Class Note for PHIL 110 at UMass(12)
Class Note for PHIL 110 at UMass(12)
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This 7 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Massachusetts taught by a professor in Fall. Since its upload, it has received 16 views.
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Date Created: 02/06/15
INTRO LOGIC DAY 10 Derivations in SL 2 Initial Modes of Reasoning Modus Ponendo Ponens affirming by affirming eC J4 C Modus Tollendo Tollens denying by denying eC C 54 Modus Tollendo Ponens 1 affirming by denying JAVZ N Z Modus Tollendo Ponens 2 affirming by denying JAVZ 23 J4 l aka disjunctive syllogism l Review Example 1 review We demonstrate show that an argument is valid by deriving deducing its conclusion from its premises using a few fundamental modes of reasoning NSR gtSRVNT PVT PsQ Q 1write all 1 NS Pr premises 2 R a s Pr 2write show 3 R V NT Pr conclusion 4 NP v T Pr 5 MP a NQ Pr 3 a rules topgvlailable 6 NQ DD lines 7 NR 12 MT 8 NT 37 MTP1 439 b x and 9 MP 48 MTP2 cancel highfive 10 Q 5399 MP other highfives 75r Tek gives Manny a highfive Tek gives ARod a highfive Rules discussed Today m ampo A A AampE AampE E E 7 7 7 7 A E AampE EampA m w A A AVE AVE NA NE AVE EVA E A gtI seeCD gt0 AgtC AgtC A NC C NA m DN A A A A Rule Sheet provided on exams available on course web page textbook keep this in front of you when doing homework Rules of Inference Basic Idea Every connective has 1 a takeout rule also called an eliminationrule OUTrule 2 a putin rule also called an introductionrule lNrule AmpersandOut ampO if you have a conjunction J4 amp B then you are entitled to infer its first conjunct J4 if you have a conjunction J4 amp B then you are entitled to infer its second conjunct B have means have as a whole line rules apply only to whole lines not pieces of lines WedgeOut v0 if you have a disjunction A v B and you have the negation of its 1st disjunct NA 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 2quotd disjunct NB then you are entitled to infer its first disjunct J4 what we earlier called modus tollendo ponens 11 AmpersandIN ampI if you have a formula J4 and you have a formula B then you are entitled to infer their first conjunction J4 amp B if you have a formula J4 and you have a formula B then you are entitled to infer their second conjunction B amp J4 WedgeIN vI if you have a formula J4 then you are entitled to infer its disjunction with any formula to its right J4 v B if you have a formula J4 then you are entitled to infer its disjunction with any formula to its left B v J4 10 12 ArrowOut gtO DoubleNegation DN ifyou have a conditional A a C 39fy have a f rm 39a A and you have its antecedent A then you are entitled to infer then you are entitled to infer its doublenegative NNA its consequent C ifyou have a conditional A a C quotYou have a dOUb39e39 ega Ve Mg and you have the negation of its consequent NC the YOU are entitled to infer then you are entitled to infer the formula A the negation of its antecedent NA What we earlier called rUIeS apply only to WhOIe lines modus ponens and modus tollens not pieces of lines 13 ArrowIntroduction Direct Derivation The Original and Fundamental SHewRule THERE IS NO SUCH THING AS ARROWIN I H suevv 54 W g o X 0 what we have instead is CONDITIONAL DERIVATION A CD In Direct Derivation DD one directly arrives at the very formula 14 one is trying to show which we examine later Example 2 NSRaSNRaNTPaTNPaNQINQ 1 Ms Pr 2 R s Pr 3 NR NT Pr 4 P sT Pr 5 MP a NQ Pr 6 NQ DD 7 NR 12 so 8 NT 37 so 9 MP 48 so 10 NO 59 so Example 3 NSRgtSRVNTNPVTNPgtNQINQ 17 1 Ms Pr 2 R a s Pr 3 R v NT Pr 4 P v T Pr 5 MP a NQ Pr 6 NQ DD 7 NR 12 so 8 NT 37 v0 9 MP 48 v0 10 NO 59 so 19 ArrowOut Strategy If you have a line of the form aC then try to apply arrowout a0 which requires a second formula as input in particular either J4 or NC have AaC find A NC or deduce C 21 18 WedgeOut Strategy If you have a line of the form AVE then try to apply wedgeout v0 which requires a second formula as input in particular either NA or NZ have AvB nd 21 13 or deduce B A 20 Example 4 PaQvRPQvRNR PaQaQaRINQ WedgeIn Strategy If you need AVE then look for either disjunct find A OR find B then apply vI to get AvB 23 1 P a Q V R Pr 2 P a Q V R a NR Pr 3 P a Q a Q a R Pr 4 NQ DD 5 NR 12 0 6 P a Q 15 Vo 7 Q a R 36 0 8 R 57 0 21 Example 5 PvQ sRVS TPT 1 P v Q S Pr 2 R v S T Pr 3 P Pr 4 T DD 5 P v Q 3 VI 6 S 1 5 0 7 R v S 6 VI 8 T 27 0 22 Example6 l PaQRvQaSRaTNTampPIQampS 1 PaQ Pr 2 RVQ 3 Pr 3 RaT Pr 4 NTampP Pr 5 QampS DD 6 NT 7 P 4ampo 8 Q 17 0 9 NR 36 0 10 Qas 29 v0 11 s 810 o 12 QampS 811ampI 24 AmpersandOut Strategy If you have a line of the form A83 then apply ampersandout 80 which can be applied immediately to produce A and B 25 AmpersandIn Strategy If you are trying to find or show A83 then look for both conjuncts nd A AND nd 3 then apply 81 to get A83 26
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