"Nixon might not have one the election" is about Nixon, not some other non-physical...whatever — Banno

Sure, but "Nixon might not have won the election" is obviously a blatant falsity.

And, if you set up a modal model, possible worlds, within which Nixon might not have won the election then this is "some other non-physical...whatever". It's nothing other than a conceptual structure.

What baffles me is that you and I spent weeks hammering out the fact that there is a real difference, and significant separation, between the "actual world" of the conceptual modal model, and the real independent "actual world". And, when the difference was finally made clear, and agreed upon by both of us, you repeatedly accused me of not respecting that difference. Now, you are firmly in that position of refusing to respect the difference.

How can you repeatedly accuse me of making the error of ignoring this difference, and now you insist that there is no difference? In the other thread you insisted that "actual world" could refer to the metaphysically independent world, and also that "actual world" could refer to a conceptual model modal, and it is a significant error to confuse these two meanings. Now you claim the exact opposite, that there is no such duality of meaning for "Nixon". What's going on?

So "
I want the one where I'm the same fellow who won the lottery.
— Metaphysician Undercover
isn't about you, but about the circumstances...

Ok. :meh: — Banno

Changing the quote doesn't help you, because now the statement is about what the actual I, in the actual physical world of here and now, wants. That's why you left that part of my statement out, in the first place, to make it look like the phrase you quoted was about an imaginary "I".

Your idiocy never ceases to amaze me. Names like "Nixon" and "I" have real physical referents. Obviously though, if we create a fictitious context, and use those same names within that fictitious scenario, we are not referring to those same physical things. Trying to pass that off would be deception, lying, plain and simple.

Odd. Who is "...the one where I'm the same fellow who won the lottery" about, if not you??

Basic grammar. — Banno

Jesus Banno! Did you not take English in school? The subject of that phrase is "the one", and this refers to the "circumstances". The phrase is about that set of circumstances, not about me.

See what I mean about your unusual straw man habits? You take your own error (faulty grammar in this case), and project it onto the other person in a false representation, as if it is the other person's error.

What we can do is note this warning and proceed with the article. Is that ok with you? — frank

OK, so here's the warning from the SEP

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. — ibid

And as I explained, extensional definitions have the fundamental problem of being self-referential.

This is exactly the problem which the extensionality of "possible worlds" produces. It creates the illusion that we are talking about a bunch of different worlds, similar to the world which we actually live in, full of fellows and other things with describable properties. This might mislead the naive. In reality we are not talking about any worlds, or fellows, or things like that, we are talking about conceptual possibilities

Is it difficult for you to understand that we're not talking about a fellow at all, we're talking about a complex concept?

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

Uh-hu, tell me another one bro. Can you tell me how I can get myself into some of these different circumstances? I want the one where I'm the same fellow who won the lottery.

Lets' use the definition of knowledge in the SEP article... — Banno

I don't see a definition of "knowledge" there. It says "The operative concept of 'knowability' remains elusive...". That's the problem I mentioned, why the appearance of paradox is created, there's too much ambiguity in key terms like "know" and "true".

So, I'll tell you again. Approaching with clear definitions for these terms, and adhering to them will resolve any apparent paradoxes. I proposed definitions already, which would dissolve the appearance of a paradox. You did not accept them. Now it's time for you to propose some definitions.

I will not proceed without definitions, because my thesis is that it is a lack of definition which is the problem. Therefore we need definitions to try my thesis. If every time that we try a set of definitions, the paradox disappears, this is good evidence for my thesis.

We can plainly talk about what the world would be like were Nixon not re-elected, without thereby committing ourselves to supposing that he had indeed in the actual world not been re-elected. — Banno

Obviously, and I agreed.

The one were you repeatedly conflated metaphysics and semantics? I remember it well. You are making the same mistake here. — Banno

You have a very strange form of straw manning, in which you project your own errors on to someone else. You equivocate, and blame the interpreter for not being able to distinguish the different meanings you give to the same word. Interesting psychology.

Can we get on to Fitch now? — Banno

Sure, you reject my definitions, as I knew you would. So, what does "know" mean to you in this context, and what does "true" mean in this context? Then we can look at interpreting Fitch's argument under these conditions.

You don't appear to be available for learning at the moment. — frank

Yes, I think I'm more in the mood for calling out bullshit than for learning.

Twaddle. Both sentences are about Nixon. The same Nixon in two different worlds, each of which is evaluated extensionally without contradiction. The basic modal view that you have not understood. — Banno

It appears I went through weeks of discussion with you in the other thread, where we hammered out the difference between referencing the metaphysical world, and referencing the modal world, to no avail. Do you have an extremely short memory? Please remember the distinction we made between what "Nixon" refers to in the real, independent metaphysical world, and what "Nixon" refers to in the modal model. Or were you just pretending to understand in that other thread?

Are you suggesting that a definition of red things that includes all red things is circular? You want a definition that leaves some of them out? — Banno

No, I want an intensional definition, because a purely extensional definition is a free-floating self-referential definition.

Notice that "both" Nixons would be different "individuals" by Leibniz's definition even though each refers to the same person as that person is rigidly defined. — NotAristotle

Yes, I think you could say that, that they must be different individuals, but I prefer to think that they are not even individuals at all. They are just conceptual structures, ideas, descriptions without any real thing being described. This is the issue of Platonism. Is an abstraction an object, or is it something else. In the possible worlds context, we can ask whether the ideas which the symbols refer to are properly called individuals or not. What is referenced is ideas, not physical things or individuals.

Again, I think the key is that Nixon's other properties are just possible properties and that being the case, there is no contradiction with them being alongside his actual properties. The fact that Metaphysician Undercover talks about them as if they were other actual properties introduces a problem that is not really there. — NotAristotle

I don't see what you are saying. Having possible properties along side actual properties is a problem. That's why in modalism they must all be modeled as possible. As Banno argued in the other thread, we can stipulate that some properties are actual and give them special status in this way, by stipulation, but that does not mean that we are talking about a real independent individual named "Nixon". It is all still modal, conceptual, and we must maintain the separation between having the words reference ideas, and having words that reference physical individuals.

Why not? — NotAristotle

Because that's what a predication is, to state that a subject has a specified property. Predication is not to say that it might have the property.

How does this prevent reference? The reasoning is unclear here. We can consider what the world might have been like if Nixon were unelected, and that is a speculation about Nixon, and not someone else. The name does refer in such counterfactual cases. — Banno

If, at time t, in one possible world Nixon is president, and at t in another possible world Nixon is not president, then what "Nixon" refers to, is not the same thing, by the law of identity, without contradiction. In other words, it is contradictory to say that the same individual is president, and is not president, at the same time.

To avoid this, we must accept that the two individuals referred to by "Nixon" in the two separate worlds, are not the same thing. What I proposed in the prior post, is that we consider this to be a relation of equality rather than identity. This is how it is stipulated in the axiom of extensionality in mathematics, as a statement of equality. However, most mathematicians tend to interpret this as identity, not recognizing the difference between equality and identity, producing a form of "identity" which is contrary to the law of identity.

Two things seem to be missing here. The first is an account of why talking about different properties at the same time prevents reference, and the second is how it is that sentences like "Nixon might not have won the 1972 election" are not about Nixon... — Banno

The above answers the first. Saying that the same individual has contrary properties at the same time is a violation of the law of non-contradiction. To answer the second, it is a well known fact that Nixon won that election. To say that Nixon might not have won this election is to doubt that fact. There is nothing inherently wrong with that sort of skepticism. But to say "Nixon won that election, and Nixon might not have won that election" is to contradict oneself. The latter part of the statement allows that Nixon might not have won, while the former stipulates that Nixon won, therefore contradiction is implied.

So to deal with such counterfactuals you may put them into separate possible worlds, and establish an equality relation between the two imaginary things named "Nixon". We are not talking about an actual physical individual in the physical world, named "Nixon", we are talking about two imaginary ideas, in two possible worlds, each named "Nixon" with a relation of equality between them.

This is the difference between the "metaphysical world", and the "modal world" which we hammered out in the other thread. In the metaphysical world we are talking about an individual named "Nixon". In the modal world, we are not talking about a thing named Nixon, we are talking about some sort of model.

If you like. The definition is pretty straight forward. We us "=" for identity, and
x = y ⇔ For every formula ϕ, substituting y for x in ϕ preserves truth. — Banno

OK, so as I say, it's a clear violation of the law of identity.

No it doesn't. The Law of Identify is just U(x)(x=x). Substituting any individual for x here results in a valid form: a=a, b=b, and so on. — Banno

Please do some simple research. The law of identity states that a thing is the same as itself.

Your account amounts to us not being able to ask "what if Nixon lost the election?" — Banno

That's false. My account validates the statements "it is true that Nixon won the election", and "it is false that Nixon did not win the election. In your imagination, you can ask "what if Nixon lost the election" all you want. I have nothing against creating imaginary scenarios.

So here's the extensional definition of "...is red"
Red:={a,b,c,d,…}⊆D
It simply lists all things in the domain D that are red. It is not self-referential. On the left, we have "red",a and on the right, the set of red things. It is objective, because anyone can check to see if the individual a is an element in the extension given, independently of their opinion. The contents of the extension might well change over time, or between possible worlds - that's exactly the point of possible world semantics. — Banno

Sorry, you do not have the set of red things on the right, you have "...". It is self-referential because every red thing must be on the list, meaning that nothing else could be red. What does "red" mean? It means that it's one of the things on the list. The list says "I am what red is, and nothing else is red". It is self referential.

In the rest of this, you confuse intension and extension, like when you say the extension might change between possible world. You incorrectly call it "the extension". Each world has its own extension.

Of corse we can. "Nixon was not elected president" attributes a predicate to Nixon - in sme other possible world. — Banno

Yes, that's the point I was making, it requires a separate world.

Now you have moved on to excluded middle. In the same way as identity is evaluated within a single world, so is excluded middle. It remains valid. — Banno

I've moved on to exclude muddle. I haven't a clue what you're trying to say here.

@Banno
So, are we ready to proceed, or are you back to ignoring me?

The basic point of extensionality is substitutivity. Extension and intension are ways to define an expression. — frank

That's not what the SEP article says, and I've provided quotes. I suggest you reread the part on extensionality.

We are talking about possible "properties" of a thing, the referent, in this case "Nixon." Insofar as those properties are merely "possible" I don't see why they can't be attributed to Nixon, even at the same time, as Nixon's actual properties. — NotAristotle

Well, when it is a possibility, we cannot say that the predication is made. And we cannot attribute a property as a possibility, that would defy the law of excluded middle. And if we simply attribute "possibility", this would be infinite. So we need principles to limit the possibility which will be attributed in a logical way.

This is what you said. But you presumably also agree that the same thing can have different properties over time. If the same thing can have different properties over time, then the same thing can have different properties and still be the same thing. Therefore, different possible attributes of Nixon can refer to the very same Nixon, as would be the case whether Nixon was actually fat or actually skinny.

EDIT: Or put another way, the fact that different possible Nixons have different properties does not render them different Nixons. — NotAristotle

Sure, but as I said, with possible worlds we are talking about different properties at the same time. That is what prevents the name from referring to the same thing.

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

The truth or falsity of this statement depends on how one would define "identity". By the law of identity, identity is a relation between a thing and itself, stating that the thing is the same as itself. Mathematics, specifically set theory, has produced a distinct form of identity, which is based in the concept of equality, rather than the empirical observations of "a thing".

Standard possible worlds semantics appears to borrow this form of "identity", from mathematics, allowing that individuals in possible worlds have the same identity through an equality relation. This form of "identity" is in violation of the law of identity. And if the equivalent individuals, in distinct possible worlds, have contradictory properties, at what is said to be the same time, and are also said to be the same individual (have the same identity), this would violate the law of non-contradiction. Therefore it is best for proper understanding, to recognize this violation of the law of identity, and that the individuals within distinct worlds who bear the same name, have an equality relation rather than an identity relation, so that the law of non-contradiction is not violated.

Modal logic is intensional: truth cannot be determined by reference in the actual world alone. But Tarski-style extensional semantics can be applied within each world. The intension of a term or predicate is a function from worlds to extensions, and this intension determines the extension in each world. Extensions still define truth inside a world, while intensions describe how extensions vary across worlds. Modal operators (□, ◇) are intensional because they quantify over extensions in multiple worlds. This is the account given in the SEP article. — Banno

The way I see it, and as described by the SEP, any logic has intensional and extensional aspects. There are very good reasons why logic could not exist as just one of these.

Nothing "semantic" or "intensional" is needed inside the world. The evaluation is purely extensional. — Banno

You are not paying close attention to what the SEP is saying:

By contrast, the intension of an expression is something rather less definite — its sense, or meaning, the semantical aspect of the expression that determines its extension. For purposes here, let us say that a logic is a formal language together with a semantic theory for the language, that is, a theory that provides rigorous definitions of truth, validity, and logical consequence for the language. — SEP

Rules of extension are intensional. So the rules of Tarskian semantics which you stated, are intensional, and they apply specifically "inside the world".

But it's not the case that extentionality produces good logic, and intensionality produces bad logic, or anything like that, as they are both necessary aspects of logic. The way I see it is that intensionality provides the creative aspect required for what the SEP calls "rigorous definitions of truth", while extensionality provides the demonstrative aspect, to show, or prove to others, the usefulness of those intensional definitions. If you are interested in reading further, my perspective on this, check my reply to frank below.

Well, no. In formal Kripke semantics, extensionality inside a world is real and exact. Nothing “artificial” or “intentionally produced” is involved inside the world. Possible-world semantics does not care whether the individuals are “real” or “fictional", since the extension of a predicate in a world is always a well-defined set of individuals in that world. Intensions tell us how the extension changes across worlds, but inside each world, extensionality is fully Tarskian, such that the truth of a sentence depends only on the domain and the extension in that world. Intension is a tool for cross-world reasoning, not a replacement for extensional truth inside a world. — Banno

I don't think you are understanding what I meant. Being a "possible world", the entire world is intentionally produced, and it is imaginary in the sense that it is a description which does not necessarily describe anything "real", as in independent, in the physical world. This is why the semantics are such that it doesn't matter if things are real or fictional, because everything is treated as fictional. That's the same as pure mathematics, the axioms are assumed to be fictionalbecause this provides for the required freedom.

So the extensions within a world are produced intensionally, through a set of rules, Tarskian in this case. They are not "real" extensions in the sense of being demonstrated or proven through reference to "real" empirical objects in the physical world, they are proven through reference to the rules, which you say in Kripke semantics are "real and exact".

John asked if Frosty the Snowman is a Christmas themed character.

The extension of "is a Christmas-themed character" is

{Santa Claus, Mrs. Claus, Reindeer (especially Rudolph), Snowmen (like Frosty), Elves, Belsnickel & Befana, The Grinch, Jack Skellington, Ebenezer Scrooge}

C(x) = "Is a Christmas-themed character."

C(Frosty the Snowman) is true.

It doesn't matter that Frosty the Snowman isn't real. — frank

I think you are missing out on the foundation, or basic point of "extension". Notice, all your examples of "Christmas-themed characters" are intensional concepts. Not one is a physical "thing" which you can point to, and say that is an example of a Christmas-themed character. Even "snowmen" is a concept, and you would need to point to individual snowmen, as an extensional demonstration of what a snowman is.

Let's take an example, the concept "red", and I'll try to draw this out threw some historical references.

Suppose we say that the meaning of the concept "red" is demonstrated by all the things in the world that are red, that is the extension. So we might be inclined to define "red" that way. If it's the colour of any of these things, then its red. There would be a problem with this definition because it self-referential, and lacks objectivity. And, even if we have agreement from the majority of people which things are red, the things referred to as "red" could shift over time, and we could be adding gold things, orange things, whatever. So conventional agreement on extensionality does not suffice for objectivity. And extension is therefore not a good base or foundation for logic.

Pythagoras got around this problem with the theory of participation, which we now know as Platonism. Every red thing is correctly called "red", or "is red", because it partakes in the Idea of red. Notice that this inverts the situation, giving priority to intension, meaning, rather than empirical observations. From this perspective it is not the case that the idea of "red" is derived from the extension (seeing, and calling things red), but the idea of what it means to be red is prior to there being red things, and we call things "red" because they fulfil the criteria of this intension.

Giving priority to the semantic idea, intension, opens the door to the very productive ideas of the empty set, zero, and possibility in general. Notice that if "red" is defined extensionally, through reference to red things, there cannot be a "red" if there is no red thing. Giving priority to the idea, intensionality, allows that "red" may be a defined concept, without having any red things. This principle allows for "zero", and "possibility" in general. We can say that we have found zero red things, while maintaining the possibility that we may find some red things.

So logic is fundamentally intensional. Logicians produce axioms, definitions and rules for logical proceedings, and these are intensional. However, philosophers are by nature skeptical, and they will doubt these logical principles, requesting demonstrations. This forces the logicians to produce extensions to demonstrate the usefulness of the principles. The philosopher says to the logician, you have an idea of red, and an empty set of red things, prove to me that this is a valid idea. So the logician must formulate extensions, ways in which "red" is useful. Aristotle for example, was very strict in his demands, insisting that the extensions must ultimately refer to substance.

The same thing cannot have different properties at different times? — NotAristotle

We're talking about at the same time, in different possible worlds. If you start trying to describe the difference between one possible world and another as a difference in time (i.e. same object at a different time), you'll open a real can of worms.

That gives us extension within worlds, but not across worlds. — frank

Yes, but even the extension within worlds is artificial, because the worlds (possibilities) are imaginary.

1. What “extensional” means hereA logic is extensional when:
To know whether a sentence is true, you only need to know the extensions (the things the predicates apply to). — Banno

Right, now you're on board with the SEP definition. Notice "the things" which the predicates apply to. Traditionally these would be objects with an identity by the law of identity.

So “Algol is John’s pet” is true just because Algol ∈ that set. Nothing else matters. That’s extensionality. — Banno

If there is a thing called Algol, and it is John's pet, then it fulfils that extension. In the case of possible worlds, Algol can be an imaginary thing, a thing which does not have an identity by the law of identity. then the supposed "thing" is not even a thing. I suggest to you that this is a very significant matter.

. Why modal logic is intensionalModal logic contains operators like □ “necessarily” and ◇ “possibly.”
Now the truth of “□φ” does not depend only on what is true in the actual world. It depends on what happens in other worlds (other interpretations).
That is why modal logic is intensional.
We need more information than just the extension in the actual world.
This is exactly what the SEP says. — Banno

No it is not exactly what the SEP says about intension. It says that while extension establishes relations with things, intension provides the semantics which determines the extension. Please look again:

By contrast, the intension of an expression is something rather less definite — its sense, or meaning, the semantical aspect of the expression that determines its extension. For purposes here, let us say that a logic is a formal language together with a semantic theory for the language, that is, a theory that provides rigorous definitions of truth, validity, and logical consequence for the language.
...
In an intensional logic, the truth values of some sentences are determined by something over and above their forms and the extensions of their components and, as a consequence, at least one classical substitutivity principle is typically rendered invalid. — SEP

Here is the key point Meta missed:
Even though modal logic is intensional globally, each individual world is fully extensional in the plain Tarskian sense.
Inside any world w
The domain is fixed
Predicate extensions are fixed
Truth is evaluated purely extensionally, just like ordinary first-order logic — Banno

Like I explain above, the extensionality inside any world is fixed by intensionality. This is because a possible world may contain fictional, imaginary things. Therefore the extensionality is not fixed through reference to real things, it is fixed by semantics.

Meta insists that failure of substitution “proves” intensionality between worlds. But that is exactly the point of possible-world semantics: — Banno

Please don't misquote me. I have said nothing about substitution. You keep insisting that extensionality is about, or defined by substitution. In reality substitution is a logical consequence, relying also of intension.

Each world has its own extensions. Therefore substituting co-referential terms across worlds need not preserve truth.
That is not a problem — it is the definition of intensionality.
There is no “illusion” here. — Banno

Really? This is the definition of intension? You really need to pay closer attention to the reading instead of just assuming your preconceptions.

Again, what must you make of the heading "1.2 Extensionality Regained"? — Banno

As I explained, the extensionality regained is an artificial extensionality, produced intensionallly, rather than through reference to real physical things with an identity. That is required, because we need to allow that a possible world has imaginary, fictional things. Since we cannot rely on true extensions ("things the predicates apply to") in the imaginary world, the referents are really a semantical (intensional) recreation of extensionality.

Kripke postulates "rigid designators," I think. So if Nixon is the referent of the term "Nixon" in any given possible world, maybe that alone solves extensionality without having to worry about the existence of "possible worlds." What do you think? — NotAristotle

That doesn't really make sense. Since the properties of the thing named "Nixon" in this case, are different in the different possible worlds, we cannot say that there is a single referent, the subject is different in each different world. The "Nixon" in one world would not be the same person as the "Nixon" in another. There could be some semantic rules about the use of the name, making it a "rigid designator", but that does not constitute a referent.

Could you quote the passage you're referring to here? — frank

I have already, here:

I think your EDIT is the proper interpretation. It makes modal logic the subject of an extensional logic. Here's a quote from the referenced supplement at the end of 1.2:

"As noted, possible world semantics does not make modal logic itself extensional; the substitutivity principles all remain invalid for modal languages under (basic) possible worlds semantics. Rather, it is the semantic theory itself — more exactly, the logic in which the theory is expressed — that is extensional." — Metaphysician Undercover

This is purely extensional. Kripke's move:

- Extensionality is preserved *within each world* (Tarski)
- Extensions can differ *across worlds*
- So substitution fails across worlds, not because modal logic is intensional,
but because predicate extensions vary from world to world.

This is exactly what necessity and possibility require. — Banno

That is contrary to what the SEP article states. Modal logic is intensional. And, it is only the expression of it, the interpretation of separate "possible worlds", which produces extensionality. There is no extensionality between possibilities because possibilities are inherently imaginary. It is only by assigning distinct "worlds", ("domains" or whatever you wish to call them), each with its own rules of extensionality, that the illusion of extensionality is artificially created.

However, the rules of extensionality cannot extend from one supposed "world" to another, to provide for the semantic modality of "possible". Therefore only intensionality relates the distinct "worlds" because the fact is that modal logic which relates possibilities is inherently intensional. This intentionality is described by you as "exactly what necessity and possibility require". Notice that the structure is ultimately designed to accommodate the intensional meaning, what necessity and possibility "require", rather than an extensional reality.

I've set it out multiple times, and you disagree with it each time. Your turn. Set it out for us, and how it goes astray. — Banno

That's right, each time I offer terms of interpretation, you assert that they are erroneous, and you reject them. Then you provide none of your own, simply insisting that your conclusion is valid and my interpretation is erroneous. So, I suggest that we proceed from clearly defined interpretive terms, whether they are produced by you or I does not matter to me, so long as we craft acceptable definitions.

Ok, then can you at least explain why Fitch and others think it a paradox? Why is it worthy of it's own article, in the Stanford Encyclopaedia, in Wikipedia, in Oxford Academic, and so on. What is it that the folk who wrote this stuff think is happening? — Banno

It seems to me like the paradox appears to these philosophers because they are making the mistake of assuming that the model modal says something about the real independent metaphysical world, when it does not. The separation between the "modal actual world", and the "independent metaphysical world" makes the semantics of terms like "know" and "true" extremely difficult and ambiguous.

So, to sort out the apparent paradox requires that we clearly define such terms, and adhere strictly to the definitions. I will make a proposal for definitions here, but you are free to reject them and offer your own. The point is to have rigorous terms of interpretation. Remember, from the SEP article on possible worlds, there is no extensionality inherent within the modal model, true extensionality is provided only by the interpretive terms. This makes interpretation extremely important, and produces the possibility of significant flexibility, if the interpretive terms are confusing or ambiguous.

Here are my proposals. "True" signifies a judgement which is made concerning a proposition. It is a very specific type of judgement which is incompatible with the judgement of "false", the opposing judgement of the very same type. To "know" a proposition means that a judgement of this type has been made, the proposition has been judged as either true or false. Note, that for the sake of the modal model we must allow for both judgements, "p is true", "p is false", to adequately represent the possibility of knowing p.

Lets now consider the meaning of the follow two propositions:
1. If p then p is possible.
2. If p is known then it is possible to know p.
The first implies that if p is true (has been so judged), then it is possible that p is true (has been judged that way). The second implies that if p has been judged as either true or false, then it is possible that p has been judged as true or false.

Due to the likelihood that you will not agree, and would prefer to use your own definitions, I will not proceed further with the analysis at this time. If you agree to the definitions, then we can continue, if not, then you can produce alternative definitions for these terms, and we can proceed from those definitions.

And yet the evidence you provide is from two quite different posts, which in context make it clear that one is about metaphysics and the other about modality. — Banno

My entire discussion with you in this thread is "the same argument". When finally persuaded to clarify your use, you admitted to equivocation. At some times you used "actual world" to talk about the metaphysical world, at other times you used "actual world" to refer to a modal world.

Then you had the gall to insist that your equivocation was my error, of not being able to distinguish when "actual world" meant metaphysical world, and when it meant modal world.

Let's start by having you demonstrate that you understand the paradox by setting it out. — Banno

Like I said, I don't understand the paradox as a paradox. This is because it appears to require that some conclusion about the independent metaphysical world, is derived from a modal model. If this is the case then it is very clearly the fallacy of equivocation, which I've charged.

So I suggest that you present it in a way which appears to make sense to you, while recognizing the separation between the actual world of modal logic, which is a human produced representation, and the independent metaphysical world. How do you propose to say something about the independent metaphysical world, from within the modal model? Or, does the paradox not say anything at all about the independent metaphysical world?

@Banno
Now, are you ready to address the so-called Fitch's paradox, and accept that it doesn't say anything about any supposed independent, metaphysical world? It says something about our representation of the actual world in the modal model. To me it says nothing other than the trivial tautology, that everything which is known is known. Where's the paradox?

Where?

Might be best to quote me. Be precise. — Banno

I just did that yesterday :

Look:

We are in the actual world.
— Banno

We are stipulating that that one world is the actual world, not deducing it. Any world can counted as w₀. It's built in, not contradictory. There is no modal difference between the actual world and other possible worlds.
— Banno — Metaphysician Undercover

There is no such equivocation. The problem is your inability to differentiate between a model-theoretic object and a metaphysical one. — Banno

That's so wrong. You, in the very same argument use "the actual world" to refer to a model-theoretic object, and also to a metaphysical object. When a person demonstrates to you that this is the fallacy of equivocation, you claim that it is that person's error for not distinguishing the two.

As I told, the argument from Fitch which you provided, fails if we maintain that separation. Are you ready to look over your argument, and see how it depends on equivocation?

I may have misunderstood, but I think the idea is that the actual world is regarded as a possible world, which does not imply that there are two worlds here. — Ludwig V

In our discussion, Banno more than once explicitly said that the actual world is the world that we live in. This is completely different from any representation of the world we live in. And, numerous times it is implied that he is referring to what he now calls the "metaphysical actual world" with "the actual world", yet other times he insists that "the actual world refers to a representation..

The reason why the argument which Banno presented from Fitch fails, is that it requires this conflating of the independent world, and the "actual world" of the modal model. It can only succeed through that fallacy of equivocation. I proposed to Banno that we revisit this argument and analyze it while maintaining the appropriate separation separation. Banno so far has refused, simply asserting that his error is mine.

We're getting sucked in to all-or-nothing positions. Ordinary language sometimes misleads and sometimes doesn't. One of the tasks for philosophy is to sort out the misleading bits and those that are not. I notice, however, that many major issues in philosophy are precisely based on misleading features of ordinary language - such as the pursuit of "Reality" and "Existence".
I don't think of language as a sort of bolt-on extra that human beings possess and other creatures don't (on the whole). In the first place, many animals have communication systems that are recognizably language-like and look very like precursors of language. In the second place, language is something that humans developed under evolutionary pressure, and hence no different from any other feature developed in the same way by other creatures. In the third place, you seem to think that our "inner intuitions" are not as liable to mislead us as language is; I see no ground for supposing that. — Ludwig V

My point was simply that when ordinary language contradicts good philosophy, we ought to accept this as a flaw in ordinary language, rather than rejecting the philosophy because it contradicts ordinary language.

Repeating the same errors over and over dosn't much help your case. — Banno

It appears like we're not as close to agreement as I thought. If you continue to insist that you can use the same term to refer to different things, within the same argument (to equivocate), and to insist that there is no logical inconsistency in doing this, and also assert that the person who points out this equivocation to you, is the one making the error, then I think there is not much point in proceeding.

H'm. How to we decide which contradictions are good philosophy and which are not? In other words, there may be a reason for it, but it does not follow that it is a good reason. The point about ordinary speech is that it is inescapable, at least as a starting-point. Specialised dialects presuppose it and develop out of it. That's because ordinary life is inescapable. — Ludwig V

I don't see that contradiction is ever good. And, I think that might be reasonable as an expressible starting principle for good philosophy.

Also I don't accept your proposal that ordinary speech is the inescapable starting point for philosophy. Human nature has inescapable features, instincts and intuitions, which go much deeper than language, and serve to guide us in decision making. The rejection of contradiction for example is a manifestation of a deeper intuition, rejecting contradiction as an impediment to the capacity to know and understand. As are infinite regress and other similar things known by intuition to be detrimental to the will to know (philosophy).

Language on the other hand is a sort of surface feature of the highly developed conscious mind. In other words, beings were living, and developing features which we've inherited, long before we learned how to speak, and these features make a more natural, therefore I believe better, starting point for philosophy. So it is natural that if common speech is producing philosophy which is deceptive and misleading to these inner intuitions which guide us in the will to know, then we ought to reject it as a poor starting point for philosophy. This is why logic is based in placing special restrictions on language, it curbs the tendency to fall back on ordinary language, which misleads.

So here's my take on this. Philosophy can be useful for digging below the metaphorical surface of our everyday speech & thoughts and can help us avoid logic errors and to think & talk more precisely. I've learned a lot from TPF. But when a philosophical statement contradicts the plain meaning of our everyday speech, there has to be a really good reason. And while I think I understand what you're saying, I just don't buy into it. — EricH

When good philosophy is contrary to everyday speech, there is a really good reason for that. It indicates a fundamental problem with everyday speech. You can say, "I don't buy into it", and decide to stick with the everyday speech, but that only indicates that you're not a good philosopher.

n modal logic, “the actual world” is a designated element of a model, usually called w₀. It is not the metaphysical world, not the planet, not the territory. — Banno

Are you saying then, that you'd prefer to use "the actual world" to refer to that aspect of the modal model in this discussion? Then what should we call the place where we live. In this discussion, we cannot say "we are in the actual world" then, because that would be equivocation, unless you are trying to say that we really live within a modal model.

Notice, the topic of the thread. It's very important to this topic that we do not conflate the two.

Hence, there is no contradiction in saying "The metaphysical actual world is mind-independent" and "A model contains a representational node we can call "the actual world". — Banno

It's not sufficient to qualify "actual world" with "metaphysical", because for the purposes of rigorous logic, "actual world" must always refer to the same thing. If we call the representation "the actual world", and then we qualify this with "metaphysical", it implies that we are using the same representation called "the actual world", and using this for metaphysical purposes. But this is not the case, metaphysics deals with something distinct which is assumed to be independent of the representation.

Meta is arguing:
Banno uses “actual world” for both the mind-independent world and the representational node w₀
Therefore Banno is equivocating.
Therefore modal logic contradicts realism.
But this rests on a category mistake. Two homonymous terms do not produce a fallacy unless they appear within the same argument, and they are treated as though they refer to the same thing. — Banno

Clearly, your argument in this thread constitutes "the same argument", and so we have a fallacy.

The question of whether modal logic contradicts realism, I readily admit, is much more complicated. Used properly it does not, because it is a principle of epistemology, and it need not, and ought not, be applied to metaphysics at all. But when it is applied to metaphysics, as you have done in this thread, contradiction with realism is inevitable. So we can keep modal logic right out of metaphysics, without a problem, or we can apply it to produce a metaphysics which will not be consistent with the type of realism we are discussing. It may be consistent with types of realism which you and I would not consider to be true realism (Platonic realism, and my example of model-dependent realism).

Meta treats representational dependence as ontological dependence. His argument is that the map is human-dependent, therefore the territory is human-dependent. — Banno

The problem is that in possible worlds semantics, the map is the territory. That's how they get extensionality. It's just like extensionality in mathematics, the sets, numbers, etc., are the objects referred to. In possible worlds semantics, the possible worlds are the things referred to (the territory) by the modal logic. Otherwise there is no territory, because the possibilities may be fictional, so there would only be intensionality, meaning, without any actual territory being referred to.

Meta claims it is contradictory to say the actual world is a possible world, but in modal semantics a “possible world” is just a node in a model, and the “actual world” is one node among others. — Banno

A "node", is a thing referred to. The possible world is the territory. It must be, to allow extensionality for something fictional. Without this there is only intensionality because there is no things referred to, only meaning, for any proposed possibility.

Keep in mind that the equation he rejects, p→◇p, is valid in both S4 and S5. — Banno

I don't necessarily reject this. I reject it in the metaphysical application you have proposed in this thread.

I reject your definition as completely different from the one in the article we are supposed to be reading, which I quoted above. Taking a definition from a different context is not helpful, only a distraction or a deliberate attempt at equivocation.

Since the middle ages at least, philosophers have recognized a semantical distinction between extension and intension. The extension of a denoting expression, or term, such as a name or a definite description is its referent, the thing that it refers to; the extension of a predicate is the set of things it applies to; and the extension of a sentence is its truth value. By contrast, the intension of an expression is something rather less definite — its sense, or meaning, the semantical aspect of the expression that determines its extension. For purposes here, let us say that a logic is a formal language together with a semantic theory for the language, that is, a theory that provides rigorous definitions of truth, validity, and logical consequence for the language.[2] A logic is extensional if the truth value of every sentence of the logic is determined entirely by its form and the extensions of its component sentences, predicates, and terms. An extensional logic will thus typically feature a variety of valid substitutivity principles. A substitutivity principle says that, if two expressions are coextensional, that is, if they have the same extension, then (subject perhaps to some reasonable conditions) either can be substituted for the other in any sentence salva veritate, that is, without altering the original sentence's truth value. In an intensional logic, the truth values of some sentences are determined by something over and above their forms and the extensions of their components and, as a consequence, at least one classical substitutivity principle is typically rendered invalid. — SEP

Your "resolved difference" is based on an equivocation. There is no logical contradiction in saying that the actual world is a possible world inside the model, while also treating the metaphysical actual world as mind-independent. — Banno

Yes, my proposal to resolve the difference is based on rectifying your equivocation. Giving the same term "the actual world" two distinct meanings within an argument, as you have done throughout this discussion, is equivocation.

In the quote above, you insist that there is "no logical contradiction" in this equivocation. And, generally equivocation, though it is a recognized fallacy, does not necessarily result in contradiction. However, in this case it does produce contradiction, like I've shown. In your usage "the actual world" refers to something independent (realism), and also something dependent (modal model). Therefore this equivocation is a very significant fallacy.

Look:

We are in the actual world. — Banno

We are stipulating that that one world is the actual world, not deducing it. Any world can counted as w₀. It's built in, not contradictory. There is no modal difference between the actual world and other possible worlds. — Banno

So, since you appear to recognize the equivocal nature here in the use of "actual world", I am proposing that we continue the discussion on better terms. Can we call the real independent world "the actual world", and the one in modal logic we will call "a representation of the actual world"? The difference being that when we talk about "the actual world" a real independent thing is referenced, but in modal logic, a representation is referenced. Therefore we need to make this difference clear.

Under these terms we can agree that the actual world is not a possible world. However, a representation of the actual world, in modal logic, can be a possible world. Do you agree?

If so, then we can go back and analyze your proposal from Fitch. Notice that if "kp" indicates "we know p", it means that we have a representation of p which we know, in this modal model. We know the representation itself. Since it could be the case that the representation, even though we know the representation as a representation, may be a wrong representation, what "kp" really means relative to the actual world, is that it is possible that we know what is represented by p. Knowing the representation does not necessitate knowing the thing represented. Therefore, relative to the actual world, "kp", and "◇Kp" really mean the same thing, they both mean that it is possible that we know what p represents. Do you see this?

To facilitate understanding, consider the difference between the actual world and the representation of the actual world. The representation may be wrong, even though it has been judged to be correct. Therefore relative to the actual world, the representation, which is employed in modal logic, is really just a possibly correct representation. So it has no intrinsic difference from all the other possible worlds, It has just been assigned a special status. That is the same with "kp". The p signified has no intrinsic ontological difference from any other p mentioned by "◇Kp", it has just been assigned a special status.

I'm not interested in your attempt to change the subject.

Take the Earth (real world) as the territory and the “Actual world” in a modal model as map (description).
You continue to conflate the two. — Banno

Are you kidding? I am the one who has repeatedly demonstrated how you equivocate between "the actual world" of realism (real and independent), and "the actual world" of modal logic.

If you are now ready to accept this difference, then you might be able to understand what I've been saying. Let's assume that the real independent world of realism is called "the actual world" and the one in modal logic is called "a representation of the real world".

Do you now agree that it would be contradictory to say that the actual world is a possible world? If so, then it may be the case that we've resolved our differences.

Step by step, Meta. Step by step. The aim here is to see what standard modal theory says before critiquing it. — Banno

Extensionality is the very first step. We ought to understand what it means before proceeding. It appeared like my interpretation was not consistent with frank's so I asked frank to clarify what he was saying.

The kind of expression we're talking about is:

Necessarily, all John's pets are mammals.

There's no mention of possible worlds in this expression. So no, it's not that we give "worlds" a referent by modal logic. — frank

I don't understand your argument here frank. How, in your mind, does possible worlds semantics establish extensionality for modal logic? Is it not the case that "necessarily" means true in all possible worlds, and that these "worlds", which are supposed to be the referent objects, provide the foundation for extensionality?

This is nonsense because numbers are already abstract objects. — frank

That abstractions such as numbers are "objects" is a specific ontological claim. That ontology is known as Platonism or Platonic realism. Abstractions are not necessarily understood as objects though. They are considered to be objects if the Platonist perspective is accepted. Set theory stipulates that these abstractions are objects, by axiom. These objects provide the foundation for extensionality. In modal logic "possible worlds" provide the objects for extensionality.

Yeah I am still confused about why modal logic itself is not extensional, but possible world semantics is apparently extensional. — NotAristotle

The possible worlds semantics creates the illusion of extensional objects, "worlds" as a referent. This is the same tactic used by mathematical set theory. They use the concept of "mathematical objects" to create the illusion of extensional referents. It's Platonic realism. The problem is that the reality of these "objects" is not well supported ontologically.

All you have done here is restate your thesis. — Banno

Yes, since you are having such trouble understanding, and continue to double down on your contradictory nonsense, I have to keep thinking of different ways to tell you the same thing.

That a model of gravity talks about the Earth does not entail that the Earth is human-dependent. That a modal modal talks about the actual world does not entail that the actual world is human dependent.

You continue to confused the metaphysical actual world with our representation of it inside a modal model. That confusion is the whole mistake, and repeating it does not amount to an argument. — Banno

How are these two statements consistent for you. In the first you speak about an object called "the Earth". Then, you talk about something "inside a modal model". Obviously, the thing inside the modal model is not the thing we talk about as "the earth" This is your mistake, your confusion, not mine. I am trying to relieve you of this mistaken attitude. You have "the actual world" within a modal model, and you talk about it as if it is a real independent thing.

I'm extremely surprised that a seemingly intelligent person like yourself, really cannot see the difference here. This is so hard for me to grasp, that it inclines me to believe that you are intentionally rejecting the reality, as a form of denial, because the reality of the situation is contrary to what you already believe.

So, here is a simple explanation. I assume that you understand the map/territory analogy. When someone "talks about the Earth", there is an aspect of the territory which is being talked about, and it's named "the Earth". When a modal model talks about the actual world, what "actual world" refers to is a description which is known as a possible world. You tell me that you recognize this distinction, so please adhere to it.

That description of the actual world is the map, not the territory. This is what is referred to within the modal model, as "the actual world" a description. That specific description is known as the actual world. And, it cannot be anything more than a description, because all the other possible worlds are descriptions. It you assume that "actual world" here refers to something other than a description, an independent object, then you produce inconsistency within the modal logic, because all the other possible worlds are descriptions, and you'd be claiming that this refers to something other than a description.

One way of alleviating this problem is to assume that the maps themselves (the descriptions), are actually a part of the territory. This is known as Platonism, and it is the route that set theory takes. The descriptive ideas are real objects in the world. This provides extensionality to mathematics. That's also the route that possible worlds semantics takes, the possible worlds (descriptive ideas) are real objects, and this provide extensionality.

However, the possible worlds semantics is much more problematic than the set theory semantics of mathematical objects. This is because we now have two very distinct things which are called "the actual world". One is the real physical reality (the territory), and the other is the descriptive idea (the map), which is supposed to be "the actual world" as one of the possible worlds. Obviously we need to distinguish between these two senses of "actual world", to avoid equivocation, and the contradiction which I have demonstrated is the inevitable consequence. Therefore one or the other cannot be called 'the actual world. Some ontologies like model-dependent realism (which I would say are unacceptable) deny a real world beyond the descriptive "actual world". My approach is to deny that the descriptive so-called "actual world" of modal logic ought to be called by that name.

What to make of this nonsense. Numbers are extensional, but you do not appear to have a firm grasp of what extensionality is. Extensionality in logic and in mathematics is simply defined in terms of substitution. Intensional contexts are those in which substitution fails. Extensionality is substitutivity of co-referential terms without changing truth. Numbers are extensional by this definition. Modal statements are intensional because substitution can change truth. Possible world semantics provdes an extensional model of this this intentionality. You conflates the two, which is the source of your confusion. — Banno

I have much experience discussing extensionality with mathematicians in this forum. It forms the basis of the equality relation, what mathematicians incorrectly (contrary to the law of identity) know as identity. This is how they know mathematical objects as "objects", they give them identity. But this is contrary to the law of identity which was designed to distinguish between so-called "Platonic objects", and physical things.

You on the other hand demonstrate here, a very inadequate understanding. Here is what the SEP article of Possible Worlds says:

The extension of a denoting expression, or term, such as a name or a definite description is its referent, the thing that it refers to; the extension of a predicate is the set of things it applies to; and the extension of a sentence is its truth value. By contrast, the intension of an expression is something rather less definite — its sense, or meaning, the semantical aspect of the expression that determines its extension. — SEP

Sounds right to me. To use the language of the article, I think "possible world semantics" is supposed to change "modal logic" from an "intensional" into an "extensional" language (EDIT: Or as I read further, to subject modal logic to an "extensional semantic theory"). — NotAristotle

I think your EDIT is the proper interpretation. It makes modal logic the subject of an extensional logic. Here's a quote from the referenced supplement at the end of 1.2:

"As noted, possible world semantics does not make modal logic itself extensional; the substitutivity principles all remain invalid for modal languages under (basic) possible worlds semantics. Rather, it is the semantic theory itself — more exactly, the logic in which the theory is expressed — that is extensional."

@Banno
Assuming you understood my last post, I'll address this issue you mention.

So, going back, p→◇p is valid in S4 and S5, the systems almost universally used for metaphysical speculation. These systems are reflexive, meaning that they permit us to talk about the possible world we are in. Denying p→◇p, as you do, blocks that reflexivity. — Banno

Following @frank's thread about the SEP article on possible worlds, we can understand the problem as the difference between what is extensional and what is intensional. In realism, "p" in your example has an extensional referent. In modal logic it has an intensional referent. The extensional referent is a necessary condition of realism therefore modal logic contradicts realism. That the referent of "p" is in fact intensional in modal logic, and not extensional, as required for consistency with realism, is indicated by the following passage:

Supplement to Possible Worlds
The Extensionality of Possible World Semantics
As noted, possible world semantics does not make modal logic itself extensional; the substitutivity principles all remain invalid for modal languages under (basic) possible worlds semantics. Rather, it is the semantic theory itself — more exactly, the logic in which the theory is expressed — that is extensional. — SEP

In other words, the supposed extensionality of modal logic is an illusion created by representing modal logic itself as an extensional thing. This is the same problem I covered in this forum with the proposed extensionality of mathematics. The extensionality of mathematics is an illusion created by treating numbers and other so-called "mathematical objects" as extensional referents, when they are really intensional. That is the basis of Platonic realism, which produces all sorts of problems such as eternal object etc..

One thing you have not demonstrated is that ~(p→◇p); you have simply assumed this. Indeed it's not the sort of thing that one can demonstrate. — Banno

I believe I've demonstrated this to you about four or more times already, in different ways. The "actual world" which you represent with your formulation here, is not at all consistent with (it is contradictory with) what "actual world" means for realism.

Here, I'll demonstrate it once more for you, in a slightly different way, even though I have no doubt that you will just continue to "double down" with your contradictory nonsense.

1.Metaphysical realism holds that there is some sort of real independent world.

2.We may produce statements or propositions which we judge, in our belief to be true, i.e. we judge them to be a true representation of the supposed real independent world.

3. If we take this representation, and make it a part of a structure of modal logic consisting of "possible worlds", and designate it "the actual world" amongst those possibilities, this so-called "actual world" is not consistent with the "actual world" of realism. It is as I've demonstrated, contradictory, because it is a human dependent representation rather than something independent.

Do you see the point? In realism, "the actual world" refers to something independent of human beings. In your formulation of modal logic, "the actual world" refers to a representation, which is produced by, judged to be true by, and therefore dependent on human beings.

Can you apprehend the contradiction in "actual world" here? In the case of realism "actual world" refers to something independent. In the case of your formulation of modal logic, "actual world" refers to a human construct, something dependent. Therefore the two meanings of "actual world" are contradictory.

The difference is very obvious if you consider that the human designated "actual world" which is a part of the modal construct might be mistaken. Therefore it is definitely not the same as the "actual world of realism. And as I've demonstrated countless times, in countless ways, the two are contradictory.

In an attempt to be as charitable as possible, I fed your criteria into an AI and asked it to put together a coherent account. Here's what I got.


Logic
A non-reflexive modal logic (NRML)
Actual world not in the modal domain
No p → ◇p
No Fitch paradox
Modality applies only to counterfactual models, not reality

Metaphysics
Actual world is primitive, not one among possibilities
Possible worlds are conceptual constructions
No metaphysical modality, only hypothetical modelling
No essentialism or counterfactual identity

Semantics
Two-tier structure: reality vs. fictional modal space — Banno

Sorry, I don't recognize this as my criteria. Either you, in your bad interpretation, the AI, or both, have greatly distorted things, creating the worst straw man I think I've ever seen.

What you have done is to deny that there can be systematic modal reasoning, without offering any clear alternative. — Banno

How do you ever make that conclusion? I have nothing against modal reasoning, it's very useful. What I say is that it is not consistent with realism. Many useful principles, such as my example of relativity theory yesterday, are not consistent with realism. That's just the way things are. And it does direct skepticism toward realism.

So, going back, p→◇p is valid in S4 and S5, the systems almost universally used for metaphysical speculation. These systems are reflexive, meaning that they permit us to talk about the possible world we are in. Denying p→◇p, as you do, blocks that reflexivity. — Banno

As I said, your interpretation is incorrect. The world we are in is not a possible world.

Thanks for the diagnosis, and prescription, Dr..

We are stipulating that that one world is the actual world, not deducing it. Any world can counted as w₀. It's built in, not contradictory. There is no modal difference between the actual world and other possible worlds. The difference is metaphysical, not modal. — Banno

So within the modal model there is no actual world, just possible worlds. When you stipulate that one of the worlds is the actual world, that is metaphysics. But when you stipulate an actual world, then the others are no longer (metaphysically) possible.

Within the modal model there is not consistency between actual and possible, because all are possible and there is no actual. And within the metaphysics there is not consistency between the actual and the possible. You only claim that the actual is possible by incorrectly conflating the modal with the metaphysical.

You have yet to study modal logic, but insist on your opinion. That's why you are confused. — Banno

You have yet to study metaphysics, and that's why you continually confuse me.