Let's do a little more carefully what we did in the previous post. We are taking inventories of universes of discourse and keeping track of these inventories with tables; these need to be compared, but also to be kept distinct. We need to keep track of what's going on in these comparisons of different tables, and the easiest way is also to put it in a table. We can call this the Reference Table. It's where we'll find our modal information. The Reference Table can describe different things, depending on what we are doing. The second most important things in modal logic is usually to know what your Reference Table is and how you are using it; a lot of mistakes in modal logic are caused by switching Reference Tables, or forgetting to track things consistently with it.
(Is the Reference Table just like the tables it is keeping track of? Depending on what you are doing, the Reference Table is sometimes set apart, but it can also be one of the tables we are keeping track of. For instance, you could be doing something with times, and want to make the Reference Table 'now' -- then the Reference Table is one of the tables of times. We could also set things up so that any table can be the Reference Table for other tables. There are lots of ways we can do it, and we will have to consider some of them eventually. But we need to start with the easiest case. In the example that follows, the Reference Table is just a different sort of thing from the tables it describes.)
The single most important thing in modal logic, however, is to know exactly what all your other tables are supposed to describe. Suppose you want to compare the Tolkien books on the top shelf of your bookcase; and, as it happens, you have three: The Hobbit, The Lord of the Rings, and The Silmarillion. We will therefore have one table for each of these. We can make a table for each of these, and put down some of the things we know about the characters in those books.
| Characters mentioned in the Tolkien books on the top shelf of my bookcase | ||
|---|---|---|
| TABLE 1: The Hobbit | TABLE 2: The Lord of the Rings | TABLE 3: The Silmarillion |
| Bilbo Baggins | Bilbo Baggins | Galadriel |
| Gandalf | Gandalf | Elrond |
| Elrond | Elrond | Gandalf |
| Frodo Baggins | Frodo Baggins | |
| Galadriel | ||
Now, if this is all we know about the characters, we still can put some information on our Reference Table. We could have a lot of different rules, depending on exactly what we want to track. Let's make these our rules:
(1) If we would find something on any table there might be, we can list it in our Reference Table as Box (□).
(2) If there's something definitely on some table somewhere, we can list it as Diamond (◇).
(Note that rule 1 is tentative in a way rule 2 is not, since it doesn't actually tell us that there are any other tables besides the Reference Table, while rule 2 always gives us a table; the reason for this difference is complicated, and somewhat arbitrary, but it's the way many systems are set up, so we'll go with it for now.) The Box tells us that, for the things we are talking about, something is found always or everywhere or necessarily or without exception or invariably, no matter what table we might be looking at; the Diamond tells us that something is on some table. How we translate Box or Diamond -- what it represents in practice -- depends on what we're talking about and what comparison we are making. In the above example, Box tells us about every Tolkien book on the top shelf and Diamond tells us about some Tolkien book or other among those on the top shelf.
| REFERENCE TABLE: Tolkien Books on the Top Shelf |
|---|
| □(Elrond is a character) |
| □(Gandalf is a character) |
| ◇(Bilbo Baggins is a character) |
| ◇(Frodo Baggins is a character) |
| ◇(Galdriel is a character) |
This is not an exhaustive list of statements we could make, since there are lots of others, like ◇(Elrond and Gandalf and Bilbo Baggins and Galadriel and Frodo Baggins are characters), which is a description of Table 2, but this will do for our purposes, since we're keeping things simple for now. (Note, too, why our Reference Table is separate from the other tables: the Reference Table talks about specific Tolkien books on the top shelf. But there is no Tolkien book on the top shelf that is called 'Tolkien Books on the Top Shelf'.)
So now we've taken our original tables, where there are no modal statements, and created a table of modal statements out of them. But in many cases of modal reasoning, we aren't just summing up our information with modal statements; we are working backwards from modal statements to try to see what they tell us about the things they describe.
So let's forget our first three tables for a moment, and suppose that someone else did all the work to get the Reference Table, and we are trying to reconstruct the information in the other tables just from what the Reference Table tells us, without knowing anything about what the books are. Because our Reference Table does not describe the other tables in exact terms, we will not be able to reconstruct them exactly. Let's see how close we can get, though.
| REFERENCE TABLE: Tolkien Books on the Top Shelf |
|---|
| □(Elrond is a character) |
| □(Gandalf is a character) |
| ◇(Bilbo Baggins is a character) |
| ◇(Frodo Baggins is a character) |
| ◇(Galdriel is a character) |
We'll do the Diamond statements first, because each one tells us something about some table somewhere. But, they don't tell us which tables. The statements might be true for the same table, but they might not be. And since we have three Diamond statements, we get three tables, which might be different or might not, and which might be all of the original tables or only some of them:
| some table or other (1) |
|---|
| Bilbo Baggins is a character |
| some table or other (2) |
|---|
| Frodo Baggins is a character |
| some table or other (3) |
|---|
| Galadriel is a character |
But our Box statements still need to be added! By rule 1, they tell us that any tables the Diamond statements give us have the other statements, too. So now our tables look like:
| some table or other (1) |
|---|
| Bilbo Baggins is a character |
| Elrond is a character |
| Gandalf is a character |
| some table or other (2) |
|---|
| Frodo Baggins is a character |
| Elrond is a character |
| Gandalf is a character |
| some table or other (3) |
|---|
| Galadriel is a character |
| Elrond is a character |
| Gandalf is a character |
And remember, just from what our Reference Table tells us, we don't know if these are all the same table, or if two of them are the same, or if none of them are the same; likewise, we don't know whether these are all the tables or just some of them. The information in our Reference Table was very incomplete and not very precise. But it still gave us enough information to reconstruct something about the Tolkien books on the top shelf.
Most reasoning in modal logic is like this last example: we have a Reference Table and are trying to see what its implications are. You can think of it like a puzzle, in which the Reference Table is a list of clues that someone else gave you, and you are trying to see what those clues tell you.
To move forward, we need to look at how some slightly more complicated cases work. What you'll find, though, is that we've pretty much covered all the essentials of basic modal reasoning -- it all works exactly like this, and more complicated cases are only more complicated because they give us more information to use.
Part III
