Logic Exercise: Columbo “It’s All In the Game” (1993)

Columbo “It’s All In the Game” (1993)

Columbo: About last night, when you went to Nick's apartment. And believe me, this is not important. I'm merely trying to pin down how you knew that Nick wasn't home. Last night, how you knew that Nick wasn't in his apartment….How did you know he wasn't there?

Lauren: Well, his car wasn't in the garage.

Columbo: Oh, but it's such a big garage, how could you be sure?

Lauren: No, but I parked in his space. It was empty.

Columbo: Oh, you parked in his space. Oh, that's how you knew. OK. But then when Nick came home he would've seen your car.

Lauren: Yeah, I guess he would, yes.

Columbo: Well, that's funny, then….If he thought you were in the apartment, why would he let himself in with his keys?

Lauren: Well, I would think, Lieutenant, that he rang the bell. Because I was with the manager, no one answered, and so he let himself in with his key.

Columbo: Oh, right. You were with the manager. Right. So he rang and rang and then…the trouble with that is that ringing would have warned the thieves and they would've run out.

Lauren: Well, maybe it did, but they didn't have enough time to get all the way out.

Columbo: Oh, OK, I see what you're saying. In other words, they started to run but he came in too fast. OK. Uh, no, no good. You see, the problem with that is the direction of the bullet that shot him came from his left. From someone standing in the hallway that led to the bedroom. If the thieves were running out and they shot him, the bullet would have come from the direction of the sliding glass doors. Oh, I tell you, I don't know. This case, it's It has me stumped.

Let:      A = Nick alerts the burglars

            B = Nick believes Lauren is there

            E = The burglars escaped

            G = Nick is shot from the direction of the glass door

            K = Nick enters with his key

            P = Lauren parked in Nick’s space

            R = Nick rings the bell

            S = Nick is shot

            T = The burglars partially escape

            V = Lauren visits with the manager

Represent the following six premises and prove they are inconsistent—i.e., they entail a contradiction of the form “X & ~X”:

(1)        Lauren parked in Nick’s space and visits with the manager.

(2)        If Lauren parks in Nick’s space, he believes she’s there and rings the bell.

(3)        If Nick rings the bell but she visits with the manager, then Nick enters with his key.

(4)        If Nick rings the bell, then he alerts the burglars.

(5)        If Nick alerts the burglars and enters with his key then either they escape (and Nick isn’t

            shot) or else they partially escape (and Nick is shot from the direction of the glass door).

(6)        Nick is shot but not from the direction of the glass door.

Solution:

(1)       P & V                                       Premise

(2)      P → (B & R)                            Premise

(3)      (R & V) → K                            Premise

(4)      R → A                                      Premise

(5)      (A & K) → [(E & ~S) v (T & G)] Premise

(6)      S & ~G

(7)       P                                              1, &-Elimination

(8)      B & R                                       2, 7 Modus Ponens

(9)      R                                              8, &-Elimination

(10)    V                                              1, &-Elimination

(11)    R & V                                       9, 10 &-Introduction

(12)    K                                              3, 11 Modus Ponens

(13)    A                                              4, 9 Modus Ponens

(14)    A & K                                       12, 13 &-Introduction

(15)    (E & ~S) v (T & G)                 5, 14 Modus Ponens

(16)                (E & ~S)                     Assumption for RAA

(17)                S                                  6, &-Elimination

(18)                ~S                               16, &-Elimination

(19)                S & ~S                         17, 18 &-Introduction

(20)    ~(E & ~S)                              16-19 RAA

(21)    T & G                                       15, 20 Disjunctive Syllogism

(22)    G                                              21 &-Elimination

(23)    ~G                                           6, &-Elimination

(24)    G & ~G                                    22, 23 &-Introduction

∴ Premises (1) through (6) are inconsistent.