Comparing one modal interchange with another

Question

In the normal systems of modal logic, you can have either of the following:

1. Possibly X is defined as not necessarily not X
2. Necessarily X is defined as not possibly not X

I know this is not the same phenomenon outright, but consider the following semi-normal modal logic (stipulationwise so as to have an ineliminable actuality operator):

3. Possibly X implies Possibly actually X & Actually possibly X
4. Actually X implies Possibly actually X & Actually possibly X

I'm not sure what to make of this. I was thinking of contrasting it with the box/diamond situation: whereas with the box and the diamond, their reflective definability indicates you can/may start with either one, it makes no absolute metaphysical/ethical difference, opposite this the outright identity of the counterpart implications for the actuality/possibility operators means that these operators must both be commenced from; there is no absolutely 'mere' possibility that we can appeal to, so as to escape the fortress of existence, but just as well there is no existence that is independent on the existence of (contingently) pure possibilia also. Is that a lesson worth learning from the identity at issue, or are there any (other) lessons worth learning from that at all?

Answer 1

I'm not exactly sure what you asking. But in case you weren't aware, there is an extension to modal logic in which an 'actually' operator, A, is defined. It is most commonly used in quantified modal logic. Its purpose is to express that AP is true with respect to a possible world, w, iff the embedded proposition P holds in the model's designated actual world w*.

This allows us to say things that would not otherwise be possible. For example, consider the proposition:

It is possible that all red things are shiny. Specifically, all things that are actually red, may have been shiny, and not necessarily red as well.

This cannot be represented as:

1. ◇(∀x)( Red(x) ⊃ Shiny(x) )

This would mean that it is possible for all red things to be shiny as well, i.e. there is a possible world in which all the things that are red are also shiny.

It also cannot be represented as:

2. (∀x)( Red(x) ⊃ ◇Shiny(x) )

This would mean that everything that is red is possibly shiny, but not together, i.e. for every red thing there is a possible world where it is shiny, but these could all be different possible worlds.

It can be represented as:

3. ◇(∀x)( A(Red(x)) ⊃ Shiny(x) )

This means that it is possible for everything that is actually red to be shiny, i.e. there is some possible world where everything that is red in the actual world is shiny in that possible world.

Another example is that the following is apparently false:

4. It is possible that the person who invented the zip did not invent the zip.

But the following is true:

5. It is possible that the person who actually invented the zip did not invent the zip.

There is some more stuff along these lines in Martin Davies, "Reference, Contingency, and the Two-Dimensional Framework" Philosophical Studies 118, (2004), pp. 83-131.