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INTRO LOGIC

INTRO LOGIC

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INTRO LOGIC

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  1. INTRO LOGIC Derivations in PL3 DAY 24

  2. Overview • Exam 1 Sentential Logic Translations (+) • Exam 2 Sentential Logic Derivations • Exam 3 Predicate Logic Translations • Exam 4 Predicate Logic Derivations • 6 derivations @ 15 points + 10 free points • Exam 5 very similar to Exam 3 • Exam 6 very similar to Exam 4

  3. Predicate Logic Rules Universal Derivation UD day 2 Universal-Out O day 1 Tilde-Universal-Out O today Existential-In I day 1 Existential-Out O day 2 Tilde-Existential-Out O today

  4. Rules Already Introduced – Day 1 O I ––––– ––––– OLD name OLD name a name counts as OLDprecisely if it occurs somewhereunboxed and uncancelled

  5. Rules Already Introduced – Day 2 O UD ––––– : :  NEW name NEW name a name counts as NEW precisely if it occursnowhereunboxed or uncancelled

  6. Rules to be Introduced Today Tilde-Universal Out O Tilde-Existential-Out O

  7. Tilde-Universal Out (O)  ––––––  •  is any variable •  is any (official) formula example xHx –––––– xHx not everyone is H –––––––––––––– someone is not H

  8. Tilde-Existential Out (O)  ––––––  •  is any variable •  is any (official) formula example xHx –––––– xHx no one is H –––––––––––––– everyone is un H = not anyone is H

  9. SL Rules that are Often Useful in Connection with O and O O &O • () ––––––––– •  &  • ( & ) ––––––––– • 

  10. Example 1 not every F is H / some F is un-H (1) x(Fx  Hx) Pr (2) : x(Fx & Hx) DD (3) x(Fx  Hx) 1, O (4) (Fa Ha) 3, O (5) Fa & Ha 4, O (6) x(Fx & Hx) 5, I

  11. Example 2 every F is G ; no G is H / no F is H (1) x(Fx  Gx) Pr (2) x(Gx & Hx) Pr (3) : x(Fx & Hx) D (4) x(Fx & Hx) As (5) :  DD (6) x(Gx & Hx) 2, O (7) Fa & Ha 4, O (8) Fa  Ga 1, O (9) (Ga & Ha) 6, O (10) Fa 7, &O (11) Ha (12) Ga 8,10, O (13) Ga Ha 9, &O (14) Ha 12,13, O (15)  11,14, I

  12. New Strategy/Rule; D • :  •  • :  • ° • ° •  is any variable D is any (official) formula As D is a species of ID DD

  13. Example 1 Re-done Using ID (1) x(Fx  Hx) Pr (2) : x(Fx & Hx) D (ID) (3) x(Fx & Hx) As (4) :  DD (5) x(Fx  Hx) 1, O (6) x(Fx & Hx) 3, O (7) (Fa Ha) 5, O (8) (Fa & Ha) 6, O (9) Fa & Ha 7, O (10) Fa Ha 8, &O (11) Fa 9, &O (12) Ha (13) Ha 10,11, O (14)  12,13, I

  14. Example 3a (using ID) (1) x(Fx  Gx) Pr (2) x(Fx & Hx) Pr (3) : x(Gx & Hx) D(ID) (4) x(Gx & Hx) As (5) :  DD (6) Fa & Ha 2, O (7) Fa  Ga 1, O (8) x(Gx & Hx) 4, O (9) Fa 6, &O (10) Ha (11) Ga 7,9, O (12) (Ga & Ha) 8, O (13) Ga Ha 12, &O (14) Ha 11,13, O (15)  10,14, I

  15. Example 3b (using DD) (1) x(Fx  Gx) Pr (2) x(Fx & Hx) Pr (3) : x(Gx & Hx) DD(…I) (4) Fa & Ha 2, O (5) Fa  Ga 1, O (6) Fa 4, &O (7) Ha (8) Ga 5,6, O (9) Ga & Ha 7,8, &I (10) x(Gx & Hx) 9, I

  16. Example 4 if anyone is F, then everyone is unH/ if someone is F, then no one is H (1) x(Fx yHy) Pr (2) : xFx yHy CD (3) xFx As (4) : yHy D (5) yHy As (6) :  DD (7) Fa 3, O (8) Fa yHy 1, O (9) Hb 5, O (10) yHy 7,8, O (11) Hb 10, O (12)  9,11, I

  17. Example 5 if someone is F, then someone is unH/ if anyone is F, then not-everyone is H (1) xFx yHy Pr (2) : x(Fx yHy) UD (3) : FayHy CD (4) Fa As (5) : yHy ID (6) yHy As (7) :  DD (8) xFx 4, I (9) yHy 1,8, O (10) Hb 9, O (11) Hb 6, O (12)  10,11 O

  18. THE END