Categorical Syllogisms

PHIL 121: Methods of Reasoning

February 25, 2013

Instructor:Karin Howe

Binghamton University

Important Definitions• Major term: the predicate term of the conclusion• Minor term: the subject term of the conclusion• Major premise: the premise containing the major

term• Minor premise: the premise containing the minor

term• Syllogism: any deductive argument containing

exactly two premises.• Categorical syllogism: a syllogism in which all

the statements in the argument are categorical propositions.

Standard Form Categorical Syllogisms

• A categorical syllogism is said to be in standard form if and only if it fulfills all of the following criteria:

1. All of the statements in the syllogism are in standard form.

2. The syllogism contains exactly three terms (the major term, minor term and middle term)

3. The premises are arranged in the correct order: major premise first, followed by minor premise, and each premise is on a separate line.

4. A line is drawn under the minor premise.5. The conclusion is written below the line.

ExamplesAll rodents are mammals.

All mice are rodents._______

All mice are mammals.

This argument is a standard form categorical syllogism.

All mice are rodents.All rodents are mammals.___All mice are mammals.

This argument is NOT a standard form categorical syllogism, because the premises are not in the right order (minor premise is first, followed by the major premise)

Mood• The mood of a standard form categorical

syllogism found by reading off the types of statements in the argument, in order (first the major premise, then the minor premise, and finally the conclusion).

• Example:Some cats are not ugly animals. (O)Some cats are not fuzzy animals. (O)No fuzzy animals are ugly animals. (E)

• Thus, the mood of this [standard form categorical] syllogism is OOE

Figure

1st figure 2nd figure 3rd figure 4th figure

M – P

S – M

S – P

P – M

S – M

S – P

M – P

M – S

S – P

P – M

M – S

S – P

There are exactly four possible arrangements of the three terms in a standard form categorical syllogism. These possible arrangements are known as the figure of the syllogism.

Example: This syllogism is an example ofSome M are not P. Figure 3.Some M are not S.No S are P.

What's the point of moods and figures?

• By specifying the mood and figure of a syllogism, we can specify the unique form of that standard form categorical syllogism.

• Example: AAA-1All small animals are cute animals. All M are PAll mice are small animals._______ All S are MAll mice are cute animals. All S are P

• In other words, every syllogism with the same mood and figure has the exact same form.

• There are 256 distinct standard form categorical syllogisms, only 15 of which are valid.

Syllogistic Fallacies

• Your text lists six (well, really seven) syllogistic fallacies.

• You will be expected to know ALL seven, and to be able to identify when they occur in an syllogism.

• Any argument that commits one of these fallacies is invalid.

• I'm just going to go over two (really, three) of them in this lecture.

Fallacies of illicit process• One of the rules of syllogistic arguments is that

any term that is distributed in the conclusion must also be distributed in the relevant premise.

• This can give rise to two different fallacies:– Illicit major: major term is distributed in the

conclusion, but not in the major premise– Illicit minor: minor term is distributed in the

conclusion, but not in the minor premise

• You will be expected to know both of these fallacies by name, and be able to specify when a syllogism commits one or the other (or both)

Fallacy of four terms• If a categorical syllogism has more than three terms, then

it commits the fallacy of four terms.– Example: All banks are edges of rivers. Some banks

are financial institutions. Thus, some financial institutions are edges of rivers.

• However, sometimes an argument appears to commit this fallacy, but really doesn't.– Example 1: No wealthy persons are vagrants, and all

lawyers are rich people, so no attorneys are tramps.– Example 2: All mammals are warm-blooded animals,

and no lizards are warm-blooded animals. Therefore all lizards are nonmammals.

Diagramming Standard Form Categorical Syllogisms using Venn Diagrams

• Step 1: Draw three interlocking circles - two circles on the top and one on the bottom.

• Label the top-leftmost circle S (for the minor term), the top-rightmost circle P (for the major term), and the bottom circle M (for the middle term)

• Step 2: Diagram the premises, just as you would when you diagrammed the premises alone, with the following exception:– When diagramming

particular propositions, if it is ambiguous which region the "x" should go in, place the "x" on the line in between the regions

Example: Some S are not M

• A related diagramming tip:– When diagramming a

syllogism with both a universal premise and a particular premise, diagram the universal premise first

• Consider the following example:No M are P

Some S are M

• Another diagramming tip:– When diagramming

two universal premises, make the lines go opposite directions when filling in the circle (makes it easier for your reader to "see" each premise)

• Example:No M are P

All M are S

Determining Validity or Invalidity from a Venn Diagram

• Once you have diagrammed both of the premises in the Venn Diagram, then you look at the diagram and ask the following question:– Based on the diagram, do you already know

that the conclusion is true?– In other words, does the diagram show that if

the premises are all true, then the conclusion must also be true?

Example: Valid Argument

Some mice are not pregnant creatures. All mice are soft creatures. Therefore some soft creatures are not pregnant creatures.

Some M are not PAll M are S______Some S are not P

Example: Invalid Argument

No pink creatures are mice. Some soft creatures are not mice. Therefore some soft creatures are not pink.

No P are MSome S are not M__Some S are not P

Let's Practice! For each of the following syllogisms, complete the

following steps:A. Put the syllogism into standard form, symbolizing

all the statements appropriately (use S to indicate the minor term, P the major term, and M the middle term)

B. Identify the mood and figure of the syllogism.C. Draw a Venn diagram representing the syllogism,

making sure to label the circles.D. Determine whether the argument is valid or invalid,

based on the diagram. If the argument is invalid, identify the relevant syllogistic fallacy.

1. Some parrots are not pests and all parrots are pets, so no pets are pests.

2. All voters are residents, because no nonresidents are citizens and all noncitizens are nonvoters.

3. All things inflammable are unsafe things, so all things that are safe are nonexplosives, because all explosives are flammable things.

4. All aardvarks are mammals, so some birds are not aardvarks, for some mammals are not birds.

5. All unsuccessful people are unmotivated people. No unmotivated people are people that can achieve their dreams. It follows that all successful people are people who can achieve their dreams.

6. Some dogs are not pit bulls, so some dogs are not Dobermans, for no Dobermans are pit bulls.

7. All mortals are imperfect beings, and no humans are immortals, whence it follows that all perfect beings are nonhumans.