Philosophy 1100 Class #8 Title: Critical Reasoning Instructor: Paul Dickey E-mail Address: pdickey2@mccneb.edu Website: http://mockingbird.creighton.edu/NCW/dickey.htm Tonight: Return Editorial Essay Student Portfolios are Due Exercise 9-5.9-6, and 9-7 Discuss Chapter Nine Next Week: Final Class Essay Is Due Be sure you have read Chapter 10, pp. 284-302, 308-312 Exercise 9-14 Final Week: No Class, Take Home Final (as before) 1

Chapter Nine Deductive Arguments: Categorical Logic 2

Categorical Logic Categorical Logic is logic based on the relations of inclusion and exclusion among classes. That is, categorical logic is about things being in and out of groups and what it means to be in or out of one group by being in or out of another group. 3

Four Basic Kinds of Claims in Categorical Logic (Standard Forms) A: All _________ are _________. (Ex. All Presbyterians are Christians. E: No ________ are _________. (Ex. No Muslims are Christians. ___________________________________ I: Some ________ are _________. (Ex. Some Arabs are Christians. O: Some ________ are not _________. (Ex. Some Muslims are not Sunnis. 4

Four Basic Kinds of Claims in Categorical Logic What goes in the blanks are terms. In the first blank, the term is the subject. In the second blank goes the predicate term. A: All ____S_____ are ____P_____. (Ex. All Presbyterians are Christians. 5

Categorical Logic Venn Diagrams The Four Basic Kinds of Claims in Categorical Logic can be represented using Venn Diagrams See page 246 in textbook. . Venn Diagrams ( The two claims that include one class or part of a class within another are the affirmative claims (I.e. the A-claims & the I-Claims. The two claims that exclude one class or part of a class from another are the negative claims (I.e. the E-claims and the O-claims. 6

Translating Claims into Standard Form for Analysis The Bottom Line? Translating Claims into Standard Form for Analysis Two claims are equivalent claims if, and only if, they would be true in all and exactly the same circumstances. Equivalent claims, in this sense, say the same thing. Equivalent claims will have the same Venn Diagram. 7

Some Tips The word “only” used by itself, introduces the predicate term of an A-claim, e.g. “Only Matinees are half-price shows” is to be translated as “All half-price shows are matinees” The phrase “the only” introduces the subject term of an A-claim, e.g Matinees are the only half-price shows” also translates to “All half-price shows are matinees.” Claims about single individuals should be treated as A-claims or E-claims, e.g. “Aristotle is left-handed” translates to either “Everybody who is Aristotle is left handed” or “No person who is Aristotle is not left-handed.” 8

Three Categorical Operations Conversion – The converse of a claim is the claim with the subject and predicate switched, e.g. The converse of “No Norwegians are Swedes” is “No Swedes are Norwegians.” Obversion – The obverse of a claim is to switch the claim between affirmative and negative (A -> E, E -> A, I -> O, and O -> I and replace the predicate term with the complementary (or contradictory) term, e.g. The obverse of “All Presbyterians are Christians” is “No Presbyterians are non-Christians.” Contrapositive – The contrapositive of a claim is the cliam with the subject and predicate switched and replacing both terms with complementary terms (or contradictory terms), e.g. The contrapositive of “Some citizens are not voters” is “Some non-voters are not noncitiizens. 9

OK, So where is the beef? By understanding these concepts, you can apply the three rules of validity for deductive arguments: Conversion – The converses of all E- and I- claims, but not A- and O- claims are equivalent to the original claim. Obversion – The obverses of all four types of claims are equivalent to their original claims. Contrapositive – The contrapositives of all A- and O- claims, but not E- and I- claims are equivalent to the original claim. 10

Class Workshop: Exercise 9-4, 9-5, & 9-6 11

Categorical Logic Translate the following claims: Everybody who is ineligible for Physics 1A must take Physical Science 1. I = “Ineligible for Physics 1A” M = “Must take Physical Science 1.” All I are M 2) No students who are required to take Physical Sciences 1 are eligible for Physics 1A. No M are non-I 12

Are these different claims or the same claim? 1) All I are M 2) No M are non-I -- Obverse is: All M are I. -- Obverse is equivalent for all claims. Draw the Venn diagrams! Or alternately, consider: The contrapositive of 2) is: No I are Non-M. The obverse of 1) is: But although the obverse of an A-claim is equivalent, the contrapositive of an E-claim is not! 13

Categorical Syllogisms A syllogism is a deductive argument that has two premises -- and, of course, one conclusion (claim). A categorical syllogism is a syllogism in which: each of these three statements is a standard form, and there are three terms which occur twice, once each in two of the statements. 14

Three Terms of a Categorical Syllogism For example, the following is a categorical syllogism: (Premise 1) No Muppets are Patriots. (Premise 2) Some Muppets are super heroes (Conclusion) Some super heroes are not Patriots The three terms of a categorical syllogism are: 1) the major term (P) – the predicate term of the conclusion (e.g. Patriots). 2) the minor term (S) – the subject term of the conclusion (e.g. Super heroes) 3) the middle term (M) – the term that occurs in both premises but not in the conclusion (e.g. Muppets). 15

USING VENN DIAGRAMS TO TEST ARGUMENT VALIDITY Identify the classes referenced in the argument (if there are more than three, something is wrong). When identifying subject and predicate classes in the different claims, be on the watch for statements of “not” and for classes that are in common. Make sure that you don’t have separate classes for a term and it’s complement. 2. Assign letters to each classes as variables. 3. Given the passage containing the argument, rewrite the argument in standard form using the variables. M = “xxxx “ S = “ yyyy“ P = “ zzzz“ No M are P. Some M are S. ____________________ Therefore, Some S are not P.

Draw a Venn Diagram of three intersecting circles. Look at the conclusion of the argument and identify the subject and predicate classes. Therefore, Some S are not P. Label the left circle of the Venn diagram with the name of the subject class found in the conclusion. (10 A.M.) Label the right circle of the Venn diagram with the name of the predicate class found in the conclusion. Label the bottom circle of the Venn diagram with the middle term.

No M are P. Some M are S. Diagram each premise according the standard Venn diagrams for each standard type of categorical claim (A,E, I, and O). If the premises contain both universal (A & E-claims) and particular statements (I & O-claims), ALWAYS diagram the universal statement first (shading). When diagramming particular statements, be sure to put the X on the line between two areas when necessary. 10. Evaluate the Venn diagram to whether the drawing of the conclusion "Some S are not P" has already been drawn. If so, the argument is VALID. Otherwise it is INVALID.

Class Workshop: Exercise 9-13, #6 More from 9-13? 19

Power of Logic Exercises: http://www.poweroflogic.com/cgi/Venn/venn.cgi?exercise=6.3B ANOTHER GOOD SOURCE: http://www.philosophypages.com/lg/e08a.htm 20

Using the Rules Method To Test Validity Background – ***If a claim refers to all members of the class, the term is said to be distributed. Table of Distributed Terms: A-claim: All S are P E-claim: No S are P I-Claim: Some S are P O-Claim: Some S are not P The bold, italic, underlined term is distributed. Otherwise, the term is not distributed.

The Rules of the Syllogism A syllogism is valid if and only if all three of the following conditions are met: The number of negative claims in the premises and the conclusion must be the same. (Remember: these are the E- and the O- claims) At least one premise must distribute the middle term. Any term that is distributed in the conclusion must be distributed in its premises.

Class Workshop: Exercise 9-15, 9-16, & 9-17, 9-18 23