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Summary Notes

These are summary notes so that you can really listen in class and not spend the entire time copying notes. These notes will not substitute for reading the chapters in the text, nor can they replace the text as there are many subtleties we will discuss in class that are also presented in the text. Use these as a supplement.

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Section 4.1| Section 4.2 | Section 4.3 | Section 4.4 |

Section 4.5 | Section 4.6 | Section 4.7 | Smartboard notes


4.1 The Components of Categorical Propositions

There are four types of categorical propositions.

Proposition Letter Name Proposition What is the assertion?
A All S are P. The whole subject class [S] is included
in the predicate class [P].
E No S are P. The whole subject class [S] is excluded
from the predicate class [P].
I Some S are P. Part of the subject class [S] is included
in the predicate class [P].
O Some S are not P. Part of the subject class [S] is excluded
from the predicate class [P].

 

5 parts of a categorical proposition

  1. Statement letter name (A, E, I or O).
  2. Quantifier: "all", "no" or "some"
  3. Subject term
  4. Copula: either "are" or "are not"; links/couples the subject term with the predicate term.
  5. Predicate term
  6. Letter Name

    Quantifier
    how much

    Subject Term

    Copula

    Predicate Term

    A

    All

    cats

    are

    animals.

    E

    No

    fish

    are

    trees.

    I

    Some

    students

    are

    graphic designers

    O

    Some

    cars

    are not

    trucks.

Things to remember:


4.2 Quality Quantity & Distribution

Affirmative Propositions: affirm class membership or put members into groups

Negative Propositions: denies class membership or remove members from groups.

Universal Propositions: assert something about every member of the S class

Particular Propositions: assert something about one or more members of the S class

Distribution: "makes an assertion about every member of a the class denoted by the term"

The best mnemonic device for memorizing distribution attributes is:

"Universals distribute Subjects.

Negatives distribute Predicates."

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4.3 Venn Diagrams & the Modern Square of Opposition

2 Ways to Interpret Categorical Propositions

  1. Aristotelian: things actually exist in all propositions
  2. Boolean: no assumptions about existence

Aristotelian and Boolean differ only in regard to A and E propositions. For I and O propositions, there is a positive claim about existence (things actually exist).

John Venn (19th century): created Venn Diagram system.

A E
I O

Contradictories: A & O; E & I

How to test arguments for validity using the Modern Square of Opposition: a step-by-step guide.

  1. Symbolize the argument.
  2. Draw a small square.
  3. Plot the truth values given for the premise and conclusion onto the square.
  4. Ask yourself if a) the statements are diagonally opposed, and b) have opposite truth values (i.e., one is true and the other is false).
  5. If the answer to either question is "no," then the argument is invalid.
  6. If you answer "yes" to both questions, then the argument is valid.

Process for testing arguments for validity with Venn diagrams: a step-by-step guide.

  1. Determine the letter names for both the premise and conclusion statements.
  2. Draw a Venn diagram for the premise.
  3. Draw a Venn diagram for the conclusion.
  4. What to do when you have a false statement: When you have a statement that begins with the phrase "it is false that..." draw the diagram for the contradictory proposition of that statement. For example, if you have an A statement, "Is is false that all S are P." [A statement false], draw the Venn diagram for the statement, "Some S are not P." [O statement True]

     

    • GIVEN STATEMENT
      DIAGRAM TO DRAW

      False A

      O

      False E

      I

      False I

      E

      False O

      A

  5. If the two diagrams express the same information (i.e., are identical), the argument is valid, otherwise the argument is invalid.

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4.4 Conversion, Obversion & Contraposition

General Notes:

  1. We are adding a new truth value for propositions in Section 4.4. It is called undetermined.
  2. Each move below can only be performed on the statements indicated within the explanation. When a move is applied to a statement according to the rules below, it is called a legal move.
  3. When you make a legal move, statements keep/retain their originally truth values. Thus, if a statement is true and you make a legal move, the statement stays true. When a statement is false and you make a legal move, the statement stays false.
  4. When a move is applied to a statement and violates one of the rules below, it is called an illegal/illicit move and the truth value for the resulting statement will be undetermined.
  5. When you make a legal move, the beginning statement and the statement generated after making the move are said to be logically equivalent (i.e., this indicates both statements mean the same thing).

Conversion: switch subject and predicate

Contraposition: two steps

Obversion: two steps

Testing arguments for validity.

Three Steps:

  1. Symbolize the argument.
  2. Determine which of the three new moves: conversion, contraposition or obversion has taken place between the premise and conclusion of the argument.
  3. If the move is a legal move, then the argument is valid. If the move is illegal/illicit, the argument is invalid.

Note: it is critical that you memorize legal versus illegal moves (or learn which statements are not logically equivalent) to do well on the quiz and subsequent tests on this material.

 

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4.5 The Traditional Square of Opposition

This square is often called the Aristotelian Square of Opposition.

  1. All of the general rules introduced at the beginning of section 4.4 still apply here.
  2. Thus when we add the three new moves below, there will be legal versus illegal moves.
  3. It is possible to have an illegal contrary statement, illegal subcontrary statement and illegal subalternation. The truth value of the resulting statements (after an illegal move) is undetermined.
  4. To understand the concepts below, you must understand the concept of a minimum condition. The minimum condition is met when the criteria for meeting a rule are fulfilled: e.g. each credit card has a minimum monthly payment for people who carry balances month to month. When you make the minimum payment, you are fulfilling the minimum condition to keep the account in good standing. Another example is that requirement that you have a minimum amount of credits to graduate along with a passing portfolio. When both requirements are fulfilled, you have met the minimum conditions for graduation.

Remember:

  1. Aristotelian: things actually exist in all propositions
  2. Boolean: no assumptions about existence

Here are the three new moves (i.e., relationships) we are adding to the square:

Points to note:

 

Testing Immediate Inferences

There are two kinds of immediate inferences:

Testing Immediate Inferences using the traditional square of opposition only

Three Steps:

  1. Assume premise is true.

  2. Determine the type of relation (i.e., the move that has occurred) that exists between the premise and conclusion.

  3. Using the basic relations from the traditional square of opposition, deduce the remaining truth values if possible.

  4. If the move is a legal move, then the argument is valid. If the move is illegal/illicit, the argument is invalid.

Three Fallacies: these fallacies occur when arguments ask us to violate the relational rules in the traditional square.

  1. Illicit contrary: argument tries to use an invalid application of the contrary relation.

  2. Illicit subcontrary: argument tries to use an invalid application of the subcontrary relation.

  3. Illicit subalternation: argument tries to use an invalid application of the subalternation relation.

The Proofs: Section 4.5 Part V

(Testing Immediate Inferences using conversion, obversion and contraposition plus the traditional square of opposition)
A Step-by-step process for doing these problems:
  1. Symbolize the argument.

  2. The object of the game is to transform the premise statement into the conclusion statement using only legal moves.

  3. Determine which terms (subject and predicate) may have to be switched and/or transformed into their complements. This determination will help you choose the moves that you will use.

  4. Some hints for choosing your moves:

    1. If only one term is to be changed to its complement, then it is likely that you will use obversion along the way.

    2. If the terms have to be switched, it is likely you will need to use conversion.

    3. If both terms must be switched and changed to their complements, then contraposition will be used.

    4. Moving around the square does not change subjects and predicates, it changes letter names and truth values only.

  5. Keep track of letter names and truth values for each statement because they will influence your next move in the proof.

Make a chart for each problem that looks like this:
(This example is from the exercises: Part V, #9.)
Letter Name Truth Value Proposition Inference Name
O

E

E

A

O

F

F

F

F

T

Some non-L are not S.

No non-L are S.

No S are non-L.

All S are L.

Some S are not L.

given

subalternation

conversion

obversion

contradictory

 

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4.6 Venn Diagrams and the Traditional Standpoint

4.6 Venn Diagrams and the Traditional Standpoint

  1. Note how the concept of existence changes the diagramming of relationships.

  2. Two diagrams change for the purpose of proving arguments valid or invalid:

    • The A diagram:

    • The E diagram: 

     

  3. Note the changes in proving validity when we add the existence symbol to the A and E propositions. There are now direct inferences that can be made from the universal propositions to their respective particular propositions (I and O) via subalternation and vice versa going upward from the particulars to the universals.

  4. When do I use the new diagrams? Use the new diagrams when you have to draw an A or E diagram for a premise statement only. When you are drawing an A or E diagram for a conclusion statement, continue to draw the old diagram.

  5. How does this change our view of an argument's validity? By using this technique you are showing that arguments can also be valid under the rule of subalternation: when A is true, I is also true and when E is true, O is also true.

 

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4.7 Translating Ordinary Language Statements into Categorical Form

Two Benefits

  1. Can manipulate using square of opposition and new argument evaluation techniques learned in this chapter.
  2. Renders statements "completely clear and unambiguous."
 

Types of Transformations:

1. Terms Without Nouns

Review the sentence beginning "Nouns and pronouns."

2. Non-standard Verbs

3. Singular Propositions

4. Adverbs and Pronouns

Words to look out for - the hot list:

Spatial Adverbs

Temporal Adverbs

where

wherever

anywhere

everywhere

nowhere

when

whenever

anytime

always

never

NOTE: There is a "switching the order" trick that must occur if one of the above words occurs in the middle of a statement.

5. Unexpressed Quantifiers

6. Nonstandard Quantifiers

7. Conditional Statements

8. Exclusive Propositions

Words to look for:

Two step Process to render statement into standard form:

  1. First phrase as a conditional statement.
  2. Transform into a categorical statement. (Remember all conditional statements are translated as universals as noted above in #7.)

9. "The Only"

10. Exceptive Propositions

Two forms:
  1. All except S are P.
  2. All but S are P.

These statements generate two standard form propositions.

Key Word

Translation Hint

  • whoever
  • wherever
  • always
  • anyone
  • never, etc.
  • use all together with persons, places and times
  • a few
  • some
  • if .. then
  • use "all" or "no"
  • unless
  • "if not"
  • only
  • none but,
  • none except, and
  • no .except.
  • use "all" switch order of terms
  • the only
  • "all"
  • all but,
  • all except,
  • few
  • two statements required
  • "not every" and
  • not all
  • "some _____ are not"
  • there is
  • "some"

 

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Smartboard Notes from Chapter 4 Lectures:

Smartboard Notes from Chapter 4 Lectures:

4.3 Venn Diagrams & the Modern Square of Opposition:

4.4 Conversion, Obversion & Contraposition:

Contraposition:

The chart Part 1 exercises:

Part III Exercises:

 

Section 4.5: The Traditional Square of Opposition:

Section 4.6 Modified Venn Diagrams:

Section 4.7 Translating Ordinary Language Statements:

 

 

 

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