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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.(200)

    Proposition Letter Name Proposition What is the assertion?
    A All S are P. The whole subject class is included
    in the predicate class.
    E No S are P. The whole subject class is excluded
    from the predicate class.
    I Some S are P. Part of the subject class is included
    in the predicate class.
    O Some S are not P. Part of the subject class is excluded
    from the predicate class.
    4 parts of a categorical proposition
    1. Quantifier: "all", "no" or "some"
    2. Subject term
    3. Copula: links/couples the subject term with the predicate term.
    4. Predicate term (200)

    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.(200)
    • 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

    • A: All S are P.

    • I: Some S are P.

    Negative Propositions: denies class membership

    • 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"(203)

     

    Affirmative Propositions

    Negative Propositions

    Universal

    Propositions

    Particular

    Propositions

    The best mnemonic device for memorizing distribution attributes is:

    "Universals distribute Subjects.

    Negatives distribute Predicates."(205)

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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).(207)

    John Venn (19th century): created Venn Diagram system.(208)

    A E
    I O

    Contradictories: A & O; E & I (211)

    • Review the modern square of opposition.(211
    • Contradictory relation = opposite truth value(211)
    • Testing arguments:
    1. Assume premise is true.
    2. Use the square to assess what truth values other propositions must necessarily have.
    3. When you have two propositions that do not have a contradictory relationship, the truth value is undetermined.
    • immediate inferences = arguments that have only one premise
    • See process in action (212)

    Process for testing arguments with Venn diagrams.

    1. Draw a Venn diagram for the premise.
    2. Draw a Venn diagram for the conclusion.
    3. If the two diagrams express the same information, the argument is valid.
    4. When you have a statement that begins with the phrase "it is false that..." diagram the contradictory proposition for 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]

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    4.4 Conversion, Obversion & Contraposition

    Conversion: switch subject and predicate

    • Review the diagrams on p. 216 to be sure you understand the relation being expressed.
    • Note that the converse of the E & I statements are logically equivalent to the starting propositions.
    • Statements are logically equivalent when they have the same truth value.(216)
    • A & O propositions are "logically unrelated as to truth value."(217)
    • Illicit conversion (see definition p. 682)

    Obversion: two steps

    1. Change quality (affirmative versus negative) without changing the quantity.
    2. Replace predicate term with its complement.(218)
    • See diagram for process on p. 219.
    • A term complement "is the word or group of words that denotes the class complement."(217)
    • All obverse propositions are logically equivalent to their original statements.
    • Review diagrams on page 218.
    • Obverting any statement twice will return you to the original given statement.

    Contraposition: two steps

    1. Switch subject and predicate terms.
    2. Replace subject and predicate terms with their term complements.(220)
    • See diagram for process on p. 220.
    • Note that A & O statements are logically equivalent.
    • Illicit contraposition: formal fallacy; see definition p. 649; occurs when the conclusion of an argument depends on the contraposition of an E or I statement.

    Testing arguments for validity.

    Note: it is critical that you memorize (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.

    Remember:

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

    Here are the new relations:

    Points to note:

    • Contrary relation: at least one is false(229)

      • 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: at least one is true(229)

      • 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: 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: (230)

    1. Assume premise is true.

    2. Determine the type of relation 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.

    Three Fallacies: (230-231): 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.

    Testing Immediate Inferences using conversion, obversion and contraposition plus the traditional square of opposition

    Three Steps: (232)

    1. Attend to the order of the terms. While placing the terms in the right order, keep the premise at its original truth value. Only use legal moves.

    2. Try to obtain the individual terms as they appear in the conclusion. (same terms, same order)

    3. Use the square of opposition to adjust the quality and quantity. After the terms in the premise and conclusion are the same and in the same order, we can use the square to determine the truth relationship between the premise and the conclusion.

    Make a chart for each problem that looks like this:
    (This example is from the exercises: Part V, #9, page 234.)
    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.(236-237)

    2. Go over diagram and modification on p.236 (big diagram with square).

    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(239).

    Testing Immediate Inferences: 2 types of problems

    Inferences Involving No Alteration of Terms:

    Inferences Involving Alteration of Terms via Conversion, Obversion & Contraposition:

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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.(241)

    2. Renders statements "completely clear and unambiguous."(241)

    Types of Transformations:

    1. Terms Without Nouns

    Review the sentence beginning "Nouns and pronouns…"(241)

    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."(242)

    3. Singular Propositions

    Watch for plural forms of nouns as they should not be translated in this matter.(243)

    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.(244)

    5. Unexpressed Quantifiers

    Here, "quantifiers are implied but not expressed."(244) The trick is figuring out if we are talking about "all" or "some" of the noun in question.

    Note the last two examples on p. 244; 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."(246)

    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 at the bottom of p. 246.

    8. Exclusive Propositions

    Words to look for:

    • only,

    • none but,

    • none except, and

    • no …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.)

    8. Exclusive Propositions continued

    Note also how individual references work (middle p. 247). 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.

    See examples on p.247.

    Only can be rendered in many ways. Thus, it is ambiguous.(247-248)

    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.

    See example on p. 248.

    "The only" is like only in that is ambiguous and has to be rendered using two statements for clarity.(248)

    10. Exceptive Propositions

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

    These statements generate two standard form propositions.(248-249)

    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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