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Deriving Predicate Logic: A Comprehensive Guide to Rules and Applications

Explore the extensive framework of Predicate Logic, which encompasses the rules and strategies of Sentential Logic. This guide offers a detailed overview of derivations, translations, and grading policies for assessments in logic. It features numerous examples that illustrate foundational concepts, including existential and universal quantifiers, and demonstrates how to perform derivations in Predicate Logic. Ideal for students preparing for exams, this resource aids in mastering the transition from Sentential Logic to Predicate Logic, fostering a clearer understanding of logical reasoning.

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Deriving Predicate Logic: A Comprehensive Guide to Rules and Applications

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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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