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Logic Unit 3 Part 2.pdf

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Niko Scharer

DERIVATIONS WITH AND OR AND BICONDITIONAL NATURAL DEDUCTION Part 2 311Derivations with AND OR and BICONDITIONAL So far weve learned how to derive any sentence from any set of sentences that entails it provided that the sentences are symbolized using the conditional and the negation signsBut often it is easier to symbolize sentences with the other three logical connectivesand or biconditional We need to extend the derivation system to include rules to cope with these new connectivesThe Extended Derivation System We will continue to use the three basic forms of derivation and subderivation Direct Derivation Conditional Derivation Indirect DerivationWe will also continue to use the rules of inference that we have already learnedModus Ponens MP or mpModus Tollens MT or mtDouble Negation DN or dnRepetition R or r We will also get a hundred new theorems out of the infinitely many But we need some new rules as well 1 Logic Unit 3 Part 2Derivations with Conjunction Disjunction and Biconditional 2011 Niko Scharer Rules for andThe new rules come in pairsan elimination rule and an introduction rule for each new connective The Greek lettersphi andpsi can represent any sentence whether atomic or molecular Simplification S or sSLSR or slsrAdjunction ADJ or adjThis rule allows us to infer either conjunct from This rule allows us to infer a conjunction sentence a sentence whose main logical connective is a from the two conjuncts conjunctionAlthough you can just use the justification S to simplify to either conjunct SL is an alternate justification for simplifying to the left conjunct SR to the right conjunct Modus Tollendo Ponens MTP or mtpAddition ADD or add This rule allows us to infer one disjunct from a This rule allows us to take a sentence and infer disjunction and the negation of the other from it a disjunction with the sentence as one disjunctThis makes senseif the butler or the disjunctThis makes senseif it will rain then it gardener did it and the gardener didnt do it will rain OR it will snow then it must have been the butler You can use ADD to join any sentence to an Modus tollendo ponens is a Latin term which available sentence creating any number of new means The mode of argument that asserts by sentencesSounds powerfulBut be careful denying By denying one disjunct of a although it can be useful ADD doesnt really get disjunction you can assert the other disjunct you anything you didnt have in the first place BiconditionalConditionalBC or bcConditionalBiconditional CB or cb This rule allows us to infer a biconditional from This rule allows us to infer either conditional two conditionalsThe two conditionals must be from a biconditional This makes sense the such that the antecedent of the first is the biconditional is really a conjunction of two consequent in the other and the consequent of the conditionals first is the antecedent in the other 2 Logic Unit 3 Part 2Derivations with Conjunction Disjunction and Biconditional 2011 Niko ScharerLike the original rules of inference these new rules are valid no matter how simple or complex the sentences being inferred are how complexandare in the rules aboveWhat matters is the logical form of the inferenceA few things to rememberJustificationsThe justification consists of line numbers andor premise numbers and the ruleFor S ADD and BC you cite one line numberFor ADJ MTP and CB you must cite two line numbers Main Logical ConnectiveTo use any rule you need to focus on the main logical connectivethat is the connective thatthe elimination rule acts on and the connective that the introduction rule introducesSo watch the main connective especially when you have been using informal notations and leaving out the parenthesesSuppose you have the sentencePQR It looks like you can use Simplification to get PQ BUTITS A MISTAKE PQRis a more informal notation of PQ RThe main connective isnot One cannot use S with that sentence 311 E1Which inference rule if any justifies the following argumentsS ADJ ADD MTP BC CB or none a b c d RPS PQS T PSQ PR S PSRSTPP SQ PRSPSRe f g h PQSP VZWY SR P R S P WY SRP RP Q VZSR3 Logic Unit 3 Part 2Derivations with Conjunction Disjunction and Biconditional 2011 Niko Scharer
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