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: Columbo TV Movie “A Friend In Deed” (1974)

From Columbo TV Movie “A Friend In Deed” (1974)

Columbo: This is complicated. See whether or not you can help me with this. Janice Caldwell died between 10:30 and 11:00. Sometime after her husband spoke to her at 10:30 and sometime before you saw the burglar run out at 11:00. Right?

Halperin: That's correct.

Columbo: That's why I can't start the report.

Halperin: Why?

Columbo: I get stuck in the beginning, the time of death.

Halperin: What are you talking about, Columbo?

Columbo: Janice Caldwell had a boyfriend. She was supposed to meet him at 9:30. That's an hour before she was supposed to have died. She never showed up.

Halperin: So she stood him up. So what?

Columbo: Yeah, but the guy called her around 9:30, too, and there was no answer on the telephone. I have to ask myself the question, why couldn't she answer the phone? You know what I wrote down? "Maybe she was dead already." You follow my thinking?

Halperin: Maybe she just didn't want to talk to the man. Did that occur to you?

Columbo: But she answered the phone later, when her husband called. You see the contradiction. If she didn't want to speak to her boyfriend at 9:30, why did she pick up the phone at 10:30? […] You see how confusing it gets. That's why I keep going back to this. "Maybe she was dead already." 'Cause if she's dead already, that explains everything. 

Let:      A  = Janice was alive

            W = Janice wanted to talk to her boyfriend

            N  = Janice answers the phone at 9:30

            T  = Janice answers the phone at 10:30

            S  = Janice stood up her boyfriend

Represent the following argument and prove the conclusion:

(1)        If Janice was alive, but didn’t want to talk to her boyfriend, then she doesn’t

            answer the phone at 9:30 nor at 10:30.

(2)        Janice answered the phone at 10:30 but not at 9:30.

(3)        If Janice was alive, then she stood up her boyfriend.

(4)        If Janice stood up her boyfriend, then she didn’t want to talk to him.

Therefore, Janice wasn’t alive

Solution:

(1)        (A & ~W) --> (~N & ~T)            Prem.

(2)        (T & ~N)                                   Prem.

(3)        (A --> S)                                    Prem.

(4)        (S --> ~W)                                 Prem.

:.           ~A

(5)                    A                                  Assumption for RAA

(6)                    S                                  3, 5 Modus Ponens

(7)                    ~W                               4, 6 Modus Ponens

(8)                    A & ~W                        5, 6 &-Introduction

(9)                    ~N & ~T                       1, 8 Modus Ponens

(10)                  ~T                                9, &-Elimination

(11)                  T                                  2, &-Elimination

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

(13)      ~A                                            5-12 RAA

Logic Exercise: Columbo TV Movie “Negative Reaction” (1974)

From Conclusion of Columbo TV Movie “Negative Reaction” (1974)

Source: https://www.youtube.com/watch?v=e1H6jZNx8Y4&t=5150

Columbo: "You see this newspaper? This is the same edition of the newspaper

that the letters and words for the ransom note were cut out of. I'm trying

to reconstruct that note. I'll tell you what the problem is. We found this

newspaper in [Deschler’s] motel room. It was sprawled over a chair.

It was left there. Even though the maid said that she cleaned up the room."

Killer: "Yes, I told you. She probably lied."

Columbo: "That's the problem, sir.  You see this terrible mess I'm making?

There's just no way [Deschler could have] cut up this paper and paste

together this note without making a mess. My point is this: If the maid

forgot to clean up the room then why didn't we find little bits and pieces

of paper somewhere? You see the problem? If she did clean up the room,

then everything gets thrown out. All the bits and pieces of paper get thrown

out and the newspaper get thrown out.  If she doesn't clean up the room

then these scraps of paper, they have to be somewhere [in the room].

You can see the contradiction."

What is the contradiction Columbo sees?

In this exercise, we represent his argument and prove its validity.

Let:      D = Deschler made the ransom note

            C = The maid cleaned the room

            N = We found the newspaper

            S = We found newspaper scraps

Represent the following argument and prove the conclusion:

(1)        If Deschler made the ransom note and the maid cleaned the room, then we

find neither the newspaper nor its scraps.

(2)        If Deschler made the ransom note and the maid didn’t clean the room, then

            we find the newspaper and its scraps.

(3)        We found the newspaper but no newspaper scraps.

Therefore: Deschler didn’t make the ransom note

Solution:

(1)        (D & C) à (~N & ~S)                Prem  

(2)        (D & ~C) à (N & S)                  Prem

(3)        N & ~S                                     Prem

:.          ~D

(4)                    D                                  Assumption for RAA

(5)                                C                      Assumption for RAA

(6)                                D & C               4,5 &-Introduction

(7)                                ~N & ~S           1, 6 Modus Ponens

(8)                                ~N                    7 &-Elimination

(9)                                  N                    3 &-Elimination

(10)                                N & ~N           8,9 &-Introduction

(11)                  ~C                                5-10 RAA

(12)                  D & ~C                         4, 11 &-Introduction

(13)                  N & S                           2, 12 Modus Ponens

(14)                  S                                  13, &-Elimination

(15)                  ~S                                3, &-Elimination

(16)                  S & ~S                          14,15 &-Introduction

(17)      ~D                                            4-16 RAA