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# Class Note for PHIL 110 at UMass(9)

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This 4 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 12 views.

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Date Created: 02/06/15

Rule Sheet provided 3 quot INTR L i available 11 on course web page online textbook Derivations in SL keeping in front of 3 you when doing 1 homework vi w u x Re e Inference R les e cer t E l We demonstrate show that J J amp3 amp3 an argument is valid a g by deriving deducing its conclusion M B M from its premises 3 m a a a V B a V B using a few 3 3 N NB fundamental modes of reasoning V V g a gtI gtO J4 C J4 C a Nc c N34 2 DirectDerivation The Original and Fundamental SHeWRule Conditional Derivation CD SHGVWJA m o o o 54 In Direct Derivation DD one directly arrives at the veryformula one is trying to show SHGW aC W J4 SHOWC As short for assumption D depends upon formula o o o Affiliated Assumption Rule Can we show the following using DD 1 PaQ Pr 2 QaR Pr 3 SHOWP R 7 We are stuck we have PaQ so to apply 0 we must nd P or find Q we also have so to apply we must find or nd QaR a0 Q R if one has a line of the form SHOW 54C then one is entitled to write the formula J4 on the very nextline as an assumption Example 1 1 P a Q Pr 2 Q a R Pr 3 P a R CD 4 As 5 R DD 6 14 0 7 26 0 ArrowOut Strategy Can we show the following using DD 1 P a Q Pr We are stuck 2 Q P Pr 3 SHOW P we have PaQ we also have QaP so to apply 0 so to apply 0 we must find P we must find Q or nd Q orfind P ArrowOut Strategy Example 2 1 P s Q Pr 2 PampQ s R Pr 3 P s R CD 4 P As 5 R DD 6 Q 14 so 7 PampQ 46 amp1 8 R 27 so 9 Example 3 1 P s Q s R Pr 2 PsQsPsR CD 3 PsQ As 4 PsR CD 5 P As 6 R DD 7 Q s R 15 so 8 Q 35 so 9 R 78 so 10 Negation Derivation D SHOW N ND 54 As SHOW X DD special symbol for absurdity always done by DD X XI see later for details Historically this method of reasoning is called REDUCTIO AD ABSURDUM Latin for reduction to absurdity In symbolic logic an absurdity is a selfcontradiction both asserting and denying the same proposition 2 Affiliated Rules ContradictionIn XI Example 5 if you have a formula J4 and you have its negation N then you are entitled to infer a contradiction absurdity X Assumption Rule If one has a line ofthe form SHOW NA then one is entitled to write the formula J4 on the very next line as an assumption 13 Example 4 1 P a Q Pr 2 Q a NP Pr 3 NP D 4 P As 5 X DD 6 Q 14 so 7 MP 26 a0 8 X 47 XI 14 1 P a Q Pr 2 NP amp NQ D 3 P amp NQ As 4 x DD 5 P 6 NO 3 ampo 7 Q 15 so 8 x 67 x1 15 Example 6 1 P a Q a R Pr 2 PsRsPsNQ CD 3 P a R As 4 P a NQ CD 5 P As 6 NQ D 7 Q As 8 x DD 9 Q a R 15 so 10 R 35 so 11 R 79 so 12 x 1011 XI 15

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