Logic Exercise: Agatha Christie, “The Idol House of Astarte” (1928)

Inspired By: Agatha Christie, “The Idol House of Astarte” (1928)

[1] Someone had the motive and opportunity to stab Sir Richard. [2] Anyone who had the opportunity to stab Sir Richard indeed stabbed him. [3] Either Sir Richard was stabbed before he fell or afterward. [4] But no one had the opportunity to stab Sir Richard if he was stabbed before he fell. [5] Elliot had the opportunity to stab Sir Richard if he was stabbed after he fell. Therefore, Elliot stabbed Sir Richard.

Let:      Mx = x had the motive to stab Sir Richard

            Ox = x had the opportunity to stab Sir Richard

            Sxy = x stabbed y

            Bx  = x was stabbed before he fell

            Ax = x was stabbed after he fell

            r = Sir Richard

Represent the argument and prove the conclusion

Solution:

(1)      (∃x)(Mx & Ox)                     Premise

(2)      (∀x) (Ox --> Sxr)                   Premise

(3)      Br v Ar                                    Premise

(4)      Br --> ~(∃x)(Ox)                   Premise

(5)      Ar --> Oe                                 Premise

∴ Ser

(6)                  Br                                Assumption for RAA

(7)                  ~(∃x)(Ox)                  4, 6 Modus Ponens

(8)                  Ma* & Oa*                  1, ∃-Elimination

(9)                  Oa*                              8, &-Elimination

(10)                (∃x)(Ox)                    9, ∃-Introduction

(11)                (∃x)(Ox) & ~(∃x)(Ox) 7, 10 &-Introduction

(12)    ~Br                                         6-11 RAA

(13)    Ar                                            3, 12 Disjunctive Syllogism

(14)    Oe                                            5, 13 Modus Ponens

(15)    Oe --> Ser                                2, ∀-Elimination

(16)    Ser                                          14, 15 Modus Ponens