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INTRO LOGIC

INTRO LOGIC. Translations in PL 3. DAY 17. REVIEW of DAY 1 and DAY 2. Existential Quantifier. someone. is happy. there is someone. who is happy. there is some x :. x is happy.  x. Hx. Existential-Negative Quantifier. someone. is unhappy. there is someone. who is not happy.

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INTRO LOGIC

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  1. INTRO LOGIC Translations in PL3 DAY 17

  2. REVIEW of DAY 1 and DAY 2

  3. Existential Quantifier someone is happy there is someone who is happy there is some x : x is happy x Hx

  4. Existential-Negative Quantifier someone is unhappy there is someone who is not happy there is some x : x is not happy x Hx

  5. Universal Quantifier everyone is happy no matter who you are you are happy no matter who x is x is happy x Hx

  6. Universal-Negative Quantifier everyone is unhappy no matter who you are you are not happy no matter who x is x is not happy x Hx

  7. Negative-Existential Quantifier no one is happy there is no one who is happy there is no x : x is happy x Hx

  8. Negative-Universal Quantifier not everyone is happy not: no matter who you are you are happy not: no matter who x is x is happy x Hx

  9. Quantifier-Specification – ‘some’ some Freshman is Happy there is someone who is F and who is H there is some x x is F and x is H x Fx ( & Hx )

  10. Quantifier-Specification – ‘no’ no Freshman is Happy 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 )

  11. Quantifier-Specification – ‘every’ every Freshman is Happy 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 )

  12. new material for day 3

  13. Multiple Quantificationsentences with more than one quantifier • GENERAL STRATEGY • (1) Count the number of quantifiers in original sentence. • (2) Determine the overall structure of the sentence. • (3) Work on constituents separately. • (4) Substitute constituents back into overall formula. • Count the number of quantifiers in final formula. • Compare (5) with (1).

  14. Example 1 • everyone is FRIENDLY, but not everyone is HAPPY everyone is F but not everyone is H &  x Fx x Hx x Fx x Hx

  15. Example 2 • every CAT is a PET, butnot every PET is a CAT every C is P but not every P is C x ( Cx  Px ) &  x ( Px  Cx ) x ( Cx  Px ) x ( Px  Cx )

  16. Example 3 • if everyone is FRIENDLY, then everyone is HAPPY if everyone is F then everyone is H xFx  xHx xFx xHx

  17. Example 4 • if every STUDENT is FRIENDLY,then every STUDENT is HAPPY if every S is F then every S is H x(Sx  Fx)  x(Sx  Hx) x(Sx  Fx) x(Sx  Hx)

  18. ‘Any’ versus ‘Every’ • Basic Principle • both ‘any’ and ‘every’ are universal quantifiers, • BUT • they are usually • notinter-changeable.

  19. Some times they are interchangeable any one can Dance xDx every one can Dance if I can Dance, then any one can Di xDx if I can Dance, then every one can

  20. Usually, they arenot interchangeable is every one here?  is any one here? Jay doesn’t respect every one  Jay doesn’t respectany one if every one can fix your car, then I can  if any one can fix your car, then I can no one respects every one  no one respects any one

  21. Difference between ‘every’ and ‘any’ • the scope of ‘every’ is narrow the scope of ‘any’ is wide

  22. Example — Not-Every Jay doesn’t respect everyone • ‘not’ () has wide scope • ‘every’ () has narrow scope not! Jay respects everyone  xRjx

  23. Example — Not-Any Jay doesn’t respect anyone • ‘any’ () has wide scope‘not’ () has narrow scope does Jay respect a? no! ?Rja Rja does Jay respect b? ?Rjb no! Rjb does Jay respect c? ?Rjc no! Rjc etc. no matter who you are Jay does not respect you no matter who x is Jay does not respect x  Rjx x

  24. Not-Any = None =Not-Some Jay respects no one there is no one whom Jay respects there is no x : Jay respects x x Rjx Recall  =  x Rjx = x  Rjx Jay respectsno one = Jay doesn’t respect anyone

  25. Example — IF-EVERY if everyone fails, then satan wins • ‘every’ has narrow scope • ‘if…then’ has wide scope if everyone fails then satan wins xFx Ws  xFx Ws

  26. How do we SHOW such a formula? (1) : xFx  Ws CD (2) xFx As (3) : Ws

  27. Example — IF-ANY if anyone fails, then satan wins if a fails then satan wins if b fails then satan wins if c fails then satan wins etc. if anyone fails then satan wins

  28. In Other Words • ‘any’ has wide scope • ‘if…then’ has narrow scope no matter who you are if you fail then satan wins no matter who x is if x fails then satan wins x Fx (  Ws )

  29. How do we SHOW such a formula? (1) : x ( Fx  Ws ) UD • UD = Universal Derivation (later!) (2) : Fa  Ws CD (3) Fa As (4) : Ws

  30. Special Note • Sometimes (but not always) • ‘if-any’ = ‘if-some’ • x ( Fx  ) • = • xFx  provided has no free occurrence of ‘x’

  31. THE END

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