It seems that the caravan has moved on while I was away. But thank you both for making me think through what I was saying.

Nuh. I reject your arguments because they are muddled. — Banno

You are still looking for epistemic truth in a semantic system. — Banno

Your ability to amuse me with your ridiculous straw manning never ceases to amaze me. Again, you take your own error "muddled" arguments (here represented as "epistemic truth in a semantic system"), and you pretend that your error is mine. For example, defining an infinite set as "complete" only creates a muddled mess of contradiction.

These sets are not "incomplete" - you trade on an ambiguity here. M is not the actual world, as you think, but an interpretation of a modal system. — Banno

You still completely ignore, and disrespect the second truth condition stated by the SEP. The one I've quoted three or four times now. The actual world of the modal system must "in fact" be the actual world. Here is the complete package of conditions:

Say that M is the “intended” interpretation of ℒ if (i) its set W of “possible worlds” is in fact the set of all possible worlds, (ii) its designated “actual world” is in fact the actual world, (iii) its set D of “possible individuals” is in fact the set of all possible individuals, and (iv) the referents assigned to the names of ℒ and the intensions assigned to the predicates of ℒ are the ones they in fact have. — SEP

@Banno, until you accept the real necessity of "(ii) its designated “actual world” is in fact the actual world", you will never understand the real epistemological problems of modal logic, and why there is so many distinct interpretations. Look:

For abstractionists, however, actuality is a special property that distinguishes exactly one possible world from all others — the actual world is the only world that happens to obtain; it is the one and only way things could be that is the way things as a whole, in fact, are. However, for most abstractionists, the distinctiveness of the actual world does not lie simply in its actuality but in its ontological comprehensiveness: the actual world encompasses all that there is. In a word: most abstractionists are actualists. — SEP 2.2.3

The reason for so many different interpretations, is because it is impossible to make "possibility" as we understand it, consistent with "the truth about the actual world of empirical observation", as we understand that. These two are fundamentally incompatible as Aristotle decisively, and irrefutably demonstrated thousands of years ago. In modern times this incompatibility is known as the uncertainty principle. A fundamental particle cannot have a true, actual location (this implies not moving), and also have the possibility of moving, at the same time.

The multitude of interpretations arise from the attempt to establish compatibility between two incompatible ideas. Human beings are very creative, and industrious, so they will keep trying more and more different ways, never succeeding. They will not succeed because the two are incompatible, and the only way to understand the both of them properly is to model them separately, in a dualist way, with a form of mediation between them.

You haven't followed what is going on in the SEP articles. — Banno

I think I've followed very well. I see section 2 as proposing three distinct interpretations of possible worlds semantics, each of these being insufficient, due to the problem described above. You seem to want to focus on one, the abstractionist interpretation, as if it is the only acceptable interpretation, not allowing for the possibility that it is just as faulty as the other two.

Again, it seems to me that what you are doing is attempting to critique modal theory, which is based on semantic theories of truth, by replacing that basis with a correspondence theory. It's no surprise that this doesn't work. — Banno

Sure, replace correspondence with some other theory of truth. That is just a move of ignorance, denial of the problem, which is the fact that the mode of "possibility" is inconsistent with "the truth about the actual world of empirical observation". Remove yourself from the relevance of the truth about the actual world of empirical observation if you like, but then what good will your logic serve? I mean, you might argue that predictive capacity is far more useful than truth about the physical world, but then why not go to a probabilistic semantics of modal logic. That makes far more sense. Instead, you want "possible worlds", which pretend to assume some sort actual world of fact, yet not respecting that as the basis for "truth". What kind of muddled nonsense is this?

Tarski's semantic theory of truth provides a rigorous, mathematical framework for understanding what makes sentences true. His famous T-schema—"Snow is white" is true if and only if snow is white—captures a correspondence intuition: a sentence is true when it corresponds to how things actually are.
However, there are some important nuances: — Relativist

The problem, as I indicate above, is that it is a pretense to correspondence. That is the problem I mentioned earlier, of replacing intension with extension. Correspondence becomes simply a stipulation, instead of criteria for judgement. The actual world of the modal model "is" the actual world of fact, because this is stipulated as a necessity for truth.

So, "snow is white " is true if and only if snow is white. Yes, now we stipulate "snow is white" (or in the case of possible worlds, the actual world of the modal model is stipulated as in fact the true actual world), and voila, "snow is white" is true by stipulated correspondence, and the actual world of the modal model is true, by stipulated correspondence. But of course, we can all see that this is just a pretense of correspondence.

Banno seems to be trying to deny the pretense of correspondence, to claim some other basis for "truth" in possible worlds semantics. Clearly this is just denial, as reference to "in fact the actual world", in the truth conditions, indicates that truth is based in correspondence. Correspondence by stipulation ("snow is white" is true because snow is white) becomes very problematic, so Banno wants to deny that it's even a part of the modal semantics of possible worlds.

As was noted in §2.1.2, for the concretist, there is no special property of the actual world — actuality — that distinguishes it, in any absolute sense, from all of the others; it is simply the world that we inhabit. For abstractionists, however, actuality is a special property that distinguishes exactly one possible world from all others — the actual world is the only world that happens to obtain; it is the one and only way things could be that is the way things as a whole, in fact, are. However, for most abstractionists, the distinctiveness of the actual world does not lie simply in its actuality but in its ontological comprehensiveness: the actual world encompasses all that there is. In a word: most abstractionists are actualists. — 2.2.3 Actuality and Actualism

Several ideas are introduced here, one being to obtain.

Obtaining is something a state of affairs does. In other words, I can conjure a state of affairs that does not obtain. The distinction between an obtaining state of affairs and a true proposition is kind of fuzzy. The early Bertrand Russell said they're the same thing:

Russell took over from Moore the conception of propositions as mind-independent complexes; a true proposition was then simply identified by Russell with a fact (cf. MTCA, 75-76). — SEP

But we generally draw a distinction between them with a proposition being the content of an expression (or hypothetical expression), and a fact, or state of affairs, being a complex of things and concepts.

So when we say abstractionists are actualists, this means they hold that any state of affairs that obtains, is a resident of the actual world. The actual world itself is a set.

It's not significantly different because Truthmaker theory is a correspondence theory. There are other correspondence theories, and the differences are subtle. Congruence theory differs in terms of the nature of the correspondence:

"A congruence theorist holds that a truthbearer and what it corresponds to are both structured complexes, and that when one corresponds to the other, there is likeness of structure, and correspondence of components to components."

Truthmaker theory allows for truthmaking to simply be logically entailed by simple existents.

As I said, it's subtle.

:up:

Several ideas are introduced here, one being to obtain. — frank

My understanding of 2.2.3

There are Concretists such as Lewis and Abstractionists such as Plantinga.

For the Abstractionsists:
There is the actual world, an actual world, a State of Affairs that exists and obtains.
There are possible worlds, non-actual worlds, States of Affairs that exist but fail to obtain.

Most Abstractionalists are Actualists.

There are two types of Actualists, Trace Actualists and No-trace Actualists.

The problem with No-trace Actualists is that they cannot explain standard modal semantics, in that it is possible that for some x, x is an Exotic. But for the No-trace Actualists, there is no x.

There are two types of Trace Actualists, New Actualists and Haecceitists.

Trace Actualists can explain standard modal semantics, in that things, whether objects or properties, do exist in possible worlds. It is possible that for some x, x is an Exotic (so different to an actual object that no actual object could be an Exotic)

It is possible that on the table is an apple. There is a possible world where the apple is on the table.

For the New Actualists, the apple necessarily exists, but is not necessarily concrete. The apple is only contingently concrete.

For the Haecceitists, such as Plantinga, even though in a possible world the apple does not exist, the property “being an apple” does exist.

For Trace Actualists, things in possible worlds can exist. This allows the modal semantics of (23) ◇∃xEx is true if there is a world in which ∃xEx

The problem, as I indicate above, is that it is a pretense to correspondence. That is the problem I mentioned earlier, of replacing intension with extension. Correspondence becomes simply a stipulation, instead of criteria for judgement. The actual world of the modal model "is" the actual world of fact, because this is stipulated as a necessity for truth. — Metaphysician Undercover

Any truth theory could be considered stipulation, although one is free to examine various theories and choose the best, or least offensive. It seems untenable to just abandon any concept of truth.

Correspondence isn't intended to be a criteria for judgement. Judgement is epistemological. Rather, correspondence is the conceptual basis for what truth is. Truthmaker theory is a specific form of correspondence theory that more specifically stipulates that the correspondence is between a proposition and an element of of the actual world, called the truthmaker.

So, "snow is white " is true if and only if snow is white. Yes, now we stipulate "snow is white" (or in the case of possible worlds, the actual world of the modal model is stipulated as in fact the true actual world), and voila, "snow is white" is true by stipulated correspondence, and the actual world of the modal model is true, by stipulated correspondence. But of course, we can all see that this is just a pretense of correspondence. — Metaphysician Undercover

There's more to the analysis:

It is true THAT snow is white IFF snow is white

The italics phrase reflects a proposition; the bold phrase represents an element of actual reality. It is assumed these meanings are clear, and that there is a distinction between a proposition and actual reality, and that truth entails that the proposition (its meaning) mirrors the element of reality.

If you don't like this view, then what view do you think better captures the concept of truth?
.

See how the sentence you keep quoting begins with "Say that...".

Why?

Here's the whole paragraph:

On the assumption that there is a (nonempty) set of all possible worlds and a set of all possible individuals, we can define “objective” notions of truth at a world and of truth simpliciter, that is, notions that are not simply relative to formal, mathematical interpretations but, rather, correspond to objective reality in all its modal glory. Let ℒ be a modal language whose names and predicates represent those in some fragment of ordinary language (as in our examples (5) and (6) above). Say that M is the “intended” interpretation of ℒ if (i) its set W of “possible worlds” is in fact the set of all possible worlds, (ii) its designated “actual world” is in fact the actual world, (iii) its set D of “possible individuals” is in fact the set of all possible individuals, and (iv) the referents assigned to the names of ℒ and the intensions assigned to the predicates of ℒ are the ones they in fact have. Then, where M is the intended interpretation of ℒ, we can say that a sentence φ of ℒ is true at a possible world w just in case φ is trueM at w, and that φ is true just in case it is trueM at the actual world. (Falsity at w and falsity, simpliciter, are defined accordingly.) Under the assumption in question, then, the modal clause above takes on pretty much the exact form of our informal principle Nec. — SEP

See how the single line you quote is part two of four of the antecedent of a mooted definition of true-in-M that is being true in any arbitrarily selected world. The conclusion is the opposite of what you suggest: any world might have been chosen to take on the place of the actual world, with the same result.

For those reading on, Meta isolates (ii) (“its designated ‘actual world’ is in fact the actual world”) and treats it as if it were doing independent semantic work. That is a mistake.

Again, there Might be a point Meta could be making, but his utter inability to understand and use the formal logic here incapacitates his expressing his view. Meta might be gesturing at a familiar philosophical concern, namely that the appeal to an “intended model” smuggles metaphysics into what is advertised as a purely semantic account. To make that objection, Meta would have to distinguish object-language truth conditions from metasemantic stipulations, recognise the difference between fixing a model and evaluating formulas within it, and understand how conditional definitions work in formal semantics. There may be a point Meta could be making, but his inability to understand and use the formal logic prevents him from expressing it.

You seem to want to focus on one, the abstractionist interpretation, as if it is the only acceptable interpretation — Metaphysician Undercover

No, Meta. I haven't moved past it because i keep answering your silly quibbles. My bad, yes, i should go back to ignoring you.

Yep.

Yep. We might be clearer about Plantinga’s view. It's not primarily about properties like “being an apple”. It is about individual essences (haecceities). For Plantinga very individual has a haecceity (e.g. being that very apple), and haecceities exist necessarily, and oddly it seems worlds contain haecceities whether or not they are exemplified. So in a world where the apple does not exist, the haecceity "being that apple" exists, and is unexemplified. That haecceity is what does the semantic work for quantification.

All of which looks quite contrived to my eye. Not keen on Plantiga's approach.

For Trace Actualists, things in possible worlds can exist. This allows the modal semantics of (23) ◇∃xEx is true if there is a world in which ∃xEx — RussellA

Isn't it more that ◇∃xEx is true if there is an accessible world in which ∃xEx can be represented? Roughly, if we can posit, or perhaps talk abuot some world in which ∃xEx?