Gen-Math-Valid-Arguments-and-Fallacies.pptx

valid arguments and fallacies

1. Valid Arguments and Fallacies General Mathematics Program

2. Objectives Define argument Define valid argument Define fallacy Define sound argument

3. Definition An argument is a compound proposition of the form (p 1 ∧ p 2 ∧ . . . ∧ p n )  q. The propositions p 1 , p 2 , . . . , p n are the premises of the argument, and q is the conclusion.

4. Definition Arguments can be written in propositional form, as above, or in column or standard form: p 1 p 2 ⁝ p n  q

5. Example 1 Write the following argument in propositional form and in standard form. If there is limited freshwater supply, then we should conserve water. There is limited freshwater supply. Therefore, we should conserve water.

6. Example 1 The premises are: p 1 : If there is limited freshwater supply, then we should conserve water. p 2 : There is limited freshwater supply. The conclusion is: q: We should conserve water. Symbol form: (p 1 ∧ p 2 )  q

7. Example 2 Standard form: p 1 p 2  q Take note that two arguments may have equivalent logical forms , even if they are different in content .

8. Example 3 p  q Consider the following: A p  q If my alarm sounds, then I will wake up p  q q  p My alarm sounded. Therefore, I woke up. B If my alarm sounds, then I will wake up I woke up. Therefore, my alarm sounded.

9. Validity Condition Is it logically impossible for the premises to be true and the conclusion false? If the answer is YES, we say that the argument satisfies the validity condition. The argument is valid.

10. Example 3 p  q p  q A If my alarm sounds, then I will wake up My alarm sounded. Therefore, I woke up. p q p  q T T T T F F F T T F F T Is it logically impossible for the premises to be true and the conclusion false?

11. Example 3 A If my alarm sounds, then I will wake up My alarm sounded. Therefore, I woke up. p q p  q T T T T F F F T T F F T Suppose the premises are both true, is it impossible for the conclusion to be false? p  q p  q There is only one possibility that both premises are true and it would show that the conclusion is also true.

12. Example 3 B If my alarm sounds, then I will wake up I woke up. Therefore, my alarm sounded. p q p  q T T T T F F F T T F F T Suppose the premises are both true, Is it impossible for the conclusion to be false? p  q q  p Yes, there is a possibility of it being false.

13. Example 3 We can now say that A satisfies the validity condition while B fails it. B failed it because it doesn’t necessarily mean that if I woke up, my alarm sounded. Maybe I woke up because my newborn child is crying. 

14. Valid Argument A valid argument satisfies the validity condition; that is, the conclusion q is true whenever the premises p 1 , p 2 ,..., p n are all true. Put another way, for a valid argument, the conditional (p 1 ∧ p 2 ∧ . . . ∧ p n )  q is a tautology.

15. Example 4 Prove that the argument ((p  q) ∧ p)  q is valid. Since ((p  q) ∧ p)  q is a tautology, then the argument is valid. This argument is called, Modus Ponens (rule of Detachment). p q p  q (p  q) ∧ p ((p  q) ∧ p)  q T T T T T T F F F T F T T F T F F T F T

16. Example 5 p Consider the following: A p  q If my alarm sounds, then I will wake up p  q My alarm sounded.  q Therefore, I woke up. B p  q If there is limited freshwater supply, then we should conserve water. There is limited freshwater supply. Therefore, we should conserve water.

17. Example 5 Notice that all three arguments are in the form ((p  q) ∧ p)  q Hence, by Modus Ponens, arguments A, B, C are valid arguments.  p  q Consider the following: C p  q If General Antonio Luna is a national hero, then he died at the hands of the Americans in 1899. General Antonio Luna is a national hero. Therefore, General Luna died at the hands of the Americans in 1899.

18. However, this does not mean that the conclusions are true. Asserting that the argument is valid simply means that the conclusion logically follows from the premises. These examples illustrate that the validity of an argument does not depend on the content of the argument, but on its form.

19. Example 6 If Antonio and Jose are friends, then they are Facebook friends. Antonio and Jose are not Facebook friends. Therefore, they are not friends. The argument is in the form ((p  q) ∧ ~q)  ~p p  q ~ q  ~ p Hence, by Modus Tollens, the argument is valid.

20. Rules of Inference Let p, q, and r be propositions.

21. Example 6 Determine whether the argument is valid If Antonio and Jose are friends, then they are Facebook friends. Antonio and Jose are not Facebook friends. Therefore, they are not friends. p: Antonio and Jose are friends q: Antonio and Jose are Facebook friends

22. Example 6 If Pan eats too much, then he will gain weight. If he gains weight, then he will work out in the gym. Therefore, if Pan eats too much, then he will work out in the gym. p: Pan eats too much. q: Pan will gain weight. r: Pan will work out in the gym. ((p  q) ∧ (q  r))  (p  r) By Law of Syllogism , the argument is valid. p  q q  r  p  r

23. Example 7 Antonio Luna and Jose Rizal like Nelly Boustead. Therefore, Antonio Luna likes Nelly Boustead. p: Antonio Luna likes Nelly Boustead q: Jose Rizal likes Nelly Boustead p ∧ q  p p  q  p

24. Example 7 If you study hard, you refine your communication skills and build up your confidence. If you refine your communication skills and build up your confidence, then your job opportunities increase. Hence, if you study hard, your job opportunities increase.

25. Fallacy An argument (p 1 ∧ p 2 ∧ . . . ∧ p n )  q which is not valid is called a fallacy . In a fallacy, it is possible for the premises p 1 , p 2 ,..., p n to be true but the conclusion q is false. Put another way, the conditional (p 1 ∧ p 2 ∧ . . . ∧ p n )  q Is NOT a tautology.

26. Example 8 Prove that the argument ((p  q) ∧ q)  p is a fallacy. We see that in the 3 rd row, the argument showed an F . This argument is called, Fallacy of the Converse. p q p  q (p  q) ∧ q ((p  q) ∧ q)  p T T T T T T F F F T F T T T F F F T F T

27. Example 9 Show that the following arguments are fallacies. A’: -If my alarm sounds, then I will wake up. -I woke up. -Therefore, my alarm sounded. B’: -If there is a limited supply of freshwater, then I will conserve water. -I will conserve water. -Therefore, there is limited supply of freshwater.

28. Table of Fallacies Let p, q, and r be propositions.

29. Table of Fallacies Let p, q, and r be propositions.

30. Example 10 Determine whether the argument is valid Given: p: Alvin sings with Nina. q: Alvin dances with Nina.

31. Example 10 Either Alvin sings or dances with Nina. Alvin sang with Nina. Therefore, Alvin did not dance with Nina. The argument is in the form ((p ∨ q) ∧ p)  ~q p  q p  ~ q Hence, by Affirming the Disjunct, the argument is a fallacy.

32. Example 10 The argument is in the form (p  q)  (q  p) Hence, by Fallacy of the Consequent, the argument is a fallacy. If Alvin sings with Nina, then Alvin dances with her, too. Therefore, if Alvin dances with Nina, he sings with her, too. p  q p  ~ q

33. Example 11 Antonio Luna is a scientist. Therefore, either Antonio Luna or Jose Rizal is a scientist. p: Antonio Luna is a scientist. q: Jose Rizal is a scientist. p  (p ∨ q) p  p  q