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: 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

Logic Exercise: Columbo, “Lovely But Lethal” (1973)

Columbo: You see, we got our poison ivy in the same place. We both touched the slide. You touched it when you picked up the microscope and hit him. That's when the slide broke. I got it when I put my hand on the floor and it touched a piece of glass. I remember because I said: "Fellows, feels like there's broken glass here." The fingerprint man, he thought the glass came from a drinking glass.

Viveca: Wow, very good, Lieutenant.

Let:      Hx = x’s hands are itching

            Txy = x touched y

            Px = poison ivy is on x

            Kx = x is the killer

            Ix =  x is investigating the killing

            c = Columbo    s = the microscope-slide   v = Viveca

Represent the following argument and prove the conclusion:

(1) Anyone whose hands are itching touched something on which there was poison ivy.

(2) Poison ivy was on the microscope-slide—and only the microscope slide.

(3) Anyone who touched the microscope-slide is either the killer or investigating the killing.

(4) Columbo and Viveca’s hands are itching.

(5) Viveca is not investigating the killing.

∴ Viveca is the killer

Solution:

(1)      (∀x)(Hx → (∃y)(Txy & Py))            Premise

(2)      Ps & (∀x)(Px → x=s)                        Premise

(3)      (∀x)(Txm → (Kx v Ix))         Premise

(4)      Hc & Hv                                  Premise

(5)      ~Iv                                          Premise

∴ Kv

(6)      Hv                                           4, &-Elimination

(7)      Hv → (∃y)(Tvy & Py)           1, ∀-Elimination

(8)      (∃y)(Tvy & Py)                     6, 7 Modus Ponens

(9)      Tva* & Pa*                             8, ∃-Elimination

(10)    Pa*                                          9, &-Elimination

(11)    (∀x)(Px → x=s)                    2, &-Elimination

(12)    Pa* → a*=s                            11, ∀-Elimination

(13)    a*=s                                        10, 12 Modus Ponens

(14)    Tva*                                        9, &-Elimination

(15)    Tvs                                          13, 14 Indiscernibility of Identicals

(16)    Tvs → (Kv v Iv)                     3, ∀-Elimination

(17)    Kv v Iv                                    15, 16 Modus Ponens

(18)    Kv                                            5, 17 v-Elimination