Part I. Part II. Part III. Part IV. Part V. (This series is a first rough draft; Part I tells what I am trying to do in it.)
So far we have recognized two defining rules:
(1) □ on the Reference Table means the statement would be found on any talked-about table there might be.
(2) ◇ on the Reference Table means that there is a table on which the statement itself is found.
And we have recognized that there are two other rules that, while not universal, nonetheless are very common:
(3) □ is interchangeable with ~◇~.
(4) ◇ is interchangeable with ~□~.
To this we added another rule that's common, although not as common as (3) and (4), the subalternation rule:
(D) □ on the Reference Table means that there is a talked-about table on which the statement itself is found.
We'll add a sixth rule, also common although not as common as (3) and (4), and that will give us quite a lot to look at before we get to a different kind of rule.
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:
(M) □ on the Reference Table means that the Reference Table is a talked-about table on which the statement itself is found.
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 all the books on the shelf mention other books; this would give us the following Reference Table:
REFERENCE TABLE: Books on the Shelf
□ (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 described by the Inventory Book, we know that the Inventory Book mentions other books.
In the next post, we'll look at more particular cases where we want to use a rule or not. Then we'll go on to look at what happens if a table a Reference Table talks about is itself another Reference Table. When we have that, we'll have most of what's needed to handle all but the most advanced tasks of modal logic.