Chapter 6 Evaluating Deductive Arguments 1: Categorical Logic Invitation to Critical Thinking First Canadian Edition Joel Rudinow Vincent E. Barry Mark Letteri

© 2008 by Nelson, a division of Thomson Canada Limited 6-2 Deductive Reasoning: Overview Formats  Mapping  Conventional  Casting  Valid Syllogism Deductive Validity  Invalidity  Testing for validity Categorical Logic  Statements to forms  Square of Opposition  Syllogisms  Figure and Mood  Venn Diagrams

© 2008 by Nelson, a division of Thomson Canada Limited 6-3 Formats All Canadians are mortal. All human beings are mortal. All Canadians are human beings. Mapping +

© 2008 by Nelson, a division of Thomson Canada Limited 6-4 Formats Mapping Conventional (1) All human beings are mortal. (2) All Canadians are human beings. ____________________ (3) All Canadians are mortal. All human beings are mortal. All Canadians are human beings. All Canadians are mortal. All human beings are mortal. All Canadians are human beings. All Canadians are mortal. +

© 2008 by Nelson, a division of Thomson Canada Limited 6-5 Formats Mapping ConventionalCasting (1) All human beings are mortal. (2) All Canadians are human beings. ____________________ (3) All Canadians are mortal. (1) + (2) ____________ (3) All human beings are mortal. All Canadians are human beings. All Canadians are mortal. +

© 2008 by Nelson, a division of Thomson Canada Limited 6-6 A Format for a Valid Syllogism Argument: Sentence Form Argument: Standard Form Major Premise Minor Premise Conclusion All human beings are mortal. All Canadians are human beings. All Canadians are mortal. All H’s are M’s All C’s are H’s All C’s are M’s subject term predicate term middle term

© 2008 by Nelson, a division of Thomson Canada Limited 6-7 Deductive Validity  Three statements Two premises that lead to a conclusion (thesis) Two premises that lead to a conclusion (thesis)  Standard form always in this order: Major premise Major premise Minor premise Minor premise Conclusion Conclusion If the premises are taken to be true, then the conclusion must also be true. If the premises are taken to be true, then the conclusion must also be true.

© 2008 by Nelson, a division of Thomson Canada Limited 6-8 Invalidity  Not all forms are valid forms Unreliable if premises do not lead to the conclusion Unreliable if premises do not lead to the conclusion Sample Invalid Format (1)All Canadians are human. (2)All Ontarians are human. (3) All Ontarians are Canadians. All C’s are B’s All O’s are B’s All O’s are C’s

© 2008 by Nelson, a division of Thomson Canada Limited 6-9 Valid and Invalid Forms VALID All human beings are mortal. All Canadians are human beings. All Canadians are mortal. INVALID All frogs are mortal. All Canadians are mortal. All Canadians are frogs. All A are B All C are A All C are B All A are B All C are B All C are A

© 2008 by Nelson, a division of Thomson Canada Limited 6-10 Testing for Deductive Validity Test 1: Ask, “Can I assert the premises and deny the conclusion without contradicting myself?” Test 2: Try to imagine a scenario in which the premises are all true and the conclusion is false. Test 3: Constructing counterexamples: using the same form (format or pattern) to construct an analogous set of statements that test the form for validity.

© 2008 by Nelson, a division of Thomson Canada Limited 6-11 Translating Categorical Statements into Standard Form Some general rules 1. Begin with a quantity indicator: some, all, no. 2. Use “are” or “are not” as the verb. 3. Subject and predicate terms must be noun phrases; they each denote a category. 4. The subject term appears before the “are” or “are not” and the predicate term appears after. 5. All + not = some—use the “some” term instead of “all” + “not”. 6. Turn adjectives into nouns or noun phrases. 7. Turn verbs into nouns or noun phrases.

© 2008 by Nelson, a division of Thomson Canada Limited 6-12 Square of Opposition AFFIRMATIVENEGATIVE A: Universal Affirmative e.g. All mothers are female. E: Universal Negative No fathers are female. I: Particular Affirmative e.g. Some women are mothers. O: Particular Negative e.g. Some women are not mothers. Total Inclusion Total Exclusion Partial Inclusion Partial exclusion

© 2008 by Nelson, a division of Thomson Canada Limited 6-13 Mood and Figure  When the syllogism is in standard form, the “mood” of a syllogism is determined by which of the four statement types appear as the major premise, the minor premise and the conclusion.  Thus, you can represent the three statements in a syllogism using statement types from the Square of Opposition: e.g. AAA, EAE, EIO, AOO, etc.  The “figure” of each syllogism is determined by the position of the middle term.

© 2008 by Nelson, a division of Thomson Canada Limited 6-14 Mood:A Mood:I Mood:E Mood:O AFFIRMATIVENEGATIVE A: Universal Affirmative e.g. All mothers are female. E: Universal Negative No fathers are female. I: Particular Affirmative e.g. Some women are mothers. O: Particular Negative e.g. Some women are not mothers. All human beings are mortal. All Canadians are human beings. All Canadians are mortal.

© 2008 by Nelson, a division of Thomson Canada Limited 6-15 Mood:A Mood:I Mood:E Mood:O AFFIRMATIVENEGATIVE A: Universal Affirmative e.g. All mothers are female. E: Universal Negative No fathers are female. I: Particular Affirmative e.g. Some women are mothers. O: Particular Negative e.g. Some women are not mothers. Some fruit are oranges. Some fruit are apples. Some oranges are apples. I:

© 2008 by Nelson, a division of Thomson Canada Limited 6-16 Mood:A Mood:I Mood:E Mood:O AFFIRMATIVENEGATIVE A: Universal Affirmative e.g. All mothers are female. E: Universal Negative No fathers are female. I: Particular Affirmative e.g. Some women are mothers. O: Particular Negative e.g. Some women are not mothers. No fruit are oranges. No fruit are apples. No oranges are apples. E:

© 2008 by Nelson, a division of Thomson Canada Limited 6-17 Mood:A Mood:I Mood:E Mood:O AFFIRMATIVENEGATIVE A: Universal Affirmative e.g. All mothers are female. E: Universal Negative No fathers are female. I: Particular Affirmative e.g. Some women are mothers. O: Particular Negative e.g. Some women are not mothers. Some fruit are not oranges. Some fruit are not apples. Some oranges are not apples. O:

© 2008 by Nelson, a division of Thomson Canada Limited 6-18 Figure— based on position of middle term S = subject M = middle term P = predicate 1 st figure M—P S—M S—P All human beings are mortal. All Canadians are human beings. All Canadians are mortal. subject predicate middle term 2 nd figure P—M S—M S—P 3 rd figure M—P M—S S—P 4 th figure P—M M—S S—P 1 st figure

© 2008 by Nelson, a division of Thomson Canada Limited 6-19 Venn Diagrams System of intersecting circles  Each circle represents a category.  A shaded area is “vacant” – an area without at least one member  An X is used to indicate a “populated” area – an area with at least one member.  Using two intersecting circles and these simple symbols, we can represent any of the four standard forms of categorical statements (A, E, I, and O).

© 2008 by Nelson, a division of Thomson Canada Limited 6-20 Venn Diagrams  All human beings are mortal.  All Canadians are human beings.  All Canadians are mortal. Valid or invalid? AM H

© 2008 by Nelson, a division of Thomson Canada Limited 6-21 Venn Diagrams  Some mysteries are entertaining.  Some books are mysteries.  Some books are entertaining. Valid or invalid? BE M XX

© 2008 by Nelson, a division of Thomson Canada Limited 6-22 Venn Diagrams  All mysteries are suspenseful.  Some books are not mysteries.  Some books are not suspenseful.  Valid or invalid? BS M X