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Chapter 16Venn Diagrams

Venn Diagrams (pp. 159-160)

- Venn diagrams represent the relationships between

classes of objects by way of the relationships

among circles. - 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.

Venn Diagrams (pp. 159-160)

- 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.

Venn Diagrams (pp. 159-160)

- 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.

Venn Diagrams for Syllogisms (pp. 162-166)

- To test a syllogism by Venn diagrams, you diagram

the premises to see whether the conclusion is

also diagrammed. - This requires three interlocking circles, one for

each term

Venn Diagrams for Syllogisms (pp. 162-166)

- This divides the diagram into eight distinct

regions (a line over a term means not)

Venn Diagrams for Syllogisms (pp. 162-166)

- Diagram the premises to see whether you have

diagrammed the conclusion. - You should always set up the diagram in the same

way upper left circle for the minor term upper

right circle for the major term bottom circle

for the middle term. - If by diagramming the premises you have

diagrammed the conclusion, the argument is valid. - If by diagramming the premises you have not

diagrammed the conclusion, the argument is

invalid.

Venn Diagrams for Syllogisms (pp. 162-166)

- If you have both a universal premise and a

particular premise, you should diagram the

universal premise first this will sometimes

force the X into a determinate region. - If you have a particular premise and the X is not

forced into a determinate section of the diagram,

it goes on the line. The line in question is

always the line of the circle not mentioned in

the premise. - It might be helpful to draw a separate,

two-circle diagram of the conclusion but never

add anything to the three-circle diagram other

than the diagrams of the premises.

Venn Diagrams Examples(pp. 162-166)

- Consider the following syllogism
- All logicians are critical thinkers.
- All philosophers are logicians.
- All philosophers are critical thinkers.
- Where L represents the middle term and C

represents the major term, and P represents the

minor term, the diagram for the major premise

looks like this

Venn Diagrams Examples(pp. 162-166)

Now you diagram the minor premise on the same

diagram

Venn Diagrams Examples(pp. 162-166)

- If youre so inclined, compare the diagram for

the conclusion alone. - Since the premises require that all of P that is

outside of C is shaded, we have diagrammed the

conclusion in diagramming the premises. The

argument is valid.

Venn Diagrams Examples(pp. 162-166)

- If you find the process a bit odd, consider an

argument of the following form - All P are M.
- No M are S.
- No S are P.
- Draw a two circle diagram for each of the

premises

Venn Diagrams Examples(pp. 162-166)

- Roll them together to form a three-circle

diagram - You have diagrammed the conclusion by diagramming

the premises. The argument form is valid.

Venn Diagrams Examples(pp. 162-166)

- 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

Venn Diagrams Examples(pp. 162-166)

Now diagram the minor premise on the same diagram

Compare the diagram for the conclusion alone, if

you wish

By diagramming the premises we have diagrammed

the conclusion. The argument is valid.

Venn Diagrams Examples(pp. 162-166)

- Consider the following syllogism
- Some lizards are reptiles. All reptiles are

beautiful beasts. Some beautiful beasts are

lizards. - Here we have a particular premise and a universal

premise. When you have both, you diagram the

universal premise first. Why you ask. The X

for representing the particular should always go

into a determinate area if possible. If you

diagram the universal first, the X is forced into

a determinate area of the diagram

Venn Diagrams Examples(pp. 162-166)

Then add the X.

The argument is valid. The diagram shows that

there is at least one thing (X) that is a

beautiful lizard, so the argument is valid.

Venn Diagrams Examples(pp. 162-166)

- If youd diagrammed the particular premise first

the X would have gone on the line, since the X

goes on the line except when the area on one side

of the line is shaded. So, if youd diagrammed

the particular premise first, the diagrams would

look like this - It is bad form to have an X on the line if the

area on one side of the line is shaded. You would

have to erase and place it in the unshaded area.

Venn Diagrams Examples(pp. 162-166)

- Most syllogistic forms are invalid. Consider the

following - 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.

Venn Diagrams Examples(pp. 162-166)

- 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 havent

diagrammed the conclusion. The argument form is

invalid

Venn Diagrams Examples(pp. 162-166)

- Consider the following syllogism
- Some aardvarks are not sheep, and no sheep are

trumpets, so all aardvarks are trumpets. - After making sure there are exactly three terms,

you could represent the form as follows - No S are T.
- Some A are not S.
- All A are T.

Venn Diagrams Examples(pp. 162-166)

- You diagram the major premise, since its

universal - Now you diagram the particular. The X has to be

in A and outside of S. Since it could be in

either of two areas, neither of which is shaded,

you place the X on the T circle that divides A

into two parts. It looks like this

Venn Diagrams Examples(pp. 162-166)

The X is on the line. That is sufficient to show

that the argument form is invalid. If you

prefer, you could compare the top two circles to

the two-circle diagram for the conclusion. Youd

notice that you have not diagrammed the

conclusion. (The diagram for a universal is

always a strictly shady affair.)

Venn Diagrams Examples(pp. 162-166)

- Consider the following
- All mice are rodents, so some mice are

bothersome beasts, since some rodents are

bothersome beasts. - There are three terms, so we may set out the form

as follows - Some R are B.
- All M are R
- Some M are B.

Venn Diagrams Examples(pp. 162-166)

- This time the major premise is a particular, and

the minor premise is a universal. So, we diagram

the minor premise first - Now we diagram the major, placing an X in the

area where B and R overlap. The X goes on the

line

Venn Diagrams Examples(pp. 162-166)

The argument is invalid.

Venn Diagrams Examples(pp. 162-166)

- 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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