Download Derivations in Sentential Logic: Validating Arguments with Modus Ponens and Modus Tollens and more Study notes Reasoning in PDF only on Docsity! 1 1 INTRO LOGIC DAY 09 2 UNIT 2 DERIVATIONS IN SENTENTIAL LOGIC 2 3 Basic Idea We start with a few argument forms, which we presume are valid, and we use these to demonstrate that other argument forms are valid. We demonstrate that a given argument form is valid by deriving (deducing) its conclusion from its premises using a few fundamental modes of reasoning. 4 Example 1 โ Modus Ponens (MP) A โ C A โโโโโโ C we can employ modus ponens (MP) to derive the conclusion from the premises. if A then C A โโโโโโโโโโ C P P โ Q Q โ R R โ S โโโโโโ S a derivative argument form 5 9 Example 3 (continued) /~Q; โผP โ โผQ; P โ T; โผR โ โผT; RโSโผS MT ~R MP ~T MT ~P MP ~Q 10 (5) (4) (3) (2) (1) : S R โ S Q โ R P โ Q P Derivations โ How to Start argument : P ; PโQ ; QโR ; RโS / S 1. write down premises 2. write down โ :โ conclusion (the goal) Pr Pr Pr Pr(emise) 6 11 (5) (4) (3) (2) (1) : S R โ S Q โ R P โ Q P Derivations โ How to Continue (goal) Pr Pr Pr Pr 3. apply rules, as appropriate, to available lines until goal is reached (8) (7) (6) 4,7,S 3,6,R 1,2,Q MP MP MP follows from lines 1 and 2 by modus ponens 12 (5) (4) (3) (2) (1) : S R โ S Q โ R P โ Q P Derivations โ How to Finish Pr Pr Pr Pr 4. Box and Cancel (8) (7) (6) 4,7,S 3,6,R 1,2,Q MP MP MP DD DD = Direct Derivation 7 13 (8) (7) (6) (5) (4) (3) (2) (1) Example 2 4,7,โผP 3,6,โผQ 1,2,โผR DD: โผP PrP โ Q PrQ โ R PrR โ S PrโผS ~S ; RโS ; QโR ; PโQ ; / ~P MT MT MT 14 (9) (8) (7) (6) (4) (3) (2) (1) Example 3 4,8,โผP 3,7,โผT 1,2,โผR DD: โผQ PrP โ T Pr~R โ ~T PrR โ S PrโผS โผS ; RโS ; ~Rโ~T ; PโT ; ~Pโ~Q / ~Q MT MP MT (5) Pr~P โ ~Q (10) 5,9,โผQ MP 10 19 Examples of MTP(1) (P&Q) โจ (RโจS) ~(P&Q) โโโโโโโโโโโโโ RโจS ~P โจ ~Q ~~P โโโโโโโโโ ~Q A โจ B ~A โโโโโโ B 20 Examples of MTP(2) (P&Q) โจ (RโจS) ~(RโจS) โโโโโโโโโโโโโ P&Q ~P โจ ~Q ~~Q โโโโโโโโโ ~P A โจ B ~B โโโโโโ A 11 21 THE END
