Logic of Quantors

Abstract

Sentences of the form “all P are Q” or “some P are Q” appearing in Syllogistic are neither primitive sentences nor are they composed of primitive sentences by means of junctors. In Antiquity and in Scholasticism, these sentences, however, have always been dealt with as if not composed, because for the composition only the junctors were taken into consideration. Only modern logic since Frege and Peirce has recognized the possibility of conceiving also of the sentences of Syllogistic as composed — and as composed of the hitherto primitive sentences in such a way that the composition is effected not only by means of junctors, but that also the logical particles “all” and “some” are used. First, we shall deal with the particular sentence: “some P are Q”. In contrast to the proposal laid down in Section 2, one can start out with the sentential form s ∈ P ∧ s ∈ Q for the purpose of interpretation. If S ∈P∧s ∈Q is true at least for one subjects then the sentence: “some P are Q.” shall be true — and only then. The truth of the particular sentence consequently signifies, then, that from among the sentences which result from s ∈ P ∧ s ∈ Q if the subject variable s is substituted by a subject, at least one sentence is true. If there are only finitely many subjects s₁s₂,... for the substitution, then consequently, “some P are Q” could be reproduced by the adjunction

$${{\mathfrak{s}}_{1}} \in P \wedge {{\mathfrak{s}}_{1}} \in Q\dot{ \vee }{{\mathfrak{s}}_{2}} \in P \wedge {{\mathfrak{s}}_{2}} \in Q\dot{ \vee } \ldots \infty \dot{ \vee }{{\mathfrak{s}}_{n}} \in P \wedge {{\mathfrak{s}}_{n}} \in Q.$$