2. 1

3. 2

4. 3 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 deriving (deducing) its conclusion from its premises using a few fundamental modes of reasoning.

5. 4      ––––––  we can employ modus ponens (MP) to derive the conclusion from the premises. ifthen if  then   ––––––––––  P P  Q Q  R R  S –––––– S Example Argument Form

6. 5 P ; P  Q ; Q  R ; R  S / S MP Q R S

7. 6 we can employ modus tollens (MT) to derive the conclusion from the premises. Example Argument Form ifthen not not if  then  not  –––––––––– not   S R  S Q  R P  Q ––––––  P         ––––––  

8. 7 SS; R  S ; Q  R ; P  Q/  P MT RR QQ PP

9. 8 we can employ a combination of MP and MT to derive the conclusion from the premises.      –––––––          –––––––    S R  S  R   T P  T  P   Q –––––––––  Q Example Argument Form

10. 9 SS ; R  S;  R   T ; P  T;  P   Q QQ MT RR MP TT MT PP MP QQ

11. 10 argument : P ; P  Q ; Q  R ; R  S ; / S 1.write down premises 2.write down “  :” conclusion (1)PPr (premise) (2)P  QPr (3)Q  RPr (4)R  SPr (5)  : Swhat one is trying to derive

12. 11 (1)PPr(premise) (2)P  QPr(premise) (3)Q  RPr(premise) (4)R  SPr(premise) (5)  : S(goal) (6) Q (7) R (8) S 1,2,MP 3,6,MP 4,7,MP 3.Apply rules, as applicable, to available lines until goal is reached. 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

13. 12 4.Box and Cancel (1)PPr (2)P  QPr (3)Q  RPr (4)R  SPr (5)  : S (6) Q1,2,MP (7) R3,6,MP (8) S4,7,MP DDDirect Derivation

14. 13 argument :  S ; R  S ; Q  R ; P  R ; /  P (1)  SPr (2)R  SPr (3)Q  RPr (4)P  QPr (5)  :  P (6)  R (7)  Q (8)  P DD 1,2,MT 3,6,MT 4,7,MT

15. 14 (1)  SPr (2)R  SPr (3)  R   TPr (4)P  TPr (5)  P   QPr (6)  :  Q (7)  R (8)  T (9)  P (10)  Q DD argument :  S ; R  S ;  R  T ; P  T ;  P  Q /  Q 1,2,MT 3,7,MP 4,8,MT 5,9,MP

16. 15 Modus Tollens      ––––––  Modus Ponens     ––––––  Modus Tollendo Ponens (1)      ––––––  Modus Tollendo Ponens (2)      –––––– 

17. 16 (P&Q)(R  S) (P&Q)  (R  S)P&Q ––––––––––––– RSRSRSRS PQP  QPQP  Q PPPP ––––––––– QQQQ       –––––– 

18. 17 (P&Q)(R  S) (P&Q)  (R  S)  (R  S) –––––––––––––  (P&Q) PQP  QPQP  Q QQQQ ––––––––– PPPP       ––––––  PQP  QPQP  QQ P valid argument BUT NOT an example of MT

19. 18 these ten pennies are worth are the dime and ten pennies the same? are they the same in value? NO YES actually, in a very important sense, they are not the same in value! $2,000,000

20. 19 (P&Q)(R  S) (P&Q)  (R  S)  (P&Q) ––––––––––––– RSRSRSRS PQP  QPQP  Q PPPP ––––––––– QQQQ       –––––– 

21. 20 (P&Q)(R  S) (P&Q)  (R  S)  (R  S) –––––––––––––P&Q PQP  QPQP  Q QQQQ ––––––––– PPPP       –––––– 

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