This document provides instructions for using Venn diagrams to determine the validity of categorical syllogisms. It explains how to represent universal and particular statements in the diagram by shading or placing an X in the relevant areas. Diagramming involves using three overlapping circles to represent the three terms and shading or marking areas based on the premises. If the conclusion is represented in the diagram after premises are mapped, the argument is valid.

1. Venn Diagram
for Categorial
Syllogisms:
Determining
Validity-Ch 9

2. Areas/Regions/Quadrants and Term Relation
• 3 overlapping circles (one for each of the 3 terms)
• 7 areas (area 8 is outside—we won’t use that).What does each one
mean?
• 2 TERMS (CIRCLES) DONE AT A TIME. ONE PREMISE AT A TIME.
• SHADE EMPTY AREAS: SHADING MEANS EMPTY
EXAMPLE: No A are B. All C are A. So, no C are B.
In “No A are B,” terms A and B: There are no As in B. SHADE THE AREAS
WHERE 2 TERMS COME TOGETHER.
No problem if the shading extends into C, or, in the case of C and A (in
the “All C are A” premise), into B.

3. Diagramming Steps
• Initials: Minor (S of conclusion), major (P of conclusion), and middle (M)
• Three overlapping circles: lower left for subject (minor), lower right for
predicate (major), and middle for the middle term.
Diagram premises only, universal statements 1st—before particular ones.
• Shading for universal statements, if overlapping area is empty.
• X for particular statements, if there are some (at least one) of one term in
another.
• If a circle line runs through the area where the x should go, place the x on the
line.
• For the Aristotelian (but not for the hypothetical) standpoint, add x in an
unshaded area next to a shaded one (to show existence)

4. Last Step of Diagramming
• After diagramming the premises, inspect the diagram.
• If the content of the conclusion is represented in the diagram, the argument is
valid (the premises guarantee the conclusion).
• If the content of the conclusion is not represented in the diagram, the
argument is invalid (the premises do not guarantee the conclusion).
• Venn diagramming is a visual test of validity. It shows whether or not the
premises guarantee the conclusion. We do not diagram the conclusion. The
diagram has to “tell” us whether the argument is valid or not.

5. Can You Diagram this?
• Some warriors are Spartans
• All Spartans are Greeks
• Some Greeks are warriors
(FYI: IAI-4)
Head start:
S= G
P=W
M=S

7. Lower left: G: Greeks (Subject of conclusion-minor term)
Lower right: W: Warriors (Predicate of the conclusion-Major term)
Top one: S: Spartans (Middle term)
1st: Diagram the universal premise: All Spartans are Greeks: Top and left
circles: Shade 1 and 4
2nd: Diagram the particular premise: Some warriors are Spartans: Left and top
circles: Place an x in area 3
Last: After diagramming the premises, can you see the conclusion (Some Greeks
are warriors? If the conclusion is represented, the argument is valid.
Shading: 1, 4
X: 3
Validity: V

8. UNIVERSAL STATEMENTS
NOTE: In the examples below, the copula should be are, not is.

9. Syllogisms with a Particular
Premise
• VALID: First example
• INVALID: 2nd and 3dr examples—more than
one area for X. All these syllogisms will be
invalid. Something exists but it is not shown
exactly where—the premises do not
guarantee the conclusion.
• SHOW INVALIDITY: An x may be placed in
each area along with a line connecting
them, as shown in 2nd. BUT USE THIS
METHOD INSTEAD: Place the x on the line,
as shown in ex. 3

10. AGAIN: The X Symbol for Particular Premises
and to show the Reality of the Subject Term
• X for particular premises: If not clear which one of two areas, place it on
the line separating the two areas
P and M
Some
• To show reality: In the case of the Aristotelian standpoint, for the subject
terms of the premises (if we can assume existence): If there is only one
region unshaded –an unshaped area next to a shaded one--(either or
both circles), place an X there

11. NOTE
• With most syllogisms, whether or not you presuppose that things exist in the three
categories doesn’t matter in assessing validity. But in a few cases – all involving
syllogisms with two universal premises and a particular conclusion – it does. If you have
a syllogism with two universal premises and a particular conclusion it will be counted
“invalid” using the hypothetical viewpoint. To tell whether it is valid assuming the
existence of things in at least one of your categories, after representing the premises,
represent the key assumption you want to make about existence and see whether the
conclusion is then represented. For instance:
• Invalid from hypothetical viewpoint: Valid from existential viewpoint,
assuming the existence of S’s

12. How to Report VD of Categorical Syllogisms
on Exam III
FIRST BOX--Shading: Identify the number/s of the region/s--just the number/s. If more than one section (more than one
number), separate the numbers with commas. Do not use the word and (before the last number), just commas.
Sample: 1, 4, 7.
If no shading, write the word none.
SECOND BOX--X: Identify the number/s of the region/s--just the numbers. If more than one section (more than one
number), separate the numbers with commas. Do not use the word and (before the last number), just commas.
Sample: 1, 4, 7.
If you have to place the x on the line, write "on the line between…(number of first area) and … (number of second area)"
Sample: on the line between 5 and 6
If an x on the line appears twice, separate the two instances with a comma:
Sample: on the line between 4 and 6, on the line between 5 and 6
Note: When diagramming from the existential viewpoint and the x goes on the line, skip it; do not add an existential x on
the line.)
If no X, write word none.
THIRD BOX--Valid or Invalid: Write the letter V for valid--not the word valid; or the letter I for invalid--not the word
invalid.