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.
The Structure of Argument: Conclusions
and Premises
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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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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.

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.

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 sexuality…in 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.

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?
•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.
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Consequently
Therefore
Thus
So
Hence
accordingly
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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

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.
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.
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Since
Because
For
whereas
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In as much as
For the reasons that
In view of the fact
As evidenced by

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.


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.

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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
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.

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:
1.
2.
3.
4.
All politicians are liars.
No politicians are liars.
Some politicians are liars.
Some politicians are not liars.

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.

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.

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.

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.

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.

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.

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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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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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."
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

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.
◦ 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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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.
Politicians
Liars
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.
Politicians
Liars
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.
Politicians
Liars
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.
Politicians
Liars
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.
Politicians
Liars
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.

Claims about single individuals, such as
“Aristotle is a logician,” can be tricky to translate
into standard form. It’s 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 isn’t a class.
(We couldn’t 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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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.”
1.
2.
In categorical logic, “some” always means “at
least one.”
“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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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.

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.

Every S is P.
Whoever is an S is a P.
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Any S is a P.
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Each S is a P.
Only P are S.
Only if something is a
P is it an S.
The only S are P.
Example:
Every dog is an animal.
Whoever is a bachelor is
a male.
Any triangle is a
geometrical figure.
Each eagle is a bird.
Only Catholics are popes.
Only if something is a dog
is it a cocker spaniel.
The only tickets available
are tickets for cheap seats.

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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No S are P.
S are not P.
Nothing that is an S
known
is a P.
No one who is an S
Republican
is a P.
All S are non-P.
Example:
No cows are reptiles.
Cows are not reptiles.
Nothing that is a
fact is a mere opinion.
No one who is a
is a Democrat.
If anything is a plant, then
it is not a mineral.
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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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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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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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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,” you’ll see it’s
a claim about times. It’s 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.

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.

“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.

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.

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.


“Unprepared Students Never Pass”
Universals distribute Subjects.
Negatives distribute Predicates.
“Any Student Earning B’s Is Not On
Probation”
◦
◦
◦
◦
A distributes Subject.
E distributes Both.
I distributes Neither.
O distributes Predicate.

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.

There are four ways in which propositions
may be opposed-as contradictories,
contraries, subcontraries, and subalterns.
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
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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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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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 proposition—universal and negative—
and the second is an I proposition—particular 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.


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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
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.


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.

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.




For example: All bananas are fruit.
Therefore, some bananas are fruit.
Or, no humans are reptiles. Therefore,
some humans are not reptiles.
However, we can’t go in reverse. We can’t
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.

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.

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.



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.




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."



The converse of“Some 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.
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.

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."



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."




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.


Obvertend
I-proposition: Some rap songs are lullabies.
Obverse
O-proposition: Some rap songs are not nonlullabies.
Obvertend
O-proposition: Some movie stars are not
geniuses.
Obverse
I-proposition: Some movie stars are nongeniuses.

As these examples indicate, obversion always
yields a valid immediate inference.

If every cartoon character is a fictional

If no current sitcoms are funny, then all of them

If some rap songs are lullabies, then those

If some movie stars are not geniuses, than they
character, then it must be true that no cartoon
character is a non-fictional character.
must be something other than funny.
particular rap songs at least must not be things
that aren't lullabies.
must be something other than geniuses.



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.





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.

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 noninteresting things to read are also nonlogic books—something other than a logic
book—and surely this must be correct.





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.




E premise:
No dentists are non-graduates.
The contrapositive is: No graduates are
non-dentists.
Obviously this is not true.


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 nonS," 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.









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.

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.

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 tonguetwisting O proposition, "Some non-brain
surgeons are not non-game show hosts."





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.
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.
Some non-P is not
non-S.
O: Some S is not P.

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.

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.


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.





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.


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.

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.
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:

To diagram the proposition that S has no
members, or that there are no S’s, 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 S’s,
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.

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:

Claims about single individuals, such as
“Aristotle is a logician,” can be tricky to translate
into standard form. It’s 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 isn’t a class.
(We couldn’t 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.




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.”
1.
2.
In categorical logic, “some” always means “at
least one.”
“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.





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.

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.

Every S is P.
Whoever is an S is a P.

Any S is a P.





Each S is a P.
Only P are S.
Only if something is a
P is it an S.
The only S are P.
Example:
Every dog is an animal.
Whoever is a bachelor is
a male.
Any triangle is a
geometrical figure.
Each eagle is a bird.
Only Catholics are popes.
Only if something is a dog
is it a cocker spaniel.
The only tickets available
are tickets for cheap seats.

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.”





No S are P.
S are not P.
Nothing that is an S
known
is a P.
No one who is an S
Republican
is a P.
All S are non-P.
Example:
No cows are reptiles.
Cows are not reptiles.
Nothing that is a
fact is a mere opinion.
No one who is a
is a Democrat.
If anything is a plant, then
it is not a mineral.






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.





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.


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.


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,” you’ll see it’s
a claim about times. It’s 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.”


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.
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.





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.

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.
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.



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.


A first order enthymeme is one in which the
syllogism’s 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.


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 aren’t
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.


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.)



No certainty should be rejected. So, no selfevident 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.


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.

But suppose we want to claim that all tuna are
fish for the simple reason that they have gills
and all fish have gills. Our syllogism would then
appear in the following form:
All fish are gilled creatures.
All tuna are gilled creatures.
All tuna are fish.

Of course, this syllogism is problematic. The
mistake seems to lie in the structure itself.
From the fact that tuna have gills we cannot
conclude that tuna must be fish, because we do
not know that only fish have gills.


Another example:
John is pro-choice, therefore John is a
Democrat. Some Republicans or Libertarians are
pro-choice. Just because John is pro-choice
does not mean that he is necessarily a
Democrat. An argument of this kind, where the
conclusion fails to follow from the premises, is
considered invalid. That is, the form of the
argument is flawed so that the reasons that are
given do not support the claim that is made.
Suppose we were to argue the following:
All trees are reptiles.
All rocks are trees.
All rocks are reptiles.
 It is true that if all trees are reptiles, and all
rocks are trees, then it logically follows that all
rocks are reptiles. The obvious problem is that
trees are not reptiles and rocks are not trees.
The logical structure of an argument can be
sound. Given the premises, the conclusion
follows necessarily from them, but the premises
are untrue.

Truth is correspondence with reality. A statement
is true if it describes things as they are. Validity,
on the other hand, applies to the structure of an
argument, not to the statements that make up its
content. As we have seen, an argument is valid if,
given the premises, the conclusion is unavoidable.