Elements of Modal Logic I

When dealing with arguments we generally have a universe of discourse; 'universe of discourse' in the sense used here is the category that includes all the things that are relevant to the argument and leaves out all the things that are not relevant to it. It is presupposed by everything in the argument. For instance, if I say, "Dragons breathe fire," it matters considerably whether we are talking about characters in a story or real-life monitor lizards.

Imagine we had a table corresponding to this universe of discourse; on this table we could write whatever premises might be used. So, for instance, we might have a very simple table, corresponding to a particular group of men, on which we've written:

TABLE 1
Socrates is strange.
Plato is not strange, but elegant.
Aristotle is not strange, but bossy.

We could then put these together in ways logically implied by these premises, and say that these things are true at Table 1.

But we don't always assume the same category. For instance, Table 1 might correspond to a particular group of men at a certain time. We might then also be interested in the same particular group of men at a different time. So we could have different tables. If Table 1 is the group of men on Tuesday, perhaps another thing we are interested in is the same group of men on Wednesday, so we can make another table corresponding to that, e.g.:

TABLE 2
Socrates is strange.
Plato is strange.
Aristotle is not strange, but bossy.

Strictly speaking, each table has its own logical operations and conclusions. For instance, on Table 1, we can conclude that neither Plato nor Aristotle are strange; but on Table 2, this is false. This is not a contradiction, since in drawing each of these conclusions we only stay on the relevant table, and don't leap from one to the other.

But we may also be interested in how they related to each other. And we certainly can say something about that. For instance, we can say, if these are our only two tables, "If we're looking at these tables, we would always find that Socrates is strange." We can also say, "We can find a table on which Plato is strange." In our example, the tables in question are interpreted as days, but they could be anything else. We could have tables that represent cities, possible worlds, stories, or whatever we please.

At its most crude and basic, this is all that modal logic is. "On any table, we would find that Socrates is strange" is a Box proposition; it is said to have a strong modality. "We can find a table on which Plato is strange" is a Diamond proposition; it is said to have a weak modality.

Even this on its own is, logically speaking, very important. But there is much more that could be done. For instance, we've been assuming that it's perfectly straightforward what you have to do to find a table on which a claim is found. But it could be that there are restrictions on how we find tables; and, as it happens, depending on how we interpret the table, we often want there to be different kinds of restrictions on which tables we can talk about under which circumstances. There might be tables that we can find if we start at one table but not if we start at another. Likewise, it's possible that one of the claims we have on a table tells us something about other tables, or about what we will find if we find any other tables. And in practice, most of what is dealt with in modal logic has not to do with general statements about tables as a whole, but about how they relate to each other given different assumptions and interpretations. I hope to begin discussing this a bit in a future post.

This is the first post in a series in which I'm experimenting with finding the simplest possible ways to teach the basics of modal logic at an undergraduate level, so that, for instance, one could introduce serious and substantive discussions of modal arguments into an intro-level class, or, for that matter, a high-school class, without taxing the students with too many technicalities more suitable to a higher-level course. The word 'experiment' is used advisedly.