Monday, June 09, 2014

Elements of Modal Logic V

Part I. Part II. Part III. Part IV.

So far we have recognized two defining rules:

(1) □ on the Reference Table means the statement would be found on any table there might be.

(2) ◇ on the Reference Table means that there is a table on which the statement 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 ~□~.

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

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. However, it may well be that the Reference Table does not give us enough information to construct any table. 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 tables at all; it just tells what has to be the case if there are any tables. There are a number of situations in which this is true, like the various systems of predicate calculus.

A very common assumption, however, is that if you have any Box statements at all, you could conclude a Diamond statement from them. 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, and which we will label (D):

(D) □ includes ◇.

We could also put it in a longer version, which means the same thing:

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

We can see what this means in practice, and why situations might often call for it, if we recall some of the examples of Box and Diamond mentioned in last post:

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

If something is true always, it's obvious that we can conclude that it is also true sometimes. If something is true everywhere, we can conclude that it is true somewhere. If something is obligatory, it is also permissible. If something is necessary, it is possible. And so forth. To be sure, we could use the terms in ways in which that these assumptions are false (for instance, we could use 'possible' to mean 'merely possible' rather than 'at least possible'), but in most situations, we do want Box and Diamond to be related in this way. It makes reasoning neater and easier.

In the next post, we'll look at one more important rule, not as common as subalternation, but still very common; after that we'll look at how these rules work in some examples, before going on to a different kind of rule that we sometimes find used in modal reasoning.

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