Class Note for PHIL 110 at UMass(10)
Class Note for PHIL 110 at UMass(10)
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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 28 views.
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Date Created: 02/06/15
INTRO LOGIC DAY 12 Derivations in SL 4 Schedule Day 09 Introductory Material Day 10 Direct Derivation DD Conditional Derivation CD Negation Derivation ND Indirect Derivation Day 12 show atomic show disjunction Day 11 Day 13 show conjunction Day 14 EXAM 2 Exam 2 Format 6 argument forms 15 points each plus 10 free points Symbolic argument forms no translations For each one you will be asked to construct a derivation of the conclusion from the premises The rule sheet will be provided 1 problem from SetD 2 problem from SetE 2 problems from SetF 1 problem from SetG9196 Inference Rules so far ampO A amp B A amp B ampI A A B B A B A amp B B amp A v0 A v B A v 3 VI A A NA N1 A v B B v A B A gtO A gt B A gt 13 DN A A A NB SHeW Rules so far DD SW A DD A CD D SHeW AAC CD SHGW A ID A As A As SHeWC SHGWX Affiliated Rules Assumption Rule CD If one has a line ofthe form SHOW AAC then one is entitled to write down the formula A on the very next line as an assumption Assumption Rule D If one has a line ofthe form SHOW A then one is entitled to write down the formula A on the very next line as an assumption ContradictionIn XI if you have a formula A and you have its negation A then you are entitled to infer a contradiction absurdity X DirectDerivation Strategy SHGVV 54 DD 0 o 54 in Direct Derivation DD one directly arrives at the very formula one is trying to show ShowConditional Strategy SHGW AAC CD 54 As SHOW C D o o ShowNegation Strategy SHOW 54 ND 54 As SHOW x DD X Can we show the following 1 P a Q Pr We are stuck 2 NP Q Pr 3 SHOW Q we have PAQ we also have NPAQ so to apply a0 so to apply a0 we must find P we must find NP or find or find Q 10 Indirect Derivation SHOW 54 ID SHOWS 54 ND 54 As 54 As SHOW X DD SHOW X DD X X this is exactly parallel to ND and is another version of the traditional mode of reasoning known as REDUCTIO AD ABSURDUM 11 Using ID the difference between ID and ND is that D applies only to negations whereas lD applies in principle to all formulas it is a generic rule like directderivation although lD can in principle be used on any formula it is best used on two types of formulas P Q R 52lva 1 atomic formulas 2 disjunctions etc 12 ShowAtomic Strategy SHGVV A ID NA As SHOW x DD X A is atomic PQR etc 13 Example 1 1 P gt Q Pr 2 P gt Q Pr 3 Q ID 4 Q As 5 x DD 6 14 gtO 7 26 gtO 8 Q x 47 x1 14 Example 2 1 P amp NQ Pr 2 P gt Q CD 3 P As 4 Q ID 5 Q As 6 x DD 7 P amp Q 35 ampI 8 x 17 x1 15 ShowDisjunction Strategy sHew 54va 54va As SHGW X DD X 16 Affiliated InferenceRule TildeWedgeOut v0 L v B L v 3 NA N B 17 Example 3 1 P gt Q Pr 2 P v Q ID 3 P v Q As 4 x DD 5 P N 6 NQ 3 vO 7 Q 15 gt0 8 X 67 x1 18 Example 4 1 P gtQvR Pr 2 Q gt P v R CD 3 Q As 4 P v R ID 5 P v R As 6 x DD 7 P N 8 NR 5 vO 9 Q v R 17 gto 1o Q 89 v0 11 x 310 x1 19 Example 5 1 PvQPampQ Pr 2 PampQvPampQ ID 3 P amp Q v P amp Q AS 4 X DD 5 P amp Q 3 N O 6 P amp Q V 7 P v Q 15 a0 lt8gt P N 9 NO 7 v0 10 PampQ 89 ampI 11 x 610 x1 20 1O
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