ࡱ> 574` cbjbjss 4&c $$?AAAAAA$h`eez:??:, Q`3:3 ?0=Rtmttee    SYMBOLIC LOGIC INTRODUCTION TO INDIRECT PROOF Indirect proof is based on the classical notion that any given sentence, such as the conclusion, must be either true or false. We do indirect proof by assuming the premises to be true and the conclusion to be false and deriving a contradiction. Getting a contradiction shows us that it is impossible for the premises to be true and the conclusion to be false. This means that the conclusion must be true, which means that the argument is valid. (A V P) ( B ~A ( P B Assume the premises to be true. Assume the conclusion to be false. Derive a contradiction. Apply the last four lines of indirect proof to derive the conclusion. To deduce: B 1. (A V P) ( B Premise 2. ~A ( P Premise ENDING AN INDIRECT PROOF (after you derive a contradiction, any contradiction) CP You need to summarize what was established after making the desired assumption (the contradiction of the conclusion). Because the formula you deduced depended on the formula you assumed, you use the rule of conditional proof to show what you have established. i.e. ~B ( (P & ~P) EMI This rule allows you to introduce the formula ~P V ~~P, which will be the contradiction of the consequent of the conditional above (once you have put it in the proper form using De Morgans Law). De M This rule allows you to transform the disjunction introduced using the EMI rule into a conjunction which is the negation of the contradiction you derived as a consequent. i.e. ~(P & ~P) MT This rule allows you to derive the negation of the formula you assumed. That is, this rule allows you to derive the conclusion formula, which concludes the proof or deduction. i.e. B WHEN TO USE INDIRECT PROOF The conclusion is a negation. The conclusion is a disjunction. You can use De Morgans Law to transform the negation of a disjunction into a conjunction, which gives you a lot of information to work with. You have processed all the information in the premises as far as they can go, and you arent clear how to derive the conclusion. Anything that can be proved can be proved with indirect proof. DEMONSTRATION PROBLEM From Elementary Symbolic Logic, by Gustason #4 (p. 117-8) To show F D Premise A ( B Premise E ( C Premise ~A ( (D ( E) Premise (B V C) ( F Premise ~F Assumption ~(B V C) 5, 6 Modus Tollens ~B & ~C 7 De M ~B 8 Simplification ~A 2, 9 MT D ( E 4, 10 Modus Ponens E 1, 11 MP C 3, 12 Mp ~C & ~B 8 Comm ~C 14 Simp C & ~C 13, 15 Conjunction ~F ( (C & ~C) 6-16 Conditional Proof ~C V ~~C EMI ~(C & ~C) 18 De M ~~F 17, 19 F 20 DN ./n $ R \ ʹm_mN jh ECJOJQJ^JaJh]CJOJQJ^JaJ&hP0 hfc5>*CJOJQJ^JaJhfcCJOJQJ^JaJh/CJOJQJ^JaJh ECJOJQJ^JaJh5CJOJQJ^JaJ hwUh/CJOJQJ^JaJ&hwUh/5>*CJOJQJ^JaJ hP0 5>*CJOJQJ^JaJ h55>*CJOJQJ^JaJ./    ' J b gd.! & F hhdh^hgd:/dhgd Egd Egd/c        A B C ɺqq]O> hwUh:/CJOJQJ^JaJh:/CJOJQJ^JaJ&hP0 hP0 5>*CJOJQJ^JaJ jh ECJOJQJ^JaJhP0 CJOJQJ^JaJh5CJOJQJ^JaJh ECJOJQJ^JaJhNxCJOJQJ^JaJh E>*CJOJQJ^JaJhNx>*CJOJQJ^JaJ) jh Eh E>*CJOJQJ^JaJ#h Eh E>*CJOJQJ^JaJ B C a b * + UVW^gdP0 gdP0 `^``gd E `^``gd:/gd Egd/C E V W ` a b e ) * + / VW\rsx߯ߝ߯ߝ߯߯߉{{g{YG#h5h56CJOJQJ^JaJh5CJOJQJ^JaJ&hP0 hP0 5>*CJOJQJ^JaJhP0 CJOJQJ^JaJ&hhP0 5>*CJOJQJ^JaJ#hYh E5CJOJQJ^JaJ hwUh ECJOJQJ^JaJ jh ECJOJQJ^JaJh:/CJOJQJ^JaJh ECJOJQJ^JaJ#h{mh E5CJOJQJ^JaJWXYZ[\rs !3Rd| & Fdhgd/ & F hhdh^hgd/gd/dk| $/W]cбббРРВВбВВВВЄВбВВh(CJOJQJ^JaJh/CJOJQJ^JaJ jhhTCJOJQJ^JaJ jh*cCJOJQJ^JaJhzCJOJQJ^JaJ hwUh/CJOJQJ^JaJ hwUh5CJOJQJ^JaJh5CJOJQJ^JaJ)#3HWc & F hhdh^hgdP0  & F hhdh^hgd/ & Fdhgd/21h:pz/ =!"#$% @@@ /NormalCJ_HaJmH sH tH DAD Default Paragraph FontRiR  Table Normal4 l4a (k(No ListH@H Y Balloon TextCJOJQJ^JaJc &./'Jb BCab*+UVWXYZ[\rs ! 3 R d | # 3 H W e 00000000000 0 0 0 0000000000000000@0@0@0@0@0@000@0@0@0@0@0@0@0000000000000 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0 0 0 0 0 0 0./Cab*+UVe h00j00h00h00h00j00h00h00j00j00j00j00h00j00j0 0h0 0h0 0h0 0h0 0h0 0 h0 0j0 0j0 0h0 0h0 0h0 0h0 0h0 0h0 0h0 0h00h0"0h0"0h0"0h0"0h0"0h0"0h0"0h00j00l  C c  Wc c 8@0(  B S  ?J Q e  e 3333BNNWsb b e e bdgkOxh ^`hH.h ^`hH.h pLp^p`LhH.h @ @ ^@ `hH.h ^`hH.h L^`LhH.h ^`hH.h ^`hH.h PLP^P`LhH.h ^`hH.h ^`hH.h pLp^p`LhH.h @ @ ^@ `hH.h ^`hH.h L^`LhH.h ^`hH.h ^`hH.h PLP^P`LhH.bkO                  &%/ % P0 (FZ.!"X#@%:/M2Pm7f= EwEhTYsafcRg{mBr]JYzNx%5D*c-V;sV5*@fc P@UnknownGz Times New Roman5Symbol3& z Arial5& zaTahoma"qhf;f;f $24d^ ^  2QIX)?M22INDIRECT PROOFtestuserpdn26  Oh+'0   @ L X dpxINDIRECT PROOF testuser Normal.dotpdn2613Microsoft Office Word@P@p0`@0S@`՜.+,0 hp  CSUS^  INDIRECT PROOF Title  !"#%&'()*+-./01236Root Entry FC]`81TableWordDocument4&SummaryInformation($DocumentSummaryInformation8,CompObjq  FMicrosoft Office Word Document MSWordDocWord.Document.89q