Today’s Topics

1. Today’s Topics • Thinking about proofs • Strategies and hints • Conditional and Indirect Proof (Reductio ad Absurdum) • Solving puzzles with IP • Common errors to avoid • Notes on symbolization and proof construction

2. Thinking About Proofs • Proofs in logic work just like proofs in geometry • The 18 rules we have allow us to manipulate a basic set of assumptions (the premises) so as to show that the conclusion is a logical consequence of them. • A proof is a set of instructions on how to get from the premises to the conclusion.

3. Constructing a proof is like giving instructions. The question is “How do I get there (the conclusion) from here (the premises)?” • The rules are the allowable moves or turns you can take. • Proceed stepwise. Suppose you want to get to D from A, B, and C. Well, if from A and B you can get to E, and from E and C you can get to D, you have your instructions. • That is all there is to constructing proofs

4. Basic Strategic Hints • Argument forms are patterns. Learn the patterns and look for them. • Inference rules can be grouped according to the types of statements on which they operate. • Short statements are your friends! • Work backwards from the conclusion. • BE FLEXIBLE. When stuck, experiment. Try steps and then search for familiar patterns.

5. Develop Goal Lines and work toward them. • Ask yourself, “What line, if I had it, would allow me to get to the conclusion?” • Make that line a goal and work towards it. • Think in terms of equivalences—ask yourself “To what is the conclusion (or the line you want) equivalent?” Can you get to that Version?

6. If you need one disjunct of a disjunction, scan the remaining lines for the negation of the other disjunct and use DS • If you need the consequent of a conditional, look for the antecedent and use MP • If you need the negation of the antecedent of a conditional, think MT • If there is a statement letter in the conclusion that occurs nowhere in the premises, use addition

7. A Few More Strategic Hints • Simplify conjucntions • Use DeMorgan to turn negations of disjunctions into conjunctions that can be simplified • Use commutation and association to isolate components that fit other patterns (DS or Simp) • To derive a conditional, think HS or MI • To derive a disjunction, think ADD or CD

8. Here’s How It Works • Consider the argument A  (B ▼ C), A, ~B  C v E • The first thing to notice is that since ‘E’ does not occur in the premises, you will have to use Addition on ‘C’ to get the conclusion. So, how to get ‘C’? • Since you have ‘~B’ if you could get the consequent of ‘A  (B ▼ C)’ then you could use DS to get ‘C’.

9. But since you have ‘A’, the antecedent of the conditional, you can get ‘B v C’. • A  (B v C) pr deduce C v E • A pr • ~B pr • (B v C) 1,2 MP • C 3,4 DS • C v E 5 addition • And that’s the proof!

10. Try a Few • Download the Handout on Constructing Proofs and work a few problems. Discuss your answers on the bulletin board. • Remember, there are ALWAYS several correct ways to construct a proof. That you see one path, and I see another says more about us psychologically than it says about us logically. Constructing proofs is NOT a mechanical activity, it requires creativity and artistry. • OK, now move on and try the Constructing Proofs (difficult) Handout.

11. Consider the following argument: • A  B  A  (A  B) • Some systems include us the rule absorption that enables us to construct a proof for this argument, but many systems of logic do not include that rule. • If there is valid argument for which one cannot construct a proof, one’s system of logic is incomplete. • Such systems need additional rules or methods to guarantee that a proof can be constructed for any valid argument. • Conditional Proof and Reductio Ad Absurdum are 2 such methods

12. Conditional Proof and Reductio Ad Absurdum • Download the Handout Conditional Proof Study Guide and read it carefully. This is a deceptively simple method, but it takes time to master. • In the history of the West, Conditional and Indirect proof are of major importance. Indirect Proof lead directly to the development of non-Euclidean geometry

13. Conditional Proof • Conditional Proof allows you to construct ANY conditional. • First, you are allowed to make new ASSUMPTIONS • You can assume anything, any time • HOWEVER, you must discharge your assumptions before your proof is complete

14. To use Conditional Proof, begin by assuming the antecedent of the conditional you want. • Then, using our 18 rules of inference and equivalence, derive a line identical to the consequent of the conditional you want. • Now, discharge the assumption by deriving a conditional whose antecedent was your assumption and whose consequent is the preceding line

15. Justify the new conditional as following from a series of lines (e.g. 2-5) and the rule CP. • The scope of the assumption you made is marked by a vertical line beginning at the assumption and ending with a horizontal line directly above the conditional you derive.

16. Here’s How It Works: 1. A  B  A  (A  B) 2. A AP 3. B 1,2 MP 4. A  B 2,3 Conj 5. A  (A  B) 2-4 CP

17. Lines 2-4 serve to justify line 5, but they cannot be used in any subsequent line of the proof, they are closed off from the rest of the proof, but you are free to use line 5 as you need it.

18. When using conditional proof, all you are doing is showing that IF a particular claim is true (the assumption) then another claim follows from it. But that is all that the derived conditional says.

19. Points to remember: • CP can only be used to justify a conditional • The antecedent of that conditional MUST be the assumption you made • The consequent of that conditional MUST be the line immediately preceding the discharge of the assumption • You can make multiple assumptions and nest them, but the assumption made last must be discharged first

20. Conditional Proof greatly simplifies the task of deriving many conditionals. 1. p  (q  r) pr prove p  q 2. ~p v (q  r) 1 MI 3. (~p v q)  (~p v r) 2 Dist 4. ~p v q 3 Simp 5. p  q 4 MI

21. Here’s the CP version: 1. p  (q  r) Premise >2. p Assumption (AP) 3. q  r 1,2 MP 4. q 3 Simp 5. p  q 2-4 CP

22. Yes, the CP takes the same number of steps, but you don’t need distribution and 2 steps of implication.

23. Reductio Ad Absurdum (RAA) • Recall that we can show an argument valid by showing that the negation of the conclusion is inconsistent with the truth of the premises. • This method of argument called Reductio ad Absurdum or Indirect Proof (IP) formalizes this insight • Indirect Proof is one of the most powerful tools available to the mathematician. We know what must be the case by showing what cannot be the case!

24. Begin an RAA by assuming the negation of the conclusion of the argument. Then, using the standard rules, derive a contradiction (a line of the form 'p  ~p'). Now, discharge the assumption and derive the conclusion of the argument by a sequence of lines beginning with the assumption of the negation of the conclusion and ending with the derived contradiction. A vertical line beginning with the assumption and ending with the contradiction marks the scope of the assumption.

25. RAA works because the derived contradiction is obviously false. Since it is impossible to derive falsity from truth (and the premises are assumed to be true), the source of the falsity obvious in the contradiction must be the assumption of the negation of the conclusion. But if that assumption is false, then the conclusion is true if the premises are, and that is just the definition of a valid argument.

26. Hints for using CP: • Remember, the line derived MUST be a conditional whose antecedent was your assumption. • Use 2 CP subproofs followed by CONJ and MEI to derive a biconditional. • You can do multiple steps of CP, including nesting assumptions

27. Hints for using RAA: • Scan the premises to identify a likely contradiction. • Remember that you are looking for a contradiction—IP provides you with an overall strategy. • Remember to discharge your assumption! Sometimes you may derive what you are looking for (your overall goal) within the scope of the assumption, but you cannot use it. • Download the Handout Conditional and Indirect Proof Exercises and work the problems.

28. RAA and Problem Solving • Many standardized intelligence or aptitude tests (e.g. the LSAT, the GRE, the MCAT) include problems which can be solved easily using indirect proof as a strategy. • Use the strategy to discover when certain claims can’t be right (namely, when they lead directly to contradictions), and then use that information to determine which claims are correct.

29. Solving Puzzles Using IP • Messrs. Fireman, Guard, and Driver are the fireman, guard, and driver on a train. Each man has only one job. When I tried to find out who was what, I was given these four "facts": • (1) Mr. Driver is not the guard. • (2) Mr. Fireman is not the driver. • (3) Mr. Driver is the driver. • (4) Mr. Fireman is not the guard. • It then transpired that, of the above four statements, only one is true. Who is what?

30. Solve this puzzle by applying IP. In order to determine which of the 4 statements is true, begin by assuming one to be true and then look for a contradiction. Finding it lets you know that statement is false. If you assume (1) to be true, it leads to the contradiction that Mr. Fireman is both the driver and the guard, which is impossible.

31. Do you have it yet? • Mr. Driver is the guard • Mr. Fireman is the driver • Mr. Guard is the fireman

32. Common Errors to AVOID: • Trying to use an inference rule on a part of a line • Errors concerning the scope of a negation • Confusing the role of tildes in WFF’s with their role in argument forms • Reluctance to use addition and distribution • Reluctance to use CP and IP • Attempting the impossible

33. Symbolizing Arguments and Constructing Proofs • Symbolize carefully—correctly identify the premises and the conclusion • Pay attention to detail, particularly when symbolizing conditionals • Symbolize in ways that suit your strengths and preferred strategies (e.g., if you like DS, symbolize ‘unless’ with a wedge)

34. Remember: You can always go back and change your symbolization.

35. HINT: Test your symbolization for validity with a truth value analysis. If you have symbolized incorrectly and the argument for which you are attempting to construct a proof is non-valid, you will lose your mind.