Philosophy 1100

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Philosophy 1100
Title Critical Reasoning Instructor Paul
Dickey E-mail Address pdickey2_at_mccneb.edu
Today Exercises 8-1 8-2 Exercise 8-4 (odd
numbered problems) Exercise 8-11, problems
1-5 Next week Exercise 8-11, problems
6-10 not done in class tonight Read Chapter 9,
pages 295-311, pp. 317-330 Exercise 9-1, all
problems Please Note Argumentative Essay
will NOT be due Next week.
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Chapter EightDeductive ArgumentsCategorical
Logic
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Four Basic Kinds of Claims in Categorical
Logic (Standard Forms)
A All _________ are _________. (Ex. All
Presbyterians are Christians. E No ________
are _________. (Ex. No Muslims are Christians.
___________________________________ I Some
________ are _________. (Ex. Some Arabs are
Christians. O Some ________ are not
_________. (Ex. Some Muslims are not Sunnis.
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Three Categorical Operations

• Conversion The converse of a claim is the claim
  with the subject and predicate switched, e.g.
• The converse of No Norwegians are Swedes is
  No Swedes are Norwegians.
• Obversion The obverse of a claim is to switch
  the claim between affirmative and negative (A -gt
  E, E -gt A, I -gt O, and O -gt I and replace the
  predicate term with the complementary (or
  contradictory) term, e.g.
• The obverse of All Presbyterians are
  Christians is No Presbyterians are
  non-Christians.
• Contrapositive The contrapositive of a claim is
  the cliam with the subject and predicate switched
  and replacing both terms with complementary terms
  (or contradictory terms), e.g.
• The contrapositive of Some citizens are not
  voters is Some non-voters are not
  noncitiizens.

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OK, So where is the beef?

• By understanding these concepts, you can apply
  the
• three rules of validity for deductive arguments
• Conversion The converses of all E- and I-
  claims, but not A- and O- claims are equivalent
  to the original claim.
• Obversion The obverses of all four types of
  claims are equivalent to their original claims.
• Contrapositive The contrapositives of all A-
  and O- claims, but not E- and I- claims are
  equivalent to the original claim.

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Categorical Syllogisms

• A syllogism is a deductive argument that has two
  premises -- and, of course, one conclusion
  (claim).
• A categorical syllogism is a syllogism in which
• each of these three statements is a standard
  form, and
• there are three terms which occur twice, once
  each in two of the statements.

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Three Terms of a Categorical Syllogism

• For example, the following is a categorical
  syllogism
• (Premise 1) No Muppets are Patriots.
• (Premise 2) Some Muppets are puppets that
  support themselves financially.
• (Conclusion) Some puppets that support
  themselves financially are not Patriots..
• The three terms of a categorical syllogism are
• 1) the major term (P) the predicate term of the
  conclusion (e.g. Patriots).
• 2) the minor term (S) the subject term of the
  conclusion (e.g. Self-supporting Puppets)
• 3) the middle term (M) the term that occurs in
  both premises but not in the conclusion (e.g.
  Muppets).

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USING VENN DIAGRAMS TO TEST ARGUMENT VALIDITY

• Identify the classes referenced in the argument
  (if there are more than three, something is
  wrong).
• When identifying subject and predicate classes
  in the different claims, be on the watch for
  statements of not and for classes that are in
  common.
• Make sure that you dont have separate classes
  for a term and its complement.
• 2. Assign letters to each classes as variables.
• 3. Given the passage containing the argument,
  rewrite the argument in standard form using the
  variables.

M xxxx S yyyy P zzzz
No M are P. Some M are S. ____________________ T
herefore, Some S are not P.
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• Draw a Venn Diagram of three intersecting
  circles.
• Look at the conclusion of the argument and
  identify the subject and predicate classes.
• Therefore, Some S are not P.
• Label the left circle of the Venn diagram with
  the name of the subject class found in the
  conclusion. (10 A.M.)
• Label the right circle of the Venn diagram with
  the name of the predicate class found in the
  conclusion.
• Label the bottom circle of the Venn diagram with
  the middle term.

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No M are P. Some M are S.

• Diagram each premise according the standard Venn
  diagrams for each standard type of categorical
  claim (A,E, I, and O).
• If the premises contain both universal (A
  E-claims) and particular statements (I
  O-claims), ALWAYS diagram the universal statement
  first (shading).
• When diagramming particular statements, be sure
  to put the X on the line between two areas when
  necessary.
• 10. Evaluate the Venn diagram to whether the
  drawing of the conclusion "Some S are not P" has
  already been drawn. If so, the argument is VALID.
  Otherwise it is INVALID.

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Power of Logic Exercises
http//www.poweroflogic.com/cgi/Venn/venn.cgi?exer
cise6.3B
ANOTHER GOOD SOURCE http//www.philosophypages.c
om/lg/e08a.htm
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Class Workshop Exercise 8-11, 1-5
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Using the Rules Method To Test Validity
Background If a claim refers to all members
of the class, the term is said to be distributed.
Table of Distributed Terms A-claim All S
are P E-claim No S are P I-Claim Some S
are P O-Claim Some S are not P The bold,
italic, underlined term is distributed.
Otherwise, the term is not distributed.
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Some Dogs are Not Poodles. Why is this a
statement about all poodles? Say a boxer is a
dog which is not a poodle. Thus, the statement
above says that all poodles are not boxers and
thus poodles is distributed.
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The Rules of the Syllogism

• A syllogism is valid if and only if all three of
  the following conditions are met
• The number of negative claims in the premises and
  the conclusion must be the same. (Remember these
  are the E- and the O- claims)
• At least one premise must distribute the middle
  term.
• Any term that is distributed in the conclusion
  must be distributed in its premises.

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Class Workshop Exercise 8-13, 8-14, 8-15,
8-16
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• You must perform all of the following
• on the given argument
• Translate the premises and conclusion to standard
  logical forms and put the argument into a
  syllogistic form.
• Identify the type of logical form for each
  statement.
• For each statement, give an equivalent statement
  and name the operation that you used to do so.
• Identify the minor, major, and middle terms of
  the syllogism.
• Draw the appropriate Venn Diagram for the
  premises.
• Identify all distributed terms of the argument
  and the number of negative claims in the premises
  and conclusion.
• What, if any, rules of validity are broken by the
  argument?
• State if the argument is valid or invalid.

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• Everything that Pete won at the carnival must be
  junk. I know that Pete won everything that Bob
  won, and all the stuff Bob won is junk.
• Translate the premises and conclusion to standard
  logical forms and put the argument into a
  syllogistic form.
• Identify the type of logical form for each
  statement.
• Define terms
• P Petes winnings at the carnival
• J Thing that are junk
• B Bobs winnings at the carnival
• A-claim All B is P
• A-claim - All B is J
• A-claim All P is J

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• Everything that Pete won at the carnival must be
  junk. I know that Pete won everything that Bob
  won, and all the stuff Bob won is junk.
• For each statement, give an equivalent statement
  and name the operation that you used to do so.
• Identify the minor, major, and middle terms of
  the syllogism.
• A-claim All B is P
• Contrapositive is equivalent All non-P are
  non-B.
• A-claim - All B is J
• Obverse is equivalent No B is non-J.
• A-claim All P is J
• Obverse is equivalent No P is non-J.
• Minor term is P Major term is J and Middle
  term is B.

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• Everything that Pete won at the carnival must be
  junk. I know that Pete won everything that Bob
  won, and all the stuff Bob won is junk.
• Draw the appropriate Venn Diagram for the
  premises.

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• Everything that Pete won at the carnival must be
  junk. I know that Pete won everything that Bob
  won, and all the stuff Bob won is junk.
• Identify all distributed terms of the argument
  and the number of negative claims in the premises
  and conclusion.
• What, if any, rules of validity are broken by the
  argument?
• State if the argument is valid or invalid.
• All B is P
• All B is J
• All P is J
• Since A-claims distribute their subject terms, B
  is
• Distributed in the premises and P is distributed
  in the
• conclusion. There are no negative claims in
  either the
• premises or the conclusion.
• Since P is distributed in the conclusion, but
  not in
• either premise rule 3 is broken. Thus, the
  argument is invalid.

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The Game

• You must perform all of the following
• on the given argument
• Translate the premises and conclusion to standard
  logical forms and put the argument into a
  syllogistic form.
• Identify the type of logical form for each
  statement.
• For each statement, give an equivalent statement
  and name the operation that you used to do so.
• Identify the minor, major, and middle terms of
  the syllogism.
• Draw the appropriate Venn Diagram for the
  premises.
• Identify all distributed terms of the argument
  and the number of negative claims in the premises
  and conclusion.
• What, if any, rules of validity are broken by the
  argument?
• State if the argument is valid or invalid.

Exercises 8-19, p. 290, Problems 8 19.
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Deductive ArgumentsTruth-Functional Logic
Philosophy 1100 Chapter Nine
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Truth Functional Logic

• Truth Functional logic is important because it
  gives us a consistent tool to determine whether
  certain statements are true or false based on the
  truth or falsity of other statements.
• A sentence is truth-functional if whether it is
  true or not depends entirely on whether or not
  partial sentences are true or false.
• For example, the sentence "Apples are fruits and
  carrots are vegetables" is truth-functional since
  it is true just in case each of its sub-sentences
  "apples are fruits" and "carrots are vegetables"
  is true, and it is false otherwise.
• Note that not all sentences of a natural
  language, such as English, are truth-functional,
  e.g. Mary knows that the Green Bay Packers won
  the Super Bowl.

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Truth Functional Logic The Basics

• Please note that while studying Categorical
  Logic, we used uppercase letters (or variables)
  to represent classes about which we made claims.
• In truth-functional logic, we use uppercase
  letters (variables) to stand for claims
  themselves.
• In truth-functional logic, any given claim P is
  true or false.
• Thus, the simplest truth table form is
• P
• _
• T
• F

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Truth Functional Logic The Basics

• Perhaps the simplest truth table operation is
  negation
• P P
• T F
• F T

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Truth Functional Logic The Basics

• Now, to add a second claim, to account for all
  truth-functional possibilities our representation
  must state
• P Q
• T T
• T F
• F T
• F F
• And the operation of conjunction is represented
  by
• P Q P Q
• T T T
• T F F
• F T F
• F F F

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Truth Functional Logic The Basics

• The operation of disjunction is represented by
• P Q P V Q
• T T T
• T F T
• F T T
• F F F
• The operation of the conditional is represented
  by
• P Q P -gt Q
• T T T
• T F F
• F T T
• F F T

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• Now, using these basic principles, we can
  construct truth tables for more complex
  statements. Consider the claim If Paula goes to
  work, then Quincy and Rogers will get a day off.
• We represent the claims like this
• P Paula goes to work
• Q Quincy gets a day off
• R Rogers gets a day off, and
• We symbolize the complex claim as P -gt (Q R)
• The truth table looks like this
• P Q R Q R P -gt (Q R)
• T T T T T
• T T F F F
• T F T F F
• T F F F F
• F T T T T
• F T F F T

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Class Workshop Exercises 9-1