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: PD James "The Victim" (1973)

Inspired by PD James, “The Victim” (1973)

[1] The threatening notes are to be untraceable and if so, then I wear gloves while composing them. [2] Either I write them or I type them or I cut-and-paste letters from cut-up newspapers. [3] But if I write them, they’re not untraceable; and I can’t both wear gloves and cut-and-paste letters from cut-up newspapers. So I must type them.

Let:      C = I cut-and-paste them

            G = I wear gloves

            T = I type them

            U = The notes are to be untraceable

            W = I write them

Represent the argument and prove the conclusioin

Solution:

(1)      U & (U → G)                           Premise

(2)      W v T v C                                Premise

(3)      (W → ~U) & ~(G & C)        Premise

∴          T

(4)      U                                              5, &-Elimination

(5)      ~~U                                        4, DN

(6)      W → ~U                                  3, &-Elimination

(7)      ~W                                          5, 6 Modus Tollens

(8)      T v C                                        2, 7 Disjunctive Syllogism

(9)                  C                                  Assumption for RAA

(10)                U → G                          1, &-Elimination

(11)                G                                  4, 10 Modus Ponens

(12)                G & C                           9, 11 &-Introduction

(13)                ~(G & C)                     3 &-Elimination

(14)                (G&C) & ~(G&C)       12, 13 &-Introduction

(15)    ~C                                           9-14 RAA

(16)    T                                              8, 15 Disjunctive Syllogism

Logic Exercise: Sherlock Holmes vs. Vincent Bugliosi

“Circumstantial evidence is a very tricky thing. It may seem to point very straight to one thing, but if you shift your own point of view a little, you may find it pointing in an equally uncompromising manner to something different” (Sherlock Holmes, “The Boscombe Valley Mystery”)

Let:      Cx =     x is circumstantial evidence

Ax =     x is ambiguous evidence (“if you shift your point of view, [the evidence can point to something different”)

Represent Holmes’ claim as:

(H)       All circumstantial evidence is ambiguous.

“For those who feel a case based on circumstantial evidence is, by definition, not a strong one, let me correct a common misperception,” writes former Los Angeles District Attorney Vincent Bugliosi. “Circumstantial evidence has erroneously come to be associated in the public mind and vernacular with an anemic case….But nothing could be further from the truth. In fact, most first degree murder cases are based on circumstantial evidence. This is so because other than eye-witness testimony (and in some jurisdictions, a confession), which is direct evidence, all other evidence, even fingerprints and DNA, is circumstantial evidence.” (Vincent Bugliosi, The Prosecution of George W. Bush for Murder [Vanguard Press: New York, 2008], page 100)

Let:      Dx =     x is direct evidence

            Ex  =    x is eyewitness testimony

            Ox  =    x is a confession

            d = DNA evidence

Can Bugliosi’s line of thought support a counter-argument to Holmes’s claim? Represent the following argument and prove its validity. Evaluate the argument.

(1) Evidence that’s neither eyewitness testimony nor a confession is not direct evidence.

(2) If something’s not direct evidence, then it’s circumstantial evidence.

(3) DNA evidence is not a confession and it’s not eyewitness testimony.

(4) DNA evidence is unambiguous

Therefore, Not all circumstantial evidence is ambiguous

 Solution:

(3) Sherlock Holmes vs. Vincent Bugliosi

(H)      (∀x)(Cx → Ax)

(1)      (∀x)[~(Ex v Ox) → ~Dx)     Premise

(2)      (∀x)(~Dx → Cx)                    Premise

(3)      ~Od & ~Ed                             Premise

(4)      ~Ad                                         Premise

:. ~(∀x)(Cx → Ax)

(5)                  (∀x)(Cx → Ax)           Assumption for RAA

(6)                  ~(Ed v Od)                 DeMorgan’s Law

(7)                  ~(Ed v Od) → ~Dd    1, ∀-Elimination

(8)                  ~Dd                            6, 7 Modus Ponens

(9)                  ~Dd → Cd                   2, ∀-Elimination

(10)                Cd                                8, 9 Modus Ponens

(11)                Cd → Ad                      5, ∀-Elimination

(12)                Ad                               10, 11 Modus Ponens

(13)                Ad & ~Ad                   4, 12 & Introduction

(14) ~(∀x)(Cx → Ax)                       5-13 RAA