Logic Exercise: Columbo, “Lovely But Lethal” (1973)

Columbo: You see, we got our poison ivy in the same place. We both touched the slide. You touched it when you picked up the microscope and hit him. That's when the slide broke. I got it when I put my hand on the floor and it touched a piece of glass. I remember because I said: "Fellows, feels like there's broken glass here." The fingerprint man, he thought the glass came from a drinking glass.

Viveca: Wow, very good, Lieutenant.

Let:      Hx = x’s hands are itching

            Txy = x touched y

            Px = poison ivy is on x

            Kx = x is the killer

            Ix =  x is investigating the killing

            c = Columbo    s = the microscope-slide   v = Viveca

Represent the following argument and prove the conclusion:

(1) Anyone whose hands are itching touched something on which there was poison ivy.

(2) Poison ivy was on the microscope-slide—and only the microscope slide.

(3) Anyone who touched the microscope-slide is either the killer or investigating the killing.

(4) Columbo and Viveca’s hands are itching.

(5) Viveca is not investigating the killing.

∴ Viveca is the killer

Solution:

(1)      (∀x)(Hx → (∃y)(Txy & Py))            Premise

(2)      Ps & (∀x)(Px → x=s)                        Premise

(3)      (∀x)(Txm → (Kx v Ix))         Premise

(4)      Hc & Hv                                  Premise

(5)      ~Iv                                          Premise

∴ Kv

(6)      Hv                                           4, &-Elimination

(7)      Hv → (∃y)(Tvy & Py)           1, ∀-Elimination

(8)      (∃y)(Tvy & Py)                     6, 7 Modus Ponens

(9)      Tva* & Pa*                             8, ∃-Elimination

(10)    Pa*                                          9, &-Elimination

(11)    (∀x)(Px → x=s)                    2, &-Elimination

(12)    Pa* → a*=s                            11, ∀-Elimination

(13)    a*=s                                        10, 12 Modus Ponens

(14)    Tva*                                        9, &-Elimination

(15)    Tvs                                          13, 14 Indiscernibility of Identicals

(16)    Tvs → (Kv v Iv)                     3, ∀-Elimination

(17)    Kv v Iv                                    15, 16 Modus Ponens

(18)    Kv                                            5, 17 v-Elimination

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