The modal logic I am considering is the "Simplest Quantified Modal Logic" which combines first-order predicate logic with identity, with S5 in the most straightforward way, described here and slightly differently here (with no accessibility relation, but with S5 it doesn't matter here). The semantics are possibilist.

Suppose in our world, w0, Pegasus is not actual, and has wings (much like he would be described in mythology, and how, I think, most people would think of him). We can describe a possible world, w1, exactly like w0 except that that Pegasus is not actual and Pegasus does not in fact have wings (perhaps the possible object was born with a mutation at w1), and so does not fall within the extension of "has wings" in w1. So in both worlds, we have a non-actual object, and an accidental property of this non-actual object which it has in one world but not the other.

Using formal notation, we can describe the situation by letting the constant p denote Pegasus, A stand for the "is actual" predicate, and W stand for the "is winged" predicate:

  1. V(p) ∉ [V(A)](w0) and V(p) ∉ [V(W)](w0)

  2. V(p) ∉ [V(A)](w1) and V(p) ∈ [V(W)](w1)

  3. Aside from this one difference, the values of all other constants and predicates are exactly the same

To me, this seems bizarre and unintuitive. In both worlds all of the facts about actual objects in the world are the same, including all of the facts about history, the myths we tell about Pegasus, anything believed about Pegasus, the evolution of the world from beginning to end, and so on. Just as a matter of fact, the non-actual object Pegasus falls within the extension of "has wings" in one world, and does not fall within the extension of "has wings" in the other. At the very least, it seems on any empiricist epistemology, the facts of the matter about the accidental properties possible objects like Pegasus seem, in principle, unknowable.

I like possibilist semantics because of its simplicity, but this really irks me for some reason. I tried googling around a bit, but couldn't find any possibilists who talk about something like this issue of accidental properties of possibile objects. Or, is it even an issue? Is it just flexibility in the possibilist semantics that possibilists accept?

  • "Wings" seems like poorly chosen example, since the usual conception of Pegasus is as a "winged horse." Or is that the point? Would your question be the same about a non-definition property such as "has a black spot on its forehead"? – Chris Sunami Oct 22 '18 at 17:31
  • I think as long as his being winged is an accidental property, the point should be the same, even if our usual conception of him is as being winged. As long as Pegasus isn't necessarily winged the example should be okay, and I'm just assuming Pegasus isn't necessarily winged by stipulation. But I see what you mean. Maybe a better example where the property is more clearly accidental would avoid potential confusions/controversies between definitions/essences/properties we use to pick out an object (even if these properties aren't essential to the object). – Adam Oct 22 '18 at 18:27
  • I am confused. I thought Pegasus being non-actual possible individual means that there is a possible world where it exists, even though it does not exist in our actual world. Not that it is somehow "in" the actual world, but "non-actually". In the worlds where it does not exist it has no properties either, does it? Even rigid designators designate the same object only in the worlds where it exists. – Conifold Oct 22 '18 at 21:39
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    @Conifold I think that's what is at stake in the possibilist-actualist debate? Possibilists conceive of a fixed domain of objects across all worlds, but include an existence predicate that varies from world to world depending on which objects actually exists (as opposed to those which possibly exist). So Pegasus (and everything else) exists necessarily (in all possible worlds as possible objects) but isn't actual (doesn't fall in the extension of the world-varying actuality predicate) in all worlds. Actualists conceive of a domain that varies from world to world. That's my understanding... – Adam Oct 22 '18 at 22:56
  • Indeed, @adam it might help if you explained us better what "actual" means in that context because it is polysemic (perhaps terms as "materialized" or "observed" would communicate better). Also, it would help us, if we had a better understanding of what practical problem this form of logic aims to solve in the here and now; or, at least, what question it would be a mind experiment for? – fralau Dec 7 '18 at 5:50

In positive free logics, non-existent objects can satisfy (atomic) predicates, so it can come out as true that Pegasus has wings. But in negative free logics, non-existent objects cannot satisfy (atomic) predicates, so it cannot come out as true that Pegasus has wings.

You can get different variable domain modal logics by combining propositional modal logics with either of these systems, so - as you say - it really does depend on what kind of semantics you adopt for your modal logic. Of course, there are many and various philosophical reasons for choosing one kind over another.

There is excellent discussion of all these issues in the second half of Graham Priest's book From If to Is: An Introduction to Non-Classical Logic.

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