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# Class Note for PHIL 110 at UMass(5)

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Date Created: 02/06/15

INTRO LOGIC DAY 10 DERIVATIONS IN SENTENTIAL LOGIC 2nd day Review from its premises using a few We demonstrate show that an argument is valid by deriving deducing its conclusion fundamental modes of reasoning Initial Modes of Reasoning Modus Ponendo Ponens affirming by affirming AaC 54 C Modus Tollendo Tollens denying by denying AAC C 54 Modus Tollendo Ponens 1 affirming by denying 52lva 54 TB Modus Tollendo Ponens 2 af rming by denying 52lva NTB 54 l aka disiunctive sylloqism l Example 1 s Ras RVT PVT PaQl Q 1write all 1 NS Pr premises 2 R 8 Pr 3 R v NT pr conclusion 4 N P V T Pr 5 N P N P 3apply rules Q r to available 6 NQ on L 7 NR 12 MT 8 NT 37 MTP1 4 box and 9 NP 4 8 MTPZ cancel highfive 10 NO 59 MP Tek gives Manny a highfive Tek gives ARod a highfive c Rule Sheet provided I j z 1 sum on exams dOnT available make on course Us i E 246 web page XaJp makea A 2 copy and x L your keep it in front of Own you when gt Exam 4 6 doing ru 3963 homework a amp0 4 6273 353 3 3 3 J4ampB 3amp E VO JAVB vB N NB vB Bv 3 1 seeCD a0 ac ac J4 NC C Na MN DN w NN Rules of Inference Basic Idea every connective has an how to breakdown a formula with this connective 0 OUTrule how to buildup a formula with this connective 0 lNrule AmpersandOut ampO if you have a conjunction 54 amp B then you are entitled to infer its first conjunct 54 if you have a conjunction 54 amp B then you are entitled to infer its second conjunct TB have means have as a whole line rules apply only to whole lines not pieces of lines 10 AmpersandIN ampI if you have a formula A and you have a formula B then you are entitled to infer their first conjunction A amp B if you have a formula A and you have a formula B then you are entitled to infer their second conjunction TB amp A 11 WedgeOut v0 if you have a disjunction A V B and you have the negation of its 1St disjunct A then you are entitled to infer its second disjunct B if you have a disjunction A V B and you have the negation of its 2nd disjunct fB then you are entitled to infer its first disjunct A what we earlier called modus tollendo ponens 12 WedgeIN vI if you have a formula A then you are entitled to infer its disjunction with any formula to its right A V B if you have a formula A then you are entitled to infer its disjunction with any formula to its left B V A 13 ArrowOut gtO if you have a conditional A a C and you have its antecedent A then you are entitled to infer its consequent C if you have a conditional A gt C and you have the negation of its consequent C then you are entitled to infer the negation of its antecedent A what we earlier called modus ponens and modus tollens 14 ArrowIntroduction THERE IS NO SUCH THING AS ARROWIN HI 0 V X what we have instead is CONDITIONAL DERIVATION CD which we examine later 15 DoubleNegation DN if you have a formula A then you are entitled to infer its doublenegative J4 if you have a doublenegative J4 then you are entitled to infer the formula J4 rules apply only to whole lines not pieces of lines 16 Direct Derivation The Original and Fundamental SHeWRule SW 54 DD 0 o o 54 In Direct Derivation DD one directly arrives at the very formula one is trying to show Example 2 SRaSRaTPaTNPANQINQ 1 NS Pr 2 R a S Pr 3 NR T Pr 4 P a T Pr 5 P a NO Pr 6 Q DD 7 R 12 gt0 8 T 37 gt0 9 P 48 gto 10 Q 59 gtO ArrowOut Strategy If you have a line ofthe form AAC then try to apply arrowout a0 which requires a second formula as input in particular either 54 or C have A gtC find C or deduce 21 19 Example 3 8RaSRvTPvTPaQQ 1 s Pr 2 R a s Pr 3 R v T Pr 4 Pv T Pr 5 P a NO Pr 6 Q DD 7 R 12 gt0 8 T 37 v0 9 P 48 v0 10 Q 59 gtO 20 1O WedgeOut Strategy If you have a line of the form Ava then try to apply wedgeout vO which requires a second formula as input in particular either 54 or TB have AVE find N l 23 or deduce B A 21 Example 4 PaQvRPaQRANR PaQaQaRQ 1 P a Q v R Pr 2 PaQvRaR Pr 3 PAQAQAR Pr 4 Q DD 5 NR 12 gt0 6 P gt Q 15 v0 7 Q gt R 36 gt0 8 Q 57 gto 22 11 Example 5 PvQ gtSRvS gtTPIT PvQaS Pr RvSaT Pr P Pr 4 T DD 5 PvQ 3 vI 6 S 15 a0 7 RvS 6 vI T 27 gtO 23 WedgeIn Strategy if you need v B then look for either disjunct find A OR find 23 then apply vI to get AVE 24 Example 6 PaQRvQaSRaTTampPQampS 1 P a Q Pr 2 R v Q a S Pr 3 R a T Pr 4 T amp P Pr 5 Q amp S DD 6 T 7 P gt 4ampO 8 Q 17 gt0 9 R 36 gtO 10 Q gt S 29 v0 11 S 810 gtO 12 QampS 811ampI AmpersandOut Strategy if you have a line of the form A843 then apply ampersandout ampO which can be applied immediately to produce A and B AmpersandIn Strategy if you are trying to find or show A843 then look for both conjuncts find A AND find 23 then apply ampI to get 1843 27 THE END 28 14

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