Free Logic and Presupposition

Classical logic assumes every singular term denotes; free logic relaxes this to handle empty names like 'Pegasus' and definite descriptions that fail to refer.

Definition

Free logic is a system of quantification whose singular terms need not denote existing objects, so that inferences such as moving from 'a is F' to 'something is F' hold only on the added premise that a exists.

Scope

This topic covers free logic — logic free of existence assumptions for its singular terms — and the closely related phenomenon of presupposition failure. It treats how free logics modify the classical rules of universal instantiation and existential generalization for non-denoting terms, the positive, negative, and neutral (supervaluational) variants, and the philosophical background in the Russell-Strawson dispute over how sentences containing non-referring descriptions get truth values.

Core questions

• How should logic treat sentences containing empty names or failed descriptions?
• Do such sentences lack a truth value (presupposition failure) or come out false?
• Which classical inference rules must be restricted when terms may not denote?
• Is existence a predicate, and how should the quantifiers relate to existence?

Key concepts

• non-denoting singular terms
• universal instantiation and existential generalization
• positive, negative, and neutral free logic
• presupposition failure
• truth-value gaps
• existence as a predicate

Key theories

Free logic

Lambert systematizes logics in which singular terms may be empty; universal instantiation and existential generalization are qualified by an existence assumption, and variants differ on the truth values of atomic sentences with non-denoting terms.

Presupposition and truth-value gaps

Strawson argues that a sentence using a non-referring description (Russell's 'the present King of France') presupposes rather than asserts existence and so is neither true nor false; van Fraassen models this with supervaluations.

History

Russell's 1905 theory of descriptions handled empty terms by analysis rather than logical revision; Strawson's 1950 reply introduced presupposition and truth-value gaps. From the 1960s Lambert coined and developed free logic as a systematic alternative, and van Fraassen supplied a supervaluational semantics for the resulting gaps.

Debates

False or truth-valueless?

Whether sentences with non-referring terms are simply false, as Russell's theory of descriptions implies, or suffer presupposition failure and lack a truth value, as Strawson and the supervaluational free logician hold.

Key figures

• Karel Lambert
• P. F. Strawson
• Bas van Fraassen
• Bertrand Russell
• Hugues Leblanc

Seminal works

• strawson1950
• lambert2003

Frequently asked questions

Why can't classical logic handle empty names?

Classical logic licenses inferring 'something is F' from 'a is F' for any term a, which fails if a does not denote anything — for example inferring that something is a winged horse from 'Pegasus is a winged horse'. Free logic restricts such inferences so they require the extra premise that the named object exists.