Exam 2 Prep

Chapter 4 Categorical Propositions

4.1 Components of Categorical Propositions

Categorical Propositions

Standard Form

Subject term

Predicate term

Quantifiers ("all," "no," "some")

Copula ("are," "are not")

4.2 Quality, Quantity, and Distribution

Quality (affirmative, negative)

Quantity (universal, particular)

A, E, I, O

Distribution of terms

Proposition	Letter Name	Quantity	Quality	Terms Distributed
All S are P.	A	universal	affirmative	S
No S are P.	E	universal	negative	S and P
Some S are P.	I	particular	affirmative	none
Some S are not P.	O	particular	negative	P

4.3 Venn Diagrams and the Modern Square of Opposition

Aristotelian and Boolean standpoints

Venn diagrams (shading to indicate empty area, "x" to indicate at least one)

Modern square of opposition

Contradictory relations (A-O, and E-I)

Logically undetermined relations (all others)

Testing immediate inferences (Venn diagrams or the modern square)

Unconditional validity

How to say that a categorical proposition is false

Existential fallacy

4.4 Conversion, Obversion, Contraposition

Conversion consists in switching the subject and predicate terms

E and I propositions are logically equivalent to their converses

(To be logically equivalent is to necessarily have the same truth value)

Illicit conversion (formal fallacy) - arguing from an A or O proposition to its converse

Obversion consists in changing the quality of a proposition and then replacing the predicate term with its term complement

Class complement

Term complement

Scope of discourse (a.k.a. universe of discourse or u.d.)

Each proposition is logically equivalent to its obverse

Contraposition consists in conversion of a proposition followed by replacing the subject and predicate terms with their term complements

A and O propositions are logically equivalent to their contrapositives

Illicit contraposition (formal fallacy) - arguing from an E or I proposition to its contrapositive

4.5 Traditional Square of Opposition

Aristotelian standpoint

Contrary (not both true) - A/E

Subcontrary (not both false) - I/O

Subalternation (truth flows down, A/I and E/O; falsity flows up, I/A and O/E)

Formal fallacies: illicit contrary, illicit subcontrary, illicit subalternation

Existential fallacy: using contrary, subcontrary, or subalternation (otherwise correctly) with propositions about things that do not exist

Conditional validity and unconditional validity

4.6 Venn Diagrams and the Traditional Standpoint

Using the circled X in A and E propositions

3 step procedure for testing immediate inferences from the Aristotelian standpoint:

1. Reduce the inference to its form and test from the Boolean standpoint. If valid, stop - it is unconditionally valid.

2. If invalid from the Boolean standpoint, adopt the Aristotelian standpoint and employ the circled X if applicable and retest.

3. If conditionally valid, determine whether the circled X represents something that exists. If so, the inference is valid from the Aristotelian standpoint. If not, the inference is invalid and it commits the existential fallacy from the Aristotelian standpoint.

4.7 Translation

See many examples and hints in the book, especially the box on p. 233

Chapter 5 Categorical Syllogisms

5.1 Standard Form, Mood, Figure

Categorical Syllogisms

Major Term (predicate of the conclusion)

Minor Term (subject of the conclusion)

Middle Term (occurs in each premise but not the conclusion)

Standard Form defined (4 clauses, p. 237)

Mood (EIO, AAI, etc.)

Figure (1, 2, 3, 4)

Form (AAA-1, OAE-4, etc.)

Tables and poem for validity

Unconditional and conditional validity

5.2 Venn Diagrams

7 guidelines (pp. 245-6)

3 more steps for the Aristotelian standpoint (p. 251)

5.3 Rules and Fallacies

These are 5 rules that a categorical syllogism must comply with, in order to be valid

If a categorical syllogism breaks none of the rules, it is unconditionally valid

If it breaks one of the first four rules, it is invalid

If it breaks rule 5 only, it is invalid from the Boolean standpoint, and conditionally valid from the Aristotelian standpoint, conditional upon the non-empty extension of a critical term

Rule 1: The middle term must be distributed

Fallacy: Undistributed middle

Rule 2: If a term is distributed in the conclusion, it must be distributed in a premise

Fallacy: Illicit major, or illicit minor, as the case may be

Rule 3: Two negative premises are not allowed

Fallacy: Exclusive premises

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, as the case may be

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

Fallacy: Existential fallacy

5.4 Reducing the number of terms

Conversion, obversion, and contraposition may be used to reduce the number of terms to three

5.5 Ordinary Language Arguments

This section is a rehash of Section 4.7

5.6 Enthymemes

An enthymeme is an argument expressible as a categorical syllogism, but with one of the statements (premise or conclusion) missing

The reader must determine what is missing (premise or conclusion), and then supply the missing statement with the aim of converting the enthymeme into a good argument (p. 270)

Any enthymeme that contains an indicator word is missing a premise

5.7 Sorites

A sorites is a chain of categorical syllogisms, missing the intermediate conclusions.

Standard form for sorites

3-step evaluation procedure

Chapter 6 Propositional Logic

6.1 Symbols and Translation

Simple and compound statements

Simple statements symbolized by capital letters

Logical operators or connectives (there are 5 of them)

Tilde (negation)

Dot (conjunction)

Wedge (disjunction)

Horseshoe (material implication)

Triple Bar (material equivalence)

Conditional and biconditional statements

Horseshoe and "only if"

DeMorgan's Rule (p. 286)

Note the boxes on pp. 286-7 for translation

WFFs

6.2 Truth Functions

Statement variables (e.g. "p," "q," etc.)

Truth tables and the definitions of logical operators

Computing truth values for propositions

6.3 Truth Tables for Propositions

L=2ⁿ

Classification of Statements (tautologous, self-contradictory, contingent)

Comparing Statements (logically equivalent, contradictory, consistent, inconsistent)

6.4 Truth Tables for Arguments

Validity and Invalidity

Corresponding Conditional

6.5 Indirect Truth Tables

6.6 Argument Forms and Fallacies

Be able to recognize the valid forms DS, HS, MP, and MT (see p. 327)

Be able to recognize the invalid/fallacious forms AC and DA

Premises may be switched (p. 328)

p is logically equivalent to ~~p (Double Negation, p. 329)

p v q is logically equivalent to q v p (Commutativity, p. 329)