Logic Exercise: Inspired by: Columbo, “Publish or Perish” (1974)

[1] If Eddie’s the killer, then he planted the key at the murder scene if and only if he used the key we found at his place. [2] But Eddie used the key we found at his place only if the murder was committed on Tuesday—[3] and it wasn’t. [4] Eddie’s the killer only if he was at the murder scene and planted a key there. Therefore, Eddie’s not the killer.

Let:      E = Eddie’s the killer

            K = Eddie used the key we found at his place

            M = Eddie was at the murder scene

            P = Eddie planted the key at the murder scene

            T = The murder was committed on Tuesday

Represent the argument and prove the conclusion.

Solution:

(1)      E → (P ≡ K))

(2)      (K → T)

(3)      ~T

(4)      E → (M & P)

∴          ~E

(5)                  E                                  Assumption for RAA

(6)                  P ≡ K                          1, 5 Modus Ponens

(7)                  M & P                          1, 4 Modus Ponens

(8)                  P                                  7 &-Elimination

(9)                  K                                  6, 8 ≡-Elimination

(10)                T                                  2, 9 Modus Ponens

(11)                T & ~T                        3, 10 &-Introduction

(12)    ~E                                           5-11 RAA