Logic Exercise: Arthur Conan Doyle, “The Boscombe Valley Mystery”

Part (a):

Sherlock Holmes: “However innocent he might be, he could not be such an absolute imbecile as not to see that the circumstances were very black against him. Had he appeared surprised at his own arrest, or feigned indignation at it, I should have looked upon it as highly suspicious because such surprise or anger would not be natural under the circumstances, and yet might appear to be the best policy to a scheming man.”

Let:      C = He schemes to appear innocent

            D = He feigns indignation

            I  = He’s an imbecile

            N = He’s innocent

            R = He feigns surprise

            S = He sees the evidence is against him

            T = He’s too innocent to see the evidence

Represent the following argument and prove its validity:

(1)        He sees how bad the evidence is against him unless he’s an imbecile or too innocent to see it.

(2)        But he’s neither an imbecile nor too innocent to see the evidence against him.

(3)        If he sees the evidence against him, then he schemes to appear innocent—unless he’s actually innocent.

(4)        If he schemes to appear innocent, then he feigns surprise or he feigns indignation.

(5)        But he feigned neither surprise nor indignation.

:. He’s innocent

Part (b)

Holmes: “His frank acceptance of the situation marks him as either an innocent man, or else as a man of considerable self-restraint and firmness.”

Let:      A = He accepts the situation

            F = He has firmness

N = He’s innocent

            S = He has self-restraint

Represent Holmes’ claim as “He accepts the situation, and if he accepts the situation, then he’s either an innocent man or else he has self-restraint and firmness.” Show (by a line of a truth table) that the inference from Holmes’ claim to the conclusion that “He’s innocent” would be an invalid inference. Prove that the inference becomes valid if we add the following premise: “If he has self-restraint, then he didn’t commit the crime; and if he didn’t commit the crime, then he’s innocent” (where: R = He committed the crime)

Solution:

Part (a):

(1)       S v (I v T)       Premise

(2)      ~(I v T)          Premise

(3)      S → (C v N)     Premise

(4)      C → (R v D)    Premise

(5)        ~(R v D)        Premise

:. N

(6)      S          1, 2 Disjunctive Syllogism

(7)      C v N   3, 6 Modus Ponens

(8)       ~C       4, 5 Disjunctive Syllogism

(9)      N         7, 8 Disjunctive Syllogism

Part (b):

Holmes’ Claim: A & {A → [N v (S & F)]}

Yet this doesn’t entail N:  Let A = S = F = True, but N = False. Then Holmes’ Claim is true but “He’s innocent” is false.

Added Premise: [(S → ~R) & (~R → N)]

(1)        A & {A → [N v (S & F)]}            Premise

(2)        [(S → ~R) & (~R → N)]             Premise

(3)        A                                              1, &-Elimination

(4)        A → [N v (S & F)]                     1, &-Elimination

(5)        N v (S & F)                               3, 4 Modus Ponens

(6)                    ~N                                Assumption for RAA

(7)                    (S & F)                          5, 6 Disjunctive Syllogism

(8)                    S                                  7, &-Elimination

(9)                    S → ~R                         2, &-Elimination

(10)                  ~R                                8, 9 Modus Ponens

(11)                  ~R → N                        2, & Elimination

(12)                  N                                  10, 11 Modus Ponens

(13)                  N & ~N                         6, 12 &-Introduction

(14)      ~~N                                          6-13 RAA

(15)      N                                              14, DNE