Logic

[ralphmelton]

I've been slowly working my way through Raymond Smullyan's book What Is the Name of This Book? over the past few weeks, since I picked it up in Half-Price Books.

Historically, which this sort of puzzle book, I have a strong tendency to wimp out. When the puzzles get too tough to solve easily, I turn to the answers. But this time, I've been trying hard to be more persistent and solve the problems myself.

One problem (appended below) really stumped me:

The problem deals with small silver and gold caskets. All caskets of interest are made by one of two families: Bellini or his sons, or Cellini or his sons. Caskets made by Bellini or his sons only have true inscriptions on them; caskets made by Cellini or his sons only have false inscriptions on them. The problem, then, is to find a pair of inscriptions that for two caskets that has the following properties:
- From reading both inscriptions, a logician is able to conclude that one was made by Bellini and one was made by Cellini, but not to determine which was made by which.
- It is not possible to draw that conclusion from reading only one of the inscriptions.

I had been gnawing on it for at least a week, possibly more. But finally, as I was going to sleep on Saturday night, I figured out an answer. I felt very proud.

Comments

[stresskitten.livejournal.com]

i presume the setup implies that the inscription on one casket can say something about the other casket?

[ralphmelton.livejournal.com]

Correct.

[stresskitten.livejournal.com]

proposed answer sent in email to avoid spoilage

[ralphmelton.livejournal.com]

It seems I wasn't entirely specific; the inscriptions are supposed to imply that the caskets were made by the two fathers, not by any of the sons.

requests for clarification

[cellio]

Ok, the two inscriptions can refer to each other, and are supposed to imply that the caskets were made by the fathers. Are these caskets a pair, in the sense that the maker of one knew that it would go with a specific one made by the other family? (Is there collaboration?)

Re: requests for clarification

[ralphmelton.livejournal.com]

Yes, there is collaboration.

When I get home, I'll post some of the simpler examples from the book; it is easier if you can get the buildup.

I remember a simple example with a single casket that says, "This casket was made by a son of Cellini." A bit of logic proves that this was made by a member of the Cellini family (since a truth-telling member of the Bellini family could not make that inscription). Since it was made by the Cellini family, the statement must be false, so it was made by Cellini pater.

one more question

[cellio]

From reading both inscriptions, a logician is able to conclude that one was made by Bellini and one was made by Cellini, but not to determine which was made by which.

Bellini and Cellini specifically, and not their sons?

Re: one more question

[ralphmelton.livejournal.com]

Yes, Bellini and Cellini specifically.