CHAPTER NINECHAPTER NINE

THE STEPPING STONES THE STEPPING STONES OF LOGIC: OF LOGIC: SYLLOGISMSSYLLOGISMS

SECOND THOUGHTS, 4SECOND THOUGHTS, 4thth ed ed..Wanda TeaysWanda Teays

McGraw-Hill Higher Ed.McGraw-Hill Higher Ed.

© 2010. Wanda Teays. All rights reserved.© 2010. Wanda Teays. All rights reserved.

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Form of the syllogismForm of the syllogism

Dfn. Syllogism.Dfn. Syllogism. This is a three-line argument with two This is a three-line argument with two premises and one conclusion in which there are only premises and one conclusion in which there are only three terms. three terms.

FOR EXAMPLE:FOR EXAMPLE:

All donuts are delicious treats.All donuts are delicious treats.Some junk foods are delicious treats. Some junk foods are delicious treats. Therefore, some junk food are donuts.Therefore, some junk food are donuts.

The three terms are: donuts, delicious treats, & junk oof.The three terms are: donuts, delicious treats, & junk oof.

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ValidityValidity

First, there is the issue of First, there is the issue of validityvalidity. The . The argument is structurally correct (so that argument is structurally correct (so that ifif the the premises were true, the conclusion could not premises were true, the conclusion could not be false). be false).

FOR EXAMPLE:FOR EXAMPLE:

All leopards have spots.All leopards have spots.All spotted animals wish they had stripes.All spotted animals wish they had stripes.Therefore, all leopards wish they had stripes.Therefore, all leopards wish they had stripes.

NOTE: If the two premises were true, the NOTE: If the two premises were true, the conclusion would have to be true too.conclusion would have to be true too.

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SoundnessSoundnessAn argument is sound if An argument is sound if

(1) the argument is valid(1) the argument is valid

(2) the premises are actually true.(2) the premises are actually true.

FOR EXAMPLE:FOR EXAMPLE:

All leopards are cats.All leopards are cats.No cat is a squirrel.No cat is a squirrel.Therefore, no leopard is a squirrel.Therefore, no leopard is a squirrel.

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Universal PropositionsUniversal PropositionsForm 1: Form 1: “All A is B.” “All A is B.” Universal positive Universal positive

“ “All cockatoos are birds that can talk.”All cockatoos are birds that can talk.”

Form 2: Form 2: “No A is B.” “No A is B.” Universal Universal negativenegative“No cockatoo is a duck.”“No cockatoo is a duck.”

Form 3: Form 3: “A is/is not B.” “A is/is not B.” Universal Universal positive/negativepositive/negative

“Australia is a place with many “Australia is a place with many cockatoos.” cockatoos.”

This includes where A has only one member This includes where A has only one member “That baby cockatoo is a darling bird.”“That baby cockatoo is a darling bird.”

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Particular Propositions Particular Propositions Form 1:Form 1: “Some A is B” “Some A is B” Particular positive Particular positive

““Some chefs are good bakers.”Some chefs are good bakers.”

Form 2: Form 2: “Some A is not B” “Some A is not B” Particular negativeParticular negative

““Some fish are not rainbow trout.”Some fish are not rainbow trout.”

Form 3:Form 3: “x% of A is/is not B” “x% of A is/is not B” Particular Particular positive/negative. positive/negative. Where Where xx100 or 0.100 or 0.

““64% of women are tea drinkers.”64% of women are tea drinkers.”

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Categorical PropositionsCategorical PropositionsIn analyzing a syllogism, it’s usually best to rewrite the In analyzing a syllogism, it’s usually best to rewrite the

premises and the conclusion in the form of premises and the conclusion in the form of categorical categorical propositionspropositions. .

These are :These are :

A:A: All P are Q.All P are Q. All basketball players are athletes.All basketball players are athletes.

E:E: No P is Q.No P is Q. No violinist is a football player.No violinist is a football player.

I:I: Some P is Q.Some P is Q. Some gymnasts are shy people.Some gymnasts are shy people.

O:O: Some P is not Q.Some P is not Q. Some mountain climbers are not Some mountain climbers are not stamp collectors.stamp collectors.

NOTE: The letters A, E, I, and O are handy ways to NOTE: The letters A, E, I, and O are handy ways to abbreviate these 4 forms.abbreviate these 4 forms.

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Categorical Categorical SyllogismsSyllogisms

A A categorical syllogism categorical syllogism is a syllogism in which the is a syllogism in which the premises and the conclusion are premises and the conclusion are categoricalcategorical claims. claims.

FOR EXAMPLE:FOR EXAMPLE:

All racoons are pesky animals.All racoons are pesky animals.No pesky animal is a good pet.No pesky animal is a good pet.Therefore, no raccoon is a good petTherefore, no raccoon is a good pet

The The standard form of a categorical syllogismstandard form of a categorical syllogism is a is a syllogism stated in the order of major premise, minor syllogism stated in the order of major premise, minor premise, and then the conclusion. premise, and then the conclusion.

This gives us a uniform way to set out syllogisms so they This gives us a uniform way to set out syllogisms so they are easy to assess, and we aren’t scrambling trying to are easy to assess, and we aren’t scrambling trying to figure out what’s what. figure out what’s what.

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Categorical Syllogism in Categorical Syllogism in Standard FormStandard Form

Here’s a categorical syllogism in standard form.Here’s a categorical syllogism in standard form.

No vampires are morning people.No vampires are morning people.Some morning people are folks who like scrambled Some morning people are folks who like scrambled eggs for breakfast.eggs for breakfast.Therefore, no folks who like scrambled eggs for Therefore, no folks who like scrambled eggs for breakfast are vampires.breakfast are vampires.

NOTE: The NOTE: The major premisemajor premise is the premise that contains is the premise that contains the predicate term (=major term) found in the the predicate term (=major term) found in the conclusion. conclusion.

The second premise is called the The second premise is called the minor premiseminor premise and it and it contains the subject term (=minor term) found in the contains the subject term (=minor term) found in the conclusion. Both premises have a linking term (= conclusion. Both premises have a linking term (= middle term) that does not appear in the conclusion. middle term) that does not appear in the conclusion.

The The middle termmiddle term is the term that is found only in the is the term that is found only in the premises, not the conclusion. premises, not the conclusion.

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Handy AbbreviationsHandy Abbreviations

P P = = PredicatePredicate of the conclusion of the conclusion Major termMajor term

SS = = Subject Subject of the conclusion of the conclusion Minor termMinor term

M =M = Term found in both premises Term found in both premises Middle termMiddle term

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The Figures of the SyllogismThe Figures of the Syllogism

MMPP PP MM MM PP PP MMSS MMSS MMMM SSMM SS

  SS PP SS PP SS PP SS PP

FIGURE 1FIGURE 1 FIGURE 2FIGURE 2 FIGURE 3FIGURE 3 FIGURE 4FIGURE 4

Step downStep down M’s on rightM’s on right M’s on leftM’s on left step upstep up

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Mood and FigureMood and FigureThe mood of the syllogism is found after the syllogism is in categorical The mood of the syllogism is found after the syllogism is in categorical standard form. Then you just read the abbreviations (A,E,I, O) of the standard form. Then you just read the abbreviations (A,E,I, O) of the universal/particular and positive/negative propostions. The figure is universal/particular and positive/negative propostions. The figure is found by the location of the middle term.found by the location of the middle term.

FOR EXAMPLE: FOR EXAMPLE:

Some vegetarians are cheese-eaters.Some vegetarians are cheese-eaters.All bicyclists are cheese-eaters.All bicyclists are cheese-eaters.Therefore, some bicyclists are vegetarians.Therefore, some bicyclists are vegetarians.

The MOOD of the syllogism is: IAI. The figure is figure 2 (M’s on right). So The MOOD of the syllogism is: IAI. The figure is figure 2 (M’s on right). So the mood and figure is written:the mood and figure is written:

IAI—(2).IAI—(2).

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DistributionDistributionDistribution of a term refers to how much of the class (the subject Distribution of a term refers to how much of the class (the subject

or the predicate) is being referred to in the propostion. or the predicate) is being referred to in the propostion.

It’s easy to find: Claims that are all-or-nothing (A and E claims) refer It’s easy to find: Claims that are all-or-nothing (A and E claims) refer to all of the subject class. Claims that are particular (I and O to all of the subject class. Claims that are particular (I and O claims) refer to only some. So the SUBJECT IS DISTRIBUTED in claims) refer to only some. So the SUBJECT IS DISTRIBUTED in universal claims—but not particular claims.universal claims—but not particular claims.

Claims that are positive (A and I) do not distribute the PREDICATE —Claims that are positive (A and I) do not distribute the PREDICATE —the predicate is only distributed in negative claims (E and O).the predicate is only distributed in negative claims (E and O).

PROPOSITIONPROPOSITION DISTRIBUTED TERM(S):DISTRIBUTED TERM(S):

All P is Q All P is Q subject subject

No P is Q No P is Q subject subject andand predicate predicate

Some P is Q Some P is Q nothing nothing

Some P is not Q Some P is not Q predicatepredicate

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RULES OF THE SYLLOGISMRULES OF THE SYLLOGISM  

Rule 1:Rule 1: The middle term must be distributed at least The middle term must be distributed at least once.once.

  Rule 2:Rule 2: If a term is distributed in the conclusion, it must If a term is distributed in the conclusion, it must also be distributed in its corresponding premisealso be distributed in its corresponding premise

Illicit majorIllicit major: When the major term is distributed in the : When the major term is distributed in the conclusion, but is not distributed in the major premise conclusion, but is not distributed in the major premise Illicit minor: Illicit minor: When the minor term is distributed in the When the minor term is distributed in the conclusion, but is not distributed in the minor premiseconclusion, but is not distributed in the minor premise

Note: Note: A valid syllogism does not requires the A valid syllogism does not requires the conclusion to have its terms distributed. But conclusion to have its terms distributed. But if if a term a term is distributed in the conclusion, then it must also be is distributed in the conclusion, then it must also be distributed in its corresponding premise.distributed in its corresponding premise.

  

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Rules of the Syllogism Rules of the Syllogism con.con.

Rule Rule 3:3: At least one premise must be positive. (Two At least one premise must be positive. (Two negative premises = invalid argument)negative premises = invalid argument)

Rule 4:Rule 4: If the syllogism has a negative premise, there If the syllogism has a negative premise, there must be a negative conclusion, and vice versa.must be a negative conclusion, and vice versa.

Rule 5:Rule 5: If If bothboth of the premises are universal, the of the premises are universal, the conclusion must also be universal. And if the conclusion must also be universal. And if the conclusion is universal, both premises must be conclusion is universal, both premises must be universal as well.universal as well.

(You cannot have two universal premises with a (You cannot have two universal premises with a particular conclusion and you cannot have a particular conclusion and you cannot have a universal conclusion unless both premises are also universal conclusion unless both premises are also universal.)universal.)