Thursday, July 06, 2017

Elements of Modal Logic, Part XI

Part X

Let's suppose that we have a number of books on a library table, and that these are special books. When we read them, we find that they are books that talk about themselves and each other. For instance, we open up Book 1 and read on one page:

Every book mentions strawberries.
Apples are pleasant fruits.
Every book has the sentence, "There's a book that talks about apples."
There's a book that talks about apples.
There's a book that does not mention oranges.
Strawberries are full of Vitamin C.
There's a book that has the sentence, "Every book has the sentence, 'There's a book that has the sentence, "Every book has at least one sentence."'"
There's a book that has the sentence, "Every book as at least one sentence."

Book 1 is functioning as the Reference Table for the books on the library table, and it is one of the books it is talking about, so this works like 1234DM so far. But we pick up Book 2 and read on one page:

Every book has the sentence, "There's a book that has the sentence, 'Every book has at least one sentence'."
There's a book that has the sentence, "Every book has at least one sentence."
Strawberries are usually red.
There's a book that talks about apples.
Every book mentions strawberries.
Every book has the sentence, "Every book mentions strawberries."

Exactly the same things we said about Book 1 are true, too: it is a Reference Table for the books on the library table, and it is one of the books that it is talking about. This is something new, and something that we have not seen before -- we can have multiple Reference Tables that talk about other Reference Tables.

What this means in practice is that we can have strings of Boxes and/or Diamonds. For instance, we can have ◇◇X. The Diamonds don't work any differently. What ◇◇X tells us is that there's a table on which we can find ◇X; and then ◇X on that table tells us that there's a table on which we can find X.


This can be tricky. In our example, every book is talking about every book, but this might not be the case. When you have more than one Reference Table, nothing requires that the tables that Reference Table 1 talks about are the same as the tables that Reference Table 2 talks about. Some of them might be, or all of them might be, or none of them might be. This is something we have to learn from the universe of discourse: what are we actually talking about?

Suppose we are talking about moments in time. Every table describes a moment in a time, but from each moment in time we learn something about other times -- for instance, what moments in time come after it. This is basically a Diamond: At Some Time in the Future. If I have a given moment, its future is different from some other moment in time. Compared to now, dawn tomorrow is at some time in the future, but once we get to dawn tomorrow, now is not in the future. Thus some of the tables describing future moments are shared when we take the table for now as our Reference Table and when we take the table for dawn tomorrow as our Reference Table, but the table of future moments for now makes comments on moments that the table for dawn tomorrow cannot.

This gives our analysis more flexibility, but does not introduce any fundamental changes. But sometimes we want to do more. For instance, if we have 'a moment that is at some time in the future from some time in the future from some time in the future from some time in the future', we would usually want to say that we can just collapse this into 'a moment that is at some time in the future'. This makes ◇◇◇◇ work as if it were just the same as ◇. And what about more complicated mixtures of Box and Diamond? Do we sometimes want to collapse them as well? These require new rules.

Part XII

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