Chapter_08_-_Complete - THE LOGIC BOOK, 4TH EDITION:...

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THE LOGIC BOOK, 4 TH EDITION: CHAPTER EIGHT Predicate Logic (PL): Semantics In SL, every sentence is either Truth-Functional or Atomic. To find out the truth-value of a truth- functional sentence, all we need is the truth-values of the atomic sentences. In PL, however, many of the atomic sentences are complex expressions composed of predicates and constants, and such expressions cannot be assigned set truth-values. Why? The truth-conditions of sentences in PL depend on the interpretation of that sentence, and such interpretations depend on the universe of discourse . Consider : UD: Living Things UD: Living Things Mx: x is a mammal Mx: x is a reptile w: Willy the Whale w: Willy the Whale Mw Mw 1
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Sentences without Quantifiers. (1) Use the interpretation to determine the truth-value of each atomic sentence. (2) Proceed to determine the truth value of the sentence on the basis of your knowledge of the connectives, just like we did in SL . (Pb Pa) Lcb UD: unrestricted Px: x is a politician. Lxy: x lives next to y a: Jean Chretien b: Celine Dion c: Queen Elizabeth If either Dion or Chretien is a politician then Queen Elizabeth lives next to Dion. Pa: true Pb: false Lcb: false (Pb Pa) Lcb: false UD: Positive integers Px: x is prime Lxy: x is less than y a: four b: three c: one If three or four is prime, then one is less than three. Pa: false Pb: true Lcb: true (Pb Pa) Lcb: true 2
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Sentences with Quantifiers A universally quantified sentence is true if and only if it is true for every single member of the universe of discourse. An existentially quantified sentence is true if and only if it is true for at least one member of the universe of discourse. (1) Use the interpretation to determine the truth-value of any unquantified atomic sentences. (2) Use the interpretation, the results of (1), and your knowledge of the connectives to determine the truth value or partial truth value of any quantified sentential components. (3) Use the interpretation, the results of (1) and the results of (2) to determine the truth-value of the sentence. On the interpretation below, is the following true or false? ( 2200 x)(Ax ~ Bx) UD: positive integers Px: x is evenly divisible by 2 Bx: x is an odd number 3
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A universally quantified sentence is false if there is at least one member of the UD for which the condition specified after the quantifier does not hold. A universally quantified sentence is true if there is no member of the UD for which the condition specified after the quantifier does not hold. An existentially quantified sentence is false if it is not the case that there is at least one member of the UD for which the condition specified after the quantifier does hold. An existentially quantified sentence is true if it is the
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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- Toronto.

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Chapter_08_-_Complete - THE LOGIC BOOK, 4TH EDITION:...

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