### Venn Diagrams for Categorical Propositions

John Venn, 1834-1923, developed a method for diagramming categorical propositions that let's us represent visually the informational content of the propositions.  What is more, Venn's diagrams allow one to test a syllogistic argument for validity without having to resort to notions like the distribution of terms.  Venn's diagrams involve 2 overlapping circles, each circle representing one of the classes mentioned in the proposition.  The diagram contains 4 distinct regions, each of which represents a type of object having certain properties.

The 4 regions in a 2 circle Venn diagram

Regions (1) and (2) represent the extension of the F's in the universe.  Regions (2) and (3) represent the extension of the G's.  Region (4) represents the extension of everything that is neither F nor G.  In this diagram the objects in region 1 are S's but not P's, those in region 2 are both S's and P's, those in region 3 are P's but not S's, and those in region 4 are neither S's nor P's.  Notice that no matter which terms S and P represent, every object in the the universe properly belongs in exactly one of the regions.

Two  rules apply to using Venn diagrams to represent categorical propositions:

1)  Shading a region indicates that that region is empty, the term in that region has no extension.

2)  Placing a bar or an x in a region indicates that the term in that region has some extension, the region is not empty.

If a region is neither shaded nor empty you cannot legitimately draw any inferences about whether the region is occupied or empty.  Using these 2 rules, we get the following basic Venn diagrams for our 4 basic categorical propositions:

1. A Proposition (All S's are P's)

Region 1 of the diagram is shaded because if "All S's are P's" then there are no S's that are non-P's, that is, region 1 is empty.

2. E Proposition (No S's are P's)

3. I Proposition (Some S's are P's)

4. O Proposition (Some S's are not P's)