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Published on November 16, 2007

Author: abdullah

Source: authorstream.com

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INTRO LOGIC:  INTRO LOGIC DAY 16 Translations in PL 2 Overview:  Overview Exam 1: Sentential Logic Translations (+) Exam 2: Sentential Logic Derivations Exam 3: Predicate Logic Translations Exam 4: Predicate Logic Derivations Exam 5: (finals week) very similar to Exam 3 Exam 6: (finals week) very similar to Exam 4   + + + + When computing your final grade, I count your four highest scores. (A missed exam counts as a zero.) Exams 5 and 6 are scheduled for Wednesday December 20 10:30 a.m. Mahar Auditorium Slide3:  REVIEW of DAY 1 Existential Quantifier:  Existential Quantifier Hx x formula x is happy there is some x (s.t.) pronoun  variable who is happy there is someone paraphrase is happy someone original sentence Universal Quantifier:  Universal Quantifier Hx x formula x is happy no matter who x is pronoun  variable you are happy no matter who you are paraphrase is happy everyone original sentence Existential + Negation:  Existential + Negation Hx x formula x is not happy there is some x (s.t.) pronoun  variable who is not happy there is someone paraphrase is not happy someone original sentence Universal + Negation :  Universal + Negation everyone is not happy is AMBIGUOUS everyone is UNhappy NOT everyone is happy Universal-Negative:  Universal-Negative Hx x formula x is not happy no matter who x is pronoun  variable you are unhappy no matter who you are paraphrase is unhappy everyone original sentence Negative-Universal:  Negative-Universal Hx x formula x is happy not: no matter who x is pronoun  variable you are happy not: no matter who you are paraphrase is happy not everyone original sentence Negative-Existential:  Negative-Existential Hx x formula x is happy there is no x (s.t.) pronoun  variable who is happy there is no one paraphrase is happy no one original sentence Equivalences:  Equivalences xHx no-one is happy xHx someone is un-happy everyone is un-happy xHx not-everyone is happy xHx  =   =   is any variable  (phi) is any formula Slide12:  new material for day 2 Quantifier Specification:  Quantifier Specification some F is un-H some one is un-H no F is H no one is H some F is H some one is H every F is un-H every one is un-H not-every F is H not-every one is H every F is H versus every one is H Specific Quantifier Generic Quantifier Example 1:  Example 1 there is someone who … paraphrase some Freshman is Happy original sentence Hx & Fx x x is H and x is F there is some x there is someone who is H and who is F ( ) DON’T FORGET PARENTHESES Example 2:  Example 2 there is no one who … paraphrase no Freshman is Happy original sentence Hx & Fx x x is H and x is F there is no x there is no one who is H and who is F ( ) DON’T FORGET PARENTHESES Example 3:  Example 3 no matter who you are … you paraphrase every Freshman is Happy original sentence Hx  Fx x IF x is F THEN x is H no matter who x is no matter who you are IF you are F THEN you are H ) ( DON’T FORGET PARENTHESES Arrow versus Ampersand:  Arrow versus Ampersand Rule of Thumb (not absolute) the connective immediately “beneath” a universal quantifier () is usually a conditional () the connective immediately “beneath” an existential quantifier () is usually a conjunction (&) (  ) ( & ) Summary of Quantifier Specification :  Summary of Quantifier Specification x(Fx & Hx) some F is un-H xHx someone is un-H x(Fx & Hx) no F is H xHx no-one is H x(Fx & Hx) some F is H xHx someone is H x(Fx  Hx) every F is un-H xHx everyone is un-H x(Fx  Hx) not-every F is H xHx not-everyone is H x(Fx  Hx) every F is H xHx everyone is H Conjunctive Predicate-Combinations:  Conjunctive Predicate-Combinations x ( [ Ax & Bx ]  Cx ) every AB is C every AMERICAN BIKER is CLEVER x is an American Biker = x is an American, and x is a Biker x is an AB = [ Ax & Bx ] x ( [ Ax & Bx ] & Cx ) some AB is C some AMERICAN BIKER is CLEVER x ( [ Ax & Bx ] & Cx ) no AB is C no AMERICAN BIKER is CLEVER Non-Conjunctive Predicates Combinations:  Non-Conjunctive Predicates Combinations alleged criminal imitation leather expectant mother experienced sailor small whale large shrimp deer hunter racecar driver baby whale killer dandruff shampoo productivity software woman racecar driver Ambiguous Examples:  Ambiguous Examples Bostonian cab driver Bostonian attorney A Pitfall :  A Pitfall Compare the following: every Bostonian Attorney is Clever every BA is C vs. every Bostonian and Attorney is Clever every B and A is C Zombie Logic:  Zombie Logic a Pet is Dog and Cat every it is a Pet THEN it is a Dog and it is a Cat IF for any thing Px  Dx & Cx x in other words every CAT-DOG is a PET } ) ( { WHAT EXACTLY IS A CAT-DOG?:  WHAT EXACTLY IS A CAT-DOG? Another Candidate:  Another Candidate “Distributive” Use of ‘And’:  “Distributive” Use of ‘And’ every Cat and Dog is a Pet every Cat and every Dog is a Pet every Cat is a Pet, and every Dog is a Pet & x ( Cx  Px ) x ( Dx  Px ) “Plural” Use of ‘And’:  “Plural” Use of ‘And’ every member of the class Cats-and-Dogs is a Pet x ( [ Cx  Dx ]  Px ) x is a member of the class Cats-and-Dogs, = x is a Cat or x is a Dog = [Cx  Dx] to be a member of the class Cats-and-Dogs IS to be a Cat or a Dog no matter who x is if x is a member of the class Cats-and-Dogs, then x is a Pet ‘Only’ as a Quantifier:  ‘Only’ as a Quantifier only  are  only Citizens are Voters only Men play NFL football employees only members only cars only right turn only only Employees are Allowed only Members are Allowed only Cars are Allowed only Right turns are Allowed examples Recall ‘only if’:  Recall ‘only if’ ‘only’ is an implicit double-negative modifier  only IF  not  IF not  IF not  THEN not       One Rendering of ‘only’ :  One Rendering of ‘only’ only  are  only if you are , are you  (no matter who you are) you are  only if you are  (no matter who you are) x is  only if x is  (no matter who x is) x is not  if x is not  (no matter who x is) if x is not , then x is not  (no matter who x is) x ( x  x ) Alternative Rendering of ‘only’ :  Alternative Rendering of ‘only’ only = no non only  are  no non- are  no one who is not  is  there is no one who is not  but who is  there is no x ( x is not  but x is  ) x (x & x ) x ( x  x ) = Slide32:  THE END

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