Deductive Arguments I: Categorical Logic

1
Chapter 8

• Deductive Arguments I Categorical Logic

2
Categorical Logic

• Categorical logic is logic based on the relations
  of inclusion and exclusion among classes or
  categories.

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Standard-Form Categorical Claims

• A All S are P. (Ex All snakes are pythons.)
• E No S are P. (Ex No snakes are pythons.)
• I Some S are P. (Ex. Some snakes are pythons.)
• O Some S are not P. (Ex. Some snakes are not
  pythons)
• Note S represents the subject term and P
  represents the predicate term.

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Subject and Predicate Terms

• In a standard-form categorical claim, both the
  subject terms and predicate terms must be nouns
  or noun phrases.
• Example All journalists are inquisitive should
  be changed to All journalists are inquisitive
  people

5
Translating Claims into Standard-Form Categorical
Claims

• A large portion of ordinary, everyday claims can
  be rewritten or translated into equivalent
  standard-form categorical claims.
• Note that two claims are equivalent if, and only
  if, they would both be true in all and exactly
  the same circumstances.

6
Translating to Standard-Form Categorical Claims
Contd

• Every dog is a mammal ? All dogs are mammals.
• (Every S is a P. ? All S are P.)
• Dogs are not friendly? No dogs are friendly
  animals.
• (S are not P. ? No S are P.)
• Not every thief is a liar. ? Some thieves are
  not liars.
• (Not every S is a P. ? Some S are not P.

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Translating Only Claims

• Rules of thumb The word only, used by itself,
  introduces the predicate term of an A-claim. The
  phrase the only introduces the subject term of
  an A-Claim.
• Only males are manly. ? All manly people are
  males.
• Only Xs are Ys ? All Ys are Xs
• Rakes are the only tools.? All tools are rakes.
  
• Xs are the only Ys ? All Ys are Xs
• Thieves are not the only criminals. ? Some
  criminals are not thieves.
• Xs are not the only Ys ? Some Ys are not Xs.

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Translating with Times and Places

• I always laugh when you cry. ? All times you cry
  are times when I laugh.
• I smile whenever you are near ? All times you are
  near are times when I smile.
• People cry wherever you go. ? All places you go
  are places where people cry.

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More Translation Techniques

• Specific people or things
• John is an artist. ? All people who are
  identical to John are artists.
• Kalamazoo is a small city. ? All cities identical
  to Kalamazoo are cities that are small.
• If you cheat, your friends will frown. ? All
  occasions of your cheating are occasions on which
  your friends will frown.

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More Translation

• Other cases
• Most Ss are Ps ? Some Ss are Ps.
• Ex. Most people are crazy.
• Some people are people who are crazy.
• There are some green apples. ? Some apples are
  things that are green.

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Practice Exercises Translation

• Dogs are not cats.
• Not all sharks are fish.
• Most birds are fliers.
• Every snake is poisonous.
• Only oranges are fruits.

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Practice Exercises Translation

• Dave is a magician.
• Bargains are not the only good things.
• Birds always whistle when youre near.
• There are some flying dinosaurs.
• I scream wherever there is danger.

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The Square of Opposition
If you know the truth value of one categorical
claim, The Square of Opposition can help to
determine truth values of other corresponding
categorical claims. Two categorical claims
correspond if they have the same subject and
predicate terms. Ex. All bears are animals
corresponds to No bears are animals.
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The Square of Opposition Contd
Suppose I know that the E-claim No spades are
hearts is true, and I want to determine the
truth-values for the corresponding categorical
claims. The I-claim must be false, because it is
contradictory to the E-claim (meaning that they
have opposite truth values). The A-claim must
then be false, since A-claims are contraries to
E-claims (meaning they cannot both be true). The
O-claim must then be true, since it is
contradictory to the A-claim.
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The Square of Opposition Contd
Suppose I know that the I-claim Some pears are
beans is false, and I want to determine the
truth-values for the corresponding categorical
claims. The E-claim must be true, because it is
contradictory to the I-claim. The O-claim must
be true, since I-claims are subcontraries to
O-claims (meaning they cannot both be
false). The A-claim must then be false, since it
is contradictory to the O-claim.
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The Square of Opposition Contd
But suppose I am given that the A-claim All
poles are sticks is false, and I want to
determine the truth-values for the corresponding
categorical claims. The O-claim must be true,
because it is contradictory to the A-claim
(meaning that they have opposite truth
values). The E-claim and the I-claims cannot be
determined from the information given.
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Practice Exercises Square of Opposition

• 1) 1. True All ponies are fast runners.
• 2. No ponies are fast runners.
  A) T B) F C) U
• 3. Some ponies are fast runners. A) T
  B) F C) U
• 4. Some ponies are not fast runners. A) T
  B) F C) U
• 2) 1. False Some surprises are not unpleasant
  events.
• 2. No surprises are unpleasant events. A)
  T B) F C) U
• 3. Some surprises are unpleasant events.
  A) T B) F C) U
• 4. All surprises are unpleasant events. A)
  T B) F C) U
• 3) 1. True Some people are Hobbits.
• 2. All people are Hobbits. A) T B) F
  C) U
• 3. No people are Hobbits. A) T B) F
  C) U
• 4. Some people are not Hobbits. A) T
  B) F C) U

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Practice Square of Opposition Contd

• 4) 1. False No doors are windows.
• 2. All doors are windows. A)
  T B) F C) U
• 3. Some doors are windows. A) T B) F
  C) U
• 4. Some doors are not windows. A) T B)
  F C) U
• 5) 1. False Some flicks are movies.
• 2. No flicks are movies. A) T B) F
  C) U
• 3. Some flicks are not movies. A) T B)
  F C) U
• 4. All flicks are movies. A) T B) F
  C) U
• 6) 1. True No acids are bases.
• 2. All acids are bases. A) T B) F
  C) U
• 3. Some acids are bases. A) T B) F
  C) U
• 4. Some acids are not bases. A) T B)
  F C) U

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Venn Diagrams
Categorical claims can be represented on Venn
Diagrams. Two intersecting circles represent the
subject and predicate terms. In the cases of
A-claims and E-claims, there will be a portion of
the intersecting circles that are shaded. The
shaded portion means it is empty.
No S are P
All S are P
S
P
S
P
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Venn Diagrams Contd
In the cases of I-claims and O-claims, there will
be a portion of the intersecting circles that
contains an X. The portion with an X
indicates that there is something in that area.
Some S are P
Some S are not P
S
P
S
P
X
X
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Categorical Syllogisms

• A syllogism is an argument that consists of three
  and only three claims, two premises and one
  conclusion.
• A standard-form categorical syllogism is a
  syllogism
• that consists of three standard-form categorical
  claims, and
• in which three terms each occur exactly twice in
  exactly two of the claims.
• Example Some boys are third-graders.
• All third-graders are schoolchildren.
• Therefore, some boys are schoolchildren.

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Categorical Syllogisms Contd

• Some boys are third-graders.
• All third-graders are schoolchildren.
• Some boys are schoolchildren.
• Schoolchildren is the major term because its
  the predicate of the conclusion.
• Boys is the minor term because it is the
  subject of the conclusion.
• Third-graders is the middle term because it
  occurs twice in the premises, but not at all in
  the conclusion.

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The Venn Diagram Method of Testing for Validity

• Consider this syllogism
• No mammals are reptiles.
• All cats are mammals.
• Therefore, no cats are
• reptiles.
• To determine whether its valid, first draw three
  overlapping circles.

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The Venn Diagram Method of Testing for Validity
1. No mammals are reptiles. 2. All cats are
mammals. 3. Therefore, no cats are
reptiles. Second, label the diagram Minor
term, upper left Major term, upper right
Middle term, bottom
C
R M
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The Venn Diagram Method of Testing for Validity
1. No mammals are reptiles. 2. All cats are
mammals. 3. Therefore, no cats are
reptiles. Third, draw the first premise onto the
three-circled Venn diagram.
R
C
M
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The Venn Diagram Method of Testing for Validity
1. No mammals are reptiles. 2. All cats are
mammals. 3. Therefore, no cats are
reptiles. Fourth, draw the second premise onto
the three-circled Venn diagram.
R
C
M
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The Venn Diagram Method of Testing for Validity
1. No mammals are reptiles. 2. All cats are
mammals. 3. Therefore, no cats are
reptiles. Fifth, and check to see if your
conclusion is represented by the diagram. If it
is, then the syllogism is valid, and if not, the
syllogism is invalid. As you can see, the
intersection of the cat are reptile circles
are empty, therefore, this syllogism is valid!
R
C
M
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8-10
E
R
All educated people respect books. Some bookstore
personnel are not truly educated. Some bookstore
personnel dont respect books.
B
All E are R Some B are not E Some B are not R
Translated into standard form
Be clear that
E Educated people R
People who respect books B Bookstore
personnel
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8-11
B
R
All E are R Some B are not E Some B are not R
Ok, draw the first premise. All E are inside R,
so we know that the rest of E is empty. We
represent this empty area by shading it.
E
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8-12
Or here?
All E are R Some B are not E Some B are not R
B
R
X
X
Should the X go here?
Now the second premise. We read some as at
least one and represent it with an X. So we
want to put an X inside the B circle but outside
of the E circle.
E
We want to say exactly what the premises say, but
no more.
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8-13
All E are R Some B are not E Some B are not R
B
R
X?
X?
Think about it. If we opt for the
purple X, we are saying some B are
not R, but this is not in the
premises and we cant draw something
that is not in the premises.
Likewise the red X would say, Some B are R,
and this is not in the premises either.
E
What we need is an X on the line which will
mean that some B are on one side of the line or
the other, or both, but were not sure which.
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8-14
B
R
All E are R Some B are not E Some B are not R
X
So, having drawn exactly what is in the two
premises and no more, is the conclusion
necessarily true? Is it true that some B are not
R?
No, this is an invalid argument.
The X shows that there may
be some B that
are not R, but not necessarily.
E
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8-15
M
H
I
No islands are part of the mainland and Hawaii is
an island. Therefore, Hawaii is not on the
mainland.
No I are M All H are I No H are M
Translated into standard form
M
H
Draw the first premise. Nothing that is an I is
inside the M circle. So, all the things inside I,
if there are any, are in the other parts of the
circle.
I
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8-16
H
M
No I are M All H are I No H are M
Now draw the second premise. Everything that is
in the H circle is also in the I circle. Thus,
the rest of the H circle is empty and should be
shaded.
I
Step 3 asks you to look at what youve drawn and
see if the conclusion is necessarily true. Is it
necessarily true from the picture that nothing in
the H circle is in the M circle?
Yes, this is a valid argument!
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8-17
M
C
Some modems are cable connections and some cable
connections are digital. Thus, some modems are
digital.
D
Some M are C Some C are D Some M are D
Translated into standard form
D
M
Draw the first premise. At least one thing in M
is also in C. Where should the X go?
X
Do you see why the X has to go on the line?
From the premise you cant tell which side of the
line is correct.
C
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8-18
M
D
Some M are C Some C are D Some M are D
X
X
C
Now the second premise. Where should the X go
to represent at least one C that is inside the
D circle? Remember you want to draw just what
the premise says, no more and no less.
Again, the X must go on the line. Our drawing
can never be more precise than the premise is. Is
it Valid?
No this is an invalid argument. There is no
guarantee, from the premises that the conclusion
is true. There may or may not be an M in the D
circle.