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LOGIC HUM 200 Syllogisms in Ordinary Language 1
Transcript
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LOGIC

HUM 200

Syllogisms in Ordinary Language

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Objectives

When you complete this lesson, you will be able to: Identify the three ways an argument in

ordinary language deviates from standard form

Reduce the number of terms in a syllogism to three terms

Translate categorical propositions into standard form

Use a parameter to conduct uniform translation

Identify three types of enthymemes Construct a sorites to test the validity of an

argument Identify disjunctive and hypothetical

syllogisms Describe three methods of responding to a

dilemma

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Syllogistic Arguments

Any 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

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Syllogistic Arguments, continued

First deviation Order of the premises and conclusion not

the same as a standard-form argument Second deviation

Premises appear to have more than three terms

Third deviation Component propositions may not be

standard-form propositions

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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.

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Reducing the Number of Terms to Three, continued

Eliminate class complements All mammals are warm-blooded animals. No lizards are warm-blooded mammals. Therefore all lizards are nonmammals.

Use immediate inferences All mammals are warm-blooded animals. No lizards are warm-blooded mammals. Therefore no lizards are mammals.

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Reducing the Number of Terms to Three, EXERCISES

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Reducing the Number of Terms to Three, EXERCISES

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Reducing the Number of Terms to Three, EXERCISES

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Reducing the Number of Terms to Three, EXERCISES

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Reducing the Number of Terms to Three, EXERCISES

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Translating Categorical Propositions into Standard Form

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

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Translating Categorical Propositions into Standard Form

Consider the following argument:

All mammals are warm-blooded animals.No snakes are warm-blooded animals.Therefore, All snakes are nonmammals.

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 nonmammals" becomes "No snakes are mammals." These two propositions are equivalent. The syllogism is now in standard-form and is known to be valid.

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Translating Categorical Propositions into Standard Form, continued

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

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Translating Categorical Propositions into Standard Form, continued

Many categorical propositions contain adjectives or adverbs as predicates instead of terms denoting a class of objects. For example, consider the following categoricals:

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." May 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, the following examples should suffice so you get the idea. The above propositions could be put into standard form:

Some animals are "things (or objects) that are mean." [Here we have a class of objects, those that are mean]

No automobiles are "things (or objects) available for lease."All our students are "handsome persons" (or objects).Mary is "a person who is always late."

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Translating Categorical Propositions into Standard Form, continued

Categorical propositions whose main verbs are other than the standard form “to be” All people seek recognition

Create a class and use the standard form “to be” All people are seekers of recognition

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Translating Categorical Propositions into Standard Form, continued

The standard copula for categorical propositions used in syllogisms is a form of the verb "to be" (such as is, was, are, etc.). Consider the following propositions:

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 copula and a term designating the class determined by that class-defining attribute. The above propositions 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 that "desire attention." The standard copula "are" is inserted. "Drinkers of lemonade" is now a class, those people that "drink lemonade." The standard copula "are" is again inserted.

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Translating Categorical Propositions into Standard Form, continued

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

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Translating Categorical Propositions into Standard Form, continued

Categorical propositions whose quantities are indicated by words other than “all,” “no,” and “some” “every” or “any” are translated to “all” “a” or “an” may mean “all” or “some,”

depending on context “the” may refer to a particular individual or

all members of the class “not every” and “not any” should be

translated with care

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Translating Categorical Propositions into Standard Form, continued

Exclusive propositions Assert that the predicate applies only to the

subject named Only citizens can vote

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

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Translating Categorical Propositions into Standard Form, continued

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

“Dogs are carnivores” becomes “All dogs are carnivores”

“Children are present” becomes “Some children are beings who are present”

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Translating Categorical Propositions into Standard Form, continued

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

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Translating Categorical Propositions into Standard Form, continued

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 categoricals All non-employees are eligible persons, and no

employees are eligible persons

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Translating Categorical Propositions into Standard Form, EXERCISES

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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 times when you have the poor

with you

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Uniform Translation, continued

Consider reducing by using a parameter Soiled paper plates 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.

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Uniform Translation, EXERCISES

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Enthymemes

An argument that contains an unstated proposition Jones is a native-born American. Therefore Jones is a citizen.

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

First-order enthymeme The proposition that is taken for granted is

the major premise

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Enthymemes, continued

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 students.

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Enthymemes, continued

Third-order enthymeme Proposition taken for granted is the

conclusion No true Christian is vain, but some churchgoers

are vain Infer the conclusion

Some churchgoers are not true Christians

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Enthymemes, EXERCISES

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Enthymemes, EXERCISES

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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 argument:

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 a premise of another. In the above syllogism, however, the conclusions of all the syllogisms except the last remain 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.

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Sorites, continued

Sorites appear in two distinct types: the Aristotelian and the Goclenian. It is the arrangement of the propositions within the sorite which determines what type it is.

In the Aristotelian, 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. Here is an example of the Aristotelian sorite, using letters to indicate its special arrangement:

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.

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Sorites, continued

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 subject and next as predicate. Here is an example of the Goclenian sorite, again using letters to indicate its special arrangement:

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 desires lacks much.A = B : One who has many vices has many desires.---------A = E : Therefore, One who has many vices is miserable.

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Sorites – exercises #1

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Sorites – exercises #2

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Sorites – exercises #3

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Sorites – exercises #4

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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

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Disjunctive and Hypothetical Syllogisms, continued

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 Mr. Smith is the brakeman’s next door neighbor or Mr. Robinson is the brakeman’s next door neighbor.

Mr. Robinson is not the brakeman’s next door neighbor.

Therefore Mr. Smith is the brakeman’s next door neighbor.

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Disjunctive and Hypothetical Syllogisms, continued

Hypothetical proposition If the first native is a politician, then the

first native lies Contains two propositions

Antecedent follows the “if” Consequent follows the “then”

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Disjunctive and Hypothetical Syllogisms, continued

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.

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Disjunctive and Hypothetical Syllogisms, continued

Mixed hypothetical syllogism One premise is conditional, the other is not Modus Ponens (VALID)

Categorical premise affirms the antecedent of the conditional premise, and 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.

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Disjunctive and Hypothetical Syllogisms, continued

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)

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Disjunctive and Hypothetical Syllogisms, continued

Mixed hypothetical syllogism Modus tollens (VALID)

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

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

The one-eyed prisoner could not tell the color of the hat on his own head.

Therefore the one-eyed prisoner did not see two red hats.

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Disjunctive and Hypothetical Syllogisms, continued

Fallacy of denying the antecedent Categorical premise denies the antecedent

of the conditional premise, rather than the consequent If Carl embezzled the college funds, then Carl is

guilty of a felony. Carl did not embezzle the college funds. Therefore Carl is not guilty of a felony.

(Carl could have committed another felony)

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Disjunctive and Hypothetical Syllogisms, EXERCISES

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The Dilemma

Claims that a choice must be made between two alternatives, both of which are usually bad

Simple dilemma (positive – from text) Conclusion is a single categorical

proposition If the blest 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 have them fully gratified; therefore they will be perfectly content.

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The Dilemma, continued

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.

One is said to be caught on “the horns” of the dilemma

There are three solutions….

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The Dilemma, continued

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 will be of any avail. But any student is either fond of learning or dislikes it. Therefore a stimulus is either needless or of no avail.

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

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The Dilemma, continued

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

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The Dilemma, continued

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

If you say what is just, 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!

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Summary

Deviations from standard-form arguments

Translating to three terms Reduction to standard form Parameters Enthymemes Sorites Disjunctive and hypothetical syllogisms The dilemma


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