So far we have considered five rules on the basis of which you might construct a modal logic -- (1) and (2) are defining rules that tell us how to understand Box and Diamond, (3) and (4) are interdefining rules that tell us how Box and Diamond are related to each other, and (D) is the subalternation rule, which tells us that what is true of Diamond will also be true of Box. As noted, one could have different rules in each case -- (1) and (2) are more or less fixed as definitions of Box and Diamond, but you could have variations that introduce qualifications or limitations; (3) and (4) are easily the most common rules for interrelating Box and Diamond, but you could have different rules; and whether or not one accepts (D) depends on what you are trying to say. But it is indeed the case that most modal reasoning uses these rules.
(1) □ applying to anything on the Reference Table means that it would be found on any table there might be.
(2) ◇ applying to anything on the Reference Table means that there is a table on which it is found.
(3) □ is interchangeable with ~◇~.
(4) ◇ is interchangeable with ~□~.
(D) □ includes ◇.
Each of these rules adds a new layer to the character of Box and Diamond. Let's focus on Box for the moment, since Box and Diamond are in many ways parallel. Every rule imposes a division or partition on our possibilities. Rule (1) divides our possibilities into at least two options: Box and Not Box. This could be all we have, but usually what we are applying Box to can be negated as well. That is, we have four possibilities:
Box
Not-Box
Box-Not
Not-Box-Not
Of these, Box is directly opposite to Not-Box -- they contradict each other; and the same is true of Box-Not and Not-Box-Not. so we get the following relationships:
The diagonal lines means 'These contradict each other' or 'These are exact opposites'. The above diagram represents Rule (1) when we are using Nots both before and after Box. You can, of course, do exactly the same thing with Rule (2) for Diamond.
Rules (3) and (4) together make our Box diagram and our Diamond diagram the same diagram, by aligning the corners -- that is, (3) tells us that the □ corners is the same as the ~◇~ corner, and (4) tells us that the ◇ corner is the same as the ~□~ corner. Since each of these has a direct opposite, we can then figure out that □~ corner of the diagram must be the same as the ~◇ corner, and that the ~□ corner must be the same as the ◇~ corner. So we can substitute the Diamond symbols for the Box symbols, and vice versa, whenever we want to do so.
With (D) we get something new. (D) tells us that we can get a ◇ from any □, but, of course, that's one-direction; it doesn't work in the opposite direction. We can represent this by adding an arrow from □ to ~□~, which (4) tells us is the same as ◇, and another arrow from □~ to ~□, for the same reason. But if □ always gives us ~□~, and ~□~ is inconsistent with □~, then □~ and □ have to be inconsistent, too. So if (D) is one of our rules, those also have to have a line between them. We then get the following diagram:
(And, of course, we have to remember that we can substitute Diamonds as long as we follow Rules (3) and (4) in doing so.) Corners separated by the diagonal bars are usually called contradictories. The corners separated by the horizontal bar at the top are usually called contraries. The corners linked by the arrows are called alternates, with the top one being the superalternate and the bottom one being the subalternate. The two bottom corners, which are neither separated by a bar nor linked by an arrow are usually called subcontraries. And, of course, any diagram like this is called a square of opposition. The relations that make it up were first discovered by Aristotle himself; and putting it in a diagram form like the one above goes back at least to the second century, and, because a form was used in a logic commentary by Boethius, became one of the most famous and influential of all logic diagrams. Medieval logicians didn't think in terms of Box and Diamond, but they did recognize you could form this kind of diagram with All and Some, and also with Necessary and Possible, when you are making statements. And you can use the square to think about how different kinds of statements are logically related to other kinds of statements.
Any Box and Diamond that follow Rules (1), (2), (3), (4), and (D) gives us a square of opposition that looks like the above diagram. And any group of concepts which you can relate to each other in the ways given by the diagram above, can be regarded 1234D Boxes and Diamonds. This is worth noting, because 1234D modalities are extremely common, probably because they are very easy to use; we literally use them on a day-to-day basis. Here is just a tiny selection of examples; you can substitute these ideas, or ideas like them, for the corners of the above diagram.
□ = ~◇~ | ~□~ = ◇ | □~ = ~◇ | ~□ = ◇~ |
| All | At Least Some | Not Even Some | Not All |
| Necessary | Possible | Impossible | Possibly Not |
| Always | Sometimes | Never | Not Always |
| Everywhere | Somewhere | Nowhere | Not Everywhere |
| Everybody | Somebody | Nobody | Not Everybody |
| Known | Not Ruled Out | Ruled Out | Not Known |
| Obligatory | Permissible | Impermissible | Omissible |
| Good | Acceptable | Unacceptably Bad | Bad Even If Tolerable |
| Great | At Least OK | Not Even OK | Not Great |
| Wholly | At Least Partly | Wholly Not | Not Wholly |
| Begins to Be | Does Not Begin Not to Be | Begins Not to Be | Does Not Begin to Be |
But 1234D modalities are not the end of the road, by any means; there are many more things yet to be done in order to understand how modal reasoning works. And we can start down that road by going back to our tables. We've been using a Reference Table to track what happens on the other tables. But we can actually do this in more than one way, and if we change the way we do things, we get something new.
Part VII
