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Chapter 5: Sections 5.1 - 5.4

5.1 Standard Form, Mood and Figure

5.2 Venn Diagrams

5.3 Rules and Fallacies

5.4 Reducing the Number of Terms

Notes for Chapter 5

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.

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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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    5.1 Standard Form, Mood and Figure

    Syllogism: deductive argument consisting of two premises and a conclusion.(252)

    Categorical syllogism: 

    deductive argument consisting of three categorical propositions that is capable of being translated into standard form.(252)

    Parts of a Categorical Syllogism: (252)

    Terms:

    Major Term: predicate of conclusion

    Minor Term: subject of the conclusion

    Middle Term: occurs once in each premise and does not occur in the conclusion

    Premise Names:

    Major Premise: top; one that contains the major term

    Minor Premise: listed second; one that contains the minor term

    Standard Form Rules (252)

    Note the need to rearrange arguments that are not in standard form.(252)

    The box featured on p.253 (middle) as a good summary of these last two pages.

    Mood: letter names of proposition that make it up (A, E, I & O)

    Figure: determined by the location of the middle term in the two premises that make it up

    Use the shirt collar model on p.255 to memorize figure forms.

    NOTE: This method of determining figure is the best way to solve the problems that follow in the next two sections.

    Unconditionally Valid Categorical Syllogisms

    See list on p. 256.  You are not responsible for memorizing the list.

    Explain what "unconditionally valid" means. This should be a review of Chapter 4 in that we also dealt with unconditionally valid syllogisms there also.

    Aristotelian Additions

    The Aristotelian viewpoint is bound up with the existence of the objects referenced in the argument. Thus, these forms are contingent based on existence.

    The chart and subsequent model are on pages 256-7.

    Contrast the idea of unconditionally valid with conditionally valid.

    Backwards Reconstruction of Arguments Given Mood & Figure

    We can build categorical syllogisms given only mood and figure information.

    The key here is remembering that deductive arguments, especially categorical syllogisms, are dependent on form for their validity rather that content (with the exception of the Aristotelian condition of existence).(258)

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    5.2 Venn Diagrams

    Venn Diagrams are on every major graduate school test known to man. Many of them have more than three variables, but we’ll learn tricks to solve the most complicated of puzzles.

    Review the method for drawing a basic Venn Diagram; note the seven areas that have to be visible.(261)

    The Rules (269-270)

    1. Mark premises only.

    2. Universal premises are entered first.

    3. Focus on the two areas (variables) addressed in the premise you are graphing and give only minimal attention (i.e., ignore) the third circle.

    4. Particular conclusions assert existence (i.e., are Aristotelian) and therefore should be evaluated for both conditions.

    5. Shade all of the area in question.

    6. Note process for entering particular propositions.(270)

    All arguments that contain premises that require us to place an ‘x’ on the line between two sections are invalid!!!

    1. The ‘x’ cannot dangle on the outside of a diagram, and also not placed on the intersection of two lines. (This one is a bit obscure until you start doing the problems.)

    Examples:

    • This is a three step process and that requires you to break each one down into a step-by-step to  construct the syllogism.

    • Also note how the form (mood and figure) give us the entire argument.

    • On p.266 example IAI-1  gives the first instance of invalidity by the ‘on line’ rule.

    • Do not overanalyze the propositions. Graph them and then interpret the graph to arrive at the solution to the exercises.

    • Focus on the famous "Barbara" syllogism on p. 270. This comes up again and again in the book and will appear on a quiz or test in the future.

    • An argument with two particular syllogisms is always invalid.

    Rationale for placing an ‘x’ on the boundary between two sections:

    Review this section quite carefully and be sure you understand why the "x" is placed on the boundary line.  If you are still scratching your head, bring questions to class and I will attempt to clarify the rationale behind this process.

    Aristotelian Diagrams(269 – 270)

    • Review the rules for taking the graphing test one step farther.(276)

    • Note the existential step to verify the existence of the thing in question (emphasized in the example on p. 269).

    • Be sure you understand special cases where two areas have circles that are shaded in more than one area.(269)

    • To review slowly deconstruct each example on p. 269-270 and do as many exercises as time permits.

     

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    5.3 Rules and Fallacies

    These five rules "may be used as a convenient cross-check against the method of Venn Diagrams."(274)

    • The first two are dependent on distribution.

    • Review distributed terms from p.274 top.

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    • Rule 1: Middle term must be distributed at least once.

    • Fallacy = Undistributed Middle

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    • Review the rationale for this fallacy on p.275.

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    • Rule 2: All terms distributed in the conclusion must be distributed in one of the premises.

    • Fallacy = Illicit major; Illicit minor

    • HINT: mark all distributed terms first (276)

    • Remember from Chapter 1 that a deductive argument may not contain more information in the conclusion than is contained in the premises. Thus, arguments that commit the fallacies of illicit major and illicit minor commit this error.(276)

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    • Rule 3: Two negative premises are not allowed.

    • Fallacy = Exclusive premises

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    • Review the explanation on p.276.

    • The key is that "nothing is said about the relation between the S class and the P class."(276)

    • Rule 4: A negative premise requires a negative conclusion, and a negative conclusion requires a negative premise.

    • Fallacy = Drawing an affirmative conclusion from a negative premise. OR

    Drawing a negative conclusion from affirmative premises. OR

    Any syllogism having exactly one negative statement is invalid.

    • Review the explanations for each case on p.277.

    • Note the following sub-rule: No valid syllogism can have two particular premises.(277).

     

    • The last rule is dependent on quantity.

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    • Rule 5: If both premises are universal, the conclusion cannot be particular.

    • Fallacy = Existential Fallacy

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    5.4 Reducing the Number of Terms

    In this section we’ll be working with arguments that contain more than three terms. In this case, the terms and their negations.(283)

    • Note how all of these syllogisms have six terms: i.e., a term and its opposite.

    • Use the example on p. 283.

    HOW TO CONVERT STATEMENTS TO REDUCE TERMS

    This is a review of Section 4.6 where we originally addressed conversion, obversion, and contraposition.

    REVIEW the operations*

    CONVERSION:

    (E & I statements only)

    No S are P.

    Some S are P.

    No P are S.

    Some P are S.

    OBVERSION:

    (all)

    All S are P.

    No S are P.

    Some S are P.

    Some S are not P.

    No S are non-P.

    All S are non-P.

    Some S are not non-P.

    Some S are non-P.

    CONTRAPOSITION:

    (A & O statements only)

    All S are P.

    Some S are not P.

    All non-P are non-S.

    Some non-P are not non-S.

    *"The most important to remember is never to use conversion or contraposition on statements for which they yield undetermined results."(284)

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