University:
California State University, Northridge
Course:
PHIL 230 | Introduction to Formal Logic
Academic year:

2022
Views:

239

Pages:

2
Author:

Nesbyenvkk

Rules of Natural Deduction Rules of Implication 1. Simplification 7. Constructive Dilemma (P  Q) & (R  S) P & Q___ P v R_____ P QvS 8. Disjunctive Syllogism Q 2. Conjunction PvQ P ~P____ Q____ Q P&Q 3. Modus Ponens PQ P______ Q Remember: Rules of Implication work only in one direction, and apply only to a whole line. Each one is a valid mini argument, with premises above the line and a conclusion below the line. The premises being true guarantee guaranteess (entails, implies) that the conclusion is true. 4. Modus Tollens PQ ~Q_____ ~P 5. Addition P______ PvQ 6. Hypothetical Syllogism PQ Q  R__ PR Indirect Proof 1. Assume anything. (x) 2. Set in brackets and conduct proof as normal. 3. Show that the assumption entails a contradiction contradiction. (y & ~y) 4. Close the brackets. 5. ~x is now a normal line in the proof. 1. p --> (q & ~q) ___________ ~p 2. | p assume 3. | q & ~q MP (1,2) 4. ~p IP (2-3) Rules of Replacement 9. Double Negation P :: ~~P 18. Material Equivalence P <--> Q :: (P  Q) & (Q  P) P <--> Q :: (P & Q) V (~P & ~Q) ~P :: ~~~P 10. Commutativity (P v Q) :: (Q v P) (P & Q) :: (Q & P) 11. Associativity [P v (Q v R)] :: [(P v Q) v R] [P & (Q & R)] :: [(P & Q) & R] 12. DeMorgan’s Law ~(P & Q) :: (~P v ~Q) Remember: Rules of Replacement work in both directions, and apply to whole or partial lines lines. The statements one either sides of the colons ( :: ) are logical equivalents equivalents: they have the exact same truth conditions (and thus the same truth tables). Some of these don’t exactly look equivalent—but fill out the truth tables and you’ll see that they’re the same. ~(P v Q) :: (~P & ~Q) 13. Distribution [P & (Q v R)] :: [(P & Q) v (P & R)] [P v (Q & R)] :: [(P v Q) & (P v R)] 14. Transposition P  Q :: ~Q  ~P 15. Exportation (P & Q)  R :: P  (Q  R) 16. Tautology P V P :: P 17. Material Implication P  Q :: ~P V Q Conditional Proof 1. Assume anything. (x) 2. Set in brackets and conduct proof as normal. 3. Stop at any point. (y) 4. Close the brackets. 5. x --> y is now a normal line of the proof. 1. q & r ___________ p --> q 2. | p 3. | q 4. p --> q assume Simp. (1) CP (2-3)

Rules of Natural Deduction

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