User talk:Stoicuta Olimpiu Costinel

Czeslaw Lejewski was a philosopher and logician. He studied under Lukaseiwicz, Karl Popper, and W.V.O. Quine.

Logic and Existence (1954-5)

In his paper "Logic and Existence" (1954-5) he presented a version of free logic. He begins by presenting the problem of non-refering nouns, and commends Quine for resisting the temptation to solve the problem by saying that non-refering 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 empiracle information, which may not be forthcoming, is so foreign to the character of logical inquiry that a thorough re-examination of the two infereences [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 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 'b' refers to either. Thus, $(Fa\lor Fb)\leftrightarrow Fd$ is true. We now introduce the predicate Dx which is true for d. We have no reason, here, to contend that $(x=c)\land (x\;exists)$, 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 $\forall x(x\;exists)$ to boot.

Lejewski calls this account the unrestricted interpretation. The restricted interpretation is then the language which does not distinguish between signs and elements, and so is forced to claim $\exists x(x\;does\;not\;exist)$ is true. It is obvious that everything expressible in the unrestricted interpretation is expressible in the restricted interpretation. A generalization to infinite domains and infinite signs is easy. A generalization to infinite predicates needs no explanation.

A convenient fact is that this logic can also accomodate a domain consisting of the null set, as universal claims will not need to assume an element in the domain. For example, $\forall xFx\rightarrow (\exists xFx)$ 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. Where it is correct to apply the predicate Fx to every sign in the domain, it is correct to apply the predicate to a given sign in the domain. Thus the conditional is true. (Hence the treatment above that distinguishes the use of the existential quantifier and the meta-linguistic statement 'x exists'.) Using the restricted interpretation, we see that the claim becomes $\forall x(x\;exists\rightarrow Fx)\rightarrow \exists x(x\;does\;not\;exist\land Fx)$ which is false. The antecedent is true and the consequent false, because where the antecedent is true the consequent tells us that something does and does not exist.

This logic was later developed more fully by Karel Lambert, who called the unrestricted interpretation "free logic". Instead of the meta-linguistic 'x exists', Lambert adopted the symbolization E!x, which can be axiomatized without existential quantification.