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