So far we've looked at Box and Diamond in a fairly abstract way. (This is deliberate -- most accounts of modal logic, I think, start too far downstream. This is like trying to figure out validity before you know very much about the structure of arguments.) But we should start looking at some broader applications than we have so far. So here's a small sample of important modal concepts:
|logical quantity||all||at least some|
|mereology||wholly||at least in part|
There are many, many more we could do, but these are just some of the more common and important ones. For each of these, our Box and Diamond Rules hold:
(1) □ applying to anything on the Reference Table means that it would be found on any table there might be.
(2) ◇ applying to anything on the Reference Table means that there is a table on which it is found.
What differs is what the tables stand for, since they could stand for times, locations, kinds of actions, or any number of other things. These concepts, though, often have more useful logical content than we find with just these two rules, and to use these concepts fully we often have to use this extra content. What exactly this is, will depend on the modality. But some kinds of content will be very common.
One very common example of extra logical information is what is known as duality. The Box Rule (1) and the Diamond Rule (2) don't tell us anything about how box and diamond relate to each other. But if you look at the concepts above, you can see that box and diamond do have something to do with each other! In particular, you can often redefine Box in terms of Diamond and Diamond in terms of Box.
If I say, "Always the birds are singing," and we want to restate this in terms of 'sometimes', one way I could do it is to say, "It is not true that sometimes it is not true that the birds are singing." This works a little like a double negative ("not not"), except the modal operator gets stuck in the middle and gets flipped by the negatives. Thus we say that 'always' and 'sometimes' are dual to each other.
We'll use the symbol ~ to mean 'Not'. Then ~◇~, for instance, is 'Not Diamond Not'. So this gives us two more rules that we can use to reason about modal concepts:
(3) □ is interchangeable with ~◇~.
(4) ◇ is interchangeable with ~□~.
The rules are less complicated than they might seem, because we actually use them all the time. You know that when you say that the whole (□) wall is red you are also saying that it's wrong (~) to think that part (◇) of the wall is not (~) red. You're using Rule (3). Likewise, if everyone in your family is annoying, it's not true that some of them are not annoying. That's Rule (4).
There are cases where we don't use these rules. One example of a common modal concept where we usually don't is validity. Validity is a Box. You could propose a corresponding Diamond that is dual to validity, if you wanted; but we don't even have a word for it, and people don't usually care about any corresponding Diamond. So we don't need to bother with any way to get from Box to Diamond -- at least for most practical purposes. This is why we started with just Rule 1 and Rule 2. But for most complex situations the duality rules tend to be very useful.
A very common context in which we do see these duality rules used very extensively is the predicate calculus. In the predicate calculus you have expressions like, "For every x, x is furry", which, if we represent 'furry' with F would be represented as something like:
The ∀, called the universal quantifier, tells us there are no exceptions. There is another operator, called the existential quantifier, which we would use to say things like "There is some x that is furry":
The universal quantifier is Box and the existential quantifier is Diamond, and they are dual to each other: ∀ is equivalent to ~∃~ and ∃ is equivalent to ~∀~. This duality is used all the time in proofs in the predicate calculus.
One of the ways this will change how we reason is if our Reference Table has a ~◇ or a ~□. Without Rules (3) and (4), this wouldn't mean much. But if we have them, then ~◇ also tells us □~ and ~□ also tells us ◇~. For instance, if your Reference Table tells you that it is not true that unicorns exist everywhere, you know there will be some table where "Unicorns exist" is not true. Suppose you say: "It is not true that everyone is here". That's a ~□. But by Rule (3) we know that this is the same as ~~◇~; and the ~~ is a double negative. So ~□ works just like ◇~. And therefore our sentence means the same as "Someone is not here". This can take some practice getting used to -- you need to try it out with multiple examples -- but it is very useful to be able to do.
There are other important assumptions that can come into play with modal reasoning, even if they are not quite as common as duality. We'll look briefly at one of the most important ones, subalternation, in the next post in the series.