Wednesday, May 03, 2017

Elements of Modal Logic, Part V

Part IV

It's perhaps worth pausing a moment to consider how different different modal concepts relate to each other. I gave a table previously of some of the more important cases:

Box Diamond
time always sometimes
location everywhere somewhere
duty obligatory permissible
truth necessary possible
logical quantity all at least some
mereology   wholly at least in part
topology interior closure

If we think about the tables we have been using, they can be interpreted in lots of ways. For instance, to explain what □x does, I could say:

It tells us that, for any table that exists, x is there.
It tells us that x is everywhere.
It tells us that x is always found when we have a table.
It tells us that, if there is a table, x ought to be on it.
It tells us that when a table exists, it's necessary for x to be on it.
It tells us that x is on every table.

And so on. All of these are in some sense a way of saying the same thing. And this is one of the things we regularly find with Box and Diamond: you can use one interpretation to talk about other interpretations. It's like a logical figure of speech. We can't move directly between them -- 'everywhere' is about places and 'always' is about time, so they aren't talking about the same thing. But if you have a way in which you can talk about places using times, or times using places, then you can move from one to the other. For instance, we often think of times using a timeline. If we think about a timeline, then we can talk about 'always' meaning the same as 'everywhere on the timeline'. The timeline represents times as places on a line; it serves as an instrument for making time-language and place-language fit each other. We can call the instrument that helps two modal languages fit together the Bridge: Events in time are like points on a line. Thus we will often be able to get from one modality to another this way:

Modality A + Bridge → Modality B

One of the most common examples is a weird one. I mentioned before that ∀ and ∃ in the predicate calculus are modal operators. But suppose you want to use them to talk about necessity and possibility? ∀ and ∃ were developed in order to talk specifically about arithmetic and thus are best adapted to be applied to distinct individuals that can be picked out precisely. But necessity and possibility don't seem normally to work this way. You need a Bridge. One way to do it is to think of a whole world, and then to think of different ways the world could be, and think of these different ways the world could be as possible worlds. Then we can use ∀ and ∃ on possible worlds to talk about possibilities. The necessary is what is true in every possible world; the possible is what is true in some possible world. It's the same kind of thing you are doing with the timeline, just with a different pair of modal concepts.

It's clear enough that the reason you can do this is that Box interpreted in one way shares logical features with Box interpreted another way, and the same with Diamond. That is to say, the rules are the same even though the universes of discourse are different. But this doesn't mean that it is always easy, because, as I've pointed out, you could have modal operators that follow different rules from the ones we usually use, and you'd have to take that into account. The more rules Box 1 and Box 2 share, the more easily one can move from one to the other by a Bridge.

So far we have recognized two defining rules:

(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.

And we have recognized that there are two other rules that, while not universal, are still extremely common:

(3) □ is interchangeable with ~◇~.

(4) ◇ is interchangeable with ~□~.

In this post we'll look at another rule, not as common as (3) or (4), but nonetheless very common.

As I noted before, □ as defined by these rules is fairly weak, because it doesn't say there are any tables at all. Given the way our rules are set up now, we start with the Reference Table and from there we construct any further tables we need. But what if we can't actually get a table to begin with? An example might be a job search. You might have a number of requirements (Box) for any acceptable candidates, but there might not be anyone meeting those requirements. Then you would have a Reference Table, but no tables following from it. The reason for this is that Box doesn't imply that there are any other tables at all; it just tells what has to be the case if there are any tables. For the Reference Table to give you another table, it needs to have a Diamond on it.

A very common assumption, however, is that if you have any Box on your Reference Table at all, there will be another table. If this assumption is true, there is never a situation in which you can have a Box statement in your Reference Table but no other tables. This gives us a new possible rule, which we can call the subalternation rule, but it is also often called D, so we'll continue to call it D:

(D) □ on the Reference Table implies that there is another table on which the statement is found.

We could also say the same thing more simply:

(D) □ includes ◇.

There are cases where this rule doesn't apply, but it makes sense for a great many different situations.

If it is always true that dragons breathe fire, it often makes sense to conclude that sometimes dragons breathe fire. If everyone in the class went on the trip, one concludes that someone went on the trip. If there is red ink everywhere, it is reasonable to say that there is red ink somewhere. If a dress is wholly blue, at least part of it is blue. If you are required to clean your room, you are allowed to clean your room. If something is necessary, we usually think it is possible. All of these common-sensical claims are particular instances of subalternation.

With D our modal reasoning begins to expand in extraordinary ways.

Part VI

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