Arguments Definition of a Symbolic Argument Validity of Arguments • An argument is an implication (conditional) • Arguments are either valid or invalid. whose “if” component consists of the • Ways of determining the validity of an Arguments conjunction of several propositions called premises (hypotheses) and whose “then” argument. component is a proposition called the 1. Truth table Determining Validity conclusion. 2. Analysis • Symbolic notation of an argument: 3. Syllogism [P1 P2 P3] → C Chapter 3 – Section 5 In-class Assignment 13 - 1 No in-class assignment problem Validity of an Argument by A Truth Example The Truth Table Table • 2 is an even number. • P1: p • Write each premise and the conclusion in • If 7 is odd then 2 is not • P2: (q → ~p) symbolic form. even. • Enclose each premise and the conclusion that is • If 7 is not odd then 8 is • P3: (~q → r) compound within parentheses. even. • _____________ • Write the conjunction of the several premises. • ________________ • C : r • Enclose the conjunction in braces. • 8 is even. • Write the conditional of the conjunction and the conclusion. Symbolic Argument: [p (q → ~p) (~q → r)] → r • Make a truth table for this symbolic implication. • Only if it is a tautology is the argument valid. The argument is invalid because the implication is not a tautology In-class Assignment 13 - 2 In-class Assignment 13 - 2 In-class Assignment 13 - 3 1 Arguments Validity by Analysis - Reminder Validity by Analysis Syllogisms • A conditional is only false if the “if” • Write the argument in From left to right component is true and the “then” symbolic form. • Identify the premises. component is false. 1. Law of Detachment • Force all premises to be 2. Fallacy of Inverse • A conjunction is true only if all true by using the rules of 3. Disjunction Law 1 logic. 1 components are true. In an argument the 4. Fallacy of Converse • If it is possible for the conjunction is the “if” component. conclusion to be false 5. Chain Rule • Therefore, all premises must be true. then the argument is 6. Law of Contraposition invalid. No in-class assignment problem. In-class assignment 13 - 4 Syllogisms Lead to Valid Recognizing Syllogisms Conclusions 1. If the sun shines then it will not rain. 1. Law of It is raining. • Mary did not go to • A form of p → q Contraposition Therefore, the sun is not shining. school or Tom went p 2. All puppies are cute. Felix is not a puppy. 2. Fallacy of to the store. Therefore, Felix is not cute. converse • Mary went to school. • Use the Law of 3. Sam is wet if he went swimming. Detachment to write the Sam is wet. 3. Law of conclusion. Therefore, Sam went swimming. syllogism (chain rule) 4. If you study hard you get a good grade. If you get a good grade then you can transfer. • Tom went to the Therefore, if you study hard you can transfer. 4. Fallacy of store. inverse In-class Assignment 13 – 5 In-class Assignment 13 - 6 2.