We are about to formalize a way of using Venn diagrams. Before we present a formalism for this system (we call this system Venn-I), let us see how Venn diagrams are used to test the validity of syllogisms:

1. Draw diagrams to represent the facts that the two premises of a syllogism convey. (Let us call one D1 and the other D2.)

2. Draw a diagram to represent the fact that the conclusion of the syllogism conveys. (Let us call this diagram D.)

3. See if we can read off diagram D from diagram D1 and diagram D2.

4. If we can, then this syllogism is valid.

5. If we cannot, then this syllogism is invalid.

Let us try to be more precise about each step. Step (1) and step (2) raise the following question: How is it possible for a diagram drawn on a piece of paper to represent the information that a premise or a conclusion conveys? These two steps are analogous to the translation from English to a first-order language. Suppose that we test the validity of a syllogism by using a first-order language. How does this translation take place? First of all, we need to know the syntax and the semantics of each language – English and the first-order language. We want to translate an English sentence into a first-order sentence whose meaning is the same as the meaning of the English sentence.