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Unit 3 Part 1Derivations for Sentential Logic 2011

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

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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 syst em 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. Conditional Derivation You can derive a conditional sentence by assuming the antecedent and deriving the consequent from it using the derivation rules.
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