Just to make it very clear, Rogers's J results were, > (Minna " 1 # ]) (Lucy "1 # ]) (Ax "1 # ]) P
> 0 0 1 0 > 0 1 1 1 which match exactly my J results, > find o #: o i. 2^4 > 0 0 1 0 > 0 1 1 1 and my associated comments were, > NB. That is, Lucy (either sane or insane) is a vampire and > NB. Minna (either sane or insane) is a human which provide the answer to the puzzle question, > From this, Craig was able to prove which of the sisters was the vampire. Which one was it? Moreover, the answer could have been more succinct: Lucy is the vampire. PS. I think Roger's coding is very neat. On Tue, Jun 12, 2018 at 11:14 AM, Raul Miller <[email protected]> wrote: > I guess I was distinguishing between set up statements and concluding > statements. And I tried to ignore set up statements which were not > Bo's. > > In other words, I saw > > NB. That is, Lucy (either sane or insane) is a vampire and > > NB. Minna (either sane or insane) is a human > > And presumed that that was meant to represent your result set. > > Thanks, > > -- > Raul > > On Tue, Jun 12, 2018 at 11:04 AM Jose Mario Quintana > <[email protected]> wrote: > > > > Raul wrote: > > > > > Hmm... here was my approach. I think I got different results: > > . > > . > > . > > > My way of summarizing this would be: > > > > > > "Lucy is a vampire", > > > "Minna is human", > > > "They are either both sane or both inane". > > > > responding to a post where Roger wrote: > > > > > Jose, I agree with you exactly. Here is my calculation: > > > > (I also agree with him) responding to a post where I wrote: > > > > > [...] they have to be either both sane or both > > > insane. [...] > > . > > . > > . > > > 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 > > > > So, what makes you think you got different results from Roger and me? (I > > cannot see any difference.) > > > > > > On Tue, Jun 12, 2018 at 7:09 AM, Raul Miller <[email protected]> > wrote: > > > > > Hmm... here was my approach. I think I got different results: > > > > > > lucy=: {."1 > > > minna=: {:"1 > > > species=: 2&| > > > diagnosis=: 1&< > > > human=: 0=species > > > vampire=: 1=species > > > sane=: 0=diagnosis > > > insane=: 1=diagnosis > > > invalid=: ~:/"1@(2 2&#:) > > > both=:*/"1 > > > > > > showSpecies=: ;@{&(;:'human vampire')@species > > > showDiagnosis=: ;@{&(;:'sane insane')@diagnosis > > > show=: showSpecies, ' and ', showDiagnosis > > > possible=: 3 :0"1 > > > echo 'Lucy is ',(show lucy y),' and Minna is ',show minna y > > > ) > > > > > > NB. all possibilities > > > S=: 4 4 #:i.16 > > > > > > NB. He knew that one was a vampire and one was a human > > > S=: S=:S#~~:/"1 species S > > > > > > NB. Lucy: "We are both insane" > > > S=:S#~(invalid lucy S)~:both insane S > > > > > > NB. Minna: "We are not both insane" > > > possible S=:S#~(invalid minna S)~:-.both insane S > > > > > > Running this script gives me two plausible seeming results: > > > > > > Lucy is vampire and sane and Minna is human and sane > > > Lucy is vampire and insane and Minna is human and insane > > > > > > In the ordinal fraction representation, the second possibility was > > > expressed as 2221 and rejected at the final step of the process: > > > > > > "Now we know that Minna is an insane human, and so they are not both > > > insane," > > > > > > But Mina's statement was that they are not both insane and insane > > > humans always make false statements according to the rules here. > > > > > > I am not sure what to make of that part of the ordinal fraction > approach. > > > > > > My way of summarizing this would be: > > > > > > "Lucy is a vampire", > > > "Minna is human", > > > "They are either both sane or both inane". > > > > > > Thanks, > > > > > > -- > > > Raul > > > > > > On Tue, Jun 12, 2018 at 4:30 AM roger stokes <[email protected] > > > > > wrote: > > > > > > > > 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 < > > > > [email protected]> 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 < > > > [email protected]> > > > > > 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 < > > > > > [email protected] > > > > > > >: > > > > > > > > > > > > > > > > > > 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 <[email protected]> > > > 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, < > [email protected] > > > > > > > > > > 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 < > > > > > [email protected] > > > > > > >: > > > > > > > > > > > > > > > > > > > > > > > > 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 > > > > > > > > [email protected] > > > > > > > > > > > > > > > > > > > > > > > > > On Jun 7, 2018, at 5:55 PM, 'Bo Jacoby' via Chat < > > > > > [email protected] > > > > > > > > > > > > > > > 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. > > > > > > > > ---------------------------------------------------------------------- > > 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
