Transcript Ch 7 Syllogisms in Modern Lang

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 terms
3. Translate categorical propositions into standard form
4. Use a parameter to conduct uniform translation
5. Identify three types of enthymemes
6. Construct a sorites to test the validity of an argument
7. Identify disjunctive and hypothetical syllogisms
8. 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
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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".
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
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 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.
Translating Categorical Propositions
into Standard Form
 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
Translating Categorical Propositions
into Standard Form
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.’
Translating Categorical
Propositions into Standard Form
 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
Translating Categorical
Propositions into Standard Form
 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.
Translating Categorical
Propositions into Standard Form
 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.
Translating Categorical
Propositions into Standard Form
 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
Translating Categorical
Propositions into Standard Form
 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
Translating Categorical
Propositions into Standard Form
 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
Translating Categorical
Propositions into Standard Form
 Propositions that do not resemble standardform categorical propositions, but can be
translated
– Nothing is both round and square
– No round objects are square objects
Translating Categorical
Propositions into Standard Form
 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.
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:
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–
–
–
–
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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 and Hypothetical
Syllogisms

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
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Disjunctive syllogism:
Either A or B
Not A
Therefore, B
Invalid disjunctive syllogism:
Either A or B
A
Therefore, not B
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Disjunctive and Hypothetical
Syllogisms
 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)
Disjunctive and Hypothetical
Syllogisms
 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.
Disjunctive and Hypothetical
Syllogisms
 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
Disjunctive and Hypothetical
Syllogisms
 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)
Disjunctive and Hypothetical
Syllogisms
 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.
Disjunctive and Hypothetical
Syllogisms
 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
Disjunctive and Hypothetical
Syllogisms
pure
hypothetical
syllogism
if A then B
if B then C
QED if A then
C
mixed hypothetical syllogism
modus
ponens
ponere = to
affirm
fallacy of
affirming the
consequent
modus
tollens
tollere = to
deny
fallacy of
denying the
antecedent
if A then B
A
QED B
invalid: if A
then B
B
QED A
if A then B
not B
QED not A
invalid: if A
then B
not A
QED not B
Disjunctive and Hypothetical
Syllogisms
Principal Kinds of Syllogisms
Categorical
Syllogisms
All M is P
All S is M
QED All S is P.
Disjunctive
Syllogisms
Either P or Q is
true
P is not true
QED Q is true
Hypothetical Syllogisms
Pure
If P is true then Q
is true
If Q is true then R
is true
QED If P is true
then R is true
Mixed
If P is true then Q
is true
P is true
QED 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