Chapter 6 Exercises: Venn Diagrams for Categorical Syllogisms (Symbolic Logic)

What does a Venn diagram for testing a categorical syllogism look like?

1. Diagram Premises

   1. If one premise is universal and another is particular, diagram the UNIVERSAL premise first.

   2. If there’s no shading on either side of a line dividing the area where an “X” needs to go, put the “X” ON THE LINE.

2. Check to see whether the information in the conclusion is ALREADY in the diagram.

3. If yes, then the syllogism is valid. If no, it’s invalid. 

What is the procedure for testing a syllogism with a Venn diagram?

Shade Areas 3 and 4.

Shade Areas 2 and 3.

Diagram the two premises below:

1. No M are P.

2. No M are S.

.X in Area 7.

Shade Areas 5 and 6.

Diagram the two premises below:

1. Some P are not M.

2. All S are M.

X on the line between Areas 3 and 4.

X on the line between Areas 5 and 6.

Diagram the two premises below:

1. Some M are P.

2. Some S are not M.

Invalid because we would shade Areas 3 and 6.

Mood and Figure: AIE-1

Examine the syllogism below and its diagram. Is the syllogism valid?

1. All knives are tools.

2. Some bayonets are knives.

3. So, no bayonets are tools.

Valid because we would shade Areas 3 and 6, and that has already been done

Mood and Figure: AEE-4

Examine the syllogism below and its diagram. Is the syllogism valid?

1. All toxins are poisons.

2. No poisons are healthful substances.

3. So, no healthful substances are toxins.

Invalid because we would shade Areas 2 and 5

Mood and Figure: AAA-3

Examine the syllogism below and its diagram. Is the syllogism valid?

1. All cats are animals.

2. All cats are multicellular organisms.

3. So, all multicellular organisms are animals.

Valid because you would put an X on the line between 2 and 5.

Mood and Figure: EIO-1

Test the syllogism below using a Venn Diagram:

1. No electrons are bosons.

2. Some leptons are electrons.

3. So, some leptons are not bosons.

Invalid because we would put an X in Area 3

Mood and Figure: AOI-2

Test the syllogism below using a Venn Diagram:

1. All vipers are snakes.

2. Some reptiles are not snakes.

3. So, some reptiles are vipers.

Invalid because we would put an X on the line between Areas 2 and 5.

• also know it’s invalid because both premises start with “some”

Mood and Figure: OOO-3 (I think)

Test the syllogism below using a Venn Diagram:

1. Some fish are not carp.

2. Some fish are not sturgeons.

3. So, some sturgeons are not carp.

Symbolized Argument:

1. All F are T

2. All T are O

3. So, some O are F

Invalid because we would put an X in Area 3.

Mood and Figure: AAI-4

Translate this argument into a standard-form categorical syllogism and test it with a Venn diagram:

Since every truck is an off-road vehicle, and given that all Fords are trucks, it necessarily follows that at least one off-road vehicle is a Ford.

Symbolized Argument:

1. All T are E

2. All D are T

3. So, all D are E

Valid because we would shade areas 2 and 5 and that’s already done.

Mood and Figure: AAA-1

Translate this argument into a standard-form categorical syllogism and test it with a Venn diagram:

Given that every didact is a teacher and that any teacher is an educator, we can infer that whoever is a didact is also an educator. 

Symbolized Argument:

1. No O are W

2. Alll J are O

3. So, some J are not W

Invalid because we would put an X in Area 2.

Mood and Figure: EAO-1

Test the syllogism below with a Venn diagram:

1. All oaks are willows.

2. All Japanese evergreens are oaks.

3. So, some Japanese evergreens are not willows.

True

Both premises have no existential import (because 1st premise is an A statement and the 2nd premise is an E statement), but the conclusion has existential import (because it is an O statement)

True or False: The syllogism below commits the existential fallacy (from the modern standpoint).

1. All moons are satellites.

2. No satellites are motionless objects.

3. So, some motionless objects are not moons.

existential fallacy

occurs when you try to derive a statement with existential import from premises that have no existential import