Categorical SyllogismsPHIL 121: Methods of Reasoning

February 25, 2013Instructor:Karin HoweBinghamton 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 categoricalpropositions.

Standard Form Categorical Syllogisms

• A categorical syllogism is said to be instandard form if and only if it fulfills all of thefollowing criteria:

1. All of the statements in the syllogism are in standardform.

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

3. The premises are arranged in the correct order: majorpremise first, followed by minor premise, and eachpremise 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 astandard formcategorical syllogism.

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

This argument is NOT astandard formcategorical syllogism,because the premises arenot in the right order(minor premise is first,followed by the majorpremise)

Mood• The mood of a standard form categorical

syllogism found by reading off the types ofstatements in the argument, in order (first themajor premise, then the minor premise, andfinally 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

P – M

M – S∴ S – P

M – P

M – S∴ S – P

P – M

S – M∴ S – P

M – P

S – M∴ S – P

4th figure3rd figure2nd figure1st figure

There are exactly four possible arrangements of the threeterms in a standard form categorical syllogism. Thesepossible arrangements are known as the figure of thesyllogism.

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 standardform 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 samemood and figure has the exact same form.

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

Syllogistic Fallacies

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

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

• Any argument that commits one of these fallaciesis invalid.

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

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

any term that is distributed in the conclusion mustalso 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 asyllogism 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 financialinstitutions are edges of rivers.

• However, sometimes an argument appears to commit thisfallacy, 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 alllizards are nonmammals.

Diagramming Standard Form CategoricalSyllogisms using Venn Diagrams

• Step 1: Draw threeinterlocking circles -two circles on the topand one on the bottom.

• Label the top-leftmostcircle S (for the minorterm), the top-rightmost circle P (forthe major term), andthe bottom circle M(for the middle term)

• Step 2: Diagram thepremises, just as youwould when youdiagrammed the premisesalone, with the followingexception:– When diagramming

particularpropositions, if it isambiguous whichregion the "x" shouldgo in, place the "x" onthe line in between theregions

Example: Some S are not M

• A relateddiagramming tip:– When diagramming a

syllogism with both auniversal premise and aparticular premise,diagram the universalpremise first

• Consider the followingexample:No M are PSome S are M

• Another diagrammingtip:– When diagramming

two universal premises,make the lines goopposite directionswhen filling in thecircle (makes it easierfor your reader to "see"each premise)

• Example:No M are PAll M are S

Determining Validity or Invalidityfrom a Venn Diagram

• Once you have diagrammed both of thepremises in the Venn Diagram, then youlook at the diagram and ask the followingquestion:– 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 conclusionmust also be true?

Example: Valid Argument

Some mice are notpregnant creatures.All mice are softcreatures. Thereforesome soft creatures arenot pregnant creatures.

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

Example: Invalid Argument

No pink creatures aremice. Some softcreatures are not mice.Therefore some softcreatures 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 indicatethe minor term, P the major term, and M themiddle 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 parrotsare pets, so no pets are pests.

2. All voters are residents, because nononresidents are citizens and allnoncitizens are nonvoters.

3. All things inflammable are unsafe things,so all things that are safe arenonexplosives, because all explosives areflammable things.

4. All aardvarks are mammals, so somebirds are not aardvarks, for somemammals are not birds.

5. All unsuccessful people are unmotivatedpeople. No unmotivated people arepeople that can achieve their dreams. Itfollows that all successful people arepeople who can achieve their dreams.

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

7. All mortals are imperfect beings, and nohumans are immortals, whence it followsthat all perfect beings are nonhumans.