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Published online by Cambridge University Press: 24 November 2009
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
Parentheses: (,)
Sentential connective symbols: ¬, ∧, ∨
Variables: x1, x2, x3, …
Equality: No
B. Parameters
Quantifier symbols: ∀, ∃
Predicate symbols: 1-place predicate symbols, P1, P2, …
Constant symbols: None
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.
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