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).