Introductory Logic PHI 120

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Introductory Logic PHI 120 Complex Problems II Introductory Logic PHI 120

Grading, attendance, accommodation issues RECITATION INSTRUCTOR REMINDER Grading, attendance, accommodation issues Dealt with by your RECITATION INSTRUCTOR

Homework Proofs Handout: Problem Set A: S1 - S27 Problem Set B: T1 – T4 Problem Set C: S34 - S49 Problem Set D: T5 - T7

Homework (1) Solve S39 – S49 from Proofs Handout T5 – T7 Come to review with specific problems (3) Study Guide 2

Mathematical Concepts -- Commutativity -- -- Distribution -- -- Associativity -- Problems from Ex. 1.5.1

Commutativity 2 x 2 Question Method Mathematical Concepts P & Q ⊣⊢ Q & P P v Q ⊣⊢ Q v P 2 x 2 Question Method P <-> Q ⊣⊢ Q <-> P

PvQ ⊣⊢ QvP

PvQ ⊣⊢ QvP P v Q ⊢ Q v P Q v P ⊢ P v Q

PvQ ⊣⊢ QvP The Two Questions Think vE (why?) P v Q ⊢ Q v P (1) P v Q A Think vI (why?) P v Q ⊢ Q v P (1) P v Q A (2) Q v P ⊢ P v Q Think vE (why?) The Two Questions

PvQ ⊣⊢ QvP P v Q ⊢ Q v P (1) P v Q A (2) A Q v P ⊢ P v Q

PvQ ⊣⊢ QvP P v Q ⊢ Q v P (1) P v Q A (2) ~P A Q v P ⊢ P v Q

PvQ ⊣⊢ QvP P v Q ⊢ Q v P (1) P v Q A 2 (2) ~P A Q v P ⊢ P v Q

PvQ ⊣⊢ QvP P v Q ⊢ Q v P (1) P v Q A (2) ~P A (3) Q v P ⊢ P v Q

PvQ ⊣⊢ QvP P v Q ⊢ Q v P (1) P v Q A (2) ~P A (3) 1,2 vE Q v P ⊢ P v Q

PvQ ⊣⊢ QvP P v Q ⊢ Q v P (1) P v Q A (2) ~P A (3) Q 1,2 vE

PvQ ⊣⊢ QvP P v Q ⊢ Q v P (1) P v Q A (2) ~P A 1,2 (3) Q 1,2 vE

PvQ ⊣⊢ QvP P v Q ⊢ Q v P (1) P v Q A (2) ~P A 1,2 (3) Q 1,2 vE (4)

PvQ ⊣⊢ QvP vI P v Q ⊢ Q v P (1) P v Q A (2) ~P A 1,2 (3) Q 1,2 vE (4) ?? Q v P ⊢ P v Q

PvQ ⊣⊢ QvP vI P v Q ⊢ Q v P (1) P v Q A (2) ~P A 1,2 (3) Q 1,2 vE

PvQ ⊣⊢ QvP vI P v Q ⊢ Q v P (1) P v Q A (2) ~P A 1,2 (3) Q 1,2 vE

PvQ ⊣⊢ QvP The Two Questions vI P v Q ⊢ Q v P (1) P v Q A (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI Q v P ⊢ P v Q The Two Questions Either ->I or RAA

PvQ ⊣⊢ QvP P v Q ⊢ Q v P (1) P v Q A (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI (5) Q v P ⊢ P v Q RAA Strategy

PvQ ⊣⊢ QvP P v Q ⊢ Q v P (1) P v Q A (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI (5) A Q v P ⊢ P v Q RAA Strategy

PvQ ⊣⊢ QvP P v Q ⊢ Q v P (1) P v Q A (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI (5) ~(Q v P) A Q v P ⊢ P v Q RAA Strategy

PvQ ⊣⊢ QvP P v Q ⊢ Q v P (1) P v Q A (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI 5 (5) ~(Q v P) A Q v P ⊢ P v Q RAA Strategy Remember: assumptions take the number of the line

PvQ ⊣⊢ QvP P v Q ⊢ Q v P (1) P v Q A (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI 5 (5) ~(Q v P) A Q v P ⊢ P v Q

PvQ ⊣⊢ QvP P v Q ⊢ Q v P (1) P v Q A (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI (5) ~(Q v P) A (6) Q v P ⊢ P v Q

PvQ ⊣⊢ QvP P v Q ⊢ Q v P (1) P v Q A (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI (5) ~(Q v P) A (6) 4,5 RAA(?) Q v P ⊢ P v Q

PvQ ⊣⊢ QvP P v Q ⊢ Q v P (1) P v Q A (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI (5) ~(Q v P) A (6) 4,5 RAA(?) Q v P ⊢ P v Q “save the best for last”

PvQ ⊣⊢ QvP P v Q ⊢ Q v P (1) P v Q A (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI (5) ~(Q v P) A (6) 4,5 RAA(2) Q v P ⊢ P v Q

PvQ ⊣⊢ QvP P v Q ⊢ Q v P (1) P v Q A (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI (5) ~(Q v P) A (6) P 4,5 RAA(2) Q v P ⊢ P v Q

PvQ ⊣⊢ QvP P v Q ⊢ Q v P (1) P v Q A (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI 5 (5) ~(Q v P) A 1,5 (6) P 4,5 RAA(2) Q v P ⊢ P v Q

PvQ ⊣⊢ QvP P v Q ⊢ Q v P (1) P v Q A (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI (5) ~(Q v P) A 1,5 (6) P 4,5 RAA(2) (7) Q v P ⊢ P v Q

PvQ ⊣⊢ QvP vI P v Q ⊢ Q v P (1) P v Q A (2) ~P A 1,2 (3) Q 1,2 vE (5) ~(Q v P) A 1,5 (6) P 4,5 RAA(2) (7) ?? Q v P ⊢ P v Q

PvQ ⊣⊢ QvP vI P v Q ⊢ Q v P (1) P v Q A (2) ~P A 1,2 (3) Q 1,2 vE (5) ~(Q v P) A 1,5 (6) P 4,5 RAA(2) (7) 6 vI Q v P ⊢ P v Q

PvQ ⊣⊢ QvP vI P v Q ⊢ Q v P (1) P v Q A (2) ~P A 1,2 (3) Q 1,2 vE (5) ~(Q v P) A 1,5 (6) P 4,5 RAA(2) (7) Q v P 6 vI Q v P ⊢ P v Q

PvQ ⊣⊢ QvP The Two Questions vI P v Q ⊢ Q v P (1) P v Q A (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI (5) ~(Q v P) A 1,5 (6) P 4,5 RAA(2) 1,5 (7) Q v P 6 vI Q v P ⊢ P v Q The Two Questions

PvQ ⊣⊢ QvP P v Q ⊢ Q v P (1) P v Q A (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI (5) ~(Q v P) A 1,5 (6) P 4,5 RAA(2) 1,5 (7) Q v P 6 vI (8) Q v P ⊢ P v Q

PvQ ⊣⊢ QvP P v Q ⊢ Q v P (1) P v Q A (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI (5) ~(Q v P) A 1,5 (6) P 4,5 RAA(2) 1,5 (7) Q v P 6 vI (8) 5,7 RAA(5) Q v P ⊢ P v Q

PvQ ⊣⊢ QvP P v Q ⊢ Q v P (1) P v Q A (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI (5) ~(Q v P) A 1,5 (6) P 4,5 RAA(2) 1,5 (7) Q v P 6 vI (8) Q v P 5,7 RAA(5) Q v P ⊢ P v Q

PvQ ⊣⊢ QvP P v Q ⊢ Q v P (1) P v Q A (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI 5 (5) ~(Q v P) A 1,5 (6) P 4,5 RAA(2) 1,5 (7) Q v P 6 vI (8) Q v P 5,7 RAA(5) Q v P ⊢ P v Q

PvQ ⊣⊢ QvP P v Q ⊢ Q v P (1) P v Q A (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI (5) ~(Q v P) A 1,5 (6) P 4,5 RAA(2) 1,5 (7) Q v P 6 vI 1 (8) Q v P 5,7 RAA(5) Q v P ⊢ P v Q

PvQ ⊣⊢ QvP P v Q ⊢ Q v P (1) P v Q A (2) ~P A 1,2 (3) Q 1,2 vE 1,2 (4) Q v P 3 vI (5) ~(Q v P) A 1,5 (6) P 4,5 RAA(2) 1,5 (7) Q v P 6 vI 1 (8) Q v P 5,7 RAA(5) Q v P ⊢ P v Q Solve on your own

Distribution Mathematical Concepts P & (Q v R) ⊣⊢ (P & Q) v (P & R) P v (Q & R) ⊣⊢ (P v Q) & (P v R)

Distribution Mathematical Concepts P & (Q v R) ⊣⊢ (P & Q) v (P & R) P v (Q & R) ⊣⊢ (P v Q) & (P v R)

P&(QvR) ⊣⊢ (P&Q)v(P&R)

P&(QvR) ⊣⊢ (P&Q)v(P&R) Do this left side one on your own (P & Q) v (P & R) ⊢ P & (Q v R)

P&(QvR) ⊣⊢ (P&Q)v(P&R) (P & Q) v (P & R) ⊢ P & (Q v R) 1 (1) (P & Q) v (P & R) A

P&(QvR) ⊣⊢ (P&Q)v(P&R) (P & Q) v (P & R) ⊢ P & (Q v R) 1 (1) (P & Q) v (P & R) A (2) ?? Recognize the RAA! If forced to add assumptions, you may begin with the proper strategy.

P&(QvR) ⊣⊢ (P&Q)v(P&R) (P & Q) v (P & R) ⊢ P & (Q v R) 1 (1) (P & Q) v (P & R) A 2 (2) ~(P & (Q v R)) A

P&(QvR) ⊣⊢ (P&Q)v(P&R) (P & Q) v (P & R) ⊢ P & (Q v R) 1 (1) (P & Q) v (P & R) A (possible vE) 2 (2) ~(P & (Q v R)) A (3) ??

P&(QvR) ⊣⊢ (P&Q)v(P&R) (P & Q) v (P & R) ⊢ P & (Q v R) 1 (1) (P & Q) v (P & R) A (disjunction) 2 (2) ~(P & (Q v R)) A 3 (3) ~(P & Q) A (denial of left disjunct)

P&(QvR) ⊣⊢ (P&Q)v(P&R) (P & Q) v (P & R) ⊢ P & (Q v R) 1 (1) (P & Q) v (P & R) A (disjunction) 2 (2) ~(P & (Q v R)) A 3 (3) ~(P & Q) A (denial of left disjunct) 1,3 (4) P & R 1, 3 vE (the other disjunct)

P&(QvR) ⊣⊢ (P&Q)v(P&R) (P & Q) v (P & R) ⊢ P & (Q v R) 1 (1) (P & Q) v (P & R) A 2 (2) ~(P & (Q v R)) A 3 (3) ~(P & Q) A 1,3 (4) P & R 1, 3 vE

P&(QvR) ⊣⊢ (P&Q)v(P&R) (P & Q) v (P & R) ⊢ P & (Q v R) 1 (1) (P & Q) v (P & R) A 2 (2) ~(P & (Q v R)) A 3 (3) ~(P & Q) A 1,3 (4) P & R 1, 3 vE 1,3 (5) P 4 &E

P&(QvR) ⊣⊢ (P&Q)v(P&R) (P & Q) v (P & R) ⊢ P & (Q v R) 1 (1) (P & Q) v (P & R) A 2 (2) ~(P & (Q v R)) A 3 (3) ~(P & Q) A 1,3 (4) P & R 1, 3 vE 1,3 (5) P 4 &E 1,3 (6) R 4 &E

P&(QvR) ⊣⊢ (P&Q)v(P&R) (P & Q) v (P & R) ⊢ P & (Q v R) 1 (1) (P & Q) v (P & R) A 2 (2) ~(P & (Q v R)) A 3 (3) ~(P & Q) A 1,3 (4) P & R 1, 3 vE 1,3 (5) P 4 &E 1,3 (6) R 4 &E 1,3 (7) Q v R 6 vI

P&(QvR) ⊣⊢ (P&Q)v(P&R) (P & Q) v (P & R) ⊢ P & (Q v R) 1 (1) (P & Q) v (P & R) A 2 (2) ~(P & (Q v R)) A 3 (3) ~(P & Q) A 1,3 (4) P & R 1, 3 vE 1,3 (5) P 4 &E 1,3 (6) R 4 &E 1,3 (7) Q v R 6 vI 1,3 (8) P & (Q v R) 5, 7 &I

P&(QvR) ⊣⊢ (P&Q)v(P&R) (P & Q) v (P & R) ⊢ P & (Q v R) 1 (1) (P & Q) v (P & R) A 2 (2) ~(P & (Q v R)) A 3 (3) ~(P & Q) A 1,3 (4) P & R 1, 3 vE 1,3 (5) P 4 &E 1,3 (6) R 4 &E 1,3 (7) Q v R 6 vI 1,3 (8) P & (Q v R) 5, 7 &I

P&(QvR) ⊣⊢ (P&Q)v(P&R) (P & Q) v (P & R) ⊢ P & (Q v R) 1 (1) (P & Q) v (P & R) A 2 (2) ~(P & (Q v R)) A 3 (3) ~(P & Q) A 1,3 (4) P & R 1, 3 vE 1,3 (5) P 4 &E 1,3 (6) R 4 &E 1,3 (7) Q v R 6 vI 1,3 (8) P & (Q v R) 5, 7 &I

P&(QvR) ⊣⊢ (P&Q)v(P&R) (P & Q) v (P & R) ⊢ P & (Q v R) 1 (1) (P & Q) v (P & R) A 2 (2) ~(P & (Q v R)) A 3 (3) ~(P & Q) A 1,3 (4) P & R 1, 3 vE 1,3 (5) P 4 &E 1,3 (6) R 4 &E 1,3 (7) Q v R 6 vI 1,3 (8) P & (Q v R) 5, 7 &I (9) 2,8 RAA(?)

P&(QvR) ⊣⊢ (P&Q)v(P&R) (P & Q) v (P & R) ⊢ P & (Q v R) 1 (1) (P & Q) v (P & R) A 2 (2) ~(P & (Q v R)) A 3 (3) ~(P & Q) A 1,3 (4) P & R 1, 3 vE 1,3 (5) P 4 &E 1,3 (6) R 4 &E 1,3 (7) Q v R 6 vI 1,3 (8) P & (Q v R) 5, 7 &I (9) P & Q 2,8 RAA(3)

P&(QvR) ⊣⊢ (P&Q)v(P&R) (P & Q) v (P & R) ⊢ P & (Q v R) 1 (1) (P & Q) v (P & R) A 2 (2) ~(P & (Q v R)) A 3 (3) ~(P & Q) A 1,3 (4) P & R 1, 3 vE 1,3 (5) P 4 &E 1,3 (6) R 4 &E 1,3 (7) Q v R 6 vI 1,3 (8) P & (Q v R) 5, 7 &I 1,2 (9) P & Q 2,8 RAA(3)

P&(QvR) ⊣⊢ (P&Q)v(P&R) (P & Q) v (P & R) ⊢ P & (Q v R) 1 (1) (P & Q) v (P & R) A 2 (2) ~(P & (Q v R)) A 3 (3) ~(P & Q) A 1,3 (4) P & R 1, 3 vE 1,3 (5) P 4 &E 1,3 (6) R 4 &E 1,3 (7) Q v R 6 vI 1,3 (8) P & (Q v R) 5, 7 &I 1,2 (9) P & Q 2,8 RAA(3)

P&(QvR) ⊣⊢ (P&Q)v(P&R) (P & Q) v (P & R) ⊢ P & (Q v R) 1 (1) (P & Q) v (P & R) A 2 (2) ~(P & (Q v R)) A 3 (3) ~(P & Q) A 1,3 (4) P & R 1, 3 vE 1,3 (5) P 4 &E 1,3 (6) R 4 &E 1,3 (7) Q v R 6 vI 1,3 (8) P & (Q v R) 5, 7 &I 1,2 (9) P & Q 2,8 RAA(3) 1,2 (10) P 9 &E

P&(QvR) ⊣⊢ (P&Q)v(P&R) (P & Q) v (P & R) ⊢ P & (Q v R) 1 (1) (P & Q) v (P & R) A 2 (2) ~(P & (Q v R)) A 3 (3) ~(P & Q) A 1,3 (4) P & R 1, 3 vE 1,3 (5) P 4 &E 1,3 (6) R 4 &E 1,3 (7) Q v R 6 vI 1,3 (8) P & (Q v R) 5, 7 &I 1,2 (9) P & Q 2,8 RAA(3) 1,2 (10) P 9 &E 1,2 (11) Q 9 &E

P&(QvR) ⊣⊢ (P&Q)v(P&R) (P & Q) v (P & R) ⊢ P & (Q v R) 1 (1) (P & Q) v (P & R) A 2 (2) ~(P & (Q v R)) A 3 (3) ~(P & Q) A 1,3 (4) P & R 1, 3 vE 1,3 (5) P 4 &E 1,3 (6) R 4 &E 1,3 (7) Q v R 6 vI 1,3 (8) P & (Q v R) 5, 7 &I 1,2 (9) P & Q 2,8 RAA(3) 1,2 (10) P 9 &E 1,2 (11) Q 9 &E 1,2 (12) Q v R 11 vI

P&(QvR) ⊣⊢ (P&Q)v(P&R) (P & Q) v (P & R) ⊢ P & (Q v R) 1 (1) (P & Q) v (P & R) A 2 (2) ~(P & (Q v R)) A 3 (3) ~(P & Q) A 1,3 (4) P & R 1, 3 vE 1,3 (5) P 4 &E 1,3 (6) R 4 &E 1,3 (7) Q v R 6 vI 1,3 (8) P & (Q v R) 5, 7 &I 1,2 (9) P & Q 2,8 RAA(3) 1,2 (10) P 9 &E 1,2 (11) Q 9 &E 1,2 (12) Q v R 11 vI 1,2 (13) P & (Q v R) 10, 12 &I

P&(QvR) ⊣⊢ (P&Q)v(P&R) (P & Q) v (P & R) ⊢ P & (Q v R) 1 (1) (P & Q) v (P & R) A 2 (2) ~(P & (Q v R)) A 3 (3) ~(P & Q) A 1,3 (4) P & R 1, 3 vE 1,3 (5) P 4 &E 1,3 (6) R 4 &E 1,3 (7) Q v R 6 vI 1,3 (8) P & (Q v R) 5, 7 &I 1,2 (9) P & Q 2,8 RAA(3) 1,2 (10) P 9 &E 1,2 (11) Q 9 &E 1,2 (12) Q v R 11 vI 1,2 (13) P & (Q v R) 10, 12 &I

P&(QvR) ⊣⊢ (P&Q)v(P&R) (P & Q) v (P & R) ⊢ P & (Q v R) 1 (1) (P & Q) v (P & R) A 2 (2) ~(P & (Q v R)) A 3 (3) ~(P & Q) A 1,3 (4) P & R 1, 3 vE 1,3 (5) P 4 &E 1,3 (6) R 4 &E 1,3 (7) Q v R 6 vI 1,3 (8) P & (Q v R) 5, 7 &I 1,2 (9) P & Q 2,8 RAA(3) 1,2 (10) P 9 &E 1,2 (11) Q 9 &E 1,2 (12) Q v R 11 vI 1,2 (13) P & (Q v R) 10, 12 &I 1 (14) P & (Q v R) 2, 13 RAA (2)

P&(QvR) ⊣⊢ (P&Q)v(P&R) (P & Q) v (P & R) ⊢ P & (Q v R) 1 (1) (P & Q) v (P & R) A 2 (2) ~(P & (Q v R)) A 3 (3) ~(P & Q) A 1,3 (4) P & R 1, 3 vE 1,3 (5) P 4 &E 1,3 (6) R 4 &E 1,3 (7) Q v R 6 vI 1,3 (8) P & (Q v R) 5, 7 &I 1,2 (9) P & Q 2,8 RAA(3) 1,2 (10) P 9 &E 1,2 (11) Q 9 &E 1,2 (12) Q v R 11 vI 1,2 (13) P & (Q v R) 10, 12 &I 1 (14) P & (Q v R) 2, 13 RAA (2) Distribution: Complex? Yes. Hard? Moderately so.

Associativity Do these problems at home Mathematical Concepts P & (Q & R) ⊣⊢ (P & Q) & R Do these problems at home P v (Q v R) ⊣⊢ (P v Q) v R

Homework (1) Solve S39 – S49 from Proofs Handout T5 – T7 Come to review with specific problems (3) Study Guide 2