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INTRO LOGIC

INTRO LOGIC. Derivations in SL 4. DAY 12. Schedule. Day 09. . Introductory Material. Day 10. . Direct Derivation (DD). Day 11. . Conditional Derivation (CD) Negation Derivation ( D). Day 12. Indirect Derivation show: atomic show: disjunction. Day 13. show: conjunction. Day 14.

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INTRO LOGIC

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  1. INTRO LOGIC Derivations in SL4 DAY 12

  2. Schedule Day 09  Introductory Material Day 10  Direct Derivation (DD) Day 11  Conditional Derivation (CD)Negation Derivation (D) Day 12 Indirect Derivationshow: atomicshow: disjunction Day 13 show: conjunction Day 14 EXAM #2

  3. Exam 2 Format • 6 argument forms, 15 points each, plus 10 free points • Symbolic argument forms (no translations) • For each one, you will be asked to construct a derivation of the conclusion from the premises. • The rule sheet will be provided. • 1 problem from Set D • 2 problem from Set E • 2 problems from Set F • 1 problem from Set G (91-96)

  4. Inference Rules (so far) &O & –––––––  & –––––––  &I   –––––– &    –––––– &  O   ––––––    ––––––  I  ––––––   ––––––  O   –––––––    –––––––  DN  ––––––   –––––– 

  5.  Rules (so far) DD : DD     D : IDAs:     CD : CDAs:   

  6. Affiliated Rules Assumption Rule (CD) If one has a line of the form : then one is entitled to write down the formula on the very next line, as an assumption. Assumption Rule (D) If one has a line of the form : then one is entitled to write down the formula on the very next line, as an assumption. Contradiction-In (I) if you have a formula  and you have its negation  then you are entitled to infer –––– a contradiction (absurdity) 

  7. Direct-Derivation Strategy • :  • ° • ° • ° •  DD In Direct Derivation (DD), one directly arrives at the very formula one is trying to show.

  8. Show-Conditional Strategy • :  •  As • :  • ° • ° • ° CD ??

  9. Show-Negation Strategy • :  •  As • :  • ° • ° •  D DD

  10. (1) P  Q Pr (2) PQ Pr (3) : Q ?? We are stuck!! Can we show the following?  we have PQ so to apply O we must find P or find Q we also have PQ so to apply O we must find P or find Q

  11. Indirect Derivation :   As :  ° °  ID D • :  •  As • :  • ° • ° •  DD DD This is exactly parallel to D, and is another version ofthe traditional mode of reasoning known as REDUCTIO AD ABSURDUM

  12. Using ID The difference between ID and D is that D applies only to negations, whereas ID applies (in principle) to all formulas; it is a generic rule, like direct-derivation. • Although ID can, in principle, be used onany formula, • it is best used on two types of formulas. • 1. atomic formulas P, Q, R, etc. • 2. disjunctions

  13. Show-Atomic Strategy • :  •  As • :  • ° • ° •  ID DD  is atomic (P,Q,R, etc.)

  14. Example 1 (1) P  Q Pr (2) P  Q Pr (3) : Q ID (4) Q As (5) :  DD (6) P 1,4, O (7) Q 2,6, O (8)  4,7, I

  15. Example 2 (1) (P &Q) Pr (2) : P  Q CD (3) P As (4) : Q ID (5) Q As (6) :  DD (7) P & Q 3,5, &I (8)  1,7, I

  16. Show-Disjunction Strategy • :  • [] As • :  • ° • ° •  ID DD

  17. Affiliated Inference-Rule Tilde-Wedge-Out (O)  ––––––––– –––––––––

  18. Example 3 (1) P  Q Pr (2) : P  Q ID (3) (P  Q) As (4) :  DD (5) P 3, O (6) Q (7) Q 1,5, O (8)  6,7, I

  19. Example 4 (1) P  (Q  R) Pr (2) : Q  (P  R) CD (3) Q As (4) : P  R ID (5) (P  R) As (6) :  DD (7) P 5, O (8) R (9) Q  R 1,7, O (10) Q 8,9, O (11)  3,10, I

  20. Example 5 (1) (P  Q)  (P & Q) Pr (2) : (P & Q)  (P & Q) ID (3) [(P & Q)  (P & Q)] As (4) :  DD (5) (P & Q) 3, O (6) (P & Q) (7) (P  Q) 1,5, O (8) P 7, O (9) Q (10) P & Q 8,9, &I (11)  6,10, I

  21. THE END

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