Czesław Lejewski (1913 in Minsk – 2001 in Doncaster) was a Polish philosopher and logician, and a member of the Lwow-Warsaw School of Logic.
He studied under Jan Łukasiewicz and Karl Popper in the London School of Economics, and W. V. O.
He began by presenting the problem of non-referring nouns, and commended Quine for resisting the temptation to solve the problem by saying that non-referring names are meaningless.
Lejewski found this unsatisfactory because there should be a formal distinction between referring and non-referring names.
The idea that some of our rules of inference should depend on empirical information, which may not be forthcoming, is so foreign to the character of logical inquiry that a thorough re-examination of the two inferences (existential generalization and universal instantiation) may prove worth our while."
He then elaborates a very creative formal language: Take a domain consisting of a and b, and two signs 'a' and 'b' which refer to these elements.
There is no need for universal or existential quantification, in the style of Quine in his Methods of Logic.
We now introduce the predicate Dx which is true for d. We have no reason, here, to contend that
The restricted interpretation is then the language which does not distinguish between signs and elements, and so is forced to claim
will be true on an empty domain using the unrestricted interpretation, where 'c' still does not refer.
The proof is that, assuming the antecedent true, we must understand the quantifiers to make no claims about the elements of the domain but only about the signs.
He thus suggests that we abandon the interpretation of existential quantification as "there exists an x" and replace it with "for some (sign) x" (parenthesis not Lejewski's).
(Hence the treatment above that distinguishes existential quantification and the meta-linguistic statement 'x exists'.)
This is because nothing exists and so, for every sign, the inner antecedent is false, and so vacuously true.
Lejewski then goes on to extend this interpretation to the language of inclusion, and presents an axiomatization of an unrestricted logic.
Instead of the meta-linguistic 'x exists', Lambert adopted the symbolization E!x, which can be axiomatized without existential quantification.