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Logic

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.

Something about you (optional) I'm issuing logic puzzle challenges

Jul 24th, 2021 - 3:26 PM

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?

Something about you (optional) I'm issuing logic puzzle challenges

Jul 24th, 2021 - 9:39 PM

I may relaunch this problem with the caveat that the person use a newer technique.

Any reason in particular why? You could use standard predicate calculus, which looks impressively mathematical and intimidating, but really adds nothing of substance to the classical syllogistic invented by Aristotle, at least, not for the purpose of solving puzzles like Dodgson's, which consist of simple categorical propositions.

Have you read any of Avi's works? I can highly recommend them. There is a chapter in 'Future Logic' in which he criticizes some aspects of modern logic - and he is right on the mark in my opinion.

A simpler, more modern technique would be to use conditionals, as shown in the site I linked to. The problem with that is that I don't see how you could use them to solve puzzles with particular premises ("Some A is B", or Some A is not B"), because, as far as I'm aware, the conditional "if A then B" can only represent universal premises. But, come to think of it, I don't recall seeing any of Dodgson's puzzles with particular premises.

Are you interested in any more challenge problems from Dodgson?

Sure! Actually, I have a copy of Symbolic Logic which contains more advanced material never previously published, including more difficult puzzles, most of which don't have answers. I'll dig it out and maybe post one or two examples here.

Edit : I guess you have that book too? I've just been looking at it and see that he talks about the method of trees, which you mentioned in your previous post.

I'm trying to drum up business for practical logic techniques

I'm not convinced that solving puzzles like these is a very useful exercise, although they are fun. In terms of everyday reasoning, the most common kinds of arguments we meet are inductive rather than deductive. So arguments by analogy, statistical generalizations, inference to the best explanation, causal arguments, fallacies, etc. It's also important to learn about cognitive biases such as confirmation bias, especially now that there is so much fake news online.

Certainly there is a place for formal deductive logic, especially in maths and computer science, but these are relatively specialized applications.

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:

Something about you (optional) I'm issuing logic puzzle challenges

Jul 25th, 2021 - 5:11 PM

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.

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.

There is one system of propositional logic which I've seen referred to as 'Modern Syllogistic Method'. There are a couple of papers online describing it, but they are not particularly easy to read. The clearest description is in a book called 'Ones and Zeros', by John Gregg. It's very powerful in that given ANY set of premises, it derives ALL possible conclusions which follow from them (and none that don't). It's completely mechanical to apply (once you have translated the premises into boolean equations). It could easily be used to solve any Lewis Carroll syllogism. The only reason I didn't use it is because it requires more knowledge of boolean algebra and is somewhat tedious to use, although it could be computerised. Anyway, I have a pdf of the book, which I'll be happy to send to you.

I've actually started work on a website about relatively unknown (or forgotten) systems of logic like this, including Avi's. The aim is to make logic more accessible - and fun - by showing that the predicate calculus isn't the only game in town. There are other systems which are more user-friendly, broader in scope in some cases, and yet don't sacrifice inferential power. Where possible I will include logic 'calculators' for doing the tedious grunt work.

@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.

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.

I didn't realize who owned it when I posted. Might be Yandex. It's been giving me more relevant results recently. I also searched a lot of pages. The search terms were: logic forum -"logic pro" . The Apple Logic Pro is the dominant result in logic today. :)

There is one system of propositional logic which I've seen referred to as 'Modern Syllogistic Method'. There are a couple of papers online describing it, but they are not particularly easy to read. The clearest description is in a book called 'Ones and Zeros', by John Gregg. It's very powerful in that given ANY set of premises, it derives ALL possible conclusions which follow from them (and none that don't).

If you think about it, for most problems, you can just write a program to test all possible settings of the variables, and it will spit out a solution. That sort of approach is much easier than thinking about a problem. Of course, if any of your rules were a little off, are conditional, or have hidden premises, you will have no idea about the stability of your solution.

I looked at the index of Gregg's book. It looks like the same old stuff. Can you be more specific about this new method in his book? Are you sure this isn't some variant of fuzzy logic, where they use a slightly different method to group things and think they've discovered a new logic? Everything I can find on the net regarding this subject is either behind a paywall or looks very suspect. I'm extremely suspicious of a method where there are no worked examples anywhere. Have you worked problems using this method?

Since you're suggesting... I will too.

I would suggest looking at Lewis Carroll's Symbolic logic by Bartley (1970s) if you haven't already. I think Dodgson has been criminally under-rated. It's not that Dodgson is the end-all and be-all, but he was definitely innovating, and this book gives you insight into his network of correspondents and a few of their discussions.

I would also suggest looking a Bernard Lonergan's book "Insight". He's a 20th century Jesuit philosopher who does a redux of classical philosophy, merging it with scientific method at the time. He has problems, but his definitions and method are the best I've found, especially concerning "truth" and "being". His approach is epistemological. His biggest problem is that he doesn't footnote his sources, because a good philosopher is supposed to know the basics. He's also an incredibly dense read if you have no formal philosophy training. Good philosophy training is even harder to come by these days than good logic training.

I've actually started work on a website about relatively unknown (or forgotten) systems of logic like this, including Avi's.

What's the site's address? You can email if it's not ready for the public.

Something about you (optional) I'm issuing logic puzzle challenges

Jul 26th, 2021 - 4:35 PM

I looked at the index of Gregg's book. It looks like the same old stuff. Can you be more specific about this new method in his book?

I'll post an example here in the next day or two, using one of Carroll's syllogisms. I don't have time today.

I would suggest looking at Lewis Carroll's Symbolic logic by Bartley (1970s) if you haven't already.

Yes, I have that book. I particularly like the introduction in which he talks about the three main periods of logic:

1. The traditional Aristotelian logic which lasted from Aristotle to Boole.

2. The 'algebraic' phase which was initiated by Boole, and includes De Morgan, Venn, Jevons, Keynes... and Carroll. It came to an end at the close of the 19th century.

3. Modern mathematical logic which started with Frege/Russell, and is now the dominant force.

He notes that while period 2 was revolutionary, it never really got a chance to take off because it was eclipsed by period 3, which was equally revolutionary. But the conception of logic and its purpose was quite different between the periods. Modern mathematical logic was intended to be a tool for analysing the foundations of mathematics, and isn't especially suitable for general philosophy or everyday use. Its main concern is with axiomatisation, proof construction, decision procedures, consistency proofs, metalogic, etc. In modern logic texts, both the premises and conclusion are given, and the aim is to examine the argument for validity.

But the main aim of the algebraic logic, as conceived by Boole and others in this period, was to extract as much information as possible with regard to a term or combination of terms, given a set of premises. Apparently, Bartley sent Carroll's Schoolboy Problem to logicians of "high distinction" over a period of 10 years, and none of them were able to solve it. He also complains that even though contemporary philosophers are given specialised education in mathematical logic, their ordinary work in philosophy is littered with elementary logical mistakes, and argues that these are just the kind of mistakes that 19th century logical algebra works to prevent, whereas modern logic is largely irrelevant to their prevention.

I would also suggest looking a Bernard Lonergan's book "Insight".

Will do, although I admit to having a bit of a love-hate relationship with philosophy and often get impatient with it. Much of what passes for philosophy these days is in my opinion pretentious drivel, or mere word salad. It's not so bad if the writing is clear, but this isn't common in my experience. That said, I don't have any formal education in philosophy (my background is electronics and mathematics), and tire easily reading long tracts, so maybe I'm being a bit harsh.

What's the site's address? You can email if it's not ready for the public.

I want to build up the content before I publish it, and what I have at the moment is little more than notes, but I'll post the link here when it goes live. This is long term project though, so it might be several months away.

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.

Something about you (optional) I'm issuing logic puzzle challenges

Jul 27th, 2021 - 6:55 PM