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INTRO LOGIC

INTRO LOGIC. Translations in PL 2. DAY 16. Overview.   + + + + . Exam 1: Sentential Logic Translations (+) Exam 2: Sentential Logic Derivations Exam 3: Predicate Logic Translations Exam 4: Predicate Logic Derivations Exam 5: (finals) very similar to Exam 3

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INTRO LOGIC

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  1. INTRO LOGIC Translations in PL2 DAY 16

  2. Overview   + + + + • Exam 1: Sentential Logic Translations (+) • Exam 2: Sentential Logic Derivations • Exam 3: Predicate Logic Translations • Exam 4: Predicate Logic Derivations • Exam 5: (finals) very similar to Exam 3 • Exam 6: (finals) very similar to Exam 4 When computing your final grade, I count your four highest scores. (A missed exam counts as a zero.)

  3. REVIEW of DAY 1

  4. Existential Quantifier original sentence someone is happy paraphrase there is someone who is happy pronoun  variable there is some x (s.t.) x is happy formula x Hx

  5. Universal Quantifier original sentence everyone is happy paraphrase no matter who you are you are happy pronoun  variable no matter who x is x is happy formula x Hx

  6. Negative-Existential Quantifier original sentence no one is happy paraphrase there is no one who is happy pronoun  variable there is no x (s.t.) x is happy formula x Hx

  7. Negative-Universal Quantifier original sentence not everyone is happy paraphrase not: no matter who you are you are happy pronoun  variable not: no matter who x is x is happy formula x Hx

  8. Equivalences xHx not-everyoneis happy someone is un-happy xHx xHx no-one is happy xHx everyoneis un-happy  =   = 

  9. new material for day 2

  10. Quantifier Specification Generic Quantifier versus Specific Quantifier every one is H every F is H not-every one is H not-every F is H every one is un-H every F is un-H some one is H some F is H no one is H no F is H some one is un-H some F is un-H

  11. Example 1 original sentence some Freshman is Happy paraphrase there is someone who … there is someone who is F and who is H there is some x x is F and x is H x Fx ( & Hx ) DON’T FORGET PARENTHESES

  12. Example 2 original sentence no Freshman is Happy paraphrase there is no one who … there is no one who is F and who is H there is no x x is F and x is H x Fx ( & Hx ) DON’T FORGET PARENTHESES

  13. Example 3 original sentence every Freshman is Happy paraphrase no matter who you are … you no matter who you are IF you are F THEN you are H no matter who x is IF x is F THEN x is H x ( Fx  Hx ) DON’T FORGET PARENTHESES

  14. 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 (&) ( & )

  15. Summary of Quantifier Specification everyone is H xHx every F is H x(Fx Hx) not-everyone is H xHx not-every F is H x(Fx Hx) everyone is un-H xHx every F is un-H x(FxHx) someone is H xHx some F is H x(Fx& Hx) no-one is H xHx no F is H x(Fx& Hx) someone is un-H xHx some F is un-H x(Fx&Hx)

  16. Conjunctive Predicate-Combinations x is an American Biker=x is an American, and x is a Biker x is an AB=[ Ax & Bx ] every AMERICAN BIKER is CLEVER every AB is C x ( [ Ax & Bx ] Cx ) some AMERICAN BIKER is CLEVER some AB is C x ( [ Ax & Bx ] & Cx ) no AMERICAN BIKER is CLEVER no AB is C x ( [ Ax & Bx ] & Cx )

  17. Non-Conjunctive Predicates Combinations • alleged criminal • imitation leather • expectant mother • experienced sailor • small whale • large shrimp • deer hunter • racecar driver woman racecar driver baby whale killer dandruff shampoo productivity software

  18. Ambiguous Examples • Bostonian cab driver • Bostonian attorney

  19. 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

  20. Zombie Logic every Cat and Dog is a Pet • in other words x { ( Cx & Dx )  Px } for any thing IF it is a Cat and it is a Dog THEN it is a Pet every CAT-DOG is a PET

  21. WHAT EXACTLY IS A CAT-DOG?

  22. Another Candidate

  23. “Distributive” Use of ‘And’ • every Cat andDog 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 )

  24. “Plural” Use of ‘And’ every member of the classCats-and-Dogs is a Pet no matter who x is if x is a member of the class Cats-and-Dogs, then x is a Pet to be a member of the class Cats-and-Dogs IS to be a Cat or a Dog x is a member of the class Cats-and-Dogs, = x is a Cat or x is a Dog = [Cx  Dx] x ( [ Cx Dx ] Px )

  25. ‘Only’ as a Quantifier only are  examples onlyCitizens are Voters onlyMen play NFL football employees only members only cars only right turn only onlyEmployees are Allowed onlyMembers are Allowed onlyCars are Allowed onlyRight turns are Allowed

  26. Recall ‘only if’ • ‘only’ is an implicit double-negative modifier onlyIF notIFnot IFnotTHENnot 

  27. One Rendering of ‘only’ • only are  • onlyif you are , are you  (no matter who you are) • you are onlyif you are  (no matter who you are) • x is onlyif 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 )

  28. 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 butwho is  there is no x ( x is not butx is  ) x (x &x ) = x ( x x )

  29. THE END

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