Unit 3 Part 2 Derivations with And Or and Biconditional 2011

Unit 3 Part 2 Derivations with And Or and Biconditional 2011

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Logic Unit 3 Part 2: Derivations with Conjunction, Disjunction and Biconditional © 2011 Niko Scharer 1 DERIVATIONS WITH AND, OR AND BICONDITIONAL NATURAL DEDUCTION Part 2 3.11 Derivations with AND, OR and BICONDITIONAL So far we’ve 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 signs. But often it is easier to symbolize sentences with the other three logical connectives – and, or, biconditional. We need to extend the derivation system to include rules to cope with these new connectives. The Extended Derivation System We will continue to use the three basic forms of derivation and subderivation: Direct Derivation Conditional Derivation Indirect Derivation We will also continue to use the rules of inference that we have already learned: Modus Ponens (MP or mp) ( φ ψ ) φ ⎯⎯⎯⎯ ψ Modus Tollens (MT or mt) ( φ ψ ) ~ ψ ⎯⎯⎯⎯ ~ φ Double Negation (DN or dn) Repetition (R or r) φ ⎯⎯⎯ ~~ φ ~~ φ ⎯⎯⎯ φ φ ⎯⎯⎯ φ We will also get a hundred new theorems! But we need some new rules as well . . .
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Logic Unit 3 Part 2: Derivations with Conjunction, Disjunction and Biconditional © 2011 Niko Scharer 2 Rules for , and The new rules come in pairs – an elimination rule and an introduction rule for each new connective. The Greek letters φ (phi) and ψ (psi) can represent any sentence, whether atomic or molecular. Simplification (S or s; SL/SR or sl/sr) Adjunction (ADJ or adj) φ ψ ⎯⎯⎯⎯ φ φ ψ ⎯⎯⎯⎯ ψ φ ψ ⎯⎯⎯⎯ φ ψ This rule allows us to infer either conjunct from a sentence whose main logical connective is a conjunction. Although 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. This rule allows us to infer a conjunction sentence from the two conjuncts. Modus Tollendo Ponens (MTP or mtp) Addition (ADD or add) φ ψ ~ φ ⎯⎯⎯⎯ ψ φ ψ ~ ψ ⎯⎯⎯⎯ φ φ ⎯⎯⎯⎯ φ ψ ψ ⎯⎯⎯⎯ φ ψ This rule allows us to infer one disjunct from a disjunction and the negation of the other disjunct. This makes sense: if the butler or the gardener did it, and the gardener didn’t do it, then it must have been the butler! ‘Modus tollendo ponens’ is a Latin term which means, ‘The mode of argument that asserts by denying.’ By denying one disjunct of a disjunction, you can assert the other disjunct. This rule allows us to take a sentence and infer from it a disjunction with the sentence as one disjunct. This makes sense: if it will rain, then it will rain OR it will snow.
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This note was uploaded on 02/01/2012 for the course PHL PHL245 taught by Professor Scharer during the Winter '11 term at University of Toronto- Toronto.

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Unit 3 Part 2 Derivations with And Or and Biconditional 2011

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