University:
Harvard University
Course:
PHIL S-12 | Deductive Logic
Academic year:

2024
Views:

55

Pages:

3
Author:

ueffenv0k

SYMBOLIC LOGIC INTRODUCTION TO INDIRECT PROOF Indirect proof is based on the classical notion that any given sentence, such as the conclusion, must be either true or false. We do indirect proof by assuming the premises to be true and the conclusion to be false and deriving a contradiction. Getting a contradiction shows us that it is impossible for the premises to be true and the conclusion to be false. This means that the conclusion must be true, which means that the argument is valid. (A V P)  B ~A  P B 1. Assume the premises to be true. 2. Assume the conclusion to be false. 3. Derive a contradiction. 4. Apply the last four lines of indirect proof to derive the conclusion. To deduce: B 1. 2. (A V P)  B ~A  P Premise Premise ENDING AN INDIRECT PROOF (after you derive a contradiction, any contradiction) CP You need to “summarize” what was established after making the desired assumption (the contradiction of the conclusion). Because the formula you deduced depended on the formula you assumed, you use the rule of conditional proof to show what you have established. i.e. ~B  (P & ~P) EMI This rule allows you to introduce the formula ~P V ~~P, which will be the contradiction of the consequent of the conditional above (once you have put it in the proper form using De Morgan’s Law). De M This rule allows you to transform the disjunction introduced using the EMI rule into a conjunction which is the negation of the contradiction you derived as a consequent. i.e. ~(P & ~P) MT This rule allows you to derive the negation of the formula you assumed. That is, this rule allows you to derive the conclusion formula, which concludes the proof or deduction. i.e. B WHEN TO USE INDIRECT PROOF The conclusion is a negation. The conclusion is a disjunction. You can use De Morgan’s Law to transform the negation of a disjunction into a conjunction, which gives you a lot of information to work with. You have processed all the information in the premises as far as they can go, and you aren’t clear how to derive the conclusion. Anything that can be proved can be proved with indirect proof. DEMONSTRATION PROBLEM From Elementary Symbolic Logic, by Gustason #4 (p. 117-8) 1. D To show F Premise 2. A  B Premise 3. E  C Premise 4. ~A  (D  E) Premise 5. (B V C)  F Premise 6. ~F Assumption 7. ~(B V C) 5, 6 Modus Tollens 8. ~B & ~C 7 De M 9. ~B 8 Simplification 10. ~A 2, 9 MT 11. DE 4, 10 Modus Ponens 12. E 1, 11 MP 13. C 3, 12 Mp 14. ~C & ~B 8 Comm 15. ~C 14 Simp 16. C & ~C 13, 15 Conjunction 17. ~F  (C & ~C) 6-16 Conditional Proof 18. ~C V ~~C EMI 19. ~(C & ~C) 18 De M 20. ~~F 17, 19 21. F 20 DN

Indirect Proof Method

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