Common Invalid Argument Forms aka Fallacies

1. Common Invalid Argument Forms (aka Fallacies) Affirming the consequent p?q q Therefore, p Here is a substitution instance that proves the argument form invalid: If Albert Pujols plays for the Cubs, then Albert Pujols plays in the National League Albert Pujols plays in the National League Therefore, Albert Pujols play for the Cubs

2. Continued Denying the antecedent p?q ~p Therefore, ~q Here is a substitution instance that proves the argument form invalid: If Tony Blair is eligible to vote in the US, then Tony Blair is 35 or older Tony Blair is not eligible to vote in the US Therefore, Tony Blair is not 35 or older

3. A No Name Fallacy Consider this argument: A v B A Therefore, ~B This is invalid because both A and B can be true when A v B is true (disjunction is inclusive), so we cannot validly infer that B is false (~B) from A and A v B

4. What About Exclusive Or? And yet, the following bit of reasoning seems good, that is, it looks like the conclusion follows from the premises: Either John goes to Illinois or he goes to Michigan John goes to Illinois Therefore, he doesn�t go to Michigan

5. Diagnosis The problem is that in order to represent exclusive or, we need to add an another premise to the argument I v M I ~(I * M) Therefore, ~M A truth table will show that this argument IS valid; in chapter 7 we will learn another way to prove its validity as well

6. Common Valid Argument Forms Truth tables are a simple and reliable way to test the validity of arguments in propositional logic However, they can be time consuming to construct We can determine the validity of many arguments simply by inspecting the form of the arguments

7. Continued Chapter 6 introduces us to six common argument forms. These forms, along with others, are discussed in chapter 7 as well Modus ponens: p?q p q

8. More Modus tollens: p?q ~q ~p Disjunctive syllogism: p v q ~p q

9. Ever More Hypothetical syllogism: p?q q?r p?r

10. And Finally Constructive dilemma: (p?q) * (r?s) p v r q v s Destructive dilemma: (p?q) * (r?s) ~q v ~s ~p v ~r

11. Note The order of premises does not matter to whether an argument has a certain form p p?q q is still modus ponens

12. More Notes Consider this argument: H v M ~M H Technically, this is not disjunctive syllogism, because in a disjunctive syllogism it is the left disjunct that is denied

13. However� The rule of commutativity allows us to switch the order of disjuncts at any time. In other words, p v q is logically equivalent to q v p So, we can replace H v M with M v H and get M v H ~M H Now we have a proper disjunctive syllogism

14. Double Negation Consider this argument: J?~F F ~J Technically, this is not modus tollens, since there is no tilde in front of the F on the second line But, the rule of double negation allows us to replace any statement of the form p with ~~p, and vice versa; they are logically equivalent

15. So We get J?~F ~~F ~J Now we have an argument with the form of modus tollens

16. Do the Following Arguments Fit one of the 6 forms? ~H?~F ~G v ~L ~H ~~G ~F ~L (M?C) * (W?B) R?D M v W D?J B v C ~~R?~~J

17. Refuting Constructive and Destructive Dilemmas The text mentions that skilled debaters sometimes give what turn out to be bad arguments by using constructive or destructive dilemma. Since these argument forms are valid, the only way to refute such arguments is to show that they are unsound

18. False Dilemmas The method of refutation we will discuss here involves showing that the disjunctive premise in a constructive or destructive dilemma is false

19. Ever Heard This Before? �America�love it or leave it� What do people typically mean when they utter this phrase?

20. Continued We can more or less capture what this says in a disjunction: Either you should agree with everything the American government does or you should leave America. But is this true? Are these the only options?

21. Continued We might say that the disjunction offers us a false dilemma: we do not need to accept the truth of either disjunct�maybe they are both false! If both disjuncts are false, the whole disjunction is false, and so any argument that contains it will be unsound Note that this does not apply to disjunctions of the form p v ~p, since all such disjunctions are tautologies.

22. Natural Deduction We can often prove that an argument is valid by deriving the conclusion from a set of premises using rules of inference Rules of inference state under what conditions you may validly infer a new claim, given a prior set of claims

23. For Example Modus ponens is a rule of inference It says, in effect, that if you begin with a statement with the form p?q and a statement with the form p, you may infer q. So, suppose you are given: 1. J?(M v B) 2. J

24. Continued By the rule modus ponens you can derive M v B on a new line 1. J?(M v B) 2. J 3. M v B (1, 2 MP) Notice that we have put in parentheses both the rule that permits us to derive M v B and the lines we used to derive it.

25. The Rules Can Be Used Together to Derive a New Claim B v (S?M) ~B S Can we derive M from these three claims? Yes! 4. S?M (1, 2 DS) 5. M (3, 4 MP) We derived line 4, and then used this line to derive 5. Because each step conforms to a valid rule of inference, the entire deduction is valid

26. Let�s Try Another How might we derive H v C from (M?H) * (J?C) ~(M v J)?D ~D ?

27. Like This (M?H) * (J?C) ~(M v J)?D ~D ~~(M v J) (2, 3 MT) M v J (4, DN) H v C (1, 5 CD) Note: DN is a rule of replacement, whereas the other rules are rules of implication�we will talk about the difference a bit later

28. One More Derive ~C from H?(C?(R * G)) B v H ~(R * G) ~B

29. Like This H?(C?(R * G)) B v H ~(R * G) ~B H (2, 4 DS) C?(R * G) (1, 5 MP) ~C (3, 6 MT)

30. A Note on the Rules of Inference The rules of inference do not tell you what to infer; they merely tell you what you can infer For example: H?(B v C) ~(B v C) The rule modus tollens tells us we can infer ~H from lines 1 and 2. But we do not have to infer ~H, and in many derivations it may not make sense to infer it.

31. For Example H?(B v C) ~(B v C) D v K ~D Suppose we are asked to derive K from these four lines. There would be no point in using MT to derive ~H; we could simply derive K from lines 3 and 4 using disjunctive syllogism

32. So How Do I Know Which Inferences to Make? Derivations can be long, requiring many steps to arrive at the desired result The book offers several strategies for applying the rules to derive the desired statement (we will talk about these) This takes a lot of practice and familiarity with the rules If all else fails, just start making permissible inferences; this might enable you to see better what to do next

33. For Example Suppose you are asked to derive ~(H * G)?D from F C v ((R?S)?D) ~C F?(~(H * G)?(R?S))

34. What to Do? Notice that the desired conclusion is a conditional claim: ~(H * G)?D When this happens, consider whether you can use hypothetical syllogism to derive it

35. Ah Ha! F C v ((R?S)?D) ~C F?(~(H * G)?(R?S)) Look at the right disjunct of line 2 and the consequent of line 4. If we could isolate those statements, then we could apply HS to get our desired conclusion. Can we isolate them?

36. You Bet! F C v ((R?S)?D) ~C F?(~(H * G)?(R?S)) Because we have ~C, we can derive 5. (R?S)?D (2, 3 DS) And because we have F, we can derive ~(H * G)?(R?S) (1, 4 MP) Now we are home free: 7. ~(H * G)?D (5, 6 HS)