Chapter_07_-_Complete

Chapter_07_-_Complete - THE LOGIC BOOK, 4TH EDITION:...

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THE LOGIC BOOK, 4 TH EDITION: CHAPTER SEVEN Predicate Logic (PL): Symbolization and Syntax. Problem : SL is not adequate to the task of symbolizing many English sentences. Consider "Every politician cheats. Mike is a politician. So, Mike cheats." In SL we might symbolize this as: P M ---- C Although the English argument is deductively valid, the symbolization in SL is not. Solution : We will introduce a new language, PL, that will allow us to symbolize many more English sentences than SL allows. 1
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Some Terminology Singular Term: A singular term is any word or phrase that designates (or purports to designate) some one thing. There are two sorts. (1) Proper Name: Bill Homer Simpson Napoleon Bonaparte (2) Definite Description: Susan's only sister Lee's mother The tallest building in Toronto * The largest prime number * The King of the U.S.A. Predicate: A predicate is a part of a sentence that can be obtained by deleting one or more singular terms from a sentence. Tim is taller than Carla _______ is taller than Carla Tim is taller than ________ ______ is taller than ______ Predicates can be one-place or many-place. (Two-place, three-place, . . . , n-place, . . . ) Depends on the number of blanks. 2
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Universe of discourse (UD): the set of things being talked about on a given occasion. Symbolization key: a list of constants, variables, predicates, and a specified universe of discourse UD: University of Toronto students Hx : x studies hard Sxy: x is shorter than y Lxy: x likes y q: Quincy c: Charles e: Eleanor Eleanor is shorter than Charles: Sec Quincy likes Charles: Lqc Eleanor studies hard: He Charles likes himself: Lcc Quincy likes Eleanor or Quincy studies hard: Lqe Hq If Eleanor is shorter than Charles, then Charles is shorter than Quincy: Sec Scq Eleanor studies hard if and only if Quincy doesn't: Se ~Sq 3
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Quantifiers Universal quantifier: 2200 each and every one of the things being talked about is of the sort specified by the expression following the quantifier. Every one studies hard: ( 2200 x)Hx Each student studies hard: ( 2200 x)Hx Existential quantifier: 5 at least one of the things talked about is of the sort specified by the expression following the quantifer . At least one person studies hard: ( 5 x)Hx Everyone likes Eleanor: ( 2200 x) Lxe Not everyone likes Eleanor: ~ ( 2200 x) Lxe Someone likes Quincy: ( 5 x) Lxq No-one likes Quincy: ~ ( 5 x) Lxq No-one is shorter than himself / herself: ~ ( 5 x) Sxx 4
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Note: any variable will do. So, 'Everyone likes Quincy' can be written as ( 2200 x) Lxq, or as ( 2200 y) Lyq. There is an important difference between: (1) Charles doesn't like every one: ~ ( 2200 x)Lcx (2) Charles doesn't like any one: ~ ( 5 x) Lcx There are two ways to say 'Not all': ~ ( 2200 x) P x ( 5 x ) ~ P x 'None': ( 2200 x) ~ P x ~ ( 5 x ) P x 5
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UD: The Simpson Family Sx: x studies hard Tx: x gets in trouble h: Homer Dx: x eats doughnuts m: Marge Lxy: x likes y b: Bart Txy: x is taller than y l: Lisa Either Bart studies hard or everyone eats doughnuts:
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Chapter_07_-_Complete - THE LOGIC BOOK, 4TH EDITION:...

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