## Venn Diagrams

It is often helpful to think of a term in a proposition as a circle containing the members of the relevant class. This is the method of Venn Diagrams, invented by the English mathematician John Venn.

### The Venn Diagrams for Categorical Propositions

To use this method, we need a way to represent each of the different forms of proposition. The simplest case is the I proposition. "Some S is P" says that at least one member of S is also a member of P. We can represent this by putting an X in the intersection of two overlapping circles that represent S and P.

In the same way, the O proposition "Some S are not P" means that there is at least one item in the region of the S circle outside the P circle.

The representation of the universal propositions is a little different. The procedure is easiest to understand if we recall that the A proposition "All S are P" is equivalent to its obverse--"No S is non-P." We can indicate this by shading out the region of the S circle outside the P circle:

Notice that the diagram of the A proposition is the exact opposite of that for the O proposition, as it should be, since there are contradictories. The O diagram says that there is something in the leftmost region; the A diagram says there is nothing there. Notice too that the A diagram does not contain an X anywhere and thus does not imply the existence of any Ss. Venn diagrams are based on the modern view that universal propositions do not have existential import.

To diagram the E proposition, finally, we must shade out the area of overlap between the circles.

Since the E and I propositions are contradictories, this diagram shades out the very region in which the I diagram places an X.

Comprehension Questions

 1. Which of the following is the Venn diagram for an A proposition? a) b) c) d) 2. Which of the following is the Venn diagram for an I proposition? a) b) c) d) 3. Which of the following is the Venn diagram for an E proposition? a) b) c) d) 4. Which of the following is the Venn diagram for an O proposition? a) b) c) d)