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INTRO LOGIC

INTRO LOGIC. DAY 22. UNIT 4 Derivations in Predicate Logic. 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 DAY 22

  2. UNIT 4DerivationsinPredicate Logic

  3. 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

  4. Grading Policy • When computing your final grade, • I count your four highest scores. • (A missed exam counts as a zero.)

  5. Predicate Logic Subsumes Sentential Logic • Every rule of Sentential Logic • is also a rule of Predicate Logic. Every strategy of Sentential Logic is also a strategy of Predicate Logic. • CD • ID • O • O • etc. • :  • :  • :  • : & • etc.

  6. SL-Example 1 if no one is H, then k is not H (1) : xHx Hk CD (2) xHx As (3) :Hk D (4) Hk As (5) :  DD (6) ??? 2,O (?)  ???

  7. SL-Example 2 if everyone is un-H, then no one is H (1) : xHx xHx CD (2) xHx As (3) :xHx D (4) xHx As (5) :  DD (6) ??? 2,O (7) ??? 4,O (?)  ???

  8. SL-Example 3 • if someone is F or H, then someone is F or someone is H (1) : x(Fx  Hx)  (xFx xHx) CD (2) x(Fx  Hx) As (3) :xFx xHx D (ID) (4) [ xFx xHx ] As (5) :  DD (6) xFx 4,O (7) xHx 4,O (8) ??? 2,O (9) ??? 6,O (10) ??? 7,O (?)  ??

  9. SL-Example 4 • if everyone is F and H, then everyone is F and everyone is H (1) : x(Fx & Hx)  (xFx & xHx) CD (2) x(Fx & Hx) As (3) :xFx & xHx &D (4) : xFx ?? (?) ?? ?? (?) ?? ?? (?) : xHx ?? (?) ?? ?? (?) ?? ??

  10. Rules of Predicate Logic (overview) Logical operators RULES IN OUT OUT & &I &O &O  I O O  CD O O  UD O O  I O O

  11. Universal-Out (O) any variable (z, y, x, w …) ––––– any formula  replaces  any name (a, b, c, d, …) numerous restrictions (later)

  12. O – Example 1a 1. remove quantifier  x H x a 2. choose name a 3. substitute name for variable

  13. O – Example 1b x 1. remove quantifier  x H b 2. choose name b 3. substitute name for variable

  14. O – Example 1c x 1. remove quantifier  x H c 2. choose name c 3. substitute name for variable

  15. O – Example 2a x a ) ( x  x F a  H a a 1. remove quantifier 2. remove parentheses 3. choose name 4. sub name for variable

  16. O – Example 2b x b ) ( x  x F b  H b b 1. remove quantifier 2. remove parentheses 3. choose name 4. sub name for variable

  17. Derivation Example 1 every F is un-H ; k is F / not every F is H (1) x(Fx Hx) Pr (2) Fk Pr (3) : x(Fx  Hx) D (4) x(Fx  Hx) As (5) :  DD (6) Fk Hk 1, O (7) Fk  Hk 4, O (8) Hk 2,6, O (9) Hk 2,7, O (10)  8,9, I

  18. Example 2 • every FR’s him/herself ; j doesn’t R anyone • / j is not F (1) x(Fx  Rxx) Pr (2) xRjx Pr (3) : Fj D (4) Fj As (5) :  DD (6) Fj  Rjj 1, O (7) Rjj 2, O (8) Rjj 4,6, O (9)  7,8, I

  19. Example 3 if anyone is F, then everyone is H j is F / k is H (1) x { Fx yHy } Pr (2) Fj Pr (3) : Hk DD (4) Fj yHy 1, O (5) yHy 2,4, O (6) Hk 5, O

  20. Existential-In (I) any name (a, b, c, d, …) any formula –––––  replaces  any variable (z, y, x, w …) numerous restrictions (later)

  21. I – Example 1 x  F x a x 1. select name/variable 2. replace name by variable 3. restore missing parentheses (if any) there aren’t any 4. insert quantifier

  22. I – Example 2 x (  F x a & H x a ) x x 1. select name/variable 2. replace name by variable 3. restore missing parentheses (if any) 4. insert quantifier

  23. I – Example 3 x  R k x k x x x 1. select name/variable 2. replace name by variable 3. restore missing parentheses (if any) there aren’t any 4. insert quantifier

  24. I – Example 4 x x  R k k x 1. select name/variable 2. replace name by variable 3. restore missing parentheses (if any) there aren’t any 4. insert quantifier

  25. I – Example 5  x R k x k x 1. select name/variable 2. replace name by variable 3. restore missing parentheses (if any) there aren’t any 4. insert quantifier

  26. Derivation Example 5 • every F is H ; k is F / someone is H (1) x ( Fx  Hx ) Pr (2) Fk Pr (3) : xHx DD O (4) Fk  Hk 1, O (5) Hk 2,4, I (6) xHx 5,

  27. Example 6 • if someone is F then everyone is Hk is F / j is H (1) xFx xHx Pr (2) Fk Pr (3) : Hj DD I (4) xFx 2, O (5) xHx 1,4, O (6) Hj 5,

  28. Example 7 • every F R’s him/herselfk is F / someone R’s k (1) x(Fx  Rxx) Pr (2) Fk Pr (3) : xRxk DD (4) Fk  Rkk 1, O (5) Rkk 2,4, O (6) xRxk 5, I

  29. Example 8 • everyone who R’s someone is R’ed by everyone • k R’s herself • / j R’s k (1) x { yRxy yRyx } Pr (2) Rkk Pr (3) : Rjk DD (4) yRky yRyk 1, O (5) yRky 2, I (6) yRyk 4,5, O (7) Rjk 6, O

  30. THE END

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