Thursday, May 25, 2017

Elements of Modal Logic, Part VIII

Part VII

We have been classifying modalities by which of the rules -- (1), (2), (3), (4), (D), (M) -- they are assigned. We also looked at squares of oppositions, although, since (3) and (4) make Box and Diamond interdefinable, we only looked at the Box versions. The 1234-Box, you recall had this square of opposition:

And the 1234D-Box looked like this:

So what happens with M? It's not an easy question. If we just add (M) to the 1234 square, we don't have any way yet of putting that on the diagram. Remember (M) just tells us that Box puts something on the Reference Table. We need something to represent 'being on the Reference Table'. This actually is a modal operator; it's often not explicitly noted, but one way to think of (M) is as introducing a new modality. We need to define it more explicitly to use it, though, because we need to be able to represent it. So we introduce a new rule whose purpose is simply to make explicit the modal operator implicitly introduced by (M), and I will call this Rule (T).

(T) T, applied to anything, places that to which it applies on the Reference Table.

We can use the definition in (T) to restate (M) in a different way:

(M) □ includes T.

And with this we have the means of making a square of opposition. This is the square of opposition for 1234M (or, we could equally call it, 1234TM):

T is obviously the contradictory of T-Not. (M) tells us that you can get from Box to T, and of course we have the corresponding arrow from Box-Not to T-Not. Those together give us three new contrariety oppositions.

What will happen if we also add Rule (D) to this? We get something like this:

Basically, as you might expect, this square is the 1234D square combined with the 1234M square; the one new thing is that when they are put together, the combined oppositions make it so that it also has to be true that T also works like ◊.

But here's an interesting question. Our square of opposition has T and T~. But what about ~T~ and ~T? When we think about the oppositions among these, we find something interesting:

If we are using a classical kind of negation (i.e., 'Not' is not being used in a weird way), then both of the left-hand modalities are contradictories of both the right-hand modalities, and our arrows between top and bottom go both ways -- from T you can get ~T~, and from ~T~ you can get T, and so forth. They are equivalent, so you can substitute them for each other whenever you want. Thus we could equally just represent this square of opposition as a line: on the left, T and ~T~; on the right, T~ and ~T; and the left and right are contradictory. This is why it shows up as a line on our squares of opposition above.

There are actually four different kinds of squares of opposition. In a degenerate square, all four corners are equivalent to the others -- they just all can be substituted for each other, and we could represent our square as if it were a single point. A semidegenerate square we can collapse to a line, and our T square of opposition is an example. A square of opposition that looks like our 1234 square above is often called a Boolean square. And a square of opposition that looks like our 1234D square is called a classical square.

What about our 1234M and 1234DM squares? They are actually a combination of squares. This is not surprising -- we've only been looking at the Box side, rules (3) and (4) tell us how to combine two different squares of opposition -- one with Box and one with Diamond. It's just that rules (3) and (4) go in both directions, from Box to Diamond and Diamond to Box, so the Boolean squares fit perfectly on top of each other. (D) and (M) only go in one direction, from Box to Diamond and from Box to T, so they complicate things slightly. Adding (D) to 1234 turns the Boolean squares into classical squares. And adding (M) gives us another square entirely. So 1234M is a Boolean square (Box) linked with a Boolean square (Diamond) and both of those linked with a semidegenerate square (T). And 1234DM is a classical square (Box) linked with a classical square (Diamond), and both of those linked with a semidegenerate square (T).

That we can fit various kinds of squares of opposition together in various ways is immensely important, and there is no limit to it. Every square of opposition is a kind of modality, and you can fit together all kinds of modalities together, if you just have the right rules for them. If you wanted to, you could have a tangle of modalities that would be represented by a thousand distinct Boolean squares combined with a thousand distinct classical squares combined with a thousand distinct semidegenerate squares combined with a thousand distinct degenerate squares. You'd need rules to link them up; but if you had the rules, there is no limit to how complicated you can get.

There is a jungle of different kinds of modal operators out there; and we've hardly begun exploring them. One thing we need to ask is what these squares of opposition have to do with our tables.

Part IX

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