Jose, I agree with you exactly. Here is my calculation:
NB. There are four independent propositions: Lucy human, Lucy sane, Minna
human, Minna sane.
NB. Therefore there are 16 possibilities, each representable as a 4-bit
string.
P =: #: i. 16 NB. set of all possibilities: 0000, 0001, 0010 etc
NB. A proposition is modelled as a function to select a single bit.
Lh =: 0&{ NB. Lucy is human
Ls =: 1&{ NB. Lucy is sane
Mh =: 2&{ NB. Minna is human
Ms =: 3&{ NB. Minna is sane
truthteller =: = NB. x= human, y = sane : both or neither
Ax =: Lh ~: Mh NB. Axiom: if Lucy human, then Minna not, and vice versa
Lucy =: (Lh truthteller Ls) = (Ls +: Ms) NB. "we are both insane"
Minna =: (Mh truthteller Ms) = (Ls +. Ms) NB. "not both insane"
(Minna " 1 # ]) (Lucy "1 # ]) (Ax "1 # ]) P
0 0 1 0
0 1 1 1
On Tue, Jun 12, 2018 at 12:12 AM, Jose Mario Quintana <
jose.mario.quint...@gmail.com> wrote:
> Friday night I decided to try to fall asleep by thinking about the puzzle
> and I thought I had solved it. This evening I wrote a wicked script to
> verify my thought process and I got the same result but it does not seem to
> match your conclusion.
>
> This is what I found...
>
>
>
>
>
> *** POTENTIAL SPOILER FOLLOWS ***
>
>
>
>
>
> Given the conditions of the puzzle and since the sisters' statements
> contradict each other then they have to be either both sane or both
> insane. That leaves four possibilities:
>
> 0. Lucy is a sane human and Mina is a sane vampire.
> This one is inconsistent with Lucy's statement "We are both insane."
>
> 1. Lucy is an insane human and Mina is an insane vampire.
> This one is inconsistent with Mina's statement "of course not!"
>
> 2. Lucy is a sane vampire and Mina is a sane human.
> This one is consistent with both Lucy's statement and Minna's
> statement.
>
> 3. Lucy is an insane vampire and Mina is an insane human.
> This one is also consistent with both Lucy's statement and Minna's
> statement.
>
> However, either sane or insane, Lucy is a vampire (and Minna is a human).
>
> This is a verification in the form of a session corresponding to a wicked
> brute-force quick-and-dirty script (yet, the derived verb is not wicked)
> using the J Wicked Toolkit [0] (beware of line-wrapping)...
>
> JVERSION
> Engine: j806/j64nonavx/windows
> Release: commercial/2017-11-06T10:01:33
> Library: 8.06.09
> Qt IDE: 1.6.2/5.6.3
> Platform: Win 64
> Installer: J806 install
> InstallPath: j:/program files/j
> Contact: www.jsoftware.com
>
> NB. (Extra parentheses for clarity)
>
> NB. Running the Wicked Tacit Toolkit first...
> NB. (0!:0)<'/.../J Wicked Toolkit.ijs
>
> 'LH LS MH MS'=. 4 Fetch
> NB. True and false are encoded as bits in the usual way (as 1 and 0
> respectively)
> NB. The bits are in the following order
> NB. Lucy Minna
> NB. human sane human sane
>
> 'or and not'=. [:+.*.-.]sb
>
> NB. He knew that one was a vampire and one was a human...
> ovoh=. (LH and not o MH) or (not o LH and MH)
>
> NB. Lucy: We are both insane...
>
> lbit=. (((LH and LS) or (not o LH and not o LS)) and not o LS and
> not o MS)
> NB. Lucy is not lying
> lbil=. (((LH and not o LS) or (not o LH and LS)) and LS or
> MS)
> NB. Lucy is lying
> lbi=. lbit or lbil
>
> NB. Minna: Of course not!
> mont=. (((MH and MS) or (not o MH and not o MS)) and LS or
> MS)
> NB. Minna is not lying
> moni=. (((MH and not o MS) or (not o MH and MS)) and not o LS and
> not o MS)
> NB. Minna is lying
> mon=. mont or moni
>
> find=. ((ovoh and lbi and mon)"1 # ])f. NB. Finding the consistent
> possibilities
>
> find o #: o i. 2^4
> 0 0 1 0
> 0 1 1 1
>
> NB. That is, Lucy (either sane or insane) is a vampire and
> NB. Minna (either sane or insane) is a human
>
> NB. The wrapped linear representation of find is...
>
> 66 (-@:[ ]\ 5!:5@<@:]) 'find'
> (((0&({::) *. -.@:(2&({::))) +. -.@:(0&({::)) *. 2&({::)) *. ((((0
> &({::) *. 1&({::)) +. -.@:(0&({::)) *. -.@:(1&({::))) *. -.@:(1&({
> ::)) *. -.@:(3&({::))) +. ((0&({::) *. -.@:(1&({::))) +. -.@:(0&({
> ::)) *. 1&({::)) *. 1&({::) +. 3&({::)) *. (((2&({::) *. 3&({::))
> +. -.@:(2&({::)) *. -.@:(3&({::))) *. 1&({::) +. 3&({::)) +. ((2&(
> {::) *. -.@:(3&({::))) +. -.@:(2&({::)) *. 3&({::)) *. -.@:(1&({::
> )) *. -.@:(3&({::)))"1 # ]
>
>
> PS. There is no need to run a Toolkit for this in Jx;
> just replace 'LH LS MH MS'=. 4 Fetch by,
> 'LH LS MH MS'=. 4 ((i.@:[ <@:(((_3?:0) (_1?:0))&)"0 _ ])(_2?:0)
> {::)
> and 'or and not'=. [:+.*.-.]sb by,
> 'or and not'=. [:+.*.-.]:
> and define o=. @:
>
>
> [0] J Wicked Toolkit
> http://www.2bestsystems.com/foundation/j/Jx.zip
> \Jx\J\J Wicked Toolkit.ijs
>
>
>
> On Sat, Jun 9, 2018 at 9:23 AM, 'Bo Jacoby' via Chat <c...@jsoftware.com>
> wrote:
>
> > The Transylvanian problem, solved using ordinal fractions.
> > sane humans and insane vampires make only true statements;
> >
> > insane humans and sane vampires make only false statements.
> > This is the coding,
> > 0001 Minna is human
> >
> > 0002 Minna is vampire
> > 0010 Minna is sane
> > 0020 Minna is insane
> > 0100 Lucy is human
> > 0200 Lucy is vampire
> > 1000 Lucy is sane
> > 2000 Lucy is insane
> > These are the possibilities: 10#.>:#:i.2 8
> >
> > 1111 1112 1121 1122 1211 1212 1221 1222
> > 2111 2112 2121 2122 2211 2212 2221 2222
> > He knew that one was a vampire and one was a human, Discard 0101 and 0202
> > leaving 8 possibilities: 1112 1122 1211 1221 2112 2122 2211 2221
> > Lucy: "We are both insane" = 2020.
> > "sane humans make only true statements"
> > If Lucy is sane the statement is false. Discard 1100, leaving 6
> > possibilities: 1211 1221 2112 2122 2211 2221
> > "insane humans make only false statements". Discard 2120, leaving 5
> > possibilities: 1211 1221 2112 2211 2221
> > "insane vampires make only true statements". Discard 2210, leaving 4
> > possibilities 1211 1221 2112 2221
> > "sane vampires make only false statements". Then "We are both insane" is
> > false in any case.
> > Minna: "We are not both insane" If Minna is a sane human then Lucy is
> > insane: Discard 1011 leaving 3 possibilities 1221 2112 2221
> > If Minna is a sane vampire then contradiction. Discard 0012 leaving 2
> > possibilities: 1221 2221
> > Now we know that Minna is an insane human, and so they are not both
> > insane,
> > and so Lucy is sane. 1221.
> > Summary:
> >
> > 1221
> >
> > 0001 Minna is human
> > 0020 Minna is insane
> > 0200 Lucy is vampire
> > 1000 Lucy is sane
> >
> >
> > Den 17:05 fredag den 8. juni 2018 skrev Don Guinn <
> dongu...@gmail.com
> > >:
> >
> >
> > When in elementary school there was a chart showing the numbers. But the
> > zero was to the right of the nine. That confused me then. No wonder kids
> > have difficulty grasping the concept of zero.
> >
> > On Fri, Jun 8, 2018, 8:34 AM Björn Helgason <gos...@gmail.com> wrote:
> >
> > > beenary numbers
> > >
> > > https://m.phys.org/news/2018-06-scientists-bees-concept.html
> > >
> > > On Fri, 8 Jun 2018 07:40 'Bo Jacoby' via Chat, <c...@jsoftware.com>
> > wrote:
> > >
> > > > Cardinal numbers (0, 1, 2, . . .) are including 0 (zero).
> > > > Ordinal numbers (1, 2, 3, . . .) are starting with 1 (first). There
> is
> > > no
> > > > "zeroth".
> > > > There is arithmetic of cardinal numbers (including the J verbs + * ^
> !
> > )
> > > ,
> > > > but there is no arithmetic of ordinal numbers.
> > > > The codes of the UDC are important numerical objects, but they are
> > > neither
> > > > integers, nor decimal fractions, nor rational numbers, nor real
> > numbers,
> > > > nor complex numbers, nor quaternions, nor vectors, nor matrices, nor
> > > > functions, nor operators. They have been neglected by mathematicians.
> > > > A new kind of numbers must be considered. I dubbed them 'ordinal
> > > > fractions' .
> > > > A cardinal number, such as 'one', counts a set.
> > > > An ordinal number, such as 'the first', identifies an element in a
> set.
> > > > A cardinal fraction, such as 'one half', measures a part of a
> totality.
> > > > An ordinal fraction, such as 'the first half', identifies a part of a
> > > > totality.
> > > > Consider for simplicity the binary, rather than the decimal,
> notation.
> > > > one = 1 = 0001. The digit positions, right to left, indicate ones,
> > twos,
> > > > fours, and eights, and the digit values are one-digit binary cardinal
> > > > numbers, 0 and 1.
> > > > the first = 1 = 0001. This is the cardinal number corresponding to
> the
> > > > ordinal number in question.
> > > > one half = 0.1 = 0.1000. The digit positions after the binary point
> > > > indicate halfs, fourths, eights, and sixteenths, and the digit values
> > are
> > > > one-digit binary cardinal numbers, 0 and 1.
> > > > the first half = ?????
> > > > My solution to this problem is
> > > > the first half = 1 = 1000
> > > > the second half = 2 = 2000
> > > >
> > > > the first fourth = 11 = 1100
> > > >
> > > > the second fourth = 12 = 1200
> > > >
> > > > the third fourth = 21 = 2100
> > > >
> > > > the fourth fourth = 22 = 2200
> > > >
> > > > the odd fourths = 01 = 0100
> > > > the even fourths = 02 = 0200
> > > > the sixteenth sixteenth = 2222
> > > > Note that the digit positions indicate halfs, fourths, eights, and
> > > > sixteenths, and the digit values are either 1 meaning first, and 2
> > > meaning
> > > > second, or 0 meaning both. 1000 means: first half, both fourths, both
> > > > eights, both sixteenths. 'both' goes without saying, just as 0 goes
> > > without
> > > > saying. 1000 = 1 = first half. That is one reason for choosing 0 for
> > > > 'both'.
> > > > I did not know the words hyponymy and hypernymy. Thanks! That is just
> > > what
> > > > I need.
> > > > In logic I let 1 and 2 represent True and False. 0 means unknown or
> > > > unimportant.
> > > >
> > > > The Transylvanian problem:
> > > > 0001 Minna is human
> > > >
> > > > 0002 Minna is vampire
> > > > 0010 Minna is sane
> > > > 0020 Minna is insane
> > > > 0100 Lucy is human
> > > > 0200 Lucy is vampire
> > > > 1000 Lucy is sane
> > > > 2000 Lucy is insane
> > > > Check the 2^4 ordinal fractions from 1111 to 2222 against the data.
> (I
> > > > have not done it)
> > > > Thanks! Bo. .
> > > >
> > > >
> > > >
> > > >
> > > >
> > > >
> > > >
> > > >
> > > >
> > > >
> > > >
> > > >
> > > >
> > > >
> > > >
> > > >
> > > > Den 5:58 fredag den 8. juni 2018 skrev Donna Y <
> dy...@sympatico.ca
> > >:
> > > >
> > > >
> > > > Can we agree on definitions
> > > >
> > > > Ordinal numbers and Cardinal numbers are Natural numbers which do no
> > > > include the number 0. The Natural numbers are well ordered.
> > > > Whole numbers are the Natural numbers and 0. 0 is the least element
> of
> > > the
> > > > Whole numbers.
> > > > Integers are Whole numbers and Negative signed Natural numbers. (-n +
> > n=
> > > > 0 Zero is not a positive or a negative integer—0 has no sign.) The
> set
> > > of
> > > > integers has no least element.
> > > > Real numbers are continuous and complete and can be rational or
> > > > irrational—rational numbers are integers and fractions, irrational
> > > numbers
> > > > cannot be expressed as a ratio of two integers.
> > > > Imaginary numbers are not real—they exist in another dimension—root
> > > > negative 1 or i, complex numbers have real and imaginary components.
> > > >
> > > > Tables are two dimensional arrays.
> > > >
> > > > The basic structure of UDC is hierarchy but it could be viewed in
> other
> > > > ways—as you said yourself.
> > > >
> > > > Are you using tables where Rows are records and Columns are
> attributes?
> > > Is
> > > > there a primary key?
> > > >
> > > > I am not sure where your Ordinal Fraction comes in but a computer
> > > > application of UDC would need full integration of information
> retrieval
> > > > (IR) features into a database management system (DBMS).
> > > >
> > > > Right truncation specifies hypernomy (is a—as in number theory is a
> > > subset
> > > > of mathematics)—I copied this table but from my view forms of higher
> > > degree
> > > > is not a subset of diophantine equations.
> > > >
> > > > Table 1:
> > > >
> > > > 5: mathematics and natural sciences
> > > > 51: mathematics
> > > > 511: number theory
> > > > 511.5: diophantine equations
> > > > 511.57: forms of higher degree
> > > >
> > > > What is the precise reason you chose to use 0 as a wild card?—why
> not *
> > > or
> > > > # or & or … ? What advantages are derived by using 0? For example
> when
> > > you
> > > > use boolean logic the 0 and 1 can represent False and True and then
> the
> > > > result vector of 0 and 1 can be used for selection.
> > > >
> > > > hat does this even mean?
> > > >
> > > > > For example: 0=0 and 0>1 and 1<0 and 10<>01 and 1><2. (meaning that
> > the
> > > > whole is equal to the whole, and the whole comprises the first part,
> > and
> > > > the first part is part of the whole, and the first half is compatible
> > > with
> > > > the odd fourths, and the first part is disjoint with the second
> part).
> > > > > The notation for ordinal fractions makes ordinal fraction
> arithmetic
> > > > easy, just as the notation for cardinal numbers makes cardinal number
> > > > arithmetic easy.
> > > > > Keep asking!
> > > >
> > > >
> > > > Maybe you can show how Ordinal Fractions can be applied to the
> problem
> > > > below:
> > > >
> > > > PUZZLE BREAK
> > > >
> > > > Inspector Craig Visits Transylvania
> > > > Inspector Craig of Scotland Yard was called to Transylvania to solve
> > some
> > > > cases of vampirism. Arriving there, he found the country inhabited
> both
> > > by
> > > > vampires and humans. Vampires always lie and humans always tell the
> > > truth.
> > > > However, half the inhabitants, both human and vampire, are insane and
> > > > totally deluded in their beliefs: all true propositions they believe
> > > false,
> > > > and all false propositions they believe true.
> > > > The other half of the inhabitants are completely sane: all true
> > > statements
> > > > they know to be true, and all false statements they know to be false.
> > > Thus
> > > > sane humans and insane vampires make only true statements; insane
> > humans
> > > > and sane vampires make only false statements.
> > > > Inspector Craig met two sisters, Lucy and Minna. He knew that one
> was a
> > > > vampire and one was a human, but knew nothing about the sanity of
> > either.
> > > > Here is the investigation:
> > > > Craig (to Lucy): Tell me about yourselves.
> > > > Lucy: We are both insane.
> > > > Craig (to Minna): Is that true?
> > > > Minna: Of course not!
> > > > From this, Craig was able to prove which of the sisters was the
> > vampire.
> > > > Which one was it?
> > > > — From Logician Raymond Smullyan
> > > >
> > > >
> > > > Donna Y
> > > > dy...@sympatico.ca
> > > >
> > > >
> > > > > On Jun 7, 2018, at 5:55 PM, 'Bo Jacoby' via Chat <
> c...@jsoftware.com
> > >
> > > > wrote:
> > > > >
> > > > >
> > > > > "You do not show how to access particular rows or columns or
> elements
> > > > from a table".
> > > > > The table, and the table name, is addressed by 00.
> > > > > The left column, and the left column header, is addressed by 01.
> > > > > The right column, and the right column header, is addressed by 02.
> > > > > The upper row, and the upper row header, is addressed by 10.
> > > > > The upper left table entry, and its data content, is addressed by
> 11.
> > > > > The upper right table entry, and its data content, is addressed by
> > 12.
> > > > > The lower row, and the lower row header, is addressed by 20.
> > > > > The lower left table entry, and its data content, is addressed by
> 21.
> > > > > The lower right table entry, and its data content, is addressed by
> > 22.
> > > > >
> > > > > "without knowing the number of rows or columns."
> > > > > In the example there are two rows and two columns.
> > > > > If you need a third row, call it 30.
> > > > >
> > > > > "Why not the concept of arrays with an index?"
> > > > > Arrays may have different shapes. Any ordinal fraction has the
> shape
> > > > (_$9).
> > > > > Arrays have elements. Ordinal fractions don't.
> > > > >
> > > > > Only array elements contains data. Any ordinal fraction may contain
> > > data.
> > > > >
> > > > > Arrays may have subarrays. Any ordinal fraction has subordinate
> > ordinal
> > > > fractions.
> > > > > Arrays and atoms have names. Ordinal fractions don't.
> > > > >
> > > > > "the set of Natural numbers begin with 1 yet we use base 10
> > > > representation that uses 0."
> > > > >
> > > > > Cardinal numbers and ordinal fractions have similarities and
> > > > differences.
> > > > > Cardinal number 0 (meaning "nothing") is not the same thing as
> > ordinal
> > > > fraction 0 (meaning "everything").
> > > > >
> > > > > Cardinal number 1 (meaning "one") is not the same thing as ordinal
> > > > fraction 1 (meaning "first part").
> > > > >
> > > > > Using "0" for wild card character does not get along with using "0"
> > for
> > > > counting to ten.
> > > > > A cardinal number is represented by a right-justified sequence of
> > > digits
> > > > with a finite number of nonzero digits. The cardinal number 1 may be
> > > > written 00001.
> > > > >
> > > > > An ordinal fraction is represented by a left-justified sequence of
> > > > digits with a finite number of nonzero digits. The ordinal fraction 1
> > may
> > > > be written 10000.
> > > > >
> > > > > "However this was not an easy system in which to do arithmetic so I
> > > > cannot see how your base 9 system could be either."
> > > > > Cardinal numbers, A and B, are ordered such that either A=B or A<B
> or
> > > > A>B. For example: 0=0 and 0<1 and 1>0 (meaning "zero is equal to
> zero"
> > ,
> > > > and "zero is fewer than one", and "one is more than zero").
> > > > >
> > > > > Ordinal fractions, A and B, are ordered such that either A=B or A<B
> > or
> > > > A>B or A<>B or A><B. For example: 0=0 and 0>1 and 1<0 and 10<>01 and
> > > 1><2.
> > > > (meaning that the whole is equal to the whole, and the whole
> comprises
> > > the
> > > > first part, and the first part is part of the whole, and the first
> half
> > > is
> > > > compatible with the odd fourths, and the first part is disjoint with
> > the
> > > > second part).
> > > > > The notation for ordinal fractions makes ordinal fraction
> arithmetic
> > > > easy, just as the notation for cardinal numbers makes cardinal number
> > > > arithmetic easy.
> > > > > Keep asking!
> > > > > Bo.
> > > > >
> > > > >
> > > > >
> > > > >
> > > > >
> > > > >
> > > > >
> > > > >
> > > > >
> > > > >
> > > > >
> > > > >
> > > > >
> > > > > Den 18:35 torsdag den 7. juni 2018 skrev Donna Y <
> > > dy...@sympatico.ca
> > > > >:
> > > > >
> > > > >
> > > > > UDC is a library classification system akin to the Dewey decimal
> > > > system—the creators collaborated with Dewy. UTC is also used for
> data.
> > > > >
> > > > >> Every number is thought of as a decimal fraction with the initial
> > > > decimal point omitted, which determines the filing order.
> > > > >
> > > > > I am not sure why you say it cannot handle tables.
> > > > >
> > > > >> Concepts are organized in two kinds of tables in UDC:[27] <
> > > >
> > > https://en.wikipedia.org/wiki/Universal_Decimal_
> > Classification#cite_note-UDC_Structure-27
> > > > >
> > > > >> Common auxiliary tables (including certain auxiliary signs). These
> > > > tables contain facets of concepts representing, general recurrent
> > > > characteristics, applicable over a range of subjects throughout the
> > main
> > > > tables, including notions such as place, language of the text and
> > > physical
> > > > form of the document, which may occur in almost any subject. UDC
> > numbers
> > > > from these tables, called common auxiliaries are simply added at the
> > end
> > > of
> > > > the number for the subject taken from the main tables. There are over
> > > > 15,000 of common auxiliaries in UDC.
> > > > >> The main tables or main schedules containing the various
> disciplines
> > > > and branches of knowledge, arranged in 9 main classes, numbered from
> 0
> > > to 9
> > > > (with class 4 being vacant). At the beginning of each class there are
> > > also
> > > > series of special auxiliaries, which express aspects that are
> recurrent
> > > > within this specific class. Main tables in UDC contain more than
> 60,000
> > > > subdivisions.
> > > > >
> > > > > Can you show exactly what is gained by your “ordinal fraction”?
> > > > >
> > > > > Also—the set of Natural numbers begin with 1 yet we use base 10
> > > > representation that uses 0.
> > > > >
> > > > > However in the 4 BCE the Greeks used their letters to represent
> > numbers
> > > > from one to nine with no zero:
> > > > >
> > > > > alpha beta gamma delta epsilon digamma zeta eta theta
> > > > >
> > > > > They didn’t need to use 0 for higher values because much like
> Romans
> > > > they had symbols for higher values.
> > > > >
> > > > > However this was not an easy system in which to do arithmetic so I
> > > > cannot see how your base 9 system could be either.
> > > > >
> > > > > You suddenly use 0 when you illustrate tables. You do not show how
> to
> > > > access particular rows or columns or elements from a table but
> instead
> > > use
> > > > vague
> > > > > terms upper, lower, left, right without knowing the number of rows
> or
> > > > columns..
> > > > >
> > > > > Why not the concept of arrays with an index?.
> > > > >
> > > > > Maybe you could explain what I am missing.
> > > > >
> > > > > Donna Y
> > > > > dy...@sympatico.ca
> > > > >
> > > > >
> > > > >> On Jun 7, 2018, at 3:47 AM, 'Bo Jacoby' via Chat <
> > c...@jsoftware.com>
> > > > wrote:
> > > > >>
> > > > >> Ordinal Fractions is an improvement to the idea behind the
> Universal
> > > > Decimal Classification (UDC).
> > > > >
> > > > > ------------------------------------------------------------
> > ----------
> > > > > For information about J forums see http://www.jsoftware.com/
> > forums.htm
> > > > >
> > > > >
> > > > > ------------------------------------------------------------
> > ----------
> > > > > For information about J forums see http://www.jsoftware.com/
> > forums.htm
> > > >
> > > > ------------------------------------------------------------
> ----------
> > > > For information about J forums see http://www.jsoftware.com/
> forums.htm
> > > >
> > > >
> > > > ------------------------------------------------------------
> ----------
> > > > For information about J forums see http://www.jsoftware.com/
> forums.htm
> > > ----------------------------------------------------------------------
> > > For information about J forums see http://www.jsoftware.com/forums.htm
> > ----------------------------------------------------------------------
> > For information about J forums see http://www.jsoftware.com/forums.htm
> >
> >
> > ----------------------------------------------------------------------
> > For information about J forums see http://www.jsoftware.com/forums.htm
> >
> ----------------------------------------------------------------------
> For information about J forums see http://www.jsoftware.com/forums.htm
>
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