Necessary and antipossible worlds?

Question

So suppose that ☐A → ~(◊~A), but also that ◊A → ~(☐~A). Or, rather:

◊(☐A → ~(◊~A)) & ◊(◊A → ~(☐~A)) {i.e., either order of definition is itself possible}

Maybe I'm being a fool, but I'm finding it hard to carry out the necessity-based order of definition, however, by using "necessary worlds" talk. Recall that "possible worlds" talk involves {∃, ∀} over a set (or class) of worlds: if x is in some possible world, x is possible, and if x is in all possible worlds, it is necessary. Likewise, x is contingent if x is not in all possible worlds, impossible if it is not in some possible world (if in no, or zero-many, worlds). But so switch out for "necessary worlds." The first thing that comes to my mind is, "x is necessary if x is in some necessary world," and proceeds from there, but it sounds so absurd.

For example, then, "x is possible if x is in all necessary worlds." However, if something is necessary "at all," then isn't it possible "on the side" too? Worse, then, impossibility would amount to existing in not all necessary worlds, and contingency would be existing in none of them. (That last actually sounds fine, to my ears.)

Or what if we started out from contingent worlds, or impossible worlds? Do we still get wacky results? "Something is impossible if true in at least one impossible world." "Something is contingent if true in all impossible worlds." OK, yeah, that's a no-go.

To make matters worse, what if we bring in concepts like antipossibility, antiactuality, and antinecessity? Rather than these being usable in straight definitions of the promodalities, they would mirror the "possible" definitional orders for the promodal operators. So the obvious starting question would be to evaluate, "x is antipossible if true in some antipossible world," and continue on, though here my modal intuitions are for now failing me even more deeply than in the above. Vs. the above, at least, my intuition tells me that "this is why we'd be better off using the concept of possibility first and foremost." Regarding antimodality? No clue.

Answer 1

What is a "possible world" ?

In the Kripke semantics we consider a Frame as a set of 'worlds', a relation over that set (called the accessibility relation), and an evaluation function for a set of propositions for each world. One of these worlds is usually denoted as the 'current world', or 'actual world'.

A world is usually said to be a "possible world" if it is related to the actual world by this accessibility relation. Personally, I prefer the term "accessible world", precisely to avoid the linguistic confusion you are having.

The possibility and necessity propositional operators may then be defined in terms of this semantics :

• A proposition is said to be possible, iff there exists some accessible world where it is evaluated as true.

• A proposition is said to be necessary, iff in every accessible world, it is evaluated as true.

• A proposition is said to be impossible, iff there is no accessible world where it is evaluated as true. That is that it is not possible.

• A proposition is said to be contingent, iff there exists some accessible world where it is evaluated as true, and there exists some accessible world where it is evaluated as false. Thus it is neither necessary nor impossible. This is a subset of possibility that excludes necessity.

There is no concept of necessary world, contingent world, or impossible world that can fit this semantics. Further, there is no need for such concepts, as the operators are well defined in this manner.

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Note: A Frame may contain worlds not accessible from the actual world, but these are not generally considered "impossible worlds".