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Discrete Structures Chapter 2: The Logic of Compound Statements 2.3 Valid and Invalid Arguments “Contrawise,” continued Tweedledee, “if it was so, it might be; and if it were so, it would be; but as it isn’t, it ain’t. That’s logic.” – Lewis Carroll, 1832 – 1898 Through the Looking Glass, 1865 2.3 Valid and Invalid Arguments 1 Definitions • Argument – An argument is a sequence of statements. • Argument Form – An argument form is a sequence of statement forms. • Premises – All statements in an argument and all statement forms in an argument form are called premises except for the last one. 2.3 Valid and Invalid Arguments 2 Definitions • Conclusion – The final statement or statement form is called the conclusion. • Valid – If an argument form is valid that means no matter what particular statements are substituted for the statement variable in its premises, if the resulting premises are all true, then the conclusion is true. 2.3 Valid and Invalid Arguments 3 Testing an Argument for Validity 1. Identify the premises and conclusion of the argument form. 2. Construct a truth table showing the truth values of all the premises and the conclusion. 3. A row of the truth table in which all the premises are true is called a critical row. a. If there is a critical row in which the conclusion is false, the argument form is invalid. b. If the conclusion in every row is true, then the argument form is valid. 2.3 Valid and Invalid Arguments 4 Example – pg. 62 # 12b • Use truth tables to show that the following forms of arguments are invalid. pq p Premises Conclusion q p q 1. T T 2. T F 3. F T 4. F F pq p 2.3 Valid and Invalid Arguments q 5 Valid Argument Forms The table below summarizes the rules of inference. You are expected to read each description in your book. Name Example Name Example Modus Ponens (mode that affirms) pq p q Elimination a. Modus Tollens (mode that denies) pq q p Transitivity pq qr pr Generalization a. Proof by Division into Cases pq pr qr r p pq b. q p q Specialization Conjunction a. q pq p p q p q b. p pq q p b. pq q q q Contradictio n Rule 2.3 Valid and Invalid Arguments p c p 6 Examples • For the next three examples, use truth tables to show that the argument forms referred to are valid. Indicate which column represents the premises and which represent the conclusion, and include a sentence explaining how the truth table supports your answer. Your explanation should show that you understand what it means for a form of an argument to be valid. 2.3 Valid and Invalid Arguments 7 Example – pg. 62 # 17 pq q 1. 2. 3. 4. p q T T T F F F T F Premises Conclusion pq q 2.3 Valid and Invalid Arguments 8 Example – pg. 62 # 19 pq p q Premises 1. 2. 3. 4. p q T T F F T F T F pq Conclusion p 2.3 Valid and Invalid Arguments q 9 Example – pg. 62 # 21 pq pr qr r 2.3 Valid and Invalid Arguments 10 Example Continued – pg. 62 # 21 Premises p q r 1. T T T 2. T T T T F F F T F 7. F F F T T F T F T 8. F F F 3. 4. 5. 6. pq pr 2.3 Valid and Invalid Arguments Conclusion qr r 11 Fallacies – Converse Error This claim is most simply put as • pq q p • It's a fallacy because at no point is it shown that p is the only possible cause of q; therefore, even if q is true, p can still be false. 2.3 Valid and Invalid Arguments 12 Fallacies – Converse Error Example: – If my car was Ferrari, it would be able to travel at over a hundred miles per hour. – I clocked my car at 101 miles per hour. – my car is a Ferrari. Here the fallacy is fairly obvious; given the evidence, the car might be a Ferrari, but it might also be a Bugatti, Lamborghini, or any other model of performance car, since the ability to travel that fast is not unique to Ferraris. 2.3 Valid and Invalid Arguments 13 Fallacies – Inverse Error This claim is most simply put as • pq p q • It's a fallacy because replacement is not allowed because a conditional statement is not logically equivalent to its inverse. 2.3 Valid and Invalid Arguments 14 Fallacies – Inverse Error Example – If I hit my professor with a cream pie, he will flunk me. – I will not hit my professor with a cream pie. – he will not flunk me. • Again, it is intuitively obvious that this reasoning does not work. While many professors may consider being nailed with a cream pie a sufficient reason to assign a grade of "F" to a student, there are an overwhelming number of other reasons for which you might flunk (cheating, not studying, not showing up for tests, etc.). 2.3 Valid and Invalid Arguments 15 Example – pg. 62 # 30 • The argument might be valid or it might exhibit the converse or inverse error. Use symbols to write the logical form of each argument. If the argument is valid, identify the rule of inference that guarantees its validity. Otherwise, state whether the converse or inverse error is made. – If this computer program is correct, then it produces the correct output when run with the test data my teacher gave me. – This computer program produces the correct output when run with the test data my teacher gave me. – This computer program is correct. 2.3 Valid and Invalid Arguments 16