Investigations on a comprehension axiom without negation in the defining propositional functions.
1960; Duke University Press; Volume: 1; Issue: 1-2 Linguagem: Inglês
10.1305/ndjfl/1093956425
ISSN1939-0726
Autores Tópico(s)Logic, Reasoning, and Knowledge
ResumoBemerkungen zum Kompre hens ions axiom" in Zeitschr.f. math.Logik und Grundl.d.Math., Bd 3 (1957), p. 1-17, I showed that antinomies of the same kind as Russell's could be avoided in set theory, if this was based on a certain logic, due to Lukasiewicz, with infinitely many truth values.Indeed I proved the existence of domains such that the axiom of comprehension was satisfied for elementary propositional functions Φ, that is Φ being built from atomic propositions u € v by use of conjunction, disjunction, implication and negation only.Later I proved the same for a certain 3"valued logic as shown in a paper which will appear in Math.Scand.Here I shall show in § 1 and §2 that the same is true even for ordinary 2-valued logic, provided that only conjunction and disjunction are allowed in Φ.In §3 I prove that also the axiom of extensionality is valid for the domains constructed in §1 and §2.I call the Φ constructed in this way positive propositions, abbreviated p. pr.The words "atomic propositions" are abbreviated to at.pr. The two truth values can be 0 (false) and 1 (true). In the sequel I write the conjunction of A and B as A A B and their disjunction as A v B. Further A (x) for all x is written Λ xA (x) and A (x) for some x is written V.xA (x).Now the p. pr.can be defined inductively as follows.1.The truth constants 0 and 1 are p.pr.2. Every at.pr.x € y is a p. pr.Here x and y are free variables. IfA and B are p.pr., so are A A B and A v B. The latter have the free and bound variables occurring in A and B, 4. If A (x, x 1 ,. ., x n ) is a p. pr. with x, x j ,. ., x n as free variables Λ x A (x, x f ,. ., x u ) and V xA^,%, ,. ., x n ) are p.pr. with x as bound variable, xj ,..,x n still as free variables, while the eventually occurring bound variables in A (x, x 1 , . ., x n ) remain bound inthelatterexpressions.If a set y is such that Ax ((x ey)=U (x,Xi ,..,x n )) is true, where %,%,,. ., x n are the set variables in the p.pr. U, then y is a set function of Xi , . .x , 1 ' n
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