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From: Philip Johnson-Laird BA PhD Psychology (UCL), Stuart Professor of Psychology Emeritus at Princeton. (Author isn't a logician.) How We Reason (1st edn 2008). p. 106.

Ratchett was stabbed twelve times, and here’s a description of who helped to stab him:

If the Countess helped then Cyrus helped too.
Edward helped or the Countess helped.
If Pierre helped if Hubbard did then Greta also helped.
Greta didn’t help or Hubbard did.
If not both Hildegarde and Hector helped then Hubbard didn’t help.
Edward helped or the Princess, Mary, and the Colonel helped.
If Hector or Mary helped then, Pierre helped or the Colonel didn’t.
The Countess helped or else both Hildegarde and Hector helped. If the Countess didn’t help then Pierre, Greta, and Edward helped.
If the Count didn’t help then not both Hubbard and Hector helped.
Cyrus and the Princess helped or else the Count and the Countess helped.
If not both Mary and the Colonel helped then the Countess helped.

These premises show that all but one individual took part in the stabbings. So, who dunnit, or rather who didn’t dunnit? You’ll find the answer at the end of the chapter.

p. 134

The French poet Paul Valéry once suggested that the following question might be a good test for mathematical giftedness:

If Peter resembles Paul, and Paul resembles James, does James resemble Peter?

Those who say “yes” at once are the gifted ones. His hopes for his test were dashed when one of the best mathematicians he knew began to cogitate at length. No wonder, for it is a moot point whether the relation is transitive. I’d say that Marx resembles Lenin, and that Lenin resembles Stalin, but not that Marx resembles Stalin. Perhaps a better test would be to ask prospective mathematicians to solve this problem:

I was married to a widow who had a grown-up daughter. My father fell in love with her, and soon they too were wed. This made my dad my son-in-law and now my daughter was my mother, because she was my father’s wife. To complicate the matter, I soon became the father of a bouncing baby boy. My little baby then became a brother-in-law

p. 135

to Dad, and so became my uncle. If he was my uncle, then that also made him brother of the widow’s grown-up daughter, who, of course, was my stepmother. Father’s wife then had a son who became my grandchild, for he was my daughter’s son. My wife is now my mother’s mother, and although she is my wife, she’s my grandmother, too. Now if my wife is my grandmother, then I’m her grandchild, and I have become the strangest case you ever saw. What am I?

Assume that I had to deduce the conclusion from Gordian premises like these in real-life, like a job interview (some companies like to tease applicants' brains), or court (a lawyer states the premises from p. 106 can be the facts of a criminal case, and asks the jury to conclude).

Can a logician deduce the conclusions mentally, without writing anything?

p. 112 of this book suggests not, but doesn't clarify if it refers to laypeople or logicians:

Our model-building system constructs one model at a time, and, as far as possible, seeks not to construct more than one model. The structure and content of a model capture what is common to the different ways in which the possibility could occur. When we are forced to try to hold in mind several possibilities, the task is hard, because our working memory has a limited capacity. A superhuman intelligence wouldn’t be limited in this way. Its memory would not be a bottleneck, and so it could solve the murder on the Orient Express without benefit of paper and pencil using the same mechanism as the computer program. [I emboldened.] We don’t realize our limitations because our social world is no more complicated than our ability to think about it, and our reasoning about the physical world is good enough for us to survive and to reproduce.

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I think you're reading too much into the quote of p. 112. More below.

Can a logician deduce the conclusions mentally, without writing anything?

It's not that hard to solve p. 106 without notation of some sort. (Either that or I jumped steps and just happened to get the correct answer with the incorrect way which is ofc possible.) You can just brute force it. If we start by assuming that the countess didn't help then we'll notice after a while that everyone else can have done it. Because we're also given that there's only one solution and hence that only one non-contradictory constellation is possible we don't have to infer anything else. Of course, it's prone to error, inefficient and a fairly random procedure. But any layperson that knows propositional logic can do it that way. A logician will at least have the advantage of using a procedure that isn't random (if they want to), and, it's fair to say, will also be able to do it better "in his head". But we have to look at the quote once more:

Its memory would not be a bottleneck, and so it could solve the murder on the Orient Express without benefit of paper and pencil using the same mechanism [I emboldened] as the computer program.

I think the quote isn't only about being able to solve but instead about being able to solve in a specific way. What I argue is that any layperson can do it while using a fairly random procedure and hence not the same mechanism. I suppose a logician will scan beforehand which premises seem more important, then assume a name, then scan once more, and so on. This still isn't the mechanism that is described in the book. Instead it would be a work-around against memory limitation which a computer won't have to deal with.

I guess whether solving p. 106 with the exact same mechanism is possible is either an empirical question, or something that we can't answer straightforwardly because of problems with introspection etc. The author makes an implicature that humans can't solve with the same mechanism... but because "superhuman" is undefined maybe that much wasn't meant. Now, it's possible that the author makes some claims on other pages. No idea, haven't read the book.

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