Mr. Smith was murdered in Kansas City. The police determined that the time of death was between 11:10 pm and 11:30 pm. Four suspects were questioned: Butler, the butler; Cook, the cook; Ruby, the maid; and Irma, Mr. Smith’s secretary.

The suspects made the following statements:
Butler: I did not do it. Irma did it. Mr. Smith was blackmailing Irma. Ruby and I were watching television together from 10:10 p.m. until 12:30 a.m.
Cook: I am innocent. Irma was being blackmailed. Butler murdered Mr. Smith. I saw Irma leave the house at 10:00 p.m.
Ruby: I am innocent. Butler and I were watching television together at the time of the murder. Irma was being blackmailed. I saw Irma speaking to Mr. Smith at 9:30 p.m. on the night of the murder.
Irma: I did not kill Mr. Smith. I was not being blackmailed. I was in St. Louis during the entire night of the murder. Ruby is the murderess.

If each suspect made exactly two true statements and told exactly two lies, determine who killed Mr. Smith.

(In reply to No Subject by ollie)

The proof given on March 20 has been improved in three aspects:

The total number of steps has been shortened (from 102 to 67),
as the 44 conditional or biconditional connections from the suspects' statements (previously 'premises 12 - 55') could be reduced minus 34 to only those 10 that are necessary (now 'premises 12 - 21').
So at the start we have 21 premises (instead of 55 before).

The old step no. 58 has been rectified. It was not entirely correct to conclude from
the given premise: (B v R) -> ~B4
and the assumption: B
directly to
~B4
by the justification '-> Elimination'.

We can take the logical equivalence of the given premise right from the start:
(B -> ~B4) & (R -> ~B4)
then we get
B -> ~B4
by '& Elimination'
and now we get
~B4
by the justification '-> Elimination'.
(to find now in the steps no. 24 and 25)

In the 'subproof cook' and in the 'sub-subproof cook' we can eliminate in each of them one step (the old steps no. 80 and 87).
As we have already a contradiction by arriving at 'B', it is not necessary to create the contradiction 'B & C' by another step.

The improved proof as a whole:

Propositional variables:

B                 =    Butler is the murderer
B1 - B4        =    each for the true statements of Butler

C                 =    Cook is the murderer
C1 - C4        =    each for the true statements of Cook

I                  =    Irma is the murderess
I1 - I4          =    each for the true statements of Irma

R                 =    Ruby is the murderess
R1 - R4        =    each for the true statements of Ruby


Premises 1 - 7:
The murderer was one of the four suspects and acted alone:

 1. B v C v R v I
 2. ~(B & R)
 3. ~(B & C)
 4. ~(B & I)
 5. ~(R & C)
 6. ~(R & I)
 7. ~(C & I)


Premises 8 - 11:
Each suspect made two true statements and told two lies:
        
 8. ( ((B1 & B2) & ~(B3 v B4)) v ((B1 & B3) & ~(B2 v B4))
       v ((B1 & B4) & ~(B2 v B3)) v ((B3 & B4) & ~(B1 v B2))
       v ((B2 & B4) & ~(B1 v B3)) v ((B2 & B3) & ~(B1 v B4)) )
 9. ( ((R1 & R2) & ~(R3 v R4)) v ((R1 & R3) & ~(R2 v R4))
       v ((R1 & R4) & ~(R2 v R3)) v ((R3 & R4) & ~(R1 v R2))
       v ((R2 & R4) & ~(R1 v R3)) v ((R2 & R3) & ~(R1 v R4)) )
10. ( ((C1 & C2) & ~(C3 v C4)) v ((C1 & C3) & ~(C2 v C4))
       v ((C1 & C4) & ~(C2 v C3)) v ((C3 & C4) & ~(C1 v C2))
       v ((C2 & C4) & ~(C1 v C3)) v ((C2 & C3) & ~(C1 v C4)) )
11. ( ((I1 & I2) & ~(I3 v I4)) v ((I1 & I3) & ~(I2 v I4))
       v ((I1 & I4) & ~(I2 v I3)) v ((I3 & I4) & ~(I1 v I2))
       v ((I2 & I4) & ~(I1 v I3)) v ((I2 & I3) & ~(I1 v I4)) )


Premises 12 - 21:
Some conditional or biconditional connections from the suspects' statements:

12. (B2 <-> I)
13. (C3 <-> B)
14. (C4 -> ~I3)
15. (I2 <-> ~C2)
16. (I3 -> ~I)
17. (~B1 <-> B)
18. (B -> ~B4) & (R -> ~B4)
19. (~C1 <-> C)
20. (~I1 <-> I)
21. (~I4 <-> ~R)
____

    22. B                      SUBPROOF 'Butler'
    ---
    23. ~B1                  <-> Elim. 17,22
    24. B -> ~B4           & Elim. 18
    25. ~B4                  -> Elim. 22,24
    26. ~B1 & ~B4        & Intro 23,25
    27. B2 & B3             Tautol.Conclus. 8,26
    28. B2                     & Elim. 27
    29. I                       <-> Elim. 12,28
    30. B & I                  & Intro 22,29
    31. _|_                   Contrad.Intro 4,30

32. ~B                        ~ Intro 22 - 31

    33. C                       SUBPROOF 'Cook'
    ---
    34. ~C1                   <-> Elim. 19,33
    35. ~R                     Disj.Syll. 5,33
    36. ~I4                    <-> Elim. 21,35

        37. C4                  SUB-SUBPROOF 'Cook'
        ---
        38. ~I3                 -> Elim. 14,37
        39. ~I3 & ~I4        & Intro 36,38
        40. I1 & I2             Tautol.Conclus. 11,39
        41. I2                    & Elim. 40
        42. ~C2                 <-> Elim. 15,41
        43. ~C1 & ~C2       & Intro 34,42
        44. C3 & C4           Tautol.Conclus. 10,43
        45. C3                   & Elim. 44
        46. B                     <-> Elim. 13,45
        47. _|_                  Contrad.Intro 32,46

    48. ~C4                     ~ Intro 37 - 47

    49. ~C1 & ~C4           & Intro 34,48
    50. C2 & C3                Tautol.Conclus. 10,49
    51. C3                        & Elim. 50
    52. B                          <-> Elim. 13,51
    53. _|_                       Contrad.Intro 32,52       

54. ~C                           ~ Intro 33 - 53

    55. I                          SUBPROOF 'Irma'
    ---
    56. ~I1                      <-> Elim. 20,55
    57. ~R                       Disj.Syll. 6,55
    58. ~I4                      <-> Elim. 21,57
    59. ~I1 & ~I4             & Intro 56,58
    60. I2 & I3                  Tautol.Conclus. 11,59
    61. I3                         & Elim. 60
    62. ~I                         -> Elim. 16,61
    63. _|_                       Contrad.Intro 55,62
 
64. ~I                            ~ Intro 55-63

65. ~B & ~C & ~I           & Intro 32,54,64
66. B v C v R v I             Reiteration 1
67. R                             Tautol.Conclus. 66,65


Ruby killed Mr. Smith.
 

Edited on March 25, 2017, 11:57 am

Posted by ollie on 2017-03-25 10:53:31