Friday, April 28, 2017

Elements of Modal Logic, Part III

Part II.

Another, more complicated example. Suppose you are trying to work out your schedule, so you are putting together a to-do list. You can have a Reference Table summarizes your to-do list for a given week. This means that the Reference Table is not one of the tables described by itself, because your to-do list is not a day of the week! Suppose further that this is your Reference Table:

REFERENCE TABLE (Things I Have to Do This Week)
□ (If it is the day before trash collection day, take out the trash)
□ (Brush my teeth)
□ (Do at least one of these three: Mow the lawn, or Weed the garden, or Clean the kitchen)
◇ (Do not mow the lawn and Do not weed the garden)
◇ (It is the day before trash collection day)
◇ (Do not clean the kitchen)
◇ (Do not take out the trash)
◇ (Do not weed the garden and Do not clean the kitchen)
◇ (Do not clean the kitchen and Do not mow the lawn)
◇ (It is Monday and it is not the day before trash collection day)
◇ (Go to the movies)

The Box statements are like standing rules that I always have to keep in mind every day and not forget; and the Diamond statements are things I have to get done at some point this week or, in some cases, things that I should not do on at least some days. For instance, we can see that I should have some day on which I don't weed the garden and don't clean the kitchen.

So what should our schedule be like, 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 thing to which it is applied can be found on any table there might be.

Or we could state this more simply as "□ tells us something that is constant for any table."

And we have a rule for Diamond:

(2) ◇ on the Reference Table means that there is a table on which we can find the thing to which it applies.

Or we could state this more simply as "◇ tells us there is a table with something on it."

If we apply these rules, 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 that Diamond-ed statement on it and will also have 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, take out the trash
Brush my teeth
Do at least one of these three: Mow the lawn, or Weed the garden, or Clean the kitchen
Do not mow the lawn and Do not weed the garden

TABLE B: Some Day or Other in the Week
If it is the day before trash collection day, take out the trash
Brush my teeth
Do at least one of these three: Mow the lawn, or Weed the garden, or 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, take out the trash
Brush my teeth
Do at least one of these three: Mow the lawn, or Weed the garden, or Clean the kitchen
Do not clean the kitchen

TABLE D: Some Day or Other in the Week
If it is the day before trash collection day, take out the trash
Brush my teeth
Do at least one of these three: Mow the lawn, or Weed the garden, or Clean the kitchen
Do not take out the trash

TABLE E: Some Day or Other in the Week
If it is the day before trash collection day, take out the trash
Brush my teeth
Do at least one of these three: Mow the lawn, or Weed the garden, or Clean the kitchen
Do not weed the garden and Do not clean the kitchen

TABLE F: Some Day or Other in the Week
If it is the day before trash collection day, take out the trash
Brush my teeth
Do at least one of these three: Mow the lawn, or Weed the garden, or Clean the kitchen
Do not clean the kitchen and Do not mow the lawn

TABLE G: Monday
If it is the day before trash collection day, take out the trash
Brush my teeth
Do at least one of these three: Mow the lawn, or Weed the garden, or 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, take out the trash
Brush my teeth
Do at least one of these three: Mow the lawn, or Weed the garden, or Clean the kitchen
Go to the movies

Note that we know that Table G is Monday, because Table G is the one created by "◇ (It is Monday and it is not the day before trash collection day)"!

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, take out the trash
Brush my teeth
Do at least one of these three: Mow the lawn, or Weed the garden, or Clean the kitchen
Do not mow the lawn and Do not weed the garden
------------------------------------
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, take out the trash
Brush my teeth
Do at least one of these three: Mow the lawn, or Weed the garden, or Clean the kitchen
It is the day before trash collection day
------------------------------------
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.

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 Monday' to Table B). 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 (a little harder to see, but also true)note that H could be describing the same day as any other table -- it's not inconsistent with any of them. Given our to-do list, we still have a lot of flexibility with our schedule, but thinking through the rules for our schedule gives us a sense of what kind of schedule makes sense and what kind of schedule doesn't. None of this is difficult -- it is just a big logic puzzle -- but it's worth taking some time to get a sense of how it works, and all the kinds of conclusions you can draw, and enjoy the fact that you are doing modal logic.

**********

But where do we go from here?

Modal logic is often called intensional logic, which is just a fancy way of saying that it's a logic in which it can matter to the logic 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. Many of the most useful 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 lot more, so that's what we need to consider next.

Part IV

No comments:

Post a Comment

Please understand that this weblog runs on a third-party comment system, not on Blogger's comment system. If you have come by way of a mobile device and can see this message, you may have landed on the Blogger comment page, or the third party commenting system has not yet completely loaded; your comments will only be shown on this page and not on the page most people will see, and it is much more likely that your comment will be missed.