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INTRO LOGIC

INTRO LOGIC. DAY 23. Derivations in PL 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

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INTRO LOGIC

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  1. INTRO LOGIC DAY 23 Derivations in PL2

  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. Rule Sheet provided on exams don’t makeupyourownrules available on courseweb page(textbook) keep this in front of you when doing homework

  4. Sentential Logic Rules • DD • ID • CD • D • &D • etc. • &I • &O • vO • O • O • etc.

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

  6. Rules Already Introduced – Day 1 O I ––––– –––––  is an OLD name (more about this later)

  7. Rules to be Introduced Today Universal Derivation UD Existential-Out O

  8. Example 1 • every F is H ; everyone is F / everyone is H (1) x(Fx  Hx) Pr (2) xFx Pr (3) : xHx ?? whatisultimately involvedinshowingauniversal (3) : Ha & Hb & Hc & ……… &.&.&.D (a) : Ha ?? (?) ?? ?? (b) : Hb ?? (?) ?? ?? (c) : Hc ?? (?) ?? ?? … … …

  9. Example 1a (1) x(Fx  Hx) Pr • one down, • a zillion to go! (2) xFx Pr (3) : xHx ?? (4) : Ha DD (5) Fa  Ha 1, O (6) Fa 2, O (7) Ha 5,6, O

  10. Example 1b (1) x(Fx  Hx) Pr • two down, • a zillion to go! (2) xFx Pr (3) : xHx ?? (4) : Hb DD (5) Fb  Hb 1, O (6) Fb 2, O (7) Hb 5,6, O

  11. Example 1c (1) x(Fx  Hx) Pr • three down, • a zillion to go! (2) xFx Pr (3) : xHx ?? (4) : Hc DD (5) Fc  Hc 1, O (6) Fc 2, O (7) Hc 5,6, O

  12. But wait – the derivations all look alike! (4) : Ha DD (5) Fa  Ha 1,O (6) Fa 2,O (7) Ha 5,6,O (4) : Hb DD (5) Fb  Hb 1,O (6) Fb 2,O (7) Hb 5,6,O (4) : Hc DD (5) Fc  Hc 1,O (6) Fc 2,O (7) Hc 5,6,O

  13. The Universal-Derivation Strategy • So, all we need to do is • do one derivation with one name (say, ‘a’) • and then argue that • all the other derivations will look the same. • To ensure this, • we must ensure that the name is general, • which we can do by making sure the name we select is NEW. a name counts as NEW precisely if it occurs nowhere in the derivationunboxed or uncancelled

  14. The Universal-Derivation Rule (UD) • :  • :  • ° • ° • ° • ° is any variable UD is any (official) formula ??  replaces  must be a NEW name, i.e., one that is occurs nowhere in the derivation unboxed or uncancelled

  15. Comparison with Universal-Out O UD ––––– : :  OLD name NEW name a name counts as OLDprecisely if it occurs somewherein the derivationunboxed and uncancelled a name counts as NEWprecisely if it occursnowhere in the derivationunboxed or uncancelled

  16. Example 2 every F is H / if everyone is F, then everyone is H (1) x(Fx  Hx) Pr (2) : xFx xHx CD (3) xFx As (4) : xHx UD a new (5) : Ha DD (6) Fa  Ha 1, O a old (7) Fa 3, O a old (8) Ha 6,7, O

  17. Example 3 every F is G ; every G is H / every F is H (1) x(Fx  Gx) Pr Pr (2) x(Gx  Hx) (3) : x(Fx  Hx) UD a new (4) : Fa Ha CD As (5) Fa DD (6) : Ha (7) Fa  Ga O 1, a old (8) Ga  Ha O 2, a old 5,7, (9) Ga O 8,9, (10) Ha O

  18. Example 4 everyone R’s everyone / everyone is R’ed by everyone (1) xyRxy Pr (2) : xyRyx UD a new (3) : yRya UD b new (4) : Rba DD (5) yRby 1, O b old (6) Rba 5, O a old

  19. Existential-Out (O) any variable (z, y, x, w …) ––––– any formula  replaces  any NEW name (a, b, c, d, …)

  20. Comparison with Universal-Out O O –––––– –––––– OLD name NEW name a name counts as OLDprecisely if it occurs somewhereunboxed and uncancelled a name counts as NEWprecisely if it occurs nowhereunboxed or uncancelled

  21. Example 5 every F is un-H / no F is H (1) x(Fx Hx) Pr (2) : x(Fx & Hx) D (3) x(Fx & Hx) As (4) :  DD (5) Fa & Ha 3, O new (6) Fa Ha 1, O old (7) Fa 5, &O (8) Ha (9) Ha 6,7, O (10)  8,9, I

  22. Example 6 some F is not H / not every F is H (1) x(Fx & Hx) Pr (2) : x(Fx  Hx) ID (3) x(Fx  Hx) As (4) :  DD 1, O new (5) Fa & Ha old (6) Fa  Ha 3, O 5, (7) Fa &O (8) Ha 6,7, (9) Ha O 8,9, (10)  I

  23. Example 7 every F is G ; some F is H / some G is H (1) x(Fx  Gx) Pr (2) x(Fx & Hx) Pr (3) : x(Gx & Hx) DD (4) Fa & Ha 2, O new (5) Fa  Ga 1, O old (6) Fa 4, &O (7) Ha (8) Ga 5,6, O (9) Ga & Ha 7,8, &I (10) x(Gx & Hx) 9, I

  24. Example 8 if anyone is F then everyone is H/ if someone is F, then everyone is H (1) x(Fx xHx) Pr (2) : xFx xHx CD (3) xFx As (4) : xHx UD new (5) : Ha DD (6) Fb 3, O new old (7) Fb xHx 1, O (8) xHx 6,7, O (9) Ha 8, O old

  25. Example 9 if someone is F, then everyone is H/ if anyone is F then everyone is H (1) xFx xHx Pr (2) : x(Fx yHy) UD new (3) : FayHy CD (4) Fa As (5) : yHy UD new (6) : Hb DD (7) xFx 4, I (8) xHx 1,7, O (9) Hb 8, O old

  26. Example 10 (a fragment) someone R’s someone??missing premises??/ everyone R’s everyone (1) xyRxy Pr (2) ??? Pr (3) : xyRxy UD new new (4) : yRay UD (5) : Rab ?? (6) yRcy 1, O new (7) Rcd 6, O new (8) ?? ??

  27. THE END

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