Logic Exericse:

(1) There are three people in the story: The sister, her nephew and Doc Brady

(2) Someone in the story sent the letter to the sister

(3) Anyone who sent the letter to the sister had a motive

(4) But the nephew didn’t have a motive—and neither did Doc Brady

:. The sister send the letter to herself

Let:      s = the sister n = the nephew d = Doc Brady

            Sx = x is in the story

            Exy = x sent the letter to y

            Mx = x had a motive

Represent the argument and prove the conclusion.

(1)       (Ss & Sn & Sd) & (∀x)(Sx → (x=s v x=n v x=d)

(2)       (∃x)(Sx & Exs)

(3)       (∀x)[(Exs → Mx]

(4)       ~Mn & ~Md

:. Ess

(5)      Sa* & Ea*s                                         2, ∃-Elimination

(6)       Sa*                                                      5, &-Elimination

(7)       (∀x)(Sx → (x=s v x=n v x=d)          1, &-Elimination

(8)      Sa* → (a*=s v a*=n v a*=d)           7, ∀-Elimination

(9)      (a*=s v a*=n v a*=d)                      6, 8 Modus Ponens

(10)                a*=n                                       Assumption for RAA

(11)                Ea*s                                        5, &-Elimination

(12)                Ens                                          10, 11 =Elimination

(13)                Ens → Mn                               3, ∀-Elimination

(14)                Mn                                           12, 13 Modus Ponens

(15)                ~Mn                                        4, &-Elimination

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

(17)    ~(a*=n)                                             10-16 RAA

(18)    (a*=s v a*=d)                                    9, 17 Disjunctive Syllogism

(19)                a* = d                                     Assumption for RAA

(20)                Ea*s                                        5, &-Elimination

(21)                Eds                                          19, 20 =Elimination

(22)                Eds → Md                               3, ∀-Elimination

(23)                Md                                           21, 22 Modus Ponens

(24)                ~Md                                        4, &-Elimination

(25)                Md & ~Md                              23, 24 &-Introduction

(26)    ~(a*=d)                                             19-25 RAA

(27)    a*=s                                                    18, 26 Disjunctive Syllogism

(28)    Ess                                                      11, 27 =Elimination

Logic Exercise: From Rex Stout, “Black Orchids” (1941)

“I stood and applied logic to it. Had he taken a taxi home? Not the way he hated taxis. What, as I had left him standing there, what had been his most burning desires? That was easy. To shoot me, to sit down, and to drink beer. He couldn’t shoot me because I wasn’t there. Where might he have found a chair?”

Let:      A = He hates taxis

            B = He’s drinking beer

            C = I should look for a chair

            N = I’m not there

            H = He’s trying to shoot me

            S = He’s sitting

            T = He took a taxi

Represent the following argument and prove the conclusion:

(1)        Either he took a taxi or else he’s trying to shoot me or else he’s sitting and drinking beer.

(2)        He didn’t take a taxi if he hates taxis and he’s not trying to shoot me if I’m not there.

(3)        I’m not there and he hates taxis.

(4)        I should look for a chair if he’s sitting.

:. He’s drinking beer and I should look for a chair.

Note: The first premise can be parenthesized as “[Either he took a taxi or else (he’s trying to shoot me or else (he’s sitting and drinking beer))].” 

Solution:

(1)      [T v (H v (S & B))]                 Prem.

(2)       (A → ~T) & (~N → ~H)        Prem.

(3)      ~H & A                                    Prem.

(4)      S → C                                       Prem.

∴ B & C

(5)      A                                              3, &-Elimination

(6)      A → ~T                                   2, &-Elimination

(7)      ~T                                           5, 6 Modus Ponens

(8)      H v (S & B)                             1, 7 Disjunctive Syllogism

(9)      ~H                                           3, &-Elimination

(10)    S & B                                       8, 9 Disjunctive Syllogism

(11)    S                                              10, &-Elimination

(12)    C                                              4, 11 Modus Ponens

(13)    B                                              10, &-Elimination

(14)    B & C                                       12, 13 &-Introduction