms.

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. The quantifier rules lose their unconditional form. Universal instantiation and existential generalization both acquire an E! side condition, so every inference between a term and a quantifier carries an existence premise, and the natural deduction system no longer has the clean introduction and elimination pairs classical logic has. "Free logic" names three incompatible systems rather than one: positive, negative and neutral differ on theorems, not on notation — t = t holds in the first and fails in the second, P(t) → E!t holds in the second and fails in the first — so results do not transfer between them. Each variant gives something up: negative free logic loses reflexivity of identity, since t = t fails for non-denoting t, and makes every atomic sentence about a non-denoter false, so nothing true can be predicated of a fictional object and the cases the logic was built for become unrepresentable; neutral free logic gives up bivalence and needs a partial, three-valued or supervaluational semantics, so the connectives cease to be truth-functional. Positive free logic keeps bivalence but leaves the outer domain unconstrained, which relocates questions about fiction rather than settling them. And nothing is gained in reach: the relativization translation makes free logic interpretable in classical logic, so no new structures are definable and undecidability persists.

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