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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)**

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