Wednesday, April 26, 2017

Elements of Modal Logic, Part I

This is a reworking of an experiment I began but never quite finished a while back, on a different approach to modal logic. What is the simplest way to teach it that allows the greatest kind of flexibility for intelligent use of it? The idea is that instead of trying to jump into standard modal logic systems, one should actually get some sense of how one builds modal logic from scratch, without necessarily going into all the technicalities that are there are only to get a system of a particular shape for reasons of purely logical convenience. I actually think this is a problem with a lot of 'introductory' logic courses; they are usually actually courses in using technical formulae to solve narrow ranges of problems. This is an excellent thing to have. But it's not really introducing you to anything except very particular technical formulae.

Whenever we are reasoning, some things are relevant and some are not. The universe of discourse, in the sense we will use it here, is the category that includes all the things that are relevant and leaves out all the things that are not relevant. Everything in the reasoning presupposes it. 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 -- if I switch from one to the other, it is like we are in a completely different universe of topics. A universe of discourse can be any kind of thing that you can talk about -- a time, a place, a shelf, a ball, an animal, anything whatsoever.

Sometimes we want to take an inventory of things that are in a universe of discourse. For instance, your universe of discourse might be 'Things that are scheduled at 2 o'clock'; and it might be important to know what some of these things are. Imagine we had a table corresponding to this universe of discourse; on this table we could keep track of our inventory of the universe of discourse:

Tom's party
Gabriel's job interview
The End of the World

But in real-life discussion, things can get more complicated than this. So, for instance, we might have a very simple table, for claims about Greek philosophers, on which we've written:

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 for the Table 1 inventory' or just 'true at Table 1'.

But we aren't always considering only one category; sometimes it is necessary to compare things across categories while keeping clear about the fact that they are in different categories. For instance, Table 1 might correspond to Greek philosophers at a certain time, but we might then also be interested in the same Greek philosophers but at different times. So we could have different tables. If Table 1 is for Greek philosophers on Tuesday, perhaps another thing we are interested in is the same group on Wednesday, so we can make another table corresponding to that, e.g.:

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

Strictly speaking, each table works on its own. For instance, on Table 1, we can reasonably 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 the universes of discourse compare to each other. And we certainly can say something about that. For instance, we can say, if these are our only two tables, "'Socrates is strange' is constant for all our inventories." We can also say, "We can find an inventory with the claim that 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, because the tables just list things in universes of discourse, and anything we can talk about can be a universe of discourse.

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; we could also say it has a strong modality. "We can find a table on which Plato is strange" is a Diamond proposition, which is a weak modality.

Even this on its own is, logically speaking, very important. But we can do so much more! For instance, we've been assuming that it's easy to know what you have to do to find the right information. But you might still be learning what's in the inventories for each universe of discourse! It could be that there are restrictions on what we can know. There might be tables that we can find if we start at one table but not if we start at another. Likewise, one table might be able to teach you about other tables. There are lots of possibilities, because there are endlessly many universes of discourse; and what we need to do is to start thinking about how to handle all of these possibilities.

Part II

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