Intensional Semantics for J. Venn's Logic of Classes

Abstract

In this article we expound V. I. Shalack’s approach to the adequate semantics for different syllogistic systems construction. The point is that the formulas of propositional logic are associated with the subject and the predicate of the categorical propositions as their meanings. The propositions themselves are interpreted with the help of logical entailment. We constructed semantics of this type for the J. Venn's syllogistic with non-standard primitive propositions: “All $S$ is all $P$” ($SaaP$), “All $S$ is some $P$” ($SaiP$), “Some $S$ is all $P$” ($SiaP$), “Some $S$ is some $P$” ($SiiP$), “No $S$ is $P$” ($SeP$). Each of them corresponds to one of the Euler diagrams. There are two kinds of Venn's syllogistic formalization: one is the theory of the relations between arbitrary classes and another is the theory of the relations between non-empty classes. We construct Shalack’s type semantics for the first formalization. We introduce the function $\delta$ puts arbitrary propositional formulas in correspondence with the general terms. Let the general terms $S$ and $P$ be interpreted by propositional formulas $A$ and $B$. $SaaP$ is true under this interpretation iff $A$ entails $B$ and $B$ entails $A$; $SaiP$ is true iff $A$ entails $B$ and $B$ doesn't entail $A$; $SiaP$ is true iff $A$ doesn't entail $B$ and $B$ entails $A$; $SeP$ is true iff $A$ entails the negation of $B$; $SiiP$ is true iff $A$ doesn't entail $B$, $B$ doesn't entail $A$ and $A$ doesn't entail the negation of $B$. Truth definitions for complex syllogistic formulas are standard. The adequate semantics for the second formalization of Venn's syllogistic is constructed by changing the interpretation of general terms: we assign to them only satisfiable propositional formulas. Soundness and completeness theorems are proved for both types of the syllogistic. Also we construct the semantics where we use relevant ($\textbf{FDE}$) entailment in favour of classical entailment in the truth definitions of Venn's syllogistic formulas. We formulate the syllogistic calculus which is adequate to this semantics.
In conclusion we compare the deductive power of three Venn's type syllogistics.