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Overview of Sentential and Predicate Logic Translations and Derivations

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This overview covers key aspects of sentential and predicate logic, focusing on translations and derivations. It includes exam formats and important rules for logic derivations across multiple days. Notable strategies are introduced, including quantifier rules for universal and existential logic. The content delves into the structure and strategies of logical formulas, providing examples to illustrate how to manage various logical constructs using derived methodologies. The objective is to equip learners with essential techniques for mastering logic examinations.

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Overview of Sentential and Predicate Logic Translations and Derivations

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  1. INTRO LOGIC Derivations in PL4 DAY 25

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

  4. Rules Introduced – Day 2 O UD ––––– : :  NEW name NEW name a name counts as NEW precisely if it occurs nowhere unboxed or uncancelled

  5. Rules Introduced – Day 3 O O  –––––  ––––– • is any formula  is any variable

  6. Strategies main operator show-strategy ,, &,  SL strategy  UD  DD or D

  7. Show- Strategy (UD) • :  • :  • ° • ° • ° UD ?? must be a NEW name

  8. Show- Strategy (D) • :  •  • :  • ° • ° •  D As DD

  9. Example 1 (repeated) 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

  10. Example 2 (repeated) 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

  11. Example 3 there is someone whom everyone R’s/ everyone R’s someone or other (1) xyRyx Pr (2) : xyRxy UD (3) : yRay D(ID) (4) yRay As (5) :  DD (6) yRyb 1, O (7) yRay 4, O (8) Rab 6, O (9) Rab 7, O (10)  8,9, I

  12. Example 4 there is someone who R’s no-one/ everyone is dis-R’ed by someone or other (1) xyRxy Pr (2) : xyRyx UD (3) : yRya D(ID) (4) yRya As (5) :  DD (6) yRby 1, O (7) yRya 4, O (8) yRby 6, O (9) Rba 7, O (10) Rba 8, O (11)  9,10, I

  13. Example 5 (1) xy(Fy  Rxy) Pr • there is someone who R’s every F/ every F is R’ed by someone or other (2) : x(Fx yRyx) UD (3) : FayRya CD (4) Fa As (5) : yRya D(ID) (6) yRya As (7) :  DD (8) y(Fy  Rby) 1, O (9) yRya 6, O (10) Fa  Rba 8, O (11) Rba 9, O (12) Rba 4,10, O (13)  11,12, I

  14. Example 6 (1) x(Fx & yRxy) Pr (2) : xy(Fy & Ryx) UD • there is some F who R’s no-one/ everyone is dis-R’ed by some F or other (3) : y(Fy & Rya) D(ID) (4) y(Fy & Rya) As (5) :  DD (6) Fb & yRby 1, O (7) y(Fy & Rya) 4, O (8) Fb 6, &O (9) yRby (10) (Fb & Rba) 7, O (11) yRby 9, O (12) Fb Rba 10, &O (13) Rba 11, O (14) Rba 8,12 O (15)  13,14, I

  15. THE END

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