CHAPTER 7

CHAPTER 7

SYLLOGISMS IN ORDINARY LANGUAGE

OBJECTIVES

1. Identify the 3 ways an argument in ordinary language deviates from standard form

2. Reduce the number of terms in a syllogism to 3 terms3. Translate categorical propositions into standard form4. Use a parameter to conduct uniform translation5. Identify three types of enthymemes6. Construct a sorites to test the validity of an argument7. Identify disjunctive and hypothetical syllogisms8. Describe three methods of responding to a dilemma

SYLLOGISTIC ARGUMENTS

An argument that is a standard form categorical syllogism, or can be reformulated as a standard form categorical syllogism

Reduction to standard form results in a standard-form translation.

SYLLOGISTIC ARGUMENTS

First Deviation• Order of the premises and conclusion not the same

as standard-form argument Second Deviation

• Premises appear to have more than 3 terms Third Deviation

• Component propositions may not be standard form propositions

Reducing the Number of Terms to Three

Eliminate Synonyms– No wealthy persons are vagrants– All lawyers are rich people– Therefore no attorneys are tramps

Six terms can be reduced to three– No wealthy persons are vagrants– All lawyers are wealthy persons– Therefore, no lawyers are vagrants

Reducing the Number of Terms to Three

Eliminate Class Complements– All mammals are warm-blooded animals– No lizards are warm-blooded animals– Therefore all lizards are non-mammals

Use Immediate Inferences– All mammals are warm-blooded animals– No lizards are warm blooded animals– Therefore no lizards are mammals

– Exercises

Translating Categorical Propositions into Standard Form

Singular Propositions"I play tennis" becomes "All (the class that contains just me) play tennis" and "Some (the class that contains just me) play tennis" - (The "All ... play tennis" lacks existential import).

Adjectives as Predicates"That serve was wicked" becomes "That serve was a wicked serve".

Copula Not a Form of "To Be""That serve spins" becomes "That serve is a serve that spins".

Non-Standard Form Arrangement"Aces are all well-placed serves" becomes "All aces are well-placed serves".

Quantities not "All", "No", or "Some""A student did well" becomes "Some student did well". "Not every S is P" becomes "Some S is not P" and "Not any S is P" becomes "No S is P".

Exclusive Propositions"Only S is P" or "None but S is P" become "All P is S".

No Quantity Specified"Fit men play tennis" becomes "Some tennis players are fit men".

Do Not Resemble Standard Form"A stroke is forehand or backhand" becomes "No backhand stroke is a forehand stroke".

Exceptive Propositions"All except employees may enter" becomes both "All non-employees may enter" and "No employees may enter".


Singular Propositions– Asserts that a specific individual belongs to a

particular class– Unit class

One-member class whose only member is that object itself

“All S is P”– Issues

Existential Import (some is complicated) Fallacy of the Undistributed Middle


Consider the following argument:– All mammals are warm-blooded animals– No snakes are warm-blooded animals– Therefore, all snakes are non-mammals

If we applied our general rules for syllogisms to the above argument, we would judge it to be invalid because (1) it contains four terms; and (2) it has an affirmative conclusion drawn from a negative premise. We can, however, modify it slightly without changing the substance of the argument and see that it is perfectly valid. Consider this change:– All mammals are warm-blooded animals– No snakes are warm-blooded animals– Therefore, no snakes are mammals

We have reduced the number of terms to three by simply obverting the conclusion: ‘All snakes are non-mammals” becomes “No snakes are mammals.” These 2 propositions are equivalent. The syllogism is now in standard-form and is known to be valid.


Categorical Propositions that have adjectives or adjectival phrases as predicates– Some flowers are beautiful

Replace the adjective with a term designating the class of all objects that possess that attribute– Some flowers are beauties


Many categorical propositions contain adjectives or adverbs as predicates instead of terms denoting a class of objects. For example:• Some animals are mean• No automobiles are available for lease• All our students are handsome• Mary is always late• The predicates in the above propositions convey attributes of the subject. Some animals

are “mean.” No automobiles are ‘available for lease.” All our students are ‘handsome.’ Mary is ‘always late.’ Every attribute, however, determines a class, a group of things possessing that attribute.

• We can always change the proposition to indicate a class of objects to which the attribute applies. While there are other ways of expressing these propositions, these examples should help you get the idea. Putting the above propositions into standard form:

• Some animals are ‘things that are mean.’ – Class is now things that are mean• No automobiles are ‘things available for lease.’• All our students are ‘handsome persons.’ • Mary is a ‘person who is always late.’


Categorical Propositions whose main verbs are other than the standard form of ‘to be.’– All people seek recognition

– Create a class and use the standard form of to be

– All people are seekers of recognition


The standard cupola for categorical propositions used in syllogisms is a form of the verb ‘to be’ (such as is, was, are, etc.) Consider these:– All children desire attention– Some people drink lemonade

These propositions are easily translated into standard form by regarding all of the proposition except the subject term and the quantifier as naming a class-defining attribute, and replace it by a standard cupola and a term designating the class determined by that class-defining attribute. The above would then become:– All children are desirers of attention.– Some people are drinkers of lemonade.

“Desirers of attention” has now become a class of people (or objects), those who desire attention. The standard cupola ‘are’ is inserted. ‘Drinkers of lemonade’ is now a class, those people who drink lemonade. The standard cupola ‘are’ is again inserted here.


Standard form ingredients are all present , but not arranged in standard form order.– Racehorses are all thoroughbreds.

Decide which term is the subject term and then rearrange the words to reflect a standard form categorical proposition.– All racehorses are thoroughbreds.


Categorical propositions whose quantities are indicated by words other than ‘all’, ‘no’, or ‘some.’– ‘Every’ or ‘any’ are translated into ‘all’– ‘A’ or ‘an’ may be all or ‘some’ depending on

context of sentence– ‘The’ may refer to a particular individual or all

members of a class– ‘not every’ and ‘not any’ will also depend on

context


Exclusive propositions– Assert that the predicate applies only to the

subject named Only citizens can vote

– Reversing the subject and the predicate, and replace the only with all All those who can vote are citizens


Categorical propositions that contain no words at all to indicate quantity– Examine the content

Dogs are carnivores becomes All dogs are carnivores

Children are present becomes Some children are beings who are present


Propositions that do not resemble standard-form categorical propositions, but can be translated– Nothing is both round and square– No round objects are square objects


Exceptive Propositions– Makes two assertions: that all members of

some class, except for members of one of its subclasses, are members of some other class All but employees are eligible

– All non-employees are eligible– No employees are eligible

– Translate into an explicit conjunction of two standard form categorical propositions All non-employees are eligible persons, and no

employees are eligible persons. Exercises

Uniform Translation

Parameter– An auxiliary symbol that aids in reformulating an

assertion into standard form The poor always you have with you

– Use ‘times’ as the parameter (temporal) All times are the times when you have the poor with

you– Inserting a parameter can eliminate excess terms: "The

poor are always with us" becomes "All times are times when the poor are with us". "I always win when my serve is on" becomes "All matches that I play when my serve is on are matches that I win".

Uniform Translation Consider reducing by using a parameter

– Soiled paper places are scattered only where careless people have picnicked.

– There are soiled paper plates scattered about here.– Therefore, careless people must have been picnicking here.

Use ‘places’ as the parameter– All places where soiled paper plates are scattered are places

where careless people have picnicked– This place is a place where soiled paper plates are scattered– Therefore, this place is a place where careless people have

picnicked

– Excercises

Enthymemes An argument is enthymematic if it is incompletely stated

depending on additional information for completion. An argument that contains an unstated proposition

– Jones is a native-born American– Therefore, Jones is a citizen

Missing a premise that is though to be understood– All native-born Americans are citizens

First-order enthymeme– The proposition that is taken for granted is the major

premise

Enthymemes

Second-order enthymemes– Proposition taken for granted is the minor

premise All students are opposed to the new regulations Therefore, all sophomores are opposed to the new

regulations– Missing minor premise

All sophomores are atudents

Enthymemes

Third – order enthymeme– Proposition taken for granted is the conclusion

No true Christian is vain, but some churchgoers are vain.

– Infer the conclusion Therefore, some churchgoers are not true Christians

Exercises

Sorites Sometimes a single categorical proposition will not suffice for drawing a desired

conclusion from a group of premises. The evidence for a conclusion consists of more than two propositions. The inference is not a syllogism in such cases but a series of syllogisms. Consider the following:

– All dictatorships are undemocratic– All undemocratic governments are unstable– All unstable governments are cruel– All cruel governments are objects of hate– Therefore, all dictatorships are objects of hate

The inference (stated in the conclusion) may be tested by means of the syllogistic rules. The argument is a chain of syllogisms in which the conclusion of one becomes the premise of another. In the above syllogism, however, the conclusions of all except the last one are unexpressed.

A sorite is a chain of syllogisms in which the conclusion of one is a premise in another, in which all the conclusions except the last one are unexpressed, and in which the premises are so arranged that any two successive ones contain a common term.

Sorites Sorites, appear in 2 distinct types: the Aristotlean and the Goclenian. It

is the arrangement of the propositions within the sorites which determine what type it is.

In the Aristotlean, the first premise contains the subject of the conclusion and the common term of two successive propositions appears first as a predicate and next as a subject. An example of an Aristotlean sorite:

A=B. Aristotle is a man. B=C. All men are mammals. C=D. All mammals are living beings. D=E. All living beings are substances _____ A=E. Therefore, Aristotle is a substance.

Sorites In a Goclenian sorite, the arrangement is different. The first premise

contains the predicate of the conclusion and the common term of two successive propositions appears first as a subject and next as a predicate. An example of a Goclenian sorite:

D=E. One who has no peace of mind is miserable. C=D. One who lacks much has no peace of mind. B=C. One who has many desired lacks much. A=B. One who has many vices, has many desires. ____ A=E. Therefore, one who has many vices is miserable.

Exercises

Disjunctive and Hypothetical Syllogisms

Disjunctive Proposition– Contains two component propositions

Either she was driven by stupidity or arrogance– Disjuncts

She was driven by stupidity She was driven by arrogance


Disjunctive syllogism– Disjunction in one premise– Denial or contradictory of one of its two disjuncts in other premise– Validly infer that the other disjunct is true

Either Mrs. Smith is my next door neighbor or Mrs. Robinson is my next door neighbor.

Mrs. Robinson is not my next door neighbor Therefore, Mrs. Smith is my next door neighbor

Disjunctive syllogism: Either A or B

Not ATherefore, B

Invalid disjunctive syllogism: Either A or B

ATherefore, not B


Hypothetical Proposition– If the first native is a politician, then the first

native lies– Contains 2 propositions

Antecedent follows if Consequent follows then

Conditional proposition: if (some antecedent) then (some consequent)


Hypothetical Syllogism– Contains at least one conditional proposition as

a premise– Pure hypothetical syllogism

All premises are conditional– (if p then l) If the first native is a politician, then he lies.– (if l then denies p) If he lies, then he denies being a

politician– (therefore, if p then denies p). Therefore, if the first native is

a politician, then he denies being a politician.


Mixed hypothetical syllogism– One premise is conditional, the other is not– Modus Ponens (valid) – to affirm

Categorical premise affirms the antecedent of the conditional premise, the conclusion affirms its consequent

– If the second native told the truth, then only one native is a politician.

– The second native told the truth– Therefore, only one native is a politician


Fallacy of affirming the consequent– Categorical premise affirms the consequent of

the conditional premise rather than the antecedent If Bacon wrote Hamlet, then Bacon was a great writer Bacon was a great writer Therefore, Bacon wrote Hamlet

(Any great writer could have written Hamlet)


Mixed hypothetical syylogism– Modus tollens (valid) - to deny

Categorical premise denies the consequent of the conditional premise and the conclusion denies its antecedent

– If the one-eyed professor saw two red hats, then he could tell the color of the hat on his own head

– The one- eyed professor could not tell the color of the hat on his own head

– Therefore, the one-eyed professor did not see two red hats.


Fallacy of denying the antecedent– Categorical premise denies the antecedent of

the conditional premise, rather than the consequent If John embezzled the bank funds, then John is guilty

of a felony. John did not embezzle the bank funds Therefore, John is not guilty of a felony

(John could have committed another felony) Exercises


pure hypothetical

syllogismmixed hypothetical syllogism

modus ponensponere = to affirm

fallacy of affirming the consequent

modus tollenstollere = to deny

fallacy of denying the antecedent

if A then Bif B then CQED if A then C

if A then BAQED B

invalid: if A then BBQED A

if A then Bnot BQED not A

invalid: if A then Bnot AQED not B


Principal Kinds of Syllogisms

Categorical Syllogisms

Disjunctive Syllogisms

Hypothetical SyllogismsPure Mixed

All M is PAll S is MQED All S is P.

Either P or Q is trueP is not trueQED Q is true

If P is true then Q is trueIf Q is true then R is trueQED If P is true then R is true

If P is true then Q is trueP is trueQED Q is true

The Dilemma

The Dilemma – claims that a choice must be made between two alternatives, both of which are usually bad

Simple dilemma– Conclusion is a single categorical proposition

If the blessed in heaven have no desires, they will be perfectly content; so they will be also if their desires are fully gratified; but either they have no desires, or they have them fully gratified; therefore they will be perfectly content

The Dilemma

Complex dilemma – – Conclusion is a disjunction

Every time we talked to higher level managers, they kept saying they didn’t know anything about the problems below them… Either the group at the top didn’t know, in which case they should have known, or they did know, in which case they were lying to us.

On this one is said to ‘be caught on the horns’ of the dilemma

There are 3 solutions:

The Dilemma

First, escaping between the horns– Reject the disjunctive premise

If students are fond of learning, they need no stimulus, and if they dislike learning, no stimulus would be useless. But any student is either fond of learning or dislikes it. Therefore a stimulus is either needless or useless.

– Introduce a third type of student: one who is indifferent to learning

The Dilemma

Second, grasp the dilemma by the horns– Reject the premise that is a conjunction

If students are fond of learning, they need no stimulus

– Even the students who are fond of learning may sometimes need stimulus (grades)

The Dilemma

Third, rebut the dilemma by means of a counterdilemma– Dilemma to not enter politics

If you say what is just then men will hate you; and if you say what is unjust, the gods will hate you; but you must say either one or the other; therefore you will be hated

– Counterdilemma If I say what is just, the gods will love me; and if I say

what is unjust, men will love me; I must say either one or the other. Therefore, I shall be loved!

End

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