Wednesday, June 07, 2017

Elements of Modal Logic, Part IX


Let us take some examples to see how we can use what we have learned so far. Suppose that I, being a wizard with a green thumb, am surveying my new garden of weird flowers. I have some things that I have in every quarter of my garden. That's a Box. I have other things that are in at least one quarter. And if something is in every quarter of my garden, it is in at least one quarter, and that means Box includes Diamond. So we want at least 1234D.

[1234D] Reference Table (Weird Flowers in Quarters of the New Garden):
Box (Man-eating Dandelions)
Box (Fire-breathing Snapdragons)
Diamond (Jam-and-Butter-Cups)
Box (Really Red Roses)
Diamond (Screaming Mandrakes)
Box (More-than-deadly Nightshades)
Box-Not (Daisies)
Not-Diamond (Tulips)
Diamond-Not (Venus Fly-Traps)

Now, since Box here means 'in every quarter', Not-Box means 'not in every quarter (i.e., not in some quarters)', Box-Not means 'in every quarter not (i.e., in no quarter)', and Not-Box-Not means 'not in every quarter not (i.e., in some quarters)'. From this it is easy to see how the corresponding Diamonds work, using Rules (3) and (4). Our square of opposition is (again, I only show Box, but each corner can be translated into a Diamond version):

If we take an item in our reference table, like Box (Man-eating Dandelions), then we can use the square to say what's consistent and inconsistent with this item. For instance, Box (Man-eating Dandelions) is contrary to Box-Not (Man-eating Dandelions), and it is contradictory to Not-Box (Man-eating Dandelions); so Box (Man-eating Dandelions) rules both of these out. On the other hand, it requires Not-Box-Not (Man-eating Dandelions) -- Rule (D) tells us that Box includes Diamond. And Rule (4) tells us that Not-Box-Not (Man-Eating Dandelions) is interchangeable with Diamond (Man-Eating Dandelions), so that's required, too. We can put our reasoning in a simple form:

(i) □ (Man-Eating Dandelions)
(ii) ◇ (Man-Eating Dandelions) -- [from Rule (D)]
Which is the same as:
(iii) ~□~ (Man-Eating Dandelions) -- [from Rule (4)]

If we look at Diamond-Not (Venus Fly-Traps), this is the same corner as Not-Box (Venus Fly-Traps); it is inconsistent with both Not-Diamond-Not (Venus Fly-Traps) and Box (Venus Fly-Traps), since those both are the same. However, it doesn't tell us anything about the other two corners -- it's consistent with Not-Diamond (Venus Fly-Traps) and Diamond (Venus Fly-Traps).

It's worth taking some time to explore how each of the corners of the 1234D square, both with Box and with Diamond, relate to the others, because it is just so very common, and if you know this very well, you know a huge amount of modal logic, because it's how you fully understand what a given Reference Table means.

So let's think about what our other tables have to be, given our Reference Table.

If you are only using 1234, you should always do Diamonds first, but we are using 1234D, and both Rule (D) and our square of opposition tell us that Box includes Diamond. Note that by our rules, Not-Diamond (Tulips) [= There are no quarters with tulips] means the same as Box-Not (Tulips) [= every quarter has no tulips]. If I were just using 1234, I might not have any quarters in my garden (maybe I haven't planted any yet). But (D) tells me that if I have any Boxes in my Reference Table, I have at least one quarter in my garden that has them (which is Diamond), so I know I have at least one quarter in my garden, and since Boxes are true of every table that we have, all our Boxes tell us about that quarter:

TABLE 1: Some Quarter in the New Garden
Man-eating Dandelions
Fire-breathing Snapdragons
Really Red Roses
More-than-deadly Nightshades
No Daisies!
No Tulips!

I put 'No Daisies' because we know that there are no daisies anywhere (Box-Not). And I put 'No Tulips' because Not-Diamond means the same as Box-Not.

With the flowers that are only Diamond, though, we have to be much more careful, because while we know they each are on at least one table, we don't know which or how many! Maybe Jam-and-Butter-Cups are only in one quarter. Maybe they are in two. Maybe they are in three. It could even be that they are in all four. Our Reference Table doesn't tell us. And it's even trickier, because we have more than one Diamond, and we don't know if they are talking about the same quarters or different quarters. There are lots of possibilities. So when I put them on my table, I can't assume that every table is a different quarter. Maybe I am accidentally giving two incomplete descriptions of the same table, and they really should be on the same table! But I can't just put them on the same table, either, because maybe they are all on different tables! Since the Reference Table doesn't tell us, we have to be careful to remember that we don't know these things. Not assuming that you know something you don't is often the single most important thing in logic.

What we do know from the Diamonds, is that (whether or not they are overlapping or separate) the following things are true:

(1) I have at least one quarter with Jam-and-Butter-Cups.
(2) I have at least one quarter with Screaming Mandrakes.
(3) I have at least one quarter without Venus Fly-Traps.

Again, I don't know if these quarters are the same or different. If I put them all on one table, I might be wrong. If I put them on different tables, though, then as long as I remember that any of my tables might be incomplete, and that any two tables might be giving incomplete descriptions of the same quarter, I will be just fine.

TABLE 1: Some Quarter in the New Garden
(Don't Know Which)
TABLE 2: Some Quarter in the New Garden
(Don't Know Which)
TABLE 3: Some Quarter in the New Garden
(Don't Know Which)
Man-eating DandelionsMan-eating DandelionsMan-eating Dandelions
Fire-breathing SnapdragonsFire-breathing SnapdragonsFire-breathing Snapdragons
Really Red RosesReally Red RosesReally Red Roses
More-than-deadly NightshadesMore-than-deadly NightshadesMore-than-deadly Nightshades
No Daisies!No Daisies!No Daisies!
No Tulips!No Tulips!No Tulips!
Jam-and-Butter-CupsScreaming MandrakesNo Venus Fly-Traps!

Or, if we prefer to have it in a slightly more diagrammatic form:

Part X

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