Free Logic: A Fifty-Year Past and an Open Future

Abstract

The term ‘free logic’, as introduced by Karel Lambert in 19601, is short for `presupposition-free logic’ or, more explicitly, for ‘logic free of existential presuppositions’. Since existential presuppositions are connected with terms, this paraphrase was understood as `logic free of existential presuppositions with respect to its singular and general terms’. Modern logic in the sense of standard first order predicate or quantificational logic with identity (QL=, for short) is almost fully free with respect to its general terms or predicates, with one exception, namely universal terms or predicates like ’Px y —Px’ or ‘x = x’. In standard systems of QL=, ‘∃x(Px v —iPx)’ and ‘∃x(x = x)’ are theorems. If we read such a formula as ‘something exists’ this seems to express a matter of ontology rather than logic; taking it as a law of logic is therefore “a defect in logical purity” as already noted by Russe112. Giving up this presupposition results in a logic called inclusive logic by Quine3 or empty logic by some free logicians. The main concern of free logic was and is therefore existential presuppositions with respect to singular terms. In standard systems of QL= we usually have for every singular term t and every variable v the theorem 3∃(v = t). And in standard quantification theory without identity (QL) we still have for every formula A the theorem A(t/v) →∃vA and its dual∀vA →A(t/v).KeywordsSingular TermDefinite DescriptionValuation FunctionClosed FormulaDefinite ArticleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.