Symbolic Logic I



Symbolic Logic I Conditional Proof and Reductio Ad Absurdum (RAA) Study Guide

CONDITIONAL PROOF

Sometimes, during the construction of a proof, you may need to construct a conditional. For instance, you may have the conditional 'p ( (q ( r)' as a line in your proof, but need the conditional 'p 6 q'. Since

'p ( q' does follow from 'p ( (q ( r)' (as a truth table would show) you could construct a direct proof for

'p ( q'. Such a proof, however, is rather tricky:

1. p ( (q ( r) Premise

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

Conditional proof (CP) offers a simpler and more direct route to establishing the desired conditional. When using CP, begin by ASSUMING the antecedent of the conditional that you want, in this case, 'p'. Then, using the standard rules, derive the consequent. You then discharge the assumption (close it off) by deriving the desired conditional and justifying it by a sequence of lines beginning with the assumed antecedent and ending with the consequent. Here is how it works:

1. p ( (q ( r) Premise

┌─>2. p Assumption

│ 3. q ( r 1,2 MP

│ 4. q 3 Simp

5. p ( q 2-4 CP

The vertical line beginning at line 2 and ending in a horizontal line at line 4 marks the scope of the assumption . 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. To be sure, the CP proof takes just as many lines as does the direct proof in this case, but the CP proof does not require two steps of Imp and one step of Dist.

CP is particularly valuable when deriving a biconditional. Use CP to establish each of the component conditionals, then use Equiv to establish the desired biconditional.

Note: You may use CP more than once in a proof.

Note: You may nest assumptions but the assumption made last

must be discharged first.

REDUCTIO AD ABSURDUM (RAA)

It is possible to show that the conclusion of an argument follows form the premises by showing that the negation of the conclusion is inconsistent with the truth of the premises. This method of argument is called Reductio Ad Absurdum (RAA) or Indirect Proof (IP). Begin a 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'), then 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. As with CP, a vertical line beginning with the assumption and ending with the contradiction marks the scope of the assumption.

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, but that is just the definition of a valid argument.

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download