Wednesday, May 14, 2014

Elements of Modal Logic III

Part I. Part II.

Suppose our Reference Table summarizes your schedule for a given week. This means that the Reference Table is not one of the tables described by itself, because it is not a day of the week. Suppose further that this is your Reference Table:

REFERENCE TABLE (Schedule for the Week)
□ (If it is the day before trash collection day, I take out the trash)
□ (I brush my teeth)
□ (I mow the lawn, or I weed the garden, or I clean the kitchen)
◇ (I do not mow the lawn and I do not weed the garden)
◇ (It is the day before trash collection day)
◇ (I do not clean the kitchen)
◇ (I do not take out the trash)
◇ (I do not weed the garden and I do not clean the kitchen)
◇ (I do not clean the kitchen and I do not mow the lawn)
◇ (It is Monday and it is not the day before trash collection day)
◇ (I go to the movies)

So what do we know here, if we just go off the information in our Reference Table? Although I only stated half of each rule in the previous post, our reasoning gives us a complete rule for each modal operator. We have a rule for Box:

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

And we have a rule for Diamond:

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

So we get eight tables, in no particular order, indicated by our Diamond-ed statements (more on the number in a bit), and each of these will have the statement on it and also every statement that is Box-ed. I'll label each one with a letter.

Table A: Some Day or Other in the Week
If it is the day before trash collection day, I take out the trash
I brush my teeth
I mow the lawn, or I weed the garden, or I clean the kitchen
I do not mow the lawn and I do not weed the garden

Table B: Some Day or Other in the Week
If it is the day before trash collection day, I take out the trash
I brush my teeth
I mow the lawn, or I weed the garden, or I clean the kitchen
It is the day before trash collection day

Table C: Some Day or Other in the Week
If it is the day before trash collection day, I take out the trash
I brush my teeth
I mow the lawn, or I weed the garden, or I clean the kitchen
I do not clean the kitchen

Table D: Some Day or Other in the Week
If it is the day before trash collection day, I take out the trash
I brush my teeth
I mow the lawn, or I weed the garden, or I clean the kitchen
I do not take out the trash

Table E: Some Day or Other in the Week
If it is the day before trash collection day, I take out the trash
I brush my teeth
I mow the lawn, or I weed the garden, or I clean the kitchen
I do not weed the garden and I do not clean the kitchen

Table F: Some Day or Other in the Week
If it is the day before trash collection day, I take out the trash
I brush my teeth
I mow the lawn, or I weed the garden, or I clean the kitchen
I do not clean the kitchen and I do not mow the lawn

Table G: Some Day or Other in the Week
If it is the day before trash collection day, I take out the trash
I brush my teeth
I mow the lawn, or I weed the garden, or I clean the kitchen
It is Monday and it is not the day before trash collection day

Table H: Some Day or Other in the Week
If it is the day before trash collection day, I take out the trash
I brush my teeth
I mow the lawn, or I weed the garden, or I clean the kitchen
I go to the movies

Let's look a little more closely at Table A. One of our statements in Table A is "I mow the lawn, or I weed the garden, or I clean the kitchen," while another statement is "I do not mow the lawn and I do not weed the garden". If we put these two together, we can see that the only way they can both be put together as part of our schedule for the day is if, on whatever day A may be, I clean the kitchen. So this follows as a conclusion on Table A, and we can write it down as something we know about A; to indicate that it's a conclusion, I'll put it below a line.

Table A: Some Day or Other in the Week
If it is the day before trash collection day, I take out the trash
I brush my teeth
I mow the lawn, or I weed the garden, or I clean the kitchen
I do not mow the lawn and I do not weed the garden
------------------------------------
I clean the kitchen

We can do the same thing with other tables: look to see if what the Reference Table requires for that day gives us a conclusion specific to that day. For instance, with Table B:

Table B: Some Day or Other in the Week
If it is the day before trash collection day, I take out the trash
I brush my teeth
I mow the lawn, or I weed the garden, or I clean the kitchen
It is the day before trash collection day
------------------------------------
I take out the trash

The conclusions you can get may vary from table to table. On some tables we might not have enough information to derive these kinds of additional conclusions, but this is just a matter of how much we're told by the Reference Table.

Now one thing that I've glossed over so far is an additional bit of information that's not on our Reference Table but would likely be assumed in the real world: a week has only seven days. Let's assume this little bit of information. We already knew that our tables could be all the tables or only some of them; and we already knew that some of our tables could actually be describing the same thing without our knowing it. Knowing that there are only seven days in a week, and that our tables describe days in a week, and that there are eight tables, we know that at least two tables (maybe more, but at least two) describe the same day. What is more, we can see by the statements that some tables cannot be describing the same day, because they say contradictory things.

The following pairs of tables cannot describe the same day: A and C, A and E, B and D, B and G, C and E, C and F, E and F. If we wanted to, we could add this as an extra conclusion to all of the relevant tables (e.g., we could add 'Today is not the same day as Table C' to Table A). Any of the others might be describing the same day. And, again, at least two tables, whichever ones they might be, definitely are describing the same day. And note also that H could be describing the same day as any other table.

What we're seeing here is one of the most important aspects of modal logic. Modal logic is also often called 'intensional logic', which is just a fancy way of saying that it's a logic in which it can matter what you're talking about. In this case, since we already know that a week has seven days, we can take that into account in our reasoning, because we're talking about days in the week.

There's no limit to what assumptions you could add in this way, or to what number of assumptions you can add in this way. It just depends on what you're doing. Adding assumptions in this way usually keeps our modal reasoning very simple and very weak. There are assumptions, however, that you can add that make your modal reasoning much more powerful, capable of doing much more, and there are kinds of situations where the assumptions make sense, and kinds of problems where you need that extra power. These assumptions are assumptions about Box and Diamond themselves, and they generally fall into two groups:

1. assumptions about how Box and Diamond are related to each other
2. assumptions about what it means if you have a modal operator (either Box or Diamond) for a statement that already has a modal operator (either Box or Diamond)

We'll set the second assumption aside for right now, and focus on the first kind of assumption. So far we only have a rule for Box and a rule for Diamond; they don't really have anything directly to do with each other, at least as far as we know. But what if we could get information about Box from Diamond and/or information about Diamond from Box? This would let us do a bit more. In the next post we'll start looking at some of the most important assumptions we are sometimes allowed to make.

to be continued