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INTRO LOGIC

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This overview provides an in-depth examination of key concepts in sentential and predicate logic for PL3 students. It covers translations and derivations with a focus on various exam types, including practice questions and strategies for successful completion. Key rules such as Universal Derivation and Existential Out are outlined, alongside examples demonstrating their application. Ideal for students looking to reinforce their understanding and preparation for upcoming assessments in logical reasoning.

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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

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