Class Note for PHIL 110 at UMass(24)
Class Note for PHIL 110 at UMass(24)
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This 10 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 11 views.
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Date Created: 02/06/15
INTRO LOGIC DAY 11 Derivations in SL 3 Review We demonstrate show that an argument is valid by deriving deducing its conclusion from its premises using a few fundamental modes of reasoning Rule Sheet m In In provided z i i m on exams m w don 1 available r quot quot1 6 k on course mm l m0 8 web page Exam 245 7 up make a 2 copy and a x L J your keep it in front of own you When gt Exam 46 mg m Ies homework 4 Inference Rules excerpt amp0 8513 A851 13 13 13 851 385 Q vO A VB vB MA MB vB Bv B A 1 a0 C aC NC C NA DirectDerivation The Original and Fundamental SHeWRule SHGVVZA o o o 54 DD in Direct Derivation DD one directly arrives at the very formula one is trying to show Can we show the following using DD 1 P gtQ Pr 2 Q gtR Pr 3 SHOWP gtR we are stuck we have P gtQ so to apply gtO we must find P or find Q we also have so to apply we must find or nd Q gtR gtO Q R ArrowO ut Strategy Conditional Derivation CD SHOW AQC CD conditional derivation 54 As assumption SHOW C D depends on formula o o o Affiliated AssumptionRule ifone has a line of the form SHOW AaC then one is entitled to write the formula 54 on the very next line as an assumption Example 1 1 P a Q Pr 2 Q a R Pr 3 P a R CD 4 P As 5 R DD 6 Q 14 a0 7 R 26 so Example 2 P a Q Pr PampQa R Pr P a R CD 4 P As 5 R DD 6 Q 14 a0 PampQ 46 ampI R 27 a0 Example 3 P a Q a R Pr IPQPR CD PAQ As 4 PaR CD 5 P As 6 r R DD 7 Q a R 15 a0 Q 35 a0 R 78 a0 Can we show the following using DD 1 P gt Q Pr We are stuck 2 Q gt P Pr 3 SHOW P we have P gtQ we also have Q gtP so to apply gtO so to apply gtO we must find P we must find Q or find Q or find P ArrowOut Strategy 11 Negation Derivation D SHOW 54 ND 54 As assumption SHOW X DD special symbol for absurdity 0 always done by DD X XI see later for details to show that 54 is false 54 one assumes that 54 is true and shows that this leads to absurdity 12 Fancy Name Historically this method of reasoning is called REDUCTIO AD ABSURDUM Latin for reduction to absurdity to demonstrate that a proposition is false one assumes that it is true and demonstrates that this assumption leads to absurdity in symbolic logic an absurdity is a selfcontradiction both asserting and denying the same proposition 13 Affiliated Rules ContradictionIn XI if you have a formula 54 and you have its negation 54 then you are entitled to infer a contradiction absurdity X Assumption Rule If one has a line ofthe form SHOW 54 then one is entitled to write the formula 54 on the very next line as an assumption 15 Example 4 1 P a Q Pr 2 Q a P Pr 3 NP ND 4 P As 5 r X DD 6 Q 14 so 7 NP 26 so 8 X 47 XI 16 Example 5 17 OOI amp ax rD D 57 DIDOgtO161 N amp P ox Qow Q a P PNQX P mmmWmm 18 Example 6 OOOI aaax 1 I 1 rDSDSDSD5590 PCACANAD1371 N a DO a Q Rm N R N Q a a N XN a Q QRNX P s D D xlmnm mexwmux xobwm x x THE END
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