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Smartboard Notes from Chapter 4 Lectures

Chapter 4: Sections 4.1 - 4.7

4.1 The Components of Categorical Propositions

4.2 Quality Quantity & Distribution

4.3 Venn Diagrams & the Modern Square of Opposition

4.4Conversion, Obversion & Contraposition

4.5 The Traditional Square of Opposition

4.6 Venn Diagrams and the Traditional Standpoint

4.7 Translating Ordinary Language Statements into Categorical Form

 

Notes for Chapter 4

The following notes highlight concepts you should understand from the assigned readings. They are neither intended to replace the lectures and text, nor to substitute for a reading of the text. Lectures will add to and supplement material given here. In order to do well in this class, it is recommended that you review these notes to identify main ideas after having attended class.

Some other resources in logic that might be of interest:

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Prior Resource Listings

  • The Philosophy of Logic: this page contains good background information and plenty of links for those interested in exploring history and various approaches to theoretical logic. There is also an especially good page entitled Defining Necessity and Contingency that describes the difference between necessary and sufficient conditions.

  • Want to take a look at what logicians are thinking about? Here's a link to Analysis Web where you'll find professional logicians discussing their craft.

  • The Critical Thinking Site at Longview Community College: this site has long been an excellent resource for beginners and advanced students of logic. I advise you to explore this site if you're looking for more explanations of basic concepts.

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    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].
    4 parts of a categorical proposition
    1. Quantifier: "all", "no" or "some"
    2. Subject term
    3. Copula: either "are" or "are not"; links/couples the subject term with the predicate term.
    4. Predicate term

    Things to remember:

    • Standard-form propositions have four distinct components. You cannot combine the various parts in logic.
    • The terms "subject" and "predicate" "do note mean the same thing in logic" that they mean in grammar.
    • The form "All S are not P", is not a standard form.
      • Two ways of translating it are:
    1. No S are P. or
    2. Some S are not P.
    • In this text there are only three forms of quantifiers and two types of copulas (see above). Other texts allow for variations, but for simplicity’s sake, we’ll only deal with this limited set.

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    4.2 Quality Quantity & Distribution

    Affirmative Propositions: affirm class membership or put members into groups

    • A: All S are P.

    • I: Some S are P.

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

    • E: No S are P.

    • O: Some S are not P.

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

    • A: All S are P.

    • E: No S are P.

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

    • I: Some S are P.

    • O: Some S are not P.

    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

    • Review the modern square of opposition.
    • Contradictory relation = opposite truth value

    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.
    • immediate inferences = arguments that have only one premise
    • See process in action

    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

    • Review the diagrams to be sure you understand the relation being expressed.
    • What is it? Conversion is a move that allows us to switch subjects and predicates for E & I statements only.
    • When is it a legal move? Conversion can only be used on E & I statements.
    • How do I convert E & I statements? Switch the subject and predicate terms.
    • Examples:
      • To convert the statement: No cats are fish, we switch the subject and predicate terms to arrive @ No fish are cats.
      • Since these two statements mean the same thing, we state that they are logically equivalent. 

    Contraposition: two steps

    • Review the diagrams to be sure you understand the relation being expressed.
    • What is it? Contraposition is a move that allows us to switch the subject and predicate terms for A & O statements while also changing each term to its complement.
    • When is it a legal move? Contraposition can be used on A & O statements only.
    • How do I contrapose statements? Contraposition is a two-step process
      1. Switch subject and predicate terms.
      2. Change both the subject and predicate term to its complement (i.e., opposite).
    • Examples:
      • To contrapose the statement: All  cats are fish, we first switch the subject & predicate terms to the A statement yielding "All fish are cats. Next we change each term to its complement and the final statement is All non-fish are non-cats.
      • You do not have to show each step, just the original and final statements.
      • Since these two statements mean the same thing, we state that they are logically equivalent. 

    Obversion: two steps

    • Review the diagrams to be sure you understand the relation being expressed.
    • What is it? Obversion is a move that allows us to switch the quality and predicate terms for all four statements: A, E, I & O.
    • When is it a legal move? Obversion can be used on A, E I & O statements.
    • How do I obvert statements? Obversion is a two-step process
      1. Change the quality of the statement (i.e., move horizontally across the square from the current statement letter) and,
      2. Change the predicate term to its complement (i.e., opposite).
    • Examples:
      • To obvert the statement: No cats are fish, we first move horizontally to the A statement yielding "All cats are _____. Next we change the predicate term to its complement and the final statement is All cats are non-fish..
      • You do not have to show each step, just the original and final statements.
      • Since these two statements mean the same thing, we state that they are logically equivalent. 

    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:

    • Contrary relation (A & E statements only): at least one is false

      • Both could be false!!!

      • So, if we already know that either the A or the E proposition is false,
        the truth value of the remaining proposition is undetermined.

     

    • Subcontrary relation (I & O statements only): at least one is true

      • Both could be true!!!

      • So, if we already know that either the I or the O proposition is true, the truth value of the remaining proposition is undetermined.

    • Subalternation relation (relation between A & I statements and E & O statements): truth flows downward and falsity flows upward

      • If we are given an A or an E proposition that is false,
        the truth value of the corresponding I or O proposition is undetermined.

      • Also, if we are given an I or an O proposition that is true,
        the truth value of the corresponding A or E proposition is undetermined.

    Testing Immediate Inferences

    There are two kinds of immediate inferences:

    • those you can use the square of opposition to test, and

    • those in which you will have to reduce the number of terms through conversion, obversion and contraposition before using the square to test.

    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

    • We are working with the form of the verb "to be." Various tenses (i.e., will, will not, has, has not).
    • This involves translating all other copulas into statements that contain the phrases "are" or "are not."

    3. Singular Propositions

    • Watch for plural forms of nouns as they should not be translated in this matter.

    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

    • Here, "quantifiers are implied but not expressed." The trick is figuring out if we are talking about "all" or "some" of the noun in question.
    • Note the last two examples in this section use two different uses of the word children.

    6. Nonstandard Quantifiers

    • The form "All S are not P" is not standard form.
    • Translation determines meaning: e.g., All athletes are not superstars. The previous statement is not a universal proposition, but rather a particular claim. Read "At least one athlete is not a superstar."

    7. Conditional Statements

    • Conditional statements are always rendered as universals!
    • NOTE: When a conditional statement appears in the middle of a sentence, "the statement must be restructured so that it occurs @ the beginning."
    • Transposition: applies to conditional statements where both terms are negated.
    • Note/reread the middle paragraphs on p.246 and go over the examples again.
    • The word "unless" means "if not"Again, the transposition rule applies here.
    • Carefully go over the examples.

    8. Exclusive Propositions

    Words to look for:

    • only,
    • none but,
    • none except

    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.)
    • Note also how individual references work. Basically, these references generate two categorical propositions. Our course ignores these special cases.
    • When only and none but are in the middle of a sentence, they are transposed.
    • Only can be rendered in many ways. Thus, it is ambiguous.

    9. "The Only"

    • If the words "the only" appear at the beginning of a phrase, they can be replaced by the word "all" an no transposition is necessary.
    • But, if these words appear in the middle of a phrase, then the statement must be transposed before putting it into standard form.
    • "The only" is like only in that is ambiguous and has to be rendered using two statements for clarity.

    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"

    blue line

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    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:

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    Section 4.5: The Traditional Square of Opposition:

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    Section 4.6 Modified Venn Diagrams:

    Section 4.7 Translating Ordinary Language Statements:

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