I'm trying to drum up business for practical logic techniques by issuing some challenge problems taken from Charles Dodgson (AKA Lewis Carroll) and offering a small prize for their solution. The real question is whether these problems are actually difficult for logic/mathematics professionals to handle. My challenge problems are, in a way, market research.
The challenge is to solve (in relatively short order) a small puzzle problem taken from Lewis Carroll's symbolic logic book.
It's one of his compound deduction puzzle problems that he called "Sorities". To solve, you have to link all the terms in the puzzle together to get a grand solution linking the extreme terms. This is Dodgson's rule, not mine. The real issue is getting a proof. You can find Dodgson's solution online with a little looking, but no proofs. I'm not really interested in knowing the solution, because that's not the goal. I'm also not interested in seeing how Dodgson would solve the problem, because he's not a modern & his methods are not particularly memorable. So, neither using Dodgson's algebraic solution from his Symbolic Logic book nor using his method of trees will solve the challenge problem.
I'm offering the prize to see how a modern logician solves the puzzle, if they can. It shouldn't take more than about 5 minutes if you have a good method handy.
Problem #58
(1) I never put a cheque, received by me, on that file, unless I am anxious about it;
(2) All the cheques received by me, that are not marked with a cross, are payable to bearer;
(3) None of them are ever brought back to me, unless they have been dishonoured at the Bank;
(4) All of them, that are marked with a cross, are for amounts of over £100;
(5) All of them, that are not on that file, are marked "not negotiable";
(6) No cheque of yours, received by me, has ever been dishonoured;
(7) I am never anxious about a cheque, received by me, unless it should happen to be brought back to me;
(8) None of the cheques received by me, that are marked "not negotiable", are for amounts of over £100.
Source: Symbolic Logic Part 1, Elementary (1897, Lewis Carroll, p. 128) -AND- Lewis Carroll's Symbolic Logic (1977, Bartley, p. 175)
A proper solution will include an explanation of the techniques & notations used. It is not sufficient to simply list a proof if I can't understand it. It is also not sufficient to use terms &/or methods without defining them or referring me to a published explanation that can be read in English.
I'm currently offering $25 USB (payable via paypal) for the solution. The price will go up if nobody solves the puzzle.
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Hi J.D.
I'm intrigued by your statement 'I'm trying to drum up business for practical logic techniques...'. Are you trying to publicize a website or software?
Anyway, the puzzle can be solved in several ways, either using standard categorical logic or propositional logic using conditionals (implication). See here for an example of the latter method.
Since you are looking for a 'modern' approach, I present my solution using boolean equations. Actually, it's not really modern because W. Stanley Jevons, who revised Boole's logical system into the form we recognize today, invented the method long before Lewis Carroll (in the 1860s).
But for this puzzle, where the propositions are simple categoricals, it's very quick and intuitive; once you have the equations, it's just a matter of substituting equals for equals. The hardest part, as is usually the case, is translating the English statements into logical form suitable for mechanical manipulation.
From the problem, it's evident that the Universe of Discourse is 'cheques received by me'. Here are the terms used :
a = that are brought back to me
b = that I am anxious about
c = that are honoured
d = that are marked with a cross
e = that are marked "not negotiable"
h = that are on that file
k = that are over £100
l = that are payable to the bearer
m = that are yours
Before translating the propositions into equations, I will explain the notation. All equations have the form :
a = ab
This equation means "all a is b" (standard categorical "A" form). Think of the terms a & b as sets, and juxtaposition of them means intersection; i.e. "ab" means the intersection of the sets a & b. So "ab" means the set of cheques which are brought back to me AND that I am anxious about. The objects in that set contain both attributes.
"all a is b" means that whatever is in a is also in b, and the equation a = ab tells you that a is identical to the intersection of a & b, which makes sense, if you think about it (imagine a circle representing a entirely with a larger circle b; the intersection of both, ab, is just a itself).
Where the terms are marked with a prime ('), it denotes the negative. i.e. terms which denote objects outside the given category. e.g. so whereas a = that are brought back to me, a' means "that are NOT brought back to me" and m' = "that are NOT yours", etc.
An equation with a prime attached to a term, such as a = ab' means "No a is b" (standard categorical "E" form). This is because "ab'" is the intersection of a with everything which is NOT b. so a = ab' says that a is identical to the a's which are not b's.
Note that a' = a'b is NOT in the "E" form. It says that "all non-a's are b's".
Finally, I will be making use of contraposition, which is an immediate inference. Contraposition is the converse of the negated terms. i.e. first negate the terms and then swap them round. So "all a is b" becomes "all non-b is non-a", and the equation a = ab becomes b' = b'a'. A proposition and its contrapositive are equivalent, logically speaking.
Ok, now onto the propositions. Hopefully they will now be understandable, if not ask me for clarification.
1. h = hb
2. d' = d'l
3. a = ac'
4. d = dk
5. h' = h'e
6. m = mc
7. b = ba
8. e = ek'
To solve the puzzle, we need an inference rule. There's only one : substitution. For example, given the equations a = ab and b = bc, substitute the RHS of the 2nd equation into "b" in the RHS of the 1st equation, which gives : a = abc. And because ab = a, this can be simplified to a = ac.
Proof :
m = mc
= (by contra of 3 : c = ca')
mca'
= (by contra of 7 : a' = a'b')
mca'b'
= (by contra of 1 : b' = b'h')
mca'b'h'
= (by 5)
mca'b'h'e
= (by 8 )
mca'b'h'ek'
= (by contra of 4 : k' = k'd')
mca'b'h'ek'd'
= (by 2)
mca'b'h'ek'd'l
= (Substitution of m for mca'b'h'ek'd')
ml
"All your cheques are payable to the bearer"
Carroll's solution is "No cheque of yours is payable to order", which initially puzzled me, but I'm assuming "payable to the bearer" is the negation of "payable to order", in which case my solution is equivalent to his.
Interesting. Give me a day or so to digest it.
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If you need further elaboration on the method I used, see here.
Cheques "payable to order" are payable to the the person they are made out to, and require a signature from both parties (so far as I can tell https://www.investopedia.com/terms/p/pay-to-order.asp).
"Payable to the bearer" does not require a signature from the receiving party.
Dodgson needs to have a dichotomy between "payable to bearer" vs. "payable to order" in order to solve the problem.
We don't really do personal checks that way these days. I had considered his categories to be Payable or Not Payable (Order & Bearer both being payable). Knowing Dodgson, I would have to say that you are correct.... this is exactly the sort of trick Dodgson would play.
I think that's something of a problem with the construction of the problem. If a check is not honored or marked non-negotiable, it's actually not payable to anyone. This sort of overlooked possibility is actually a common problem with Dodgson's problems. However, I give him a lot of leeway because they're clearly puzzles and not actual problems.
Anyway, I really expected this answer to take longer and to use some technique from the 20th century. But, you've definitely earned the prize. Email me to collect.
I may relaunch this problem with the caveat that the person use a newer technique.
Are you interested in any more challenge problems from Dodgson?
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Now that you remind me, I have heard of Avi Sion before, but I think I was turned off by something. Maybe his book was just too long to read at the time. I'll give him a second look.
Most people can't solve 3 part syllogism, and you can forget about compound deduction. Practical logic is either A) Not taught, or B) Taught extremely badly to the point where you are better off inventing your own methods. I agree that pure deduction is the least used form of logic, simply because the conditions don't often occur where it is profitable. However, it shouldn't be so badly done as it is.
I've never seen a method invented in the 20th century that would be useful for solving a practical problem. People say these new methods are useful, but I never see them used in earnest. Of course, maybe I'm wrong about that. That's the market research.... seeing whether modern logicians/mathematicians think these problems are trivial and how they go about solving them. I've previously sent them out to selected individuals who are philosophy/logic professors and got crickets in return.
I really wanted to post to stackexchange or other sites like that with a wider audience, but they don't seem to allow paying people. You can only earn whatever points they use. Ever since the Drexel math forums got turned into a porno site, it's been tough to find a reasonable forum. So, I'm grateful this forum exists.
The first challenge problem was selected because I thought Dodgson's answer was wrong. I misunderstood what 19th century personal checks were like. So, Dodgson's answer stands.
The second challenge problem is the Brothers problem (I believe problem #7 in the back of Dogdson's 1897 book.) It should legitimately take a while to work. Dodgson didn't give an answer, and in fact it can have many answers. The goal is to find the longest answer that uses the most rules. A secondary goal is to understand why certain rules can't be used at different times. And I think this was even beyond what Dodgson could do. In all his writings I've seen, he really focused on getting an answer rather than understanding a problem. It's a 19th century thing, I suspect. His method of trees (and the various derivatives) will work, but I am going to rule them out. I don't think anyone even knows these methods any more. I need to phrase the problem so that people can't just dig up a method they don't know and don't use for the purpose of winning.
By the way, you may not want to post solutions of advanced problems to the internet. New people come along all the time, and these problems are like sphinxes. If young people find the answers too readily, they won't put in the time to work the problems. And I think solving these puzzle problems is an accomplishment. There should be an age requirement of 30+ years old to view these XXXX answers. :grinning:
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How did you come across this forum? It belongs to Avi, and as far as I'm aware there is only one link to it - at his site. The link isn't particularly visible either, it's near the foot of this page (discussion area). There is very little traffic here either; I'm the only one (apart from Avi) who has posted in the last few months. There are a number of philosophy forums online, but hardly any have a section dedicated exclusively to logic, sadly. Not many people are interested in the subject, but I find it fascinating.
I can understand that many would be put off by Avi's books, because they are not exactly light bedtime reading - even for logic books - and there aren't many examples and applications in them. However, they are worthy of study because they contain many new insights and systems of logic, although very little of what you would call 'modern' logic (if you take 'modern' as being 'mathematical'). The style is very much that of the traditional syllogism, but the content is not just a rehash of traditional methods.
@Avi: Thanks for the email. Nice site, but Captcha is not working. Just deleted my last reply because I tried to preview before submitting and then had no captcha, so it kicked me. Also, the quote option doesn't actually insert code that I can edit. It just quotes the whole thing and won't let me edit the quoted text. Using Chrome on Linux.
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I look forward to hearing about Gregg's work. I couldn't see anything.
I have a love/hate opinion of Bartley. I loved the fact that he made Dodgson's work available. However, I felt he disparaged Dodgson unfairly. I don't remember him being critical of the Logicist school. But, I wasn't really reading the book for Bartley's opinion on logic. I was reading to get a better look at what Dodgson was doing. I view 20th century logic as a retreat into mysticism. This sort of thing happens when progress stalls. I think Bartley entertained thoughts that modern logic represented an advance in human knowledge. I don't remember him bucking any trends.
Where did you see that Bartley sent the schoolboys problem out? I didn't know that. I know that Dodgson was annoying people all over the world with these problems. The schoolboys problem currently has a solution posted on the internet, although not a complete one. I think Froggy's problem is also solved, but I haven't checked it.
Lonergan's Insight is written in the exact style that Thomas Aquinas used. However, Lonergan doesn't mince words, and he intends to be very flexible. Every sentence is loaded with meaning and appreciation for the complexities of human thought and the physical world. The book isn't long in the sense of the number of words, but it's going to be slow reading to comprehend what he's saying. If you like it, you will probably need a tutor, or at least someone to talk with who knows classical philosophy. I had a tutor. The early chapters are a good read, but he's a Jesuit priest, so the end of the book becomes very Roman Catholic at some point. The goal of the book is to justify revealed religion using only philosophy, which is quite a stretch. For example, chapter XIX.10 is "Affirmation of God". Don't feel obligated to read the whole thing.
The real gem is his treatment of epistemology in general.
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