What I Call Some Impossible Logic Problems


I’m sorry to go another day without following up the essay I meant to follow up, but it’s been a frantically busy week on a frantically busy month and something has to give somewhere. But before I return the Symbolic Logic book to the library — Project Gutenberg has the first part of it, but the second is soundly in copyright, I would expect (its first publication in a recognizable form was in the 1970s) — I wanted to pick some more stuff out of the second part.

The big thing is that Carroll offers what sure looks like a plausible name for a particular kind of logic problem I never knew had a name. I don’t imagine the name is his origin (although apparently the word “chortle” can be credited to him), but I don’t recall coming across the name before and I hope that by saying it here I’ll either commit it to long-term memory or make it something I can find again when I want to find it. That’s sorites.

What I knew them as was this series of logic puzzles given as amusements in elementary school which were agonizing because attacked as a fourth-grade mathematically aware student they looked like “John does not live in a blue or a red house; Alan only eats pork chops when there is apple sauce; Carol cannot have a dog and a mouse simultaneously; Daniel is a piece of wood with a sort of lumpy glob at the end; and Elaine does not know when the Free State of Fiume was absorbed into Yugoslavia” and you need to use that to deduce a list of who drove what car to which murder. Mostly I remember the crossword-like blank tables which were filled in with X’s and O’s in the attempt to visually render the process of concluding that these are insane puzzles.

Of course, as I grew, the amount of frustration with these puzzles diminished, by not doing them anymore. In college I even bought a book of these puzzles and did several before realizing I didn’t have the time to figure out which store had been attacked by which giant piece of furniture. However, I was getting farther in working them out.

Actually, I’m not positive that the sorites is the right category word for this. But I know that what Carroll presents, with a long chain of points which turns into this chain-reaction string of syllogisms and eventually grinds through to a conclusion, is engaging exactly the same sort of reasoning those puzzles do. Carroll shows how to work through these using a pretty decent notation for logic puzzles that he seems to have invented. At the very least it’s a notation that can be, with two trivial modifications, done entirely on typewriter and so in ASCII, so it’s easy to reproduce on the Internet too. (The main thing is he uses a fancy \dagger where it seems to me + would do the job; there’s also some use of subscripts for 1 and 0, but since those don’t appear elsewhere they could be done on the same baseline.)

I hadn’t seen these sorts of problems attacked by Carroll’s method, but I can’t deny the power it offers, particularly since he puts together some monster sorites with as many as fifty propositions and asks what can be concluded out of all that. Even with the notes he gives on how to reduce the problem to symbols that’s a staggering workload.

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