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6.1 Symbols and Translation

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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. Translating and Symbolizing Negation, Conjunction, Disjunction, Implication and Equivalence. Operator Name Logical Used to Translate Function ~ Tilde Negation Not, it is not the case that • Dot Conjunction And, also, moreover, ˅ Wedge Disjunction Or, unless ∩ Horsesho Implication If…then…, only e if… ≡ Triple Bar equivalence If and only if Example of negation ◦ It is not the case that A: ~A. Example of conjunction ◦ Tiffany sells jewelry and Gucci sells cologne: T• G 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. Examples of Biconditionals (equivalence): ◦ JFK tightens security if and only if O’Hare does: J ≡ O. Common Confusions: 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. 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. Computing the truth value of Longer Propositions: 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: Further Comparisons with Ordinary Language: ◦ She got married and had a baby: M• B. ◦ She had a baby and got married: B • M. Constructing Truth Tables: the relationship between the columns of numbers is L = 22 Number of different Number of lines in simple propositions truth table 1 2 2 4 3 8 4 16 5 32 6 64 An Alternate Method: ◦ [(a v b) • (b a)] b a b [(a v b) • (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 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. 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 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. 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. Testing Arguments for Validity 1. Assume the argument is invalid. 2. Work backward to derive the truth values of the separate components. 3. 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. Testing Statements for Consistency is similar to the method for testing arguments. 1. Assume the statements are consistent. 2. Can you derive a contradiction? 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 ◦ Pure Hypothetical Syllogism (HS): p q q r p r ◦ Modus Ponens (MP): p q p q ◦ Modus Tollens (MT): p q ~q ~p ◦ Affirming the Consequent (AC): p q q p ◦ Denying the Antecedent (DA): p q ~p ~q ◦ Constructive Dilemma (CD): (p q) • (r s) p v r q v s ◦ Destructive Dilemma (DD): (p q) • (r s) ~q v ~s ~p v ~r Refuting Constructive and Destructive Dilemmas. 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. .
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