• Venn diagrams represent the relationships between
classes of objects by way of the relationships among
• Venn diagrams assume the Boolean interpretation
of categorical syllogisms.
• Shading an area of a circle shows that it is empty.
• Placing an X in an area of a circle shows that there
is at least one thing that is contained in the class
represented by that area.
• For universal propositions, shade (draw lines
through) the areas that are empty.
All S are P. All P are S. No S are P No P are S
• For particular propositions, place an X in the area
that is inhabited.
- Some S are P. - Some P are S.
- Some S are not P. - Some P are not S.
• To test a syllogism by Venn diagrams, you diagram
the premises to see whether the conclusion is also
• This requires three interlocking circles, one for each
• This divides the diagram into eight distinct regions
(a line over a term means “not”):
Venn Diagrams: Examples
• Consider the following syllogism:
No arachnids are cows.
All spiders are arachnids.
No spiders are cows.
• Let S represent the minor term (spiders), C represent the
major term (cows), and A represent the middle term
(arachnids). Since both premises are universals, let us
begin by diagramming the major premise. We shade the
area were S and C overlap:
Now diagram the
minor premise on
the same diagram:
diagram for the
if you wish:
Venn Diagrams: (pp. 162-166)
• Most syllogistic forms are invalid. Consider the
All P are M.
All M are S.
All S are P.
• Diagram the major premise, then diagram the minor
premise on the same diagram:
We have diagrammed “All P
are M,” which is not the
conclusion. So the argument
form is invalid.
• Consider an argument of the following form:
All M are P.
No M are S.
No S are P.
• An area has been shaded twice. So, we haven’t
diagrammed the conclusion. The argument form
• In summary:
– Make sure you have exactly three terms.
– If there is a universal premise and a particular
premise, diagram the universal premise first.
– If neither of the areas where the X could go is
shaded, the X goes on the line.
– No syllogism whose diagram places an X on the line
or results in double-shading is valid.
– It is valid if and only if shading the premises results
in shading the conclusion.
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