In this chapter, I claim that Venn-II is equivalent to a first-order language ℒ0, which I will specify in the first section. This claim is supported by two subclaims. One is that for every diagram D of Venn-II, there is a sentence ϕ of ℒ0 such that the set assignments that satisfy D are isomorphic to the structures that satisfy ϕ. The other is that for every sentence ϕ of ℒ0, there is a diagram D of Venn-II such that the structures that satisfy ϕ are isomorphic to the set assignments that satisfy D.

The language of ℒ0

Our first-order language ℒ0 is as follows:

A. Logical Symbols

1. Parentheses: (,)

2. Sentential connective symbols: ¬, ∧, ∨

3. Variables: x1, x2, x3, …

4. Equality: No

B. Parameters

1. Quantifier symbols: ∀, ∃

2. Predicate symbols: 1-place predicate symbols, P1, P2, …

3. Constant symbols: None

4. Function symbols: None

From set assignments to structures

In this section, we want to show that there is an isomorphism between the set of set assignments for Venn-II and the set of structures for ℒ0.

Because we have only one closed curve type and one rectangle, we need an extra mechanism in the semantics of this Venn system. That is a counterpart relation among tokens of a closed curve or among tokens of a rectangle. Before we make a mapping between sets and structures, we need to deal with these cp-related regions.