REVIEW: The Structure of Argument: Conclusions and Premises

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REVIEW The Structure of Argument Conclusions
and Premises

• An argument consists of a conclusion (the claim
  that the speaker or writer is arguing for) and
  premises (the claims that he or she offers in
  support of the conclusion). Here is an example
  of an argument
• Premise Every officer on the force has been
  certified, and premise nobody can be certified
  without scoring above 70 percent on the firing
  range. Therefore conclusion every officer on
  the force must have scored above 70 percent on
  the firing range.

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The Structure of Argument Conclusions and
Premises

• When we analyze an argument, we need to first
  separate the conclusion from the grounds for the
  conclusion which are called premises. Stating it
  another way, in arguments we need to distinguish
  the claim that is being made from the warrants
  that are offered for it. The claim is the
  position that is maintained, while the warrants
  are the reasons given to justify the claim.
• It is sometimes difficult to make this
  distinction, but it is important to see the
  difference between a conclusion and a premise, a
  claim and its warrant, differentiating between
  what is claimed and the basis for claiming it.

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The Structure of Argument Conclusions and
Premises

• We might make a claim in a formal argument. For
  example, we might claim that teenage pregnancy
  can be reduced through sex education in the
  schools.
• To justify our claim we might try to show the
  number of pregnancies in a school before and
  after sex education classes.
• In writing an argumentative essay we must decide
  on the point we want to make and the reasons we
  will offer to prove it, the conclusion and the
  premises.

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The Structure of Argument Conclusions and
Premises

• The same distinction must be made in reading
  argumentative essays, namely, what is the writer
  claiming and the warrant is offered for the
  claim, what is being asserted and why. Take the
  following complete argument
• Television presents a continuous display of
  violence in graphically explicit and extreme
  forms. It also depicts sexuality not as a
  physical expression of internal love but in its
  most lewd and obscene manifestations. We must
  conclude, therefore, that television contributes
  to the moral corruption of individuals exposed to
  it.

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The Structure of Argument Conclusions and
Premises

• Whether we agree with this position or not, we
  must first identify the logic of the argument to
  test its soundness. In this example the
  conclusion is television contributes to the
  moral corruption of individuals exposed to it.
  The premises appear in the beginning sentences
  Television presents a continuous display of
  violence in graphic and extreme forms, and
  (television) depicts sexualityin its most lewd
  and obscene manifestations. Once we have
  separated the premises and the claim then we need
  to evaluate whether the case has been made for
  the conclusion.

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The Structure of Argument Conclusions and
Premises

• Has the writer shown that television does corrupt
  society? Has a causal link been shown between
  the depiction of lewd and obscene sex and the
  moral corruption of society? Does TV reflect
  violence in our society or does it promote it?

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Conclusion Indicators

• Since dissection is sometimes difficult because
  we cannot always see the skeleton of the
  argument. In such cases we can find help by
  looking for indicator words. When the words in
  the following list are used in arguments, they
  usually indicate a premise has just been offered
  and that a conclusion is about to be presented.

• Consequently
• Therefore
• Thus
• So
• Hence
• accordingly

• We can conclude that
• It follows that
• We may infer that
• This means that
• It leads us to believe that
• This bears out the point that

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Conclusion Indicators II

• Example
• Sarah drives a Dodge Viper. This means that
  either she is rich or her parents are.
• The conclusion is
• Either she is rich or her parents are.
• The premise is
• Sarah drives a Dodge Viper.

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Premise Indicators
When the words in the following list are used in
arguments, they generally introduce premises.
They often occur just after a conclusion has been
given.

• Since
• Because
• For
• whereas

• In as much as
• For the reasons that
• In view of the fact
• As evidenced by

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Premise Indicators II

• Example
• Either Sarah is rich or her parents are, since
  she drives a Dodge Viper.
• The premise is the claim that Sarah drives a
  Dodge Viper the conclusion is the claim that
  either Sarah is rich or her parents are.

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• Indicator words can tell us when the theses and
  the supports appear, even in complex arguments
  that are embedded in paragraphs. We can see
  whether the person has good reasons for making
  the claim, or whether the argument is weak. We
  should keep this in mind when presenting our own
  case.
• An argument that presents a clear structure of
  premises and conclusions, without narrative
  digressions, metaphorical flights, or other
  embellishments, is much easier for people to
  follow.

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Categorical Propositions

• To help us make sense of our experience, we
  humans constantly group things into classes or
  categories. These classifications are reflected
  in our everyday language. In formal reasoning
  the statements that contain our premises and
  conclusions have to be rendered in a strict form
  so that we know exactly what is being claimed.
  These logical forms were first formulated by
  Aristotle (384-322 B.C.). They are four in
  number, carrying the designations A, E, I, O, as
  follows
• All S is P (A).
• No S is P (E).
• Some S is P (I).
• Some S is not P (O).

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Categorical Propositions II

• The letter "S" stands for the class designated by
  the subject term of the proposition. The letter
  "P" stands for the class designated by the
  predicate term. Substituting any class-defining
  words for S and P generates actual categorical
  propositions.
• In classical theory, the four standard-form
  categorical propositions were thought to be the
  building blocks of all deductive arguments. Each
  of the four has a conventional designation A for
  universal affirmative propositions E for
  universal negative propositions I for particular
  affirmative propositions and O for particular
  negative propositions.

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Categorical Propositions III

• These various relationships between classes are
  affirmed or denied by categorical propositions.
  The result is that there can be just four
  different standard forms of categorical
  propositions. They are illustrated by the four
  following propositions
• All politicians are liars.
• No politicians are liars.
• Some politicians are liars.
• Some politicians are not liars.

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Universal Affirmative

• The first is a universal affirmative proposition.
  It is about two classes, the class of all
  politicians and the class of all liars, saying
  that the first class is included or contained in
  the second class. A universal affirmative
  proposition says that every member of the first
  class is also a member of the second class. In
  the present example, the subject term
  politicians designates the class of all
  politicians, and the predicate term liars
  designates the class of all liars. Any universal
  affirmative proposition may be written
  schematically as
• All S is P.
• where the terms S and P represent the subject
  and predicate terms, respectively.

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Universal Affirmative II

• The name universal affirmative is appropriate
  because the position affirms that the
  relationship of class inclusion holds between the
  two classes and says that the inclusion is
  complete or universal All members of S are said
  to be members of P also.

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Universal Negative Propositions

• The second example
• No politicians are liars.
• Is a universal negative proposition. It denies
  of politicians universally that they are liars.
  Concerned with two classes, a universal negative
  proposition says that the first class is wholly
  excluded from the second, which is to say that
  there is no member of the first class that is
  also a member of the second. Any universal
  proposition may be written schematically as
• No S is P
• Where, again, the letters S and P represent the
  subject and predicate terms.

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Universal Negative Propositions II

• The name universal negative is appropriate
  because the proposition denies that the relation
  of class inclusion holds between the two classes
  and denies it universally No members at all of
  S are members of P.

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Particular affirmative propositions

• The third example
• Some Politicians are liars.
• is a particular affirmative proposition.
  Clearly, what the present example affirms is that
  some members of the class of all politicians are
  (also) members of the class of all liars. But it
  does not affirm this of politicians universally
  Not all politicians universally, but, rather,
  some particular politician or politicians, are
  said to be liars. This proposition neither
  affirms nor denies that all politicians are
  liars it makes no pronouncement on the matter.

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Particular affirmative propositions II

• The word some is indefinite. Does it mean at
  least one, or at least two, or at least one
  hundred? In this type of proposition, it is
  customary to regard the word some as meaning
  at least one. Thus a particular affirmative
  proposition, written schematically as
• Some S is P.
• says that at least one member of the class
  designated by the subject term S is also a member
  of the class designated by the predicate term P.
  The name particular affirmative is appropriate
  because the proposition affirms that the
  relationship of class inclusion holds, but does
  not affirm it of the first class universally, but
  only partially, of some particular member or
  members of the first class.

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Particular negative propositions

• The fourth example
• Some politicians are not liars
• is a particular negative proposition. This
  example, like the one preceding it, does not
  refer to politicians universally but only to some
  member or members of that class it is
  particular. But unlike the third example, it
  does not affirm that the particular members of
  the first class referred to are included in the
  second class this is precisely what is denied.
  A particular negative proposition, schematically
  written as
• Some S is not P.
• says that at least one member of the class
  designated by the subject term S is excluded from
  the whole of the class designated by the
  predicate term P.

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Quality and Quantity

• Every categorical proposition has a quality,
  either affirmative or negative. It is affirmative
  if the proposition asserts some kind of class
  inclusion, either complete or partial. It is
  negative if the proposition denies any kind of
  class inclusion, either complete or partial.
• Every categorical proposition also has a
  quantity, either universal or particular. It is
  universal if the proposition refers to all
  members of the class designated by its subject
  term. It is particular if the proposition refers
  only to some members of the class designated by
  its subject term.

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General Schema of Standard-Form Categorical
Propositions

• Standard-form categorical propositions consist of
  four parts, as follows
• Quantifer (subject term) copula (predicate term)
• The three standard-form quantifiers are "all,"
  "no" (universal), and "some" (particular). The
  copula is a form of the verb "to be."

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Sentence Standard Form Attribute
All apples are delicious. A All S is P. Universal affirmative
No apples are delicious. E No S is P. Universal negative
Some apples are delicious. I Some S is P. Particular affirmative
Some apples are not delicious. O Some S is not P. Particular negative
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Distribution

• Distribution is an attribute of the terms
  (subject and predicate) of propositions. A term
  is said to be distributed if the proposition
  makes an assertion about every member of the
  class denoted by the term otherwise, it is
  undistributed. In other words, a term is
  distributed if and only if the statement assigns
  (or distributes) an attribute to every member of
  the class denoted by the term. Thus, if a
  statement asserts something about every member of
  the S class, then S is distributed otherwise S
  and P are undistributed.

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All S are P

• Here is another way to look at All S are P.

The S circle is contained in the P circle, which
represents the fact that every member of S is a
member of P. Through reference to this diagram,
it is clear that every member of S is in the P
class. But the statement does not make a claim
about every member of the P class, since there
may be some members of the P class that are
outside of S.
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Exercises

• Translate the following sentences into standard
  form categorical statements
• Each insect is an animal.
• Not every sheep is white.
• A few holidays fall on Saturday.
• There are a few right handed first basemen.

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Venn Diagrams

Liars
Politicians
Anything in area 1 is a politician, but not a
liar. Anything in area 2 is both a politician
and a liar. Anything in area 3 is a liar but not
a politician. And anything in area 4, the area
outside the two circles is neither a politician
or a liar.
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Venn Diagrams II

• 
  

Liars
Politicians
The shading means that the part of the
politicians circle that does not overlap with the
liars circle is empty that is, it contains no
members. The diagram thus asserts that there are
no politicians who are are not liars. All
politicians are liars.
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Venn Diagrams III

To say that no politicians are liars is to say
that no members of the class of politicians are
members of the class of liars that is, that
there is no overlap between the two classes. To
represent this claim, we shade the portion of the
two circles that overlaps as shown above. No
politicians are liars.
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Venn Diagrams IV

In logic, the statement Some politicians are
lairs means There exists at least one
politician and that politician is a liar. To
diagram this statement, we place an X in that
part of the politicians circle that overlaps with
the liars circle.
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Venn Diagrams IV

A similar strategy is used with statements of the
form Some S are not P. In logic, the statement
Some politicians are not liars means At least
one politician is not a liar. To diagram this
statement we place an X in that part of the
politicians circle that lies outside the liars
circle.
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Claims about single individuals

• Claims about single individuals, such as
  Aristotle is a logician, can be tricky to
  translate into standard form. Its clear that
  this claim specifies a class, logicians, and
  places Aristotle as a member of that class. The
  problem is that categorical claims are always
  about two classes, and Aristotle isnt a class.
  (We couldnt talk about some of Aristotle being a
  logician.) What we want to do is treat such
  claims as if they were about classes with exactly
  one member.

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Claims about single individuals II

• One way to do this is to use the term people who
  are identical with Aristotle, which of course
  has only Aristotle as a member.
• Claims about single individuals should be treated
  as A-claims or E-claims.
• Aristotle is a logician can be translated into
  All people identical with Aristotle are
  logicians.
• Individual claims do not only involve people.
  For example, Fort Wayne is in Indiana is All
  cities identical with Fort Wayne are cities in
  Indiana.

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Two important things to remember about Some
Statements

1. In categorical logic, some always means at
   least one.
2. Some statements are understood to assert that
   something actually exists. Thus, some mammals
   are cats is understood to assert that at least
   one mammal exists and that that mammal is a cat.
   By contrast, all or no statements are not
   interpreted as asserting the existence of
   anything. Instead, they are treated as purely
   conditional statements. Thus, All snakes are
   reptiles asserts that if anything is a snake,
   then it is a reptile, not that there are snakes
   and that all of them are reptiles.

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Exercises

• Draw Venn diagrams of the following statements.
  In some cases, you may need to rephrase the
  statements slightly to put them in one of the
  four standard forms.
• No apples are fruits.
• Some apples are not fruits.
• All fruits are apples.
• Some apples are fruits.

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Translating into standard categorical form

• Do people really go around saying things like
  some fruits are not apples? Not very often.
  But although relatively few of our everyday
  statements are explicitly in standard categorical
  form, a surprisingly large number of those
  statements can be translated into standard
  categorical form.

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Common Stylistic Variants of All S are P

• Example
• Every S is P. Every dog is an animal.
• Whoever is an S is a P. Whoever is a
  bachelor is a male.
• Any S is a P. Any triangle is a
  geometrical figure.
• Each S is a P. Each eagle is a bird.
• Only P are S. Only Catholics are popes.
• Only if something is a Only if something is
  a dog
• P is it an S. is it a cocker
  spaniel.
• The only S are P. The only tickets
  available are tickets for cheap
  seats.

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ONLY

• Pay special attention to the phrases containing
  the word only in that list. (Only is one of
  the trickiest words in the English language.)
  Note, in particular, that as a rule the subject
  and the predicate terms must be reversed if the
  statement begins with the words only or only
  if. Thus, Only citizens are voters must be
  rewritten as All voters are citizens, not All
  citizens are voters. And, Only if a thing is
  an insect is it a bee must be rewritten as All
  bees are insects, not All insects are bees.

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Common Stylistic Variants of No S are P

• Example
• No S are P. No cows are reptiles.
• S are not P. Cows are not reptiles.
• Nothing that is an S Nothing that is a
  known
• is a P. fact is a mere opinion.
• No one who is an S No one who is a
  Republican
• is a P. is a Democrat.
• All S are non-P. If anything is a plant,
  then it is not a mineral.

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Common Stylistic Variants of Some S are P

• Example
• Some P are S. Some students are men.
• A few S are P. A few mathematicians are
• poets.
• There are S that are P. There are monkeys that
  are
• carnivores.
• Several S are P. Several planets in the solar
  system are gas giants.
• Many S are P. Many students are hard
  workers.
• Most S are P. Most Americans are
  carnivores.

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Common Stylistic Variants of Some S are not P

• Example
• Not all S are P. Not all politicians are
  liars.
• Not everyone who is Not everyone who is a
• an S is a P. politician is a liar.
• Some S are non-P. Some philosophers are non
  Aristotelians.
• Most S are not P. Most students are not
  binge drinkers.
• Nearly all S are Nearly all students are not
• not P. cheaters.

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Paraphrasing

• The process of casting sentences that we find in
  at ext into one of these four forms is
  technically called paraphrasing, and the ability
  to paraphrase must be acquired in order to deal
  with statements logically.
• In the processing of paraphrasing we designate
  the affirmative or negative quality of a
  statement principally by using the words no or
  not. We indicate quantity, meaning whether we
  are referring to the entire class or only a
  portion of it, by using words all or some.
  In addition, we must render the subject and the
  predicate as classes of objects with the verb
  is or are as the copula joining the halves.

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Paraphrasing II

• We must pay attention to the grammar, diagramming
  the sentences if need be, to determine the parts
  of the sentence, the group that is meant, and
  even what noun is being modified.
• The kind of thing a claim directly concerns is
  not always obvious. For example, if you think
  for a moment about the claim I always get
  nervous when I take logic exams, youll see its
  a claim about times. Its about getting nervous
  and about logic exams indirectly,of course, but
  it pertains directly to times or occasions. The
  proper translation of the example is All times I
  take logic exams are times that I get nervous.

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• Once our statement is translated into proper
  form, we can see it implications to other forms
  of the statement. For example, if we claim All
  scientists are gifted writers, that certainly
  implies that Some scientists are gifted
  writers, but we cannot logically transpose the
  proposition to All gifted writers are
  scientists. In other words, some statements
  would follow, others would not.
• To help determine when we can infer one statement
  from another and when there is disagreement,
  logicians have devised tables that we can refer
  to if we get confused.

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Distribution

• Thus, by the definition of distributed term, S
  is distributed and P is not. In other words for
  any (A) proposition, the subject term, whatever
  it may be, is distributed and the predicate term
  is undistributed.

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No S are P

• No S are P states that the S and P class are
  separate, which may be represented as follows

This statement makes a claim about every member
of S and every member of P. It asserts that
every member of S is separate from every member
of P, and also that every member of P is separate
from every member of S. Both the subject and the
predicate terms of universal negative (E)
propositions are distributed.
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Some S are P

• The particular affirmative (I) proposition states
  that at least one member of S is a member of P.
  If we represent this one member of S that we are
  certain about by an asterisk, the resulting
  diagram looks like this

Since the asterisk is inside the P class, it
represents something that is simultaneously an S
and a P in other words, it represents a member
of the S class that is also a member of the P
class. Thus, the statement Some S are P makes
a claim about one member (at least) of S and also
one member (at least) of P, but not about all
members of either class. Thus, neither S or P is
distributed.
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Some S are not P

• The particular negative (O) proposition asserts
  that at least one member of S is not a member of
  P. If we once again represent this one member of
  S by an asterisk, the resulting diagram is as
  follows

Since the other members of S may or may not be
outside of P, it is clear that the statement
Some S are not P does not make a claim about
every member of S, so S is not distributed. But,
as may be seen from the diagram, the statement
does assert that the entire P class is separated
from this one member of the S that is outside
that is, it does make a claim about every member
of P. Thus, in the particular negative (O)
proposition, P is distributed and S is
undistributed.
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Two mnemonic devices for distribution

• Unprepared Students Never Pass
• Universals distribute Subjects.
• Negatives distribute Predicates.
• Any Student Earning Bs Is Not On Probation
• A distributes Subject.
• E distributes Both.
• I distributes Neither.
• O distributes Predicate.

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The Traditional Square of Opposition

• Quality, quantity, and distribution tell us what
  standard-form categorical propositions assert
  about their subject and predicate terms, not
  whether those assertions are true. Taken
  together, however, A, E, I, and O propositions
  with the same subject and predicate terms have
  relationships of opposition that do permit
  conclusions about truth and falsity. In other
  words, if we know whether or not a proposition in
  one form is true or false, we can draw certain
  valid conclusions about the truth or falsity of
  propositions with the same terms in other forms.

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Traditional Square of Opposition II

• There are four ways in which propositions may be
  opposed-as contradictories, contraries,
  subcontraries, and subalterns.

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Contradictories

• Two propositions are contradictories if one is
  the denial or negation of the other that is, if
  they cannot both be true and cannot both be false
  at the same time. If one is true, the other must
  be false. If one is false, the other must be
  true.
• A propositions (All S is P) and O propositions
  (Some S is not P), which differ in both quantity
  and quality, are contradictories.

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Contradictories II

• All logic books are interesting books.Some logic
  books are not interesting books.
• Here we have two categorical propositions with
  the same subject and predicate terms that differ
  in quantity and quality. One is an A proposition
  (universal and affirmative). The second is an O
  proposition (particular and negative).
• Can both of these propositions be true at the
  same time? The answer is "no." If all logic books
  are interesting, than it can't be true that some
  of them are not. Likewise, if some of them are
  not interesting, then it can't be true that all
  of them are.
• Can both propositions be false at the same time?
  Again, the answer is "no". If it's false that all
  logic books are interesting, then it must be true
  that some of them are not interesting. Likewise
  if it's false that some of them are not
  interesting, then all of them must be
  interesting.
• Like this pair, all A and O propositions with the
  same subject and predicate terms are
  contradictories. One is the denial of the other.
  They can't both be true or false at the same
  time.

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Contradictories III

• E propositions (No S is P) and I propositions
  (Some S is P) likewise differ in quantity and
  quality and are contradictories.
• Example No presidential elections are contested
  elections. Some presidential elections are
  contested elections.
• Here again we have two categorical propositions
  with the same subject and predicate terms that
  differ in both quantity and quality. In this
  case, the first is an E propositionuniversal and
  negativeand the second is an I
  propositionparticular and positive.
• Can both be true at the same time? The answer is
  "no." If no presidential elections are contested,
  then it can't be true that some are. Likewise is
  some are contested, then it can't be true that
  none are.

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Contradictories IV

• Can both be false at the same time? Again the
  answer is "no." If it's false that no
  presidential elections are contested, then it
  must be true that some of them are. Likewise if
  it's false that some are contested, then it must
  be the case that none are.
• Like this pair, all E and I propositions with the
  same subject and predicate terms are
  contradictories. One is the denial of the other.
  They can't both be true or false at the same
  time.

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Contraries

• Two propositions are contraries if they cannot
  both be true that is, if the truth of one
  entails the falsity of the other. If one is true,
  the other must be false. But if one is false, it
  does not follow that the other has to be true.
  Both might be false.
• A (All S is P) and E (No S is P)
  propositions-which are both universal but differ
  in quality-are contraries unless one is
  necessarily (logically or mathematically) true.
• For example
• All books are written by Stephen King.
• No books are written by Stephen King.
• Both are false.

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Subcontraries

• Two propositions are subcontraries if they cannot
  both be false, although they both may be true.
• I (Some S is P) and O (Some S is not P)
  propositions-which are both particular but differ
  in quality-are subcontraries unless one is
  necessarily false.
• For example
• Some dogs are cocker spaniels.
• Some dogs are not cocker spaniels.

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Subalternation

• Subalternation is the relationship between a
  universal proposition (the superaltern) and its
  corresponding particular proposition (the
  subaltern).
• According to Aristotelian logic, whenever a
  universal proposition is true, its corresponding
  particular must be true. Thus if an A proposition
  (All S is P) is true, the corresponding I
  proposition (Some S is P) is also true. Likewise
  if an E proposition (No S is P) is true, so too
  is its corresponding particular (Some S is not
  P). The reverse, however, does not hold. That is,
  if a particular proposition is true, its
  corresponding universal might be true or it might
  be false.

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Subalternation II

• For example All bananas are fruit. Therefore,
  some bananas are fruit.
• Or, no humans are reptiles. Therefore, some
  humans are not reptiles.
• However, we cant go in reverse. We cant say
  some animals are not dogs. Therefore, no animals
  are dogs.
• Or, some guitar players are famous rock
  musicians. Therefore, all guitar players are
  famous rock musicians.

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Conversion

• The first kind of immediate inference, called
  conversion, proceeds by simply interchanging the
  subject and predicate terms of the proposition.
• Conversion is valid in the case of E and I
  propositions. No women are American
  Presidents, can be validly converted to No
  American Presidents are women.
• An example of an I conversion Some politicians
  are liars, and Some liars are politicians are
  logically equivalent, so by conversion either can
  be validly inferred from the other.

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Conversion II

• One standard-form proposition is said to be the
  converse of another when it is formed by simply
  interchanging the subject and predicate terms of
  that other proposition. Thus, No idealists are
  politicians is the converse of No politicians
  are idealists, and each can validly be inferred
  from the other by conversion. The term
  convertend is used to refer to the premise of an
  immediate inference by conversion, and the
  conclusion of the inference is called the
  converse.

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Conversion III

• Note that the converse of an A proposition is not
  generally valid form that A proposition.
• For example All bananas are fruit, does not
  imply the converse, All fruit are bananas.
• A combination of subalternation and conversion
  does, however, yield a valid immediate inference
  for A propositions. If we know that "All S is P,"
  then by subalternation we can conclude that the
  corresponding I proposition, "Some S is P," is
  true, and by conversion (valid for I
  propositions) that some P is S. This process is
  called conversion by limitation.

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Conversion IV

• Convertend A proposition All IBM computers are
  things that use electricity. Converse A
  proposition All things that use electricity are
  IBM computers.
• Convertend A proposition All IBM computers are
  things that use electricity. Corresponding
  particular I proposition Some IBM computers
  are things that use electricity. Converse (by
  limitation) I proposition Some things that use
  electricity are IBM computers.
• The first part of this example indicates why
  conversion applied directly to A propositions
  does not yield valid immediate inferences. It is
  certainly true that all IBM computers use
  electricity, but it is certainly false that all
  things that use electricity are IBM computers.
• Conversion by limitation, however, does yield a
  valid immediate inference for A propositions
  according to Aristotelian logic. From "All IBM
  computers are things that use electricity" we
  get, by subalternation, the I proposition "Some
  IBM computers are things that use electricity."
  And because conversion is valid for I
  propositions, we can conclude, finally, that
  "Some things that use electricity are IBM
  computers."

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Conversion V

• The converse ofSome S is not P, does not yield
  an valid immediate inference.
• Convertend O proposition Some dogs are not
  cocker spaniels.Converse O proposition Some
  cocker spaniels are not dogs.
• This example indicates why conversion of O
  prepositions does not yield a valid immediate
  inference. The first proposition is true, but its
  converse is false.

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Conversion Table
Does not convert to A A All men are wicked creatures. All wicked creatures are men.
Does convert to E E No men are wicked creatures. No wicked creatures are men.
Does convert to I I Some wicked men are creatures. Some wicked creatures are men.
Does not convert to O O Some men are not wicked creatures. Some wicked creatures are not men.
69
Obversion

• Obversion - A valid form of immediate inference
  for every standard-form categorical proposition.
  To obvert a proposition we change its quality
  (from affirmative to negative, or from negative
  to affirmative) and replace the predicate term
  with its complement. Thus, applied to the
  proposition "All cocker spaniels are dogs,"
  obversion yields "No cockerspaniels are nondogs,"
  which is called its "obverse." The proposition
  obverted is called the "obvertend."

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Obversion II

• The obverse is logically equivalent to the
  obvertend. Obversion is thus a valid immediate
  inference when applied to any standard-form
  categorical proposition.
• The obverse of the A proposition "All S is P" is
  the E proposition "No S is non-P."
• The obverse of the E proposition "No S is P" is
  the A proposition "All S is non-P."

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Obversion III

• The obverse of the I proposition "Some S is P" is
  the O proposition "Some S is not non-P."
• The obverse of the O proposition "Some S is not
  P" is the I proposition "Some S is non-P."
• Obvertend A-proposition All cartoon characters
  are fictional characters. Obverse
  E-proposition No cartoon characters are
  non-fictional characters.
• Obvertend E-proposition No current sitcoms are
  funny shows. Obverse A-proposition All current
  sitcoms are non-funny shows.

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Obversion IV

• Obvertend I-proposition Some rap songs are
  lullabies. Obverse O-proposition Some rap
  songs are not non-lullabies.
• Obvertend O-proposition Some movie stars are
  not geniuses. Obverse I-proposition Some movie
  stars are non-geniuses.

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Obversion V

• As these examples indicate, obversion always
  yields a valid immediate inference.
• If every cartoon character is a fictional
  character, then it must be true that no cartoon
  character is a non-fictional character.
• If no current sitcoms are funny, then all of them
  must be something other than funny.
• If some rap songs are lullabies, then those
  particular rap songs at least must not be things
  that aren't lullabies.
• If some movie stars are not geniuses, than they
  must be something other than geniuses.

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Contraposition

• Contraposition is a process that involves
  replacing the subject term of a categorical
  proposition with the complement of its predicate
  term and its predicate term with the complement
  of its subject term.
• Contraposition yields a valid immediate inference
  for A propositions and O propositions. That is,
  if the proposition
• All S is P is true, then its contrapositive
• All non-P is non-S is also true.

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Contraposition II

• For example
• Premise
• A proposition All logic books are interesting
  things to read.
• Contrapositive
• A proposition All non interesting things to read
  are non logic books.

76
Contraposition III

• The contrapositive of an A proposition is a valid
  immediate inference from its premise. If the
  first proposition is true it places every logic
  book in the class of interesting things to read.
  The contrapositive claims that any
  non-interesting things to read are also non-logic
  bookssomething other than a logic bookand
  surely this must be correct.

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Contraposition IV

• Premise
• I-proposition Some humans are non-logic
  teachers.
• Contrapositive
• I-proposition Some logic teachers are not human.
  
• As this example suggests, contraposition does
  not yield valid immediate inferences for I
  propositions. The first proposition is true, but
  the second is clearly false.

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Contraposition V

• E premise
• No dentists are non-graduates.
• The contrapositive is No graduates are
  non-dentists.
• Obviously this is not true.

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Contraposition VI

• The contrapositive of an E proposition does not
  yield a valid immediate inference. This is
  because the propositions "No S is P" and "Some
  non-P is non-S" can both be true. But in that
  case "No non-P is non-S," the contrapositive of
  "No S is P," would have to be false.
• A combination of subalternation and
  contraposition does, however, yield a valid
  immediate inference for E propositions. If we
  know that "No S is P" is true, then by
  subalternation we can conclude that the
  corresponding O proposition, "Some S is not P,"
  is true, and by contraposition (valid for O
  propositions) that "Some non-P is not non-S" is
  also true. This process is called contraposition
  by limitation.

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Contraposition VII

• Premise
• E-proposition No Game Show Hosts are Brain
  Surgeons.
• Contrapositive
• E proposition No non-Brain Surgeons are non-Game
  show hosts.
• Premise
• E proposition No game show hosts are brain
  surgeons.
• Corresponding particular O proposition Some game
  show hosts are not brain surgeons.
• Contrapositive
• O proposition Some non-brain surgeons are not
  non-game show hosts.

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Contraposition VIII

• The first part of this example indicates why
  contraposition applied directly to E propositions
  does not yield valid immediate inferences. Even
  if the first proposition is true then the second
  can still be false. This may be hard to see at
  first, but if we take it apart slowly we can
  understand why. The first proposition, if true,
  clearly separates the class of game show hosts
  from the class of brain surgeons, allowing no
  overlap between them. It does not, however, tell
  us anything specific about what is outside those
  classes. But the second proposition does refer to
  the areas outside the classes and what it says
  might be false. It claims that there is not even
  one thing outside the class of brain surgeons
  that is, at the same time, a non-game show host.
  But wait a minute. Most of us are neither brain
  surgeons nor game show hosts. Clearly the
  contrapositive is false.

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Contraposition IX

• Contraposition by limitation, however, does yield
  a valid immediate inference for E propositions
  according to Aristotelian logic. By
  subalternation from the first proposition we get
  the O proposition "Some game show hosts are not
  brain surgeons." And then by contraposition,
  which is valid for O propositions, we get the
  valid, if tongue-twisting O proposition, "Some
  non-brain surgeons are not non-game show hosts."

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Contraposition X

• O proposition.
• Premise
• Some flowers are not roses.
• Some non-roses are not non-flowers.
• This is valid. Thus we can see that
  contraposition is a valid form of inference only
  when applied to A and O propositions.
  Contraposition is not valid at all for I
  propositions and is valid for E propositions only
  by limitation.

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Contraposition XI
Table of Contraposition Table of Contraposition
Premise Contrapositive
A All S is P. A All non-P is non-S.
E No S is P. O Some non-P is not non-S. (by limitation)
I Some S is P. Contraposition not valid.
O Some S is not P. Some non-P is not non-S.
85
Existential Import and the Interpretation of
Categorical Propositions

• Aristotelian logic suffers from a dilemma that
  undermines the validity of many relationships in
  the traditional Square of Opposition.
  Mathematician and logician George Boole proposed
  a resolution to this dilemma in the late
  nineteenth century. This Boolean interpretation
  of categorical propositions has displaced the
  Aristotelian interpretation in modern logic.

86
Existential Import and the Interpretation of
Categorical Propositions II

• The source of the dilemma is the problem of
  existential import. A proposition is said to have
  existential import if it asserts the existence of
  objects of some kind. I and O propositions have
  existential import they assert that the classes
  designated by their subject terms are not empty.
  But in Aristotelian logic, I and O propositions
  follow validly from A and E propositions by
  subalternation. As a result, Aristotelian logic
  requires A and E propositions to have existential
  import, because a proposition with existential
  import cannot be derived from a proposition
  without existential import.

87
Existential Import and the Interpretation of
Categorical Propositions III

• A and O propositions with the same subject and
  predicate terms are contradictories, and so
  cannot both be false at the same time. But if A
  propositions have existential import, then an A
  proposition and its contradictory O proposition
  would both be false when their subject class was
  empty.
• For example
• Unicorns have horns. If there are no unicorns,
  then it is false that all unicorns have horns and
  it is also false that some unicorns have horns.

88
Existential Import and the Interpretation of
Categorical Propositions IV

• The Boolean interpretation of categorical
  propositions solves this dilemma by denying that
  universal propositions have existential import.
  This has the following consequences
• I propositions and O propositions have
  existential import.
• A-O and E-I pairs with the same subject and
  predicate terms retain their relationship as
  contradictories.
• Because A and E propositions have no existential
  import, subalternation is generally not valid.
• Contraries are eliminated because A and E
  propositions can now both be true when the
  subject class is empty. Similarly, subcontraries
  are eliminated because I and O propositions can
  now both be false when the subject class is
  empty.

89
Existential Import and the Interpretation of
Categorical Propositions V

• Some immediate inferences are preserved
  conversion for E and I propositions,
  contraposition for A and O propositions, and
  obversion for any proposition. But conversion by
  limitation and contraposition by limitation are
  no longer generally valid.
• Any argument that relies on the mistaken
  assumption of existence commits the existential
  fallacy.

90
Existential Import and the Interpretation of
Categorical Propositions VI

• The result is to undo the relations along the
  sides of the traditional Square of Opposition but
  to leave the diagonal, contradictory relations in
  force.

91
Symbolism and Diagrams for Categorical
Propositions

• The relationships among classes in the Boolean
  interpretation of categorical propositions can be
  represented in symbolic notation. We represent a
  class by a circle labeled with the term that
  designates the class. Thus the class S is
  diagrammed as shown below

92
Symbolism and Diagrams for Categorical
Propositions II

• To diagram the proposition that S has no members,
  or that there are no Ss, we shade all of the
  interior of the circle representing S, indicating
  in this way that it contains nothing and is
  empty. To diagram the proposition that there are
  Ss, which we interpret as saying that there is
  at least one member of S, we place an x anywhere
  in the interior of the circle representing S,
  indicating in this way that there is something
  inside it, that it is not empty.

93
Symbolism and Diagrams for Categorical
Propositions III

• To diagram a standard-form categorical
  proposition, not one but two circles are
  required. The framework for diagramming any
  standard-form proposition whose subject and
  predicate terms are abbreviated by S and P is
  constructed by drawing two intersecting circles

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Claims about single individuals

• Claims about single individuals, such as
  Aristotle is a logician, can be tricky to
  translate into standard form. Its clear that
  this claim specifies a class, logicians, and
  places Aristotle as a member of that class. The
  problem is that categorical claims are always
  about two classes, and Aristotle isnt a class.
  (We couldnt talk about some of Aristotle being a
  logician.) What we want to do is treat such
  claims as if they were about classes with exactly
  one member.

98
Claims about single individuals II

• One way to do this is to use the term people who
  are identical with Aristotle, which of course
  has only Aristotle as a member.
• Claims about single individuals should be treated
  as A-claims or E-claims.
• Aristotle is a logician can be translated into
  All people identical with Aristotle are
  logicians.
• Individual claims do not only involve people.
  For example, Fort Wayne is in Indiana is All
  cities identical with Fort Wayne are cities in
  Indiana.

99
Two important things to remember about Some
Statements

1. In categorical logic, some always means at
   least one.
2. Some statements are understood to assert that
   something actually exists. Thus, some mammals
   are cats is understood to assert that at least
   one mammal exists and that that mammal is a cat.
   By contrast, all or no statements are not
   interpreted as asserting the existence of
   anything. Instead, they are treated as purely
   conditional statements. Thus, All snakes are
   reptiles asserts that if anything is a snake,
   then it is a reptile, not that there are snakes
   and that all of them are reptiles.

100
Exercises

• Draw Venn diagrams of the following statements.
  In some cases, you may need to rephrase the
  statements slightly to put them in one of the
  four standard forms.
• No apples are fruits.
• Some apples are not fruits.
• All fruits are apples.
• Some apples are fruits.

101
Translating into standard categorical form

• Do people really go around saying things like
  some fruits are not apples? Not very often.
  But although relatively few of our everyday
  statements are explicitly in standard categorical
  form, a surprisingly large number of those
  statements can be translated into standard
  categorical form.

102
Common Stylistic Variants of All S are P

• Example
• Every S is P. Every dog is an animal.
• Whoever is an S is a P. Whoever is a
  bachelor is a male.
• Any S is a P. Any triangle is a
  geometrical figure.
• Each S is a P. Each eagle is a bird.
• Only P are S. Only Catholics are popes.
• Only if something is a Only if something is
  a dog
• P is it an S. is it a cocker
  spaniel.
• The only S are P. The only tickets
  available are tickets for cheap
  seats.

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ONLY

• Pay special attention to the phrases containing
  the word only in that list. (Only is one of
  the trickiest words in the English language.)
  Note, in particular, that as a rule the subject
  and the predicate terms must be reversed if the
  statement begins with the words only or only
  if. Thus, Only citizens are voters must be
  rewritten as All voters are citizens, not All
  citizens are voters. And, Only if a thing is
  an insect is it a bee must be rewritten as All
  bees are insects, not All insects are bees.

104
Common Stylistic Variants of No S are P

• Example
• No S are P. No cows are reptiles.
• S are not P. Cows are not reptiles.
• Nothing that is an S Nothing that is a
  known
• is a P. fact is a mere opinion.
• No one who is an S No one who is a
  Republican
• is a P. is a Democrat.
• All S are non-P. If anything is a plant,
  then it is not a mineral.

105
Common Stylistic Variants of Some S are P

• Some P are S.
• A few S are P.
• There are S that are P.
• Several S are P.
• Many S are P.
• Most S are P.

106
Common Stylistic Variants of Some S are not P

• Not all S are P.
• Not everyone who is
• an S is a P.
• Some S are non-P.
• Most S are not P.
• Nearly all S are
• not P.

107
Paraphrasing

• The process of casting sentences that we find in
  at ext into one of these four forms is
  technically called paraphrasing, and the ability
  to paraphrase must be acquired in order to deal
  with statements logically.
• In the processing of paraphrasing we designate
  the affirmative or negative quality of a
  statement principally by using the words no or
  not. We indicate quantity, meaning whether we
  are referring to the entire class or only a
  portion of it, by using words all or some.
  In addition, we must render the subject and the
  predicate as classes of objects with the verb
  is or are as the copula joining the halves.

108
Paraphrasing II

• We must pay attention to the grammar, diagramming
  the sentences if need be, to determine the parts
  of the sentence, the group that is meant, and
  even what noun is being modified.
• The kind of thing a claim directly concerns is
  not always obvious. For example, if you think
  for a moment about the claim I always get
  nervous when I take logic exams, youll see its
  a claim about times. Its about getting nervous
  and about logic exams indirectly,of course, but
  it pertains directly to times or occasions. The
  proper translation of the example is All times I
  take logic exams are times that I get nervous.

109

• Once our statement is translated into proper
  form, we can see it implications to other forms
  of the statement. For example, if we claim All
  scientists are gifted writers, that certainly
  implies that Some scientists are gifted
  writers, but we cannot logically transpose the
  proposition to All gifted writers are
  scientists. In other words, some statements
  would follow, others would not.
• To help determine when we can infer one statement
  from another and when there is disagreement,
  logicians have devised tables that we can refer
  to if we get confused.

110
Conversion Table
Does not convert to A A All men are wicked creatures. All wicked creatures are men.
Does convert to E E No men are wicked creatures. No wicked creatures are men.
Does convert to I I Some wicked men are creatures. Some wicked creatures are men.
Does not convert to O O Some men are not wicked creatures. Some wicked creatures are not men.
111
Syllogisms

• Syllogism a deductive argument in which a
  conclusion is inferred from two premises.
• In a syllogism we lay out our train of reasoning
  in an explicit way, identifying the major premise
  of the argument, the minor premise, and the
  conclusion.
• The major premise consists of the chief reason
  for the conclusion, or more technically, it is
  the premise that contains the term in the
  predicate of the conclusion.
• The minor premise supports the conclusion in an
  auxiliary way, or more precisely, it contains the
  term that appears in the subject of the
  conclusion.
• The conclusion is the point of the argument, the
  outcome, or necessary consequence of the premise.

112
Syllogisms II

• Example in an argumentative essay
• In determining who has committed war crimes we
  must ask ourselves who has slaughtered unarmed
  civilians, whether as reprisal, ethnic
  cleansing, terrorism, or outright genocide.
  For along with pillaging, rape, and other
  atrocities, this is what war crimes consist of .
  In the civil war in the former Yugoslavia,
  soldiers in the Bosnian Serb army committed
  hundreds of murders of this kind. They must
  therefore be judged guilty of war crimes along
  with the other awful groups in our century, most
  notably the Nazis.

113
Syllogisms III

• The conclusion to this argument is that soldiers
  in the Bosnian Serb army are guilty of war
  crimes. The premises supporting the conclusion
  are that slaughtering unarmed civilians is a war
  crime, and soldiers in the Bosnian Serb army have
  slaughtered unarmed civilians. The following
  syllogism will diagram this argument.
• All soldiers who slaughter unarmed civilians are
  guilty of war crimes.
• Some Bosnian Serb soldiers are soldiers who
  slaughter unarmed civilians
  
  
• Some Bosnian Serb soldiers are guilty of war
  crimes.

114
Enthymeme

• Enthymeme - An argument that is stated
  incompletely, the unstated part of it being taken
  for granted. An enthymeme may be the first,
  second, or third order, depending on whether the
  unstated proposition is the major premise, the
  minor premise, or the conclusion of the argument.
  
• Enthymemes traditionally have been divided into
  different orders, according to which part of the
  syllogism is left unexpressed.

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Enthymeme II

• A first order enthymeme is one in which the
  syllogisms major premise is not stated.
• For example, suppose someone said, We must
  expect to find needles on all pine trees they
  are conifers after all. Once we recognize this
  as an enthymeme we must provide the unstated
  (major) premise, namely, All conifers have
  needles. Then we need to paraphrase the
  statements and arrange them in a syllogism,
  indicating by parentheses which one we added was
  not in the text
• (All conifers are trees that have needles.)
• All pine trees are conifers.
• All pine trees are trees that have needles.

116
Enthymeme III

• A second - order enthymeme is one in which only
  the major premise and the conclusion are stated,
  the minor premise being suppressed.
• For example, Of course tennis players arent
  weak, in fact, no athletes are weak. Obviously,
  the missing premise is Tennis players are
  athletes, so the syllogism would appear this
  way.
• No athletes are weak.
• (All tennis players are athletes.)
• No tennis players are weak.

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Enthymeme IV

• A third order enthymeme is one in which both
  premises are sated, but the conclusion is left
  unexpressed.
• For example, All true democrats believe in
  freedom of speech, but there are some Americans
  who would impose censorship on free expression.
  The reader is left to draw the conclusion that
  some Americans are not true democrats. The
  syllogism
• All true democrats are people who believe in
  freedom of speech.
• Some Americans are not people who believe in
  freedom of speech.
• (Some Americans are not true democrats.)

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Exercises

• No certainty should be rejected. So, no
  self-evident propositions should be rejected.
• Some beliefs about aliens are not rational, for
  all rational beliefs are proportional to the
  available evidence.
• John is a member of the police force and all
  policemen carry guns.

119
Validity and Truth

• No matter how diligent we are in constructing our
  argument in proper form, our conclusion can still
  be mistaken if the conclusion does not strictly
  follow from the premises, that is, if the logic
  is not sound.
• For example,
• All fish are gilled creatures.
• All tuna are fish.
• All tuna are gilled creatures.
• This seems correct.

120
Validity and Truth II

• But suppose we want to claim that all tuna are