Friday, May 19, 2017

Elements of Modal Logic, Part VII

Part VI

Sometimes when we are reasoning, we are taking something as a reference point that is part of what we are talking about. For instance, if we are talking about time, we might talk about now; if we are talking about location, we might talk about here. 'Now' is a time, albeit a special one; 'here' is a location, although it is a special one. None of our rules so far take this kind of thing into account. In everything we've done so far, our Reference Table is not assumed to be itself one of the tables we are talking about. But sometimes we want our Reference Table to be one of the tables described in the Reference Table; we want it to include itself. This brings us to our next rule, which we might call the reflexivity rule. Rules like it are often called T, or sometimes M; I will call it M:

(M) □ on the Reference Table means that what it applies to is on the Reference Table.

Suppose we are thinking about books on your shelf. We can represent each of them as a table. Suppose you keep track of the books on your shelf by describing them all in a book, which you keep on that shelf. We can call it your Inventory Book. Your Inventory Book is working in a way that can be represented by a Reference Table. We might look at a line in the Inventory Book and discover that it says every book on the shelf mentions other books; this would give us the following Reference Table:

□ (Other books are mentioned.)

Since our Inventory Book is one of the books the Inventory Book itself describes, and □ here tells us that the statement following it is true of every book on the shelf, and thus every book described by the Inventory Book, we know that the Inventory Book mentions other books.

We use this kind of rule quite a bit. So, for instance, we can all see that if something is always true, it is true now, and that if something is found everywhere, it is found here, and that if everybody has a man-eating lion for a pet, I have a man-eating lion for a pet. These are all straightforward uses of the reflexivity rule. And it doesn't matter what the reference point is; perhaps my reference point is not now but the beginning of the world, not here but there, not me but you: it all works the same.

Since our rules are independent, we may use both (D) and (M) or only one. We've seen already that there are many cases in which (D) on its own will suffice; 1234D modalities are very common. 1234M modalities are much less common. The reason is that they are a little odd. They pick a reference point, and doing this makes it so that the Reference Table can talk about itself, (M) on its own is not enough to make the Reference Table a real table; for instance, it might just be something that could be a real table. So, for instance, if my Reference Table is The Day Pigs Fly, and it is true on that day that □p, where p is some claim about the way things are, (M) tells us that p is true on the day pigs fly; but it doesn't tell us that there is actually any real day that is the day pigs fly. (D) tells us that there is a real table somewhere; (M) on its own doesn't -- it just tells us what will be true on the Reference Table if the Reference Table actually exists.

While 1234M modalities seem to be found only in unusual cases, 1234DM modalities are much more common, because in 1234DM, the modalities work just like they do in the easy-to-use 123D modalities, but now, thanks to (M), you have the ability to pick a special reference point like 'here' or 'now'. This makes them very powerful and flexible. Just a few examples of cases in which we often like to use them:

reference point

But, again, even with the same Box and Diamond, we can often pick different reference points. Any point in time can be a reference point for times, and so forth. This raises the question: Can we use more than once reference point at a time? And the answer is a very definite yes, and that gets us into a lot of interesting things. But before we get there, we should take a look at how (M) affects the square of opposition, and also at some of the things 1234DM modalities let us easily do.


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