But, for some conversations, we can use modal logic and take the metaphysically actual world as the modally actual world, and look that the accessibility relations that originate in the metaphysically actual world. — Banno

We cannot "take the metaphysically actual world as the modally actual world" because the difference between these two is the difference you insisted that we must respect. The "metaphysically actual world" is the world we live and act in. The "modally actual world" is a representation. That is the difference which you accepted in the other thread, and agreed that we must respect. By the same principle, "Nixon" refers to something different in the metaphysically actual world, from what it refers to in the modally actual world. And we must respect the fact that there is a significant difference between these two, to allow for the reality of incomplete, mistaken, or otherwise misguided representations posing as the "modally actual world". As you say, the modally actual world is just another possible world, though it is assigned special status.

This is why I emphasized in the other thread that truth is a judgement. The representation is judged to be adequate, and given the name "actual world", but it is still just a representation which could be mistaken. But "truth" does not mean correspondence in modal logic. It is very important to respect this difference between the representation within the modal model which is called "the actual world", and the real "metaphysically actual world", because "the actual world" in modal logic can be created from a variety of different principles which do not necessarily require rigorous criteria of "truth" in the sense of correspondence. The "modally actual world" does not necessarily correspond with the metaphysically actual world. That's the deficiency of assigning "truth" an entirely extensional meaning, which the article refers to in section 1.2. Truth is arbitrary. This is required to make the modal model effective. Instead of an intensional criteria for "truth", there is an extensional stipulation.

Possible world semantics, therefore, explains the intensionality of modal logic by revealing that the syntax of the modal operators prevents an adequate expression of the meanings of the sentences in which they occur. Spelled out as possible world truth conditions, those meanings can be expressed in a wholly extensional fashion. — SEP

Another way to think of a intension is the rule we apply in order to decide, say, if that bird is a swan or not. But the truth of "That bird is a swan" is completely determined by the extension of "That bird" and the extension of "...is a swan": it will be true if and only if "That bird" satisfies "...is a swan" — Banno

This is a good example, I suggest you take a good close look. With the intensional definition we have criteria, "the rule" by which we judge whether or not "that bird is a swan" is true. We follow the rule and make the judgement. By the extensional definition however, "that bird is a swan" is true if that bird is a swan, i.e. is a member of that set. In this case the judgement may be completely arbitrary. Without an intensional definition, we can decide for whatever reason we want, whether or not the bird is a swan, we place it in the set of swans or not, and this forms the grounds for whether or not the proposition is true.

However, as Plato showed, Pythagoras avoided that arbitrariness by assuming real independent Ideas, and the theory of participation. The independent Idea serves as the criteria for "swan" which human beings don't necessarily know, making the bird a member of the set or not, without any human being needing to judge. This is Platonism. Now there is an eternal objective Idea of "swan", and it is true that all the birds who are swans, are swans because they partake in this Idea. So "is a swan" is satisfied if the bird partakes in the Idea of swan, whether or not a human being makes that judgement. Furthermore, "'that bird is a swan' is true if that bird is a swan", implies that the latter "is a swan" means partakes in the Platonic Idea of swan.

We cannot "take the metaphysically actual world as the modally actual world" because the difference between these two is the difference you insisted that we must respect. — Metaphysician Undercover

All I did was point to the difference between metaphysics and modality.

And this is not my account. The account here is the standard account of logicians.

But you twist and swivel.

There are things that you could say here that would be interesting. But your inability to understand modal logic prevents you from framing them in anything like a coherent fashion.

Step by step.

1. The core mistake: reifying the “modally actual world”
Your opening move is this: We cannot "take the metaphysically actual world as the modally actual world" because … the "modally actual world" is a representation. This misfires because in possible-world semantics, “the modally actual world” is not a representation of the metaphysically actual world. It just is the world designated by the model as actual. There is no further ontological claim being made.

As I have said, within a modal model, we stipulate a world as actual, and then examine accessibility relations from it. That stipulation does not compete with metaphysical actuality; it is a modelling device.
You are treating the model as if it were trying — and possibly failing — to represent reality. But modal semantics is not representational in that sense. It is instrumental. So the objection attacks a position that isn’t there.

2. Confusion between semantic stipulation and epistemic judgement
You write "This is why I emphasized … that truth is a judgement". That is false, or at least badly equivocal. In modal semantics, truth-at-a-world is not a judgement, nor is it an epistemic act. It is a semantic relation defined by the model. No one is “judging” that Nixon exists at a world; the valuation function assigns extensions at that world. That’s it. You slide illicitly from truth-in-a-model to truth-as-human-judgement. This is a category mistake.

You are psychologising semantics.

3. The Nixon move fails for the same reason. You say that "Nixon" refers to something different in the metaphysically actual world, from what it refers to in the modally actual world. Again: no. Within a model, “Nixon” has an extension at each world in which it exosts. Across models, reference is fixed by interpretation. None of this implies that the model’s Nixon is a representation that might be mistaken.
Mistake only arises if you assume the model is making a claim about the world. It isn’t. It’s a tool.
This is exactly the point Kripke, Lewis, and the SEP article are making — and which you are resisting by importing epistemology where it does not belong.

4. Misreading SEP on extensionality
You quote SEP as spelled out as possible world truth conditions, those meanings can be expressed in a wholly extensional fashion, then respond that truth is arbitrary. That is simply incorrect.
Truth is not arbitrary; it is stipulated relative to a model. That is not arbitrariness in the philosophical sense, any more than choosing a coordinate system is arbitrary in physics. Extensionality ≠ lack of constraint. Instead, once the model is fixed, truth values follow mechanically.

You are conflating “not grounded in metaphysical correspondence” with “arbitrary”. Those are very different claims.

5. The swan example: a serious error
You write that without an intensional definition, we can decide for whatever reason we want … whether or not the bird is a swan. This is flatly false. In extensional semantics, membership is fixed by the interpretation function. There is no discretion left to the user once the model is set up. You are smuggling human judgement back in again, where it explicitly does not belong.

Intensions explain how extensions vary across worlds, not that extensions are chosen on a whim.

6. The Platonic turn is a non sequitur
Your appeal to Plato and Ideas does no work here. Possible-world semantics is neutral on whether universals are Platonic, Aristotelian, nominalist, or fictional. Introducing Forms does not “solve” a problem — because there was no problem to begin with. You move from “extensions are stipulated in a model” to “therefore we need eternal Ideas” That inference is invalid.

Modal logic does not require metaphysical grounding to function, any more than arithmetic requires Platonism to be usable.

You are repeatedly:

• mistaking semantic machinery for metaphysical representation
• mistaking stipulation for arbitrariness
• importing epistemology into model theory
• and then trying to fix the resulting pseudo-problem with Platonism

The critique dissolves once the role of possible-world semantics is properly understood.

Platonism is common in mathematical interpretations. The Idea of "two" for example, is supposed to have real meaning, independent from human minds, so the symbol stands for that intensional package of meaning, as an object. — Metaphysician Undercover

As you say, an extensional definition is trapped by limited empirical observations.

Suppose there is an extensional definition of S, where S = {two red books, two green trees, two black thoughts}

You say that the elements of the set S are there because they are part of the Platonic “Idea” of S, where the “Idea” is the intensional meaning of S, and is objective and independent of the human mind.

The problem is, how can you go from the extensional definition of S existing in your mind to an intensional definition of S existing independently of your mind?

IE, what is the intensional definition of S?

I think the same sorts of questions could be asked about ordinary language use. If Paul talks about x, what guarantees that Paul knows what his words mean?

Btw, this:

The Idea of "two" for example, is supposed to have real meaning, independent from human minds, — Metaphysician Undercover

is straight up incorrect. This is the type of thing Meta just pulls out of his butt.

If Paul talks about x, what guarantees that Paul knows what his words mean? — frank

At this exact moment in time, when I write “swan”, I know without doubt what my concept of a swan is.

However, with time, as I learn new things about swans, my concept of a swan will change. However, I will still use the same word “swan”.

IE, what I mean when I write “swan” will inevitably change with time.

Therefore, one can say that Paul knows without doubt what he means when he talks about "x", even though what he means by "x" must inevitably change with time, as he learns new things about "x".

If what I mean by “swan” inevitably changes with time and what Paul means by “swan” inevitably changes with time, it is perhaps surprising that communication using language is possible at all.

Can whether a statement has a truth value be answered only by whether the statement has terms that have extension?

I am thinking of the following statement: "Necessarily, Frosty the Snowman does not exist." I think you would say the statement lacks any terms with extension. However, it appears to be a true statement.

If the statement is true, by reference to what is it true?

I think if you have defined "swan" correctly by reference to its essential properties, your definition may change by addition, but the essential properties you have identified will not have changed. Ifessential properties are not acknowledged, any kind of definition would appear to be rendered meaningless.

Notice that it is an intensional definition: it does not list the very things that are swans, but gives a rule for deciding of something is a swan. — Banno

I think it would be right to say that "Swan" defined in such an such a way, would be a de dicto proposition.

I think if you have defined "swan" correctly by reference to its essential properties, your definition may change by addition, but the essential properties you have identified will not have changed. — NotAristotle

That’s the problem. What are the essential properties of a “swan”. This sounds like Aristotle’s essences.

The web site - https://birdsology.com/25-fascinating-characteristics-of-swans/ - lists 25 characteristics of swans, including 1 distinctive long necks 2 powerful flight capabilities 3 pristine black and white plumage…….. 25. Limited Global Species but Wide Range

Are these necessary or contingent properties of being a “swan”?

That’s the problem. — RussellA

I do not understand what the problem is; can you elaborate?

The problem is, how can you go from the extensional definition of S existing in your mind to an intensional definition of S existing independently of your mind? — RussellA

Good question.

1.3

Approximate definitions of de re an de dicto:

De re = a specific thing that is meant.
De dicto = a nonspecific thing that is meant.

Intensionality appears to be something like --> the picking out of "something" as the referent; extensionality appears to be the "thing" that is picked out by reference, where "something" and "thing" may not need to be physical substances in the actual world.

Intensional definitions (properties, relations, and propositions) are defined by possible world semantics independent of any specific language.

The de re and de dicto distinction helps articulate essential as opposed to accidental properties by using modal logic. De re classification anticipates identity "across worlds."

What we have is a rigorous account of what it is for "Swans are white" to be true. But it doesn't tell us if swans are white...................To work that out we are gong to have to go out and take a look. — Banno

Hopefully summarising your posts correctly and adding my own understanding.

In possible world 5 (England)

Step one - First we start with many observations of things that are waterfowl, flighted and white. We accept that somewhere hidden could have been a solitary black flighted waterfowl.

Step two - Someone in authority, along the lines of JL Auston’s performative utterance, officially declares that “swans” are {waterfowls, flighted, white}. This fits in with Russell’s and Kripke's idea that proper names are descriptions, and not references to Aristotelian essences. Such an authority could be a person, such as a King, or general consensus.

Step three - In the event that a solitary black flighted waterfowl was discovered, by definition, it could not be a “swan” but must be something else.

Step four - At least there is now a foundation for Wittgenstein's language game, enabling a public language with a measure of agreement as to the meaning of words used.

It is clear that an extensional definition as found in a public dictionary can never give us the intensional meaning of a word. It may be that “Swan” = {waterfowl, flighted, white}, where “white” = {low saturation, reflects almost all wavelengths of light}. But then “light” = {electromagnetic radiation, composed of photons}, ad infinitum

For Wittgenstein, on the one hand, all thoughts are spoken and on the other hand there can be no private language. This implies that all thoughts are necessarily part of a public language. Thereby, there is no separation between the private concept of a swan and the public word “swan”. Such is the consequence that the meaning of swan, the intension of swan, becomes knowable within a public language game. In summary, private concepts are not particular to the individual but general within the public language game and necessarily apply within the language game.

But, is it really the case that Wittgenstein enables a public understanding of the intensional meaning of words!?

In possible world semantics, linguistic truth is not arbitrary. In possible world 5 (England) there may be a language game whereby “swans are white” is true, and in possible world 6 (Australia) there may be a different language game whereby “swans are black” is true. Truth is relative between different language games, between different possible worlds.

In possible world semantics, logical truth is not arbitrary. As it is true that the number 3 is a prime in all possible worlds, the material truth tables are true in all possible worlds, even though the particular forms of life expressed within these material truth tables may be different in different possible worlds.

I do not understand what the problem is; can you elaborate? — NotAristotle

How can one ever know what is essential to something.

For example, what is essential to something being a “game”.

What is essential to “a table”, “love”, “pain”, “a house”, etc.

Truth is relative between different language games, between different possible worlds. — RussellA

If swan = frog, in possible world 6, then intensionally truth is relative in a manner of speaking, but only because the words refer to different things. Truth is evaluated by what is meant, which can change between speakers or speakers in different possible worlds.

On the other hand, I would think that truth would not be relative in an extensional sense. So "that is a swan" will be true in one possible world and false in another, but only because swan means frog in one of the possible worlds. Extensionally, the referent is unchanged.

a house — RussellA

I don't think artifacts like houses have an essence.

How can one ever know what is essential to something. — RussellA

It seems to me that: knowing that something has an essence, and for example that the essence of a dog is different from that of a swan, is not the same as knowing what those essences are.

At this exact moment in time, when I write “swan”, I know without doubt what my concept of a swan is.

However, with time, as I learn new things about swans, my concept of a swan will change. However, I will still use the same word “swan”. — RussellA

This wouldn't be a problem for first order logic. When your concept of a swan changes, the interpretation in your model changes. No biggie.

ruth is evaluated by what is meant, which can change between speakers or speakers in different possible worlds..............On the other hand, I would think that truth would not be relative in an extensional sense. — NotAristotle

In possible world 5
Let the word “swan” have both the intension and extension {waterfowl, flighted, white}
Let the word “frog” have both the intension and extension {short bodied, protruding eyes, strong hind legs}
Then
It is true that “swans” are white
It is true that “frogs” have strong hind legs
It is false that “swans” have strong hind legs
It is false that “frogs” are white

In possible world 6
Let the word “swan” have both the intension and extension {short bodied, protruding eyes, strong hind legs}
Let the word “frog” have both the intension and extension {waterfowl, flighted, white}
Then
It is true that “swans” have strong hind legs
It is true that “frogs” are white
It is false that “swans” are white
It is false that “frogs” strong hind legs

Therefore, what is true in possible world 5 is false in possible world 6, and what is false in possible world 5 is true in possible world 6

Truth is relative in both an intensional and extensional sense.
====================================================================

I don't think artifacts like houses have an essence. — NotAristotle

Then what makes a house not a table?
====================================================================

It seems to me that: knowing that something has an essence, and for example that the essence of a dog is different from that of a swan, is not the same as knowing what those essences are. — NotAristotle

If you don't know the essence of a dog, and you don't know the essence of a swan, then how do you know that they are different?

Truth is relative in both an intensional and extensional sense. — RussellA

I do not think so. You have changed the word that is used, and each utterance refers to a different thing, intensionally. But the thing that belongs to the predicates enumerated does not itself change. In other words, the thing of which "has strong hind legs" is predicated does not change.

I guess you can do possible worlds without rigid designators or across world identity, but then I am not sure what can meaningfully be asserted about these possible worlds other than that something can be anything else in some possible world. The range of possibility seems too "wide" in a way that would prevent meaningful discourse.

Things have gone a bit astray, as well as awry. I'm going to step away from the text for a bit, and consider what logic does.

Logic is about what we can coherently say. At its best it applies not only to the world around us but to other things we talk about, setting out such things as validity and implication in the most general terms.

I used the example of Middle Earth previously. IF logic did not apply to Middle Earth, the books would be unreasonable. Our logic ought apply in such cases. And indeed it does.

Here's an example from propositional logic. Frodo walked into Mordor. Samwise also walked into Mordor. And we can use a logical rule that allows us to introduce a conjunction. We can write "Frodo walked into Mordor AND Samwise also walked into Mordor."

We can move on to first order logic. Since Frodo walked in to Mordor, we can conclude that Something walked in to Mordor. This is an instance of the rule of Existential Generalisation. Formally, it's fa → ∃x(fx) — If a is f, then there is an x such that x is f.

Have we proved, by this, that Frodo exists? Not at all. We introduced Frodo when we set up the Domain of Middle Earth. His existence is not a consequence of our deductions, but a presumption or stipulation.

The domain is in a sense a list of the things we are talking about. In first order logic and basic modal logic it is static. (There are variable-domain modal logics.)

As you say, an extensional definition is trapped by limited empirical observations. — RussellA

I'm not at all sure what this might mean, and it may well be a good rendering of something Meta has said. However not all extensional definitions are empirical. We can set up the extension of "Creatures who walked in to Mordor" as {Frodo, Samwise} without doing empirical observations of the borders in Middle Earth. The extension of some predicate can be any of the members of the Domain. It can be arbitrary, but of course it that does not mean that it is always arbitrary. We put {Frodo, Samwise} into the extension of "...walked into Mordor" because that's what happens in the book.

There is nothing here about having to be empirically verifiable.

Also, an even more pedantic point. S = {two red books, two green trees, two black thoughts} is not an extensional definition - it's intensional. It doesn't list five individuals but gives instructions for picking out five individuals. An extensional definition would list which books, which trees and which thoughts. So "what is the intensional definition of S" is already answered by the definition you gave...

Strange, isn't it. Logic does not, and ought not, presume Platonism or realism or any other philosophical doctrine. If it did, then using it to decide between these doctrines would be begging the question - as if it were reasonable to presume Platonism in order to prove Platonism. Logic is ontologically neutral.

So here I might go back to the formal definition of "intensional" given previously.

Speaking roughly, the intension of π is the rule that tells you what π’s truth-value would be in every possible world. — Banno

So what would be the intension of S, which in some world is S = {two red books, two green trees, two black thoughts}, in every possible world? Well, there will be worlds in which one of the two red books is green, and worlds in which one of the black thoughts didn't happen. The extension of S in other possible worlds is not given, so the intension of S per se remains unsettled. So we have an intensional definition of S in some world wₙ, but find that its intension can not be analysed across other possible worlds. It has as yet no intension per se.

Now as a side issue, take a look at how long this reply is. It's not difficult, I hope, but it is intricate. The formal language will pay out, if we stick to it. We know that because we know that modal logic is consistent, and hence that if we stick to it we will get a consistent result.

And it follows that attempts to show global inconsistency in standard modal logic will not work.

This wouldn't be a problem for first order logic. When your concept of a swan changes, the interpretation in your model changes. No biggie. — frank

Indeed. The extension will be different in different interpretations.

Notice that swan, frog, book, tree and so on are kinds, not individuals.

The analysis of kinds differs from the analysis of individuals. For kinds, we look to criteria of membership rather than identity and persistence conditions across time and possible worlds. Identity, reference and existence relative to a domain are aspects of individuals.

It's often an error to look at identity, reference and existence in kinds.

Necessarily, Frosty the Snowman does not exist — NotAristotle

This is problematic. See my comments about existential quantification and domains, above. If we set up a domain that includes Frosty, then we can use existential generalisation: "Frosty is a snowman, therefore something is a snowman" and suppose that we have proven that snowmen exist when what we have actually done is to assume that snowmen exist when we set out the domain.

So, if Frosty is treated as a constant that denotes an individual in the domain, we might parse "Necessarily, Frosty the Snowman does not exist" as □¬∃x(x = Frosty). But this is incoherent, because the interpretation already assigns Frosty to an element of the domain. You cannot then say, necessarily, that Frosty does not exist. (de re reading)

If the claim means that in the actual world, no individual satisfies the description “Frosty the Snowman”, and this holds in every accessible world, then this is a claim about the emptiness of a predicate, not about the non-existence of a named individual. (de dicto reading)

We should avoid Meta's error of thinking that logic must imply metaphysics, the confusion between existence in the model, which amounts to domain membership, and existence simpliciter, which logic says little about. (But which folk seem to think must be dependent on empirical observation alone, a point of contention outside of this thread.)

1. The core mistake: reifying the “modally actual world”
Your opening move is this: We cannot "take the metaphysically actual world as the modally actual world" because … the "modally actual world" is a representation. This misfires because in possible-world semantics, “the modally actual world” is not a representation of the metaphysically actual world. It just is the world designated by the model as actual. There is no further ontological claim being made. — Banno

Yes, I see you understand the issue very well then. Every time you refer to "Nixon" or "the actual world" as if this is a representation of the real person, or the real world, within the modal model, this is incorrect. You are wrong in doing this because as you clearly state here, it "is not a representation of the metaphysically actual world. It just is the world designated by the model as actual. There is no further ontological claim being made."

Since you understand this separation very well, could you please, in the future, refrain from making statements like the following:

The claim that individuals in possible worlds might lose identity is false in standard semantics. — Banno

The name does refer in such counterfactual cases. — Banno

Both sentences are about Nixon. The same Nixon in two different worlds, each of which is evaluated extensionally without contradiction. — Banno

These are both Nixon. The Nixon who did not get elected is not a different Nixon to the one who was. They are the very same fellow, but under different circumstances. — Banno

Notice, "identity" names an individual, and insisting that the name refers to the same individual across different possible worlds violates what you insist that you understand as #1 above.

As I have said, within a modal model, we stipulate a world as actual, and then examine accessibility relations from it. That stipulation does not compete with metaphysical actuality; it is a modelling device.
You are treating the model as if it were trying — and possibly failing — to represent reality. But modal semantics is not representational in that sense. It is instrumental. So the objection attacks a position that isn’t there. — Banno

Here you go, projecting your own error on to me, in your usual straw man way. Clearly, with the evidence of the quotes above, you are the one treating the model as if it is trying to represent a real person, the one called "Nixon".

No one is “judging” that Nixon exists at a world; the valuation function assigns extensions at that world. — Banno

You sure as hell were, insisting that "Nixon" refers to "the same fellow" in different possible worlds.

The Nixon move fails for the same reason. You say that "Nixon" refers to something different in the metaphysically actual world, from what it refers to in the modally actual world. Again: no. — Banno

This clearly contradicts your #1, which says: "This misfires because in possible-world semantics, “the modally actual world” is not a representation of the metaphysically actual world. It just is the world designated by the model as actual."

In #1 you are saying that there is no relationship between the metaphysically actual world and the modal actual world. In #3 you say that "Nixon" refers to the same thing in both.

Truth is not arbitrary; it is stipulated relative to a model. That is not arbitrariness in the philosophical sense, any more than choosing a coordinate system is arbitrary in physics. — Banno

That looks like "arbitrary" to me. I don't know why you would argue against this. One stipulates "truth" according to one's needs, or purposes, just like in physics one stipulates the rest frame according to what is required for the purpose. That is arbitrariness. In other words, there is no set system of rigorous criteria by which truth is determined. Would you prefer if I used "subjectivity" instead of "arbitrary"?

This is flatly false. In extensional semantics, membership is fixed by the interpretation function. — Banno

That is intensionality. It is intensionality entering into the extensions which curbs the arbitrariness. But, as the SEP indicates, the meaning of the operators is lacking in rigor. This allows the influence of subjectivity.

Modal logic does not require metaphysical grounding to function, any more than arithmetic requires Platonism to be usable. — Banno

OK, then please quit doing things like talking about "Nixon" as if this refers to a metaphysically grounded fellow. You cannot have it both ways, insist that modal logic is not metaphysically grounded, yet speak about the items within possible worlds as if they are grounded in a metaphysical world.

The problem is, how can you go from the extensional definition of S existing in your mind to an intensional definition of S existing independently of your mind? — RussellA

That is a problem, addressed by Aristotle. And analysis of this problem leads to his refutation of Platonism. It is sort of like the interaction problem. We do not have direct access to the independent Ideas, so we can never really know if our intensional definitions are correct. This renders the Platonic ideas epistemically useless. That's why Banno claims, above, that Platonism is irrelevant. But without assuming the Ideas we have no assurance that there is such a thing as "truth". So every time someone claims an independent truth, Platonism is implied.

So Banno claims that truth is not arbitrary, and also claims that Platonism is irrelevant. This leaves "truth" as either completely arbitrary, or rescued from arbitrariness by subjectivity.

is straight up incorrect. This is the type of thing Meta just pulls out of his butt. — frank

Why would you have a problem with that? It's commonly understood that "2" is a numeral which represents a mathematical object, known as the number two.

I am thinking of the following statement: "Necessarily, Frosty the Snowman does not exist." I think you would say the statement lacks any terms with extension. However, it appears to be a true statement.

If the statement is true, by reference to what is it true? — NotAristotle

I don't think I really understand the question here. Wouldn't we have to check every snowman, and make sure that it is not Frosty before we conclude that Frosty the Snowman does not exist. Or could we go through a process of intensional definition, and deductive logic, to make that conclusion?

I used the example of Middle Earth previously. IF logic did not apply to Middle Earth, the books would be unreasonable. Our logic ought apply in such cases. And indeed it does.

Here's an example from propositional logic. Frodo walked into Mordor. Samwise also walked into Mordor. And we can use a logical rule that allows us to introduce a conjunction. We can write "Frodo walked into Mordor AND Samwise also walked into Mordor."

We can move on to first order logic. Since Frodo walked in to Mordor, we can conclude that Something walked in to Mordor. This is an instance of the rule of Existential Generalisation. Formally, it's fa → ∃x(fx) — If a is f, then there is an x such that x is f.

Have we proved, by this, that Frodo exists? Not at all. We introduced Frodo when we set up the Domain of Middle Earth. His existence is not a consequence of our deductions, but a presumption or stipulation.

The domain is in a sense a list of the things we are talking about. In first order logic and basic modal logic it is static. (There are variable-domain modal logics.) — Banno

This is incorrect. The Domain spoken about here is clearly not a list of the "things". Things exist and you have explicitly stated that you have not proven the existence of what the words refer to. Until then, it is wrong to claim that your words refer to things. Otherwise we could prove all sorts of inductive conclusions to be wrong, by talking about imaginary things. Someone claims all swans are white, all I have to do is talk about a black one as a "thing" and I've proven that proposition to be wrong.

Using a name does not imply that there is a thing which corresponds to that name. Simply put, things have identity. And things are confined to the metaphysically actual world. Your claim that a name in a modal model refers to a thing with an identity is simply incorrect, as is evident from the fact that a "possible world" is not a thing itself, it is an interpretive tool.

We should avoid Meta's error of thinking that logic must imply metaphysics, the confusion between existence in the model, which amounts to domain membership, and existence simpliciter, which logic says little about. — Banno

It's not the case that logic necessarily implies metaphysics, but using metaphysical terms like "thing" and "identity" do imply metaphysics. And if you believe that epistemology can be separated from its metaphysical grounding you are mistaken.

's confusion comes from collapsing two very different discussions:

The model-theoretic discussion: In possible-world semantics, the “actual world” is just a designated world in the model. It has no necessary connection to metaphysical reality. The model doesn’t aim to represent the real world; it simply stipulates a world as actual for the purposes of evaluating modal claims.

The metaphysical discussion: That’s the world we inhabit. Whether “Nixon” exists here or not is a separate question. The modal model doesn’t care; it just assigns extensions to names and predicates according to the interpretation function.

I dunno. There's a madness to Meta's responses. I've mostly given up trying to make sense of his posts. It's pretty much incomprehensible.

Again, this is not my account that I am giving. It is the standard account.

Logic itself is formal and syntactic. Modal logic, for example, manipulates symbols and operators according to rules. It does not by itself make claims about what exists in reality. There’s no necessary metaphysical commitment in saying “□P → P” or in using quantifiers. Using words like “thing” or “identity” does not automatically import metaphysics into the formal system. These are often placeholders in a logical model. In possible-world semantics, “identity” can just mean having the same extension in a world according to the valuation function, not metaphysical sameness. Treating formal labels as metaphysically loaded is precisely the error I was critiquing in the Nixon example. Claiming that epistemology must always be grounded in metaphysics is false. You can study knowledge, belief, or justification in a formal or model-theoretic setting without assuming that the objects of knowledge exist metaphysically in a particular way.

@Frank's description is accurate.

The direction of this is ok, I think, but the detail... again, it's intricate. As I explained above, “swans” are {waterfowls, flighted, white} is an intensional definition. An extensional definition would list every swan.

There might be a reversal in your account, in that you have an "official" definition of "swan" and work through to the language game. but that's not what Austin or Wittgenstein might say. The game comes first, the definition is post hoc. We call those birds "swans", and later invent the definition "waterfowls, flighted, white", and later on finding a black swan we drop the "white". The game has priority.

And "those birds", as a list, is the extension of "swan". The very individuals...

Kripke’s Possible World Semantics is logic demanding purity in language and purity in reality. However, neither is pure when studied closely and neither need be.

Kripke asserts that science discovers that “Water is H2O” is a statement of identify. This is not the case. Utilizing the empirical implications of Atomic Theory of Matter, science discovers the molecular composition of everyday objects that we name like water”, “air”, “soil”, etc. One finding that science discovered is naturally occurring samples named “water” consistently will be composed of “H2O” molecules and other isotopes, gases, organics, and dissolved salts. Science is concerned with statements of composition not identity.

This poses a problem for the process of rigid designation. In rigid designation, the term “water” must refer to H2O in all possible worlds, because that is the structure it referred to when the name was first introduced (or “baptized”) in our world. This is quite a feat given that this was prior to any understanding of Atomic Theory. Somehow when naming any liquid we called “water” we somehow miraculously only referred to H2O and selectively excluded any other molecules that may have beenpresent, like “D2O”, “NaCl”, etc.

Kripke may respond that "water" fixes its reference to the dominant underlying chemical structure in our world. But this sounds more like linguistic legislation, the stipulation of essence. Consider the term “air”, another term that was used well before the development of Atomic Theory. What is the dominant underlying chemical structure that was reference when the term was first introduced? Nitrogen? Oxygen? Argon? According to Kripke, science discovers that “Air is N2” because it is the dominant underlying chemical structure. But this seems misaligned with how we typically use this term, “We took a trip to the mountains to breathe fresh air.” We could counter with another stipulation, and consider biological function to fix its reference, “Air is O2”.

Kripke imagines humans selectively baptizing a particular microscopic molecule and ignoring others by decreeing what are the dominant underlying chemical structure. What he really seems to be doing is performing a metamorphosis of the term “water” into the term “H2O” to gain that purity. “Water is H2O” is not what he is talking about, but “H2O is H2O”. This is the purity he longs for and easily fits into his Possible World Semantics. However, it seems at a cost, what was once a posteriori knowledge now turns into a priori analytical truth.

The domain is in a sense a list of the things we are talking about. In first order logic and basic modal logic it is static. (There are variable-domain modal logics.) — Banno

Here's a proposal for a compromise. Since you insist that names like "Frodo" which actually refer to ideas, refer to "things", and you presume the existence of these things, would you agree that this is Platonism? I think there may be some coherence to your interpretations if you maintain Platonism.

Again, this is not my account that I am giving. It is the standard account. — Banno

Do you agree, that what you call "the standard account", is a Platonist account?

There are other accounts which are not necessarily Platonist, such as the denial of identity that I am trying to bring to your attention. However, you prefer a Platonist account.

Here's a proposal for a compromise. — Metaphysician Undercover

There's no space for a compromise. I'm engaged in giving the standard account of how modal logic and possible world semantics function. You are up the garden path.

Since you insist that names like "Frodo" which actually refer to ideas, refer to "things" — Metaphysician Undercover

"Frodo" refers to Frodo, a fictional character in LOTR. It does not refer to the idea of Frodo.. We have two different things - Frodo, who carried the one ring, and the idea of Frodo, which never carried anything. "Frodo" is the name of Frodo, not the name of the-idea-of-Frodo.

Do you agree, that what you call "the standard account", is a Platonist account? — Metaphysician Undercover

After I've spent a hour or so explaining that logic does not have metaphysical consequences, and specifically pointed out that

Logic does not, and ought not, presume Platonism or realism or any other philosophical doctrine. If it did, then using it to decide between these doctrines would be begging the question - as if it were reasonable to presume Platonism in order to prove Platonism. — Banno

Well, no.