Unit 3 Part 1Derivations for Sentential Logic 2011

Unit 3 Part 1Derivations for Sentential Logic 2011 - UNIT...

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Logic Unit 3 Part 1: Derivations with Negation and Conditional 2011 Niko Scharer 1 UNIT 3: DERIVATIONS FOR SENTENTIAL LOGIC NATURAL DEDUCTION Part 1 3.1 What is a derivation? A derivation is a proof or demonstration that shows how a sentence or sentences can be derived (obtained 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 inconsistent. In a derivation, you are proving that the conclusion logically follows from the premises. We‟ll be using a natural deduction system for first-order logic that uses the symbolic language that we have learned. Our system is based on that presented by Kalish and Montague in their text, Techniques of Formal Reasoning . 1 All of our derivation rules are truth-preserving, so that if we follow the rules and the premises are true, we can only derive true conclusions. Every sentence that is logically entailed by a set of sentences can be derived from that set of sentences using our derivation system – the system is complete. Every one of the infinite number of valid theorems and valid arguments (within the scope of first-order sentential logic) can be proven. Every argument arrived at through our sentential derivation system will be deductively valid – our system is consistent. Thus, true premises will always lead to a true conclusion. 1 Kalish, Donald, and Montague, Richard, 1964. Logic: Techniques of Formal Reasoning . Harcourt, Brace, and Jovanovich. Our system is based on Kalish and Montague‟s natural deduction system for first-order logic, an elegant logical system that treats negation and material conditional as primary. It uses three types of derivation which we will be learning in this unit: direct derivation, indirect derivation and conditional derivation. Except for a single rule of inference ( modus ponens) no other logical symbol or rule of inference is necessary to express any sentence (for any possible truth-value assignment) or to complete a derivation in sentential logic. However, the other logical operators and rules of inference will make things a little easier and more intuitive! 2 This illustration is the property of Gerald Grow, Professor of Journalism, Florida A&M University. http://www.longleaf.net/ggrow/CartoonPhil.html Our natural deduction system should be a little less painful than this! ©1996 Gerald Grow 2
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Logic Unit 3 Part 1: Derivations with Negation and Conditional 2011 Niko Scharer 2 Three Types of Derivation for Sentential Logic: Direct Derivation Using the premises, you derive the sentence that you want to prove through the application of the derivation rules.
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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 1Derivations for Sentential Logic 2011 - UNIT...

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