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INTRO LOGIC

INTRO LOGIC. DAY 09. UNIT 2 DERIVATIONS IN SENTENTIAL LOGIC. Basic Idea. We start with a few argument forms, which we presume are valid, and we use these to demonstrate that other argument forms are valid. We demonstrate (show) that a given argument form is valid by

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INTRO LOGIC

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  1. INTRO LOGIC DAY 09

  2. UNIT 2DERIVATIONS INSENTENTIAL LOGIC

  3. Basic Idea • We start with a few argument forms, • which we presume are valid, • and we use these • to demonstrate that other argument forms are valid. We demonstrate (show) that a given argument form is valid by deriving (deducing) its conclusion from its premises using a few fundamental modes of reasoning.

  4. Example 1 – Modus Ponens (MP) •   –––––– if  then   –––––––––– Example Argument Form P P  Q Q  R R  S –––––– S we can employ modus ponens (MP) to derive the conclusion from the premises.

  5. Example 1 (continued) MP Q MP R MP S

  6. Example 2 – Modus Tollens (MT)   –––––– if  then  not  ––––––––––not  Example Argument Form S R  S Q  R P  Q ––––––P we can employ modus tollens (MT) to derive the conclusion from the premises.

  7. Example 2 (continued) MT R MT Q MT P

  8. Example 3 – using both MP and MT • we can employ a combination of MP and MT to derive the conclusion from the premises.   –––––––   ––––––– Example Argument Form S R  SR T P  TP Q –––––––––Q

  9. Example 3 (continued) MT R MP T MT P MP Q

  10. Derivations – How to Start • argument : P; PQ ; QR ; RS ;/ S 1. write down premises 2. write down “:” conclusion (1) P Pr (premise) (2) P  Q Pr (3) QR Pr (4) R  S Pr :Swhat one is trying to derive

  11. Derivations – How to Continue 3. Apply rules, as applicable, to available lines until goal is reached. • (1) P Pr (premise)(2) P  Q Pr (premise)(3) QR Pr (premise)(4) R  S Pr (premise)(5) : S (goal) (6) Q (7) R (8) S 1,2,MP 3,6,MP 4,7,MP follows from lines 1 and 2 by modus ponens follows from lines 3 and 6 by modus ponens follows from lines 4 and 7 by modus ponens

  12. Derivations – How to Finish 4. Box and Cancel • (1) P Pr(2) P  Q Pr(3) QR Pr(4) R  S Pr(5) : S(6) Q 1,2,MP(7) R 3,6,MP(8) S 4,7,MP DD Direct Derivation

  13. Derivation – Example 2 • argument : S; RS ; QR ; PR ;/ P (1) S Pr (2) R  S Pr (3) QR Pr (4) P  Q Pr (5) :P (6) R (7) Q (8) P DD 1,2,MT 3,6,MT 4,7,MT

  14. Derivation – Example 3 argument : S ; RS ; RT ; PT ; PQ / Q (1) S Pr (2) R  S Pr (3) RT Pr (4) P  T Pr (5) P  Q Pr (6) :Q (7) R (8) T (9) P (10) Q DD 1,2,MT 3,7,MP 4,8,MT 5,9,MP

  15. Initial Inference Rules Modus Ponens –––––– Modus Tollens–––––– Modus Tollendo Ponens (1)–––––– Modus Tollendo Ponens (2)––––––

  16. (P&Q)(RS) P&Q ––––––––––––– RS PQ P ––––––––– Q Examples of Modus Ponens   –––––– 

  17. (P&Q)(RS) (RS) ––––––––––––– (P&Q) PQ Q ––––––––– P Examples of Modus Tollens   ––––––  PQ Q ––––––––– P valid argument BUT NOT an example of MT

  18. Form versus Content/Value are the dime and ten pennies the same? NO are they the same in value? YES actually, in a very important sense, they are not the same in value! these ten pennies are worth $2,000,000

  19. (P&Q)(RS) (P&Q) ––––––––––––– RS PQ P ––––––––– Q Examples of MTP(1)   –––––– 

  20. (P&Q)(RS) (RS) ––––––––––––– P&Q PQ Q ––––––––– P Examples of MTP(2)   –––––– 

  21. THE END

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