Venn Diagram Test

A Venn diagram can represent pictorially the logical form of a categorical syllogism. Each term is represented by a circle. So three overlapping circles are necessary to represent all Boolean conjunctions of the three terms of the categorical syllogism.

Area | Represents
3 | SPM
6 | SP~M
2 | S~PM
5 | S~P~M
4 | ~SPM
7 | ~SP~M
1 | ~S~PM
8 | ~S~P~M

The three overlapping circles provide a means of testing the validity of a categorical syllogism. To do this one needs to know:

- How to label the diagram.
- What mark to place in the diagram.
- Where to put the mark.
- How to read the result.

Following is a diagram which will help find where to mark the diagram.

Premise | Where
PM | 3, 4
P~M | 6,7
M~P | 1, 2
SM | 2, 3
S~M | 5, 6
M~S | 1,4
Conclusion | Where
SP | 3, 6
S~P | 2, 5
P~S | 4, 7

Examples:

All labor leaders are true liberals.
No wealthy men are true liberals.
No wealthy men are labor leaders.

Old form | Boolean form | Venn Diagram | Mapping
All L are T No W are T No W are L | L~T = 0 (6, 7) WT = 0 (2, 3) WL = 0 (3, 6) |  | S = W P = L M = T

- How to label the Venn diagram:
  - Have the bottom left circle, labeled S, represent the subject term of the conclusion.
  - The bottom right circle, labeled P, represents the predicate term of the conclusion.
  - The top circle, labeled M, represents the middle term, the term that appears once in each premise but never in the conclusion.
- How to mark the diagram.
  - Only diagram the two premises-NEVER the conclusion.
  - When the Boolean form for the premise is an equation (= 0) shade the designated areas.
- Where to put the mark.
  - Concentrate on only two of the circles representing the premise. When shading, the third circle can usually be ignored.
  - If you have difficulty finding the two areas to be shaded, use the table above:
    - Mapping (L~T = 0) to P and M results in (P~M = 0).
    - Looking up P~M in the table above shows areas 6 and 7.
    - Shade both areas.
    - Mapping (WT = 0) to S and M results in (SM = 0).
    - Looking up SM in the above table shows areas 2 and 3.
    - Shade both areas.
- How to read the resulting diagram to determine whether the syllogism is valid or not.
  - See if the premises implied the information of the conclusion. Did diagramming the premises automatically diagram the conclusion?
  - Mapping the conclusion (WL= 0) to S and P results in (SP = 0 ).
  - Looking up SP in the table above shows the areas 3 and 6.
  - If the argument is valid the areas 3 and 6 should have been shaded by the premises. An examination of the diagram shows that areas 3 and 6 are shaded. The argument is valid.

All democrats are for Medicare.
All liberals are for Medicare.
So all democrats are liberals.

Old Form | Boolean Form | Venn Diagram | Mapping
All L are F All D are F All D are L | L~F = 0 (6, 7) D~F = 0 (5, 6) D~L = 0 (2, 5) |  | S = D P = L M = F

- Diagram the premises to see if they imply the conclusion. If you have trouble locating the areas use the table above.
  - Mapping (L~F = 0) to P and M results in (P~M = 0).
  - In the table above P~M designates areas 6 and 7. Shade these two areas.
  - Mapping (D~F = 0) to S and M results in (S~M = 0).
  - In the above table S~M designates areas 5 and 6. Shade these two areas.
  - Mapping the conclusion (D~L = 0) to S and P results in ( S~P = 0).
  - In the above table S~P designates areas 2 and 5.
  - If the argument is valid, the premises should have shaded areas 2 and 5.
  - An examination of the Venn diagram shows that this did not happen: area 2 is not shaded. The premises did not imply the conclusion. The argument is invalid.

Some juvenile delinquents are products of broken homes.
All juvenile delinquents are maladjusted individuals.
Some maladjusted individuals are products of broken homes.

Old Form | Boolean Form | Venn Diagram | Mapping
Some J are B All J are I Some I are B | JB != 0 (3, 4) J~I = 0 (1, 4) IB != 0 (3, 6) |  | S = I P = B M = J

- Placing an X in the diagram:
  - The X always goes in an area that is divided into two parts by the usually ignored third circle.
  - If neither of the two areas is shaded the X goes on the arc of the third circle that divides the two parts.
  - If one of the two parts is shaded, the X goes in the unshaded part.
  - HINT: Always diagram the (= 0) premise before the (!= 0) one.
- Diagram the premises to see if they imply the conclusion.
  - Mapping (J~I = 0) to S and M results in (M~S = 0).
  - In the table above M~S designates areas 1 and 4. Shade both areas.
  - Mapping the premise (JB != 0) to P and M results in (MP != 0).
  - In the table above areas MP designates the areas 3 and 4.
  - since the premise is (!= 0) place an X in either 3 or 4. In this case place the X in area 3 since area 4 is shaded.
  - Mapping the conclusion (IB != 0) to S and P results in (SP != 0).
  - In the above table SP designates areas 3 and 6.
  - So if the argument is valid the premises should have placed an x in either 3 or 6. There is an X in 3 so the argument is valid.

Some wealthy men are not Republicans.
No Republicans are Democrats.
So some Democrats are wealthy men.

Old Form | Boolean Form | Venn Diagram | Mapping
Some W are not R No R are D Some D are W | WR != 0 (3 ,4) RD = 0 (2, 3) DW != 0 (3,6) |  | S = D P = W m = R

- Diagram the premises to see if they imply the conclusion.
  - Mapping (RD = 0) to S and M results in (MS = 0).
  - In the table above MS designates areas 2 and 3. Shade both areas.
  - Mapping premise (WR != 0) to P and M results in (PM != 0).
  - In the table above PM designates areas 3 and 4.
  - Area three is shaded so the X has to go in area 4.
  - Mapping the conclusion (DW != 0) to S and P results in (SP != 0).
  - In the table above SP designates areas 3 and 6.
  - If the argument is valid the premises should have placed an X in either 3 or 6. Area 3 is shaded, and area 6 is blank. There is no X in either area. The argument is invalid.