Today’s Topics

1. Today’s Topics • Conditional Proof • Indirect Proof (Reductio ad Absurdum)

2. Consider the following argument: • A  B  A  (A  B) • Clearly this is a valid argument (use a truth table test) • HOWEVER, we could not construct a proof for this argument in our system, that is, our 18 rules are incomplete. • We need more rules (some systems call this pattern Absorption)

3. 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

4. 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

5. 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.

6. 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

7. 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.

8. 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.

9. 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

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

11. 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

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

13. INDIRECT PROOF 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 is called Indirect Proof (IP) or Reductio ad Absurdum formalizes this insight

14. Begin an IP 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.

15. IP 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.

16. Your brains are probably fried by now, so I won’t show you how this works today.