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Logic Unit 3 Part 1.pdf

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University of Toronto St. George
Niko Scharer

UNIT 3DERIVATIONS FOR SENTENTIAL LOGIC NATURAL DEDUCTION Part 1 31What is a derivation A derivation is a proof or demonstration that shows how a sentence or sentences can be derivedobtained by making valid inferences from a set of sentences A derivation can be used to demonstrate that an argument is valid that a sentence is a tautology or that a set of sentences is inconsistentIn a derivation you are proving that the conclusion logically follows from the premises Well be using a natural deduction system for firstorder logic that uses the symbolic language that we have learnedOur system is based on that presented by Kalish and Montague in their text Techniques 1of Formal Reasoning All of our derivation rules are truthpreserving so that if we follow the rules and the premises are true we can only derive true conclusionsEvery sentence that is logically entailed by a set of sentences can be derived from that set of sentences using our derivation systemthe system is completeEvery one of the infinite number of valid theorems and valid arguments within the scope of firstorder sentential logic can be provenEvery argument arrived at through our sentential derivation system will be deductively validour system is consistentThus true premises will always lead to a true conclusion Our natural deduction system should be a little less painful than this21996 Gerald Grow 1 Kalish Donald and Montague Richard 1964 Logic Techniques of Formal Reasoning Harcourt Brace and JovanovichOur system is based on Kalish and Montagues natural deduction system for firstorder logic an elegant logical system that treats negation and material conditional as primaryIt uses three types of derivation which we will be learning in this unitdirect derivation indirect derivation and conditional derivationExcept for a single rule of inference modus ponensno other logical symbol or rule of inference is necessary to express any sentence for any possible truthvalue assignment or to complete a derivation in sentential logicHowever the other logical operators and rules of inference will make things a little easier and more intuitive2 This illustration is the property of Gerald Grow Professor of Journalism Florida AM University httpwwwlongleafnetggrowCartoonPhilhtml 1 Logic Unit 3 Part 1Derivations with Negation and Conditional2011 Niko Scharer Three Types of Derivation for Sentential LogicDirect Derivation Using the premises you derive the sentence that you want to prove through the application of the derivation rules Conditional Derivation You can derive a conditional sentence by assuming the antecedent and deriving the consequent from it using the derivation rules Indirect Derivation You can derive a sentence of any form by assuming its negation and deriving a contradiction from itIf you can show that one sentence leads to a contradiction then you can infer that that sentence is false and hence you can infer its negationTo derive a contradiction you derive a sentence and its negation using the derivation rulesDerivation Rules The derivation rules determine what sentences you can derive from the sentences that you already have either premises or other sentences that you have derived and provides the justification for each step in the derivationEach rule is itself a valid argument form or a pattern of valid reasoning Basic RulesThere are ten basic rules in sentential logicThe rules for conjunctiondisjunctionand biconditionalcome in pairs we need rules that allow us to move from sentences without these connectives to sentences that have connectives as the main logical operators introduction rules and we need rules that allow us to move from sentences for which these are the main logical operators to sentences that do not elimination rulesThe basic rules for negationand conditionalthe primary logical operators are a little different because conditional and indirect derivations are the primary methods for introducing the conditional sign and both introducing and eliminating the negation sign Derived RulesIn addition to the basic rules there are a potentially endless number of derived rulesA derived rule is a rule that is developed out of a pattern of valid reasoningonce youve shown that the pattern of reasoning is valid given certain starting conditions you can validly deduce the end product without going through the whole reasoning processIts a shortcutWe will be introducing five derived rules each of which you will prove with the basic rules Theorems as RulesFinally the derivation system allows us to derive an infinite number of theoremssentences that can be validly derived from the empty set from no premises at allThese are tautologies such asPP or It will rain tomorrow or it wont It is not logically possible for such sentences to be falseAnd every tautology can also be used as a derived derivation rule Once youve proved it you can use the theorem as a short cut2 Logic Unit 3 Part 1Derivations with Negation and Conditional2011 Niko Scharer 32The Basic Rules forandThe basic rules concern the first two logical operators we discussednegation and conditional The Greek lettersphi andpsi can represent any sentence whether atomic or molecularModus Ponens mp or MPModus Tollens mt or MTThis rule allows us to infer the consequent of a This rule allows us to infer the negation of the conditional from a conditional sentence and the antecedent from a conditional sentence and antecedent the negation of the consequent Modus ponens is a Latin termIts short for Modus tollens is short for Modus tollendo Modus ponendo ponens which means The tollens which means The mode of argument mode of argument that asserts by asserting that denies by denyingBy denying the By asserting the antecedent you can assert the consequent you can conclude the denial of consequent as your conclusion the antecedent This makes sensewith a conditional sentence This makes sensewith a conditional if the antecedent is true so is the consequentsentence the antecedent cannot be true and the We just follow the direction of the arrow consequent falsethus if the consequent is false the antecedent must be false as wellDouble Negation dn or DNRepetition r or Rrom a sentence the This rule allows us to infer a sentence from This rule allows us to infer fsame sentence with two negations in front of it itself This may look too trivial to be a rule or to infer the unnegated sentence from the but it we will need it sentence with two negations in front of itThis makes sensea doubly negated sentence is logically equivalent to the unnegated sentence 3 Logic Unit 3 Part 1Derivations with Negation and Conditional2011 Niko Scharer
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