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2 We demonstrate (show) that an argument is valid deriving by deriving (deducing) its conclusion from its premises using a few fundamental modes of reasoning.

3 available on course web page (on-line textbook) provided on exams keep this in front of you when doing homework don’t make up your own rules 

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5  :  ° ° °  DD In Direct Derivation (DD), one directly arrives at the very formula one is trying to show. The Original and Fundamental  -Rule

6 (1)P  QPr (2)Q  RPr (3)  : P  R?? We are stuck!!   we haveP  Q  O so to apply  O we must findP or find  Q  we also haveQ  R  O so to apply  O we must findQ or find  R Arrow-Out Strategy

7  :     As  :  ° ° ° CD ?D Affiliated Assumption Rule if one has a line of the form  :    then one is entitled to write the formula  on the very next line, as an assumption. short for ‘assumption’ depends upon formula

8 (7) (6) (5) (4) (3) (2) (1) 2,6, R 1,4, Q DD  : R As P CD  : P  R Pr Q  R Pr P  Q OO OO

9 (8) (7) (6) (5) (4) (3) (2) (1) 2,7, R 4,6, P&Q 1,4, Q DD  : R As P CD  : P  R Pr (P&Q)  R Pr P  Q OO &I&I OO

10 (8) (9) (7) (6) (5) (4) (3) (2) (1) 3,5, Q 7,8, R 1,5, Q  R DD  : R As P CD  : P  R As P  Q CD  : (P  Q)  (P  R) Pr P  (Q  R) OO OO OO

11 (1)P  QPr (2)Q   PPr (3)  :  P?? We are stuck!!  we haveP  Q  O so to apply  O we must findP or find  Q  we also haveQ   P  O so to apply  O we must findQ or find  P  Arrow-Out Strategy

12  :    As   :  ° °  I DD DD short for ‘assumption’ special symbol for “absurdity” Historically, this method of reasoning is called REDUCTIO AD ABSURDUM (Latin for “reduction to absurdity”) self-contradiction assertingdenying In symbolic logic, an absurdity is a self-contradiction – both asserting and denying the same proposition. always done by DD see later for details

13 Assumption Rule If one has a line of the form  :   then one is entitled to write 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) 

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

15 (8) (7) (6) (5) (4) (3) (2) (1) 6,7,  1,5, Q  Q 3, P DD  :  As P &  Q DD  :  (P &  Q) Pr P  Q II OO &O

16 (11) (8) (9) (10) (12) (7) (6) (5) (4) (3) (2) (1) 7,9,  R DD  :  1,5, Q   R 3,5, R 10,11,  As Q DD  :  Q As P CD  : P   Q As P  R CD  : (P  R)  (P   Q) Pr P  (Q   R) OO OO OO II

17 Not all absurdities are self-contradictions; some are just plain wacked.

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