Logic Exercise: Inspired by Murder, She Wrote, “Capitol Offense” (1985)

[1] If the suspect killed the victim, then there would be blood and makeup on his hands. [2] But if the suspect killed the victim, then he also moved the victim. [3] The suspect moved the victim’s corpse only if he touched the victim’s shirt. [4] But if the suspect touched the victim’s shirt, then anything on the suspect’s hands would be on the victim’s shirt. [5] But there’s no makeup on the victim’s shirt. So the suspect didn’t kill the victim

Let:      s = the suspect, h = the suspect’s hands, i = the victim’s shirt

Bx = x is blood

Kx = x killed the victim

            Mx = x is makeup

            Vx  = x moved the victim

            Oxy = x is on y

            Txy = x touched y

Represent the argument and prove the conclusion.

Solution:

[1] If the suspect killed the victim, then there would be blood and makeup on his hands. [2] But if the suspect killed the victim, then he also moved the victim. [3] The suspect moved the victim’s corpse only if he touched the victim’s shirt. [4] But if the suspect touched the victim’s shirt, then anything on the suspect’s hands would be on the victim’s shirt. [5] But there’s no makeup on the victim’s shirt. So the suspect didn’t kill the victim

(1)       Ks → (∃x)(∃y)(Bx & My & Oxh & Oyh)       Premise

(2)      Ks → Vs                                                           Premise

(3)      Vs → Tsi                                                          Premise

(4)      Tsi → (∀x)(Oxh → Oxi)                                 Premise

(5)      ~(∃x)(Mx & Oxi)                                           Premise

:. ~Ks

(6)                   Ks                                                        Assumption for RAA

(7)                  (∃x)(∃y)(Bx & My & Oxh & Oyh)     1, 6 Modus Ponens

(8)                  (∃y)(Ba* & My & Oa*h & Oyh)         7, ∃-Elimination

(9)                  Ba* & Mb* & Oa*h & Ob*h                8, ∃-Elimination

(10)                Vs                                                        2, 6 Modus Ponens

(11)                Tsi                                                       3, 10 Modus Ponens

(12)                (∀x)(Oxh → Oxi)                                4, 11 Modus Ponens

(13)                Ob*h                                                   9, &-Elimination

(14)                Ob*h → Ob*i                                       12, ∀-Elimination

(15)                Ob*i                                                     13, 14 Modus Ponens

(16)                Mb*                                                     9, &-Elimination

(17)                Mb* & Ob*i                                         15, 16 &-Introduction

(18)                (∃x)(Mx & Oxi)                                  17, ∃-Introduction

(19)                (∃x)(Mx & Oxi) & ~(∃x)(Mx & Oxi)            5. 18 &-Introduction

(20)    ~Ks                                                                 6-19 RAA