6.1 Symbols and Translation Simple Statements: James Joyce wrote Ulysses. Compound Statements: Contain at least one simple statement. It is not the case that Al Qaeda is a humanitarian organization translated to: It is not the case that A.

6.1 Continued Translating and Symbolizing Negation, Conjunction, Disjunction, Implication and Equivalence. Operator Name Logical Function Used to Translate ~ Tilde Negation Not, it is not the case that • Dot Conjunction And, also, moreover, ˅ Wedge Disjunction Or, unless ∩ Horseshoe Implication If…then…, only if… ≡ Triple Bar equivalence If and only if

6.1 Continued Example of negation Example of conjunction It is not the case that A: ~A. Example of conjunction Tiffany sells jewelry and Gucci sells cologne: T• G

6.1 Continued Example of disjunction: Aspen sells snowboards or Teluride does: A ˅ T. Examples of conditionals (implication): If Purdue raises tuition, then so does Notre Dame: P  N.

6.1 Continued Examples of Biconditionals (equivalence): JFK tightens security if and only if O’Hare does: J ≡ O. Common Confusions:

6.2 Truth Functions A Truth Function is any compound proposition whose truth value is completely determined by the truth values of its components. The definitions of the Logical Operators are presented in the forms of Statement Variables, which are lowercase letters that can stand for any statement.

6.2 Continued A Truth Table is an arrangement of truth values that shows in every possible case how the truth value of a compound proposition is determined by the value of its simple components.

6.2 Continued Computing the truth value of Longer Propositions: F If A, B, and C are true and D, E and F are false, then : (A v D)  E with these truth values below: (T v F)  F. Next: T  F. Finally: F

6.2 Continued Further Comparisons with Ordinary Language: She got married and had a baby: M• B. She had a baby and got married: B • M.

6.3 Truth Tables for Propositions Constructing Truth Tables: the relationship between the columns of numbers is L = 22 Number of different simple propositions Number of lines in truth table 1 2 4 3 8 16 5 32 6 64

6.3 Continued An Alternate Method: [(a v b) • (b  a)]  b a b T T T T T T T F T T T F F T T F F T F F F F T T

6.3 Continued Classifying Statements: A Tautologous or Logically True Statement is true regardless of the truth values of its components. A Self-contradictory or Logically False Statement is false regardless of the truth values of its components. A Contingent Statement’s truth value varies according to the truth values of its components.

6.3 Continued Comparing Statements: Logically Equivalent statements have the same truth value on each line under their main operators. Contradictory Statements have the opposite truth value on each line under their main operators. Consistent Statements have at least one line on which both (or all) the truth values are true. Inconsistent Statements have no line on which both (or all) the truth values are true. Compare main operator columns; logically equivalent and contradictory statements take precedence over consistent and inconsistent ones

6.4 Truth Tables for Arguments Constructing a Truth Table for an Argument: Symbolize the argument using letters for simple propositions. Write out the argument using a single slash between premises and a double slash between the last premises and the conclusion. Draw a truth table for the symbolized argument as if it were a proposition broken into parts. Look for a line where all the premises are true and the conclusion false. If there is no such line, the argument is valid. If there is not, it is invalid.

6.5 Indirect Truth Tables They are shorter and faster than ordinary truth tables and work best for arguments with a large number of simple propositions. However, you must be able to work backward from the truth value of the main operator of a compound statement to the truth values of its simple components.

6.5 Continued Testing Arguments for Validity Assume the argument is invalid. Work backward to derive the truth values of the separate components. Can you derive a contradiction? If you can, it is not possible for the premises to be true and the conclusion false, so the argument is valid.

6.5 Continued Testing Statements for Consistency is similar to the method for testing arguments. Assume the statements are consistent. Can you derive a contradiction?

6.6 Argument Forms and Fallacies An Argument Form is an arrangement of statement variables and operators so that uniformly substituting statements in place of variables results in arguments. Common forms are as follows: Disjunctive Syllogism (DS) p v q ~p q

6.6 Continued Pure Hypothetical Syllogism (HS): Modus Ponens (MP): p  q q  r p  r Modus Ponens (MP): p q

6.6 Continued Modus Tollens (MT): Affirming the Consequent (AC): p  q

6.6 Continued Denying the Antecedent (DA): Constructive Dilemma (CD): p  q ~p ~q Constructive Dilemma (CD): (p  q) • (r  s) p v r q v s Destructive Dilemma (DD): ~q v ~s ~p v ~r

6.6 Continued Refuting Constructive and Destructive Dilemmas.

6.6 Continued Note on Invalid Forms: any substitution instance of a valid argument form is a valid argument. However, this result does not apply to invalid forms.