Lejewski

Czesław Lejewski (1913–2001) 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. Quine.

In his paper "Logic and Existence" (1954–55) he presented a version of free logic. He begins by presenting the problem of non-referring nouns, and commends Quine for resisting the temptation to solve the problem by saying that non-referring names are meaningless. Quine's solution, however, was that we must first decide whether our name refers before we know how to treat it logically. Lejewski found this unsatisfactory because we should have a formal distinction between referring and non-referring names. He goes on to write, "This state of affairs does not seem to be very satisfactory. 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." (parenthesis not Lejewski's).

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 one predicate, Fx. There is no need for universal or existential quantification, in the style of Quine in his Methods of Logic. The only possible atomic statements are Fa and Fb. We now introduce new signs but no new elements in the domain. 'c' refers to neither element and 'd' refers to either. Thus, ${\displaystyle (Fa\lor Fb)\leftrightarrow Fd}$ is true. We now introduce the predicate Dx which is true for d. We have no reason, here, to contend that ${\displaystyle (x=c)\land (x{\text{ exists}})}$, and thus to claim that there is something which does not exist. We simply do not have good reason to make existential claims about the referent of every sign, since that would assume that every sign refers. Instead, we should remain agnostic until we have better information. By the stipulations given here, however, we have downright good reason to be atheists about c, and have good reason to still claim ${\displaystyle \forall x(x{\text{ exists}})}$ to boot.

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