Chapter_05 - THE LOGIC BOOK 4TH EDITION CHAPTER FIVE SL...

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THE LOGIC BOOK, 4 TH EDITION: CHAPTER FIVE: SL DERIVATIONS Whales breathe by lungs and whales are warm-blooded. If whales breathe by lungs, then whales are not fish. Whales are warm-blooded and whales are not fish. 1. Whales breathe by lungs and whales are warm-blooded Assumption. 2. If whales breathe by lungs, then whales are not fish. Assumption. 3. Whales breathe by lungs. From 1 4. Whales are not fish. From 2, 3 5. Whales are warm-blooded. From 1 6. Whales are warm-blooded and whales are not fish. From 4, 5. Assumption. 2. L ~ F Assumption. 3. L from 1 4. ~F from 2,3 5. W from 1 from 4, 5 1
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Semantics : the interpretation of a language. Syntax: the formal properties of a language. So, the reasoning in our example is the application of a series of syntactic rules for deriving sentences. In this chapter, we will learn derivation rules. Since these are syntactic, we will be able to use them without having any interpretation of the symbols in mind. Derivation: A proof or demonstration that shows how a sentence or sentences (conclusion) can be derived (obtained through reasoning) from a set of sentences (premises). SD: SD (sentential derivation) is a natural deduction system for SL (sentential logic). SD uses truth-preserving derivation rules, so if we follow those rules and the premises are true, we will only be able to derive true conclusions. Every sentence that is entailed by a set of sentences in SL can be derived from that set of sentences in SD. Every argument arrived at through SD will deductively valid. 2
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Here are the elements of a derivation : Line # Sentences Justifications (must include line # and derivation rule) 1 Assumption 2 Assumption 3 4 5 Derived sentences go here 6 7 8 9 Vertical line is the scope line 3
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Reiteration (R) : P This rule lets us derive any sentence from itself. P Example : 1 ( S K ) Q Assumption 2 A ~ B Assumption 3 ( S K ) Q 1 R Line 3 is the application of the reiteration rule (R). The justification shows the line and rule being used. We must reiterate the entire line, not just parts of it. 4
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Conjunction Introduction P This allows us to join two sentences Q together into a conjunction . P
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This note was uploaded on 09/28/2009 for the course PHL 245 taught by Professor Bangu during the Winter '05 term at University of Toronto.

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Chapter_05 - THE LOGIC BOOK 4TH EDITION CHAPTER FIVE SL...

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