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.

The thing is, there is a system of modal logic which, I understand works reasonably well by the relevant standards. — Ludwig V

I agree.

However, it seems that it is not a question of two worlds, with a difference between them, but a difference in the same world. If the difference involved here is not a difference in the description of the possible world, it must be a difference in status of that same world. — Ludwig V

Ok, let's consider this perspective then. The representations of modal logic, are different possible descriptions of an independent actual world. It should be clear to you that none of the possibilities is the actual, independent world. Therefore it should also be clear to you that none of the possibilities is the actual. To think such would be a case of what is commonly called confusing the map with the terrain. We can establish some principles for judgement, criteria of truth or whatever, but that produces a judgement of truth, it does not make one of the possibilities into the actual.

H'm. Maybe. I agree, however, that more would need to be said about what "discover" means. But I like the implication that discovery presupposes an independent pre-existing something. It's not difficult with the empirical, but the a priori needs careful handling. — Ludwig V

Well, the issue is what exactly is the nature of what is termed here as "an independent pre-existing something". If we talk about discovering knowledge, then we imply that the independent something which we discover is knowledge. If we talk about discovering true statements, then it is implied that the independent something discovered is statements. If we talk about discovering information than it is implied that the independent something is information.

Conventionally, we would assign "matter" to the independent something. But Aristotle demonstrated that matter on its own is completely unintelligible, therefore unknowable. It is actually the form which the matter is in which is intelligible. But this poses the question of what exactly is "form", and how does it make something unknowable, matter, into something knowable.

It doesn't. — Banno

You yourself said:

"There is no modal difference between the actual world and the other possible worlds... "

Therefore if we assign to one of the possible worlds the status of "actual world" by realist principles, (which would constitute a modal difference), we would be attributing a difference to this world which violates the modal system which dictates "no modal difference".

No wonder I'm so confused, you keep making contradictory assertions without backing any of them up. Without the proper support for these assertions, I can 't tell which of the contrary claims you actually believe. Therefore I can only conclude that you just don't understand what you're talking about.

I take your point. Perhaps we should restrict ourselves to talking of "the unknown". It might clearer to change tack and only talk about the possibilities of discovering new knowledge. — Ludwig V

This is good. Now we have the basic separation between Platonic realism and non-Platonic types of realism to navigate. Notice you mention "the possibilities of discovering new knowledge". That knowledge is something independent from human beings, which is "discovered" by us implies Platonism. This form of realism is conducive to the idea that there is unknown true propositions, which exist independently from us, which we "discover".

The alternative, non-Platonic realism would say that we create, produce or "construct" knowledge while something other than knowledge is what is independent from us. There are also forms of realism which blur the boundary between these two by invoking concepts like "information".

That's an example of using thing in a generously vague way. It is useful because it avoids annoying debates about what is a thing and what is not, etc; — Ludwig V

It might avoid such debates, but if we want to understand the metaphysics, and the possibility of the reality of an independent world, we need to engage these difficult subjects. Mundane life, and common language use in general, for communication, has no need for metaphysics. So habits of language usage are developed in ways of ambiguity, the ambiguity providing for a difference in underlying world-views. The ambiguity is effective in allowing me to interpret by my world-view, and you to interpret by your world-view, such that we can effectively communicate and move along in our day to day projects without the need for consistency between our ontological foundations.

So we can avoid "annoying debates about what is a thing and what is not", and move along with our mundane communications without the need to address metaphysical differences. If however, metaphysics is the subject of discussion, then avoiding these annoying discussions is a mistake conducive to misunderstanding.

Risible.

One of the possible worlds is the actual world.

Either that, or the actual world is not possible. — Banno

Exactly.

Within the modal model, we now have two possibilities, "one of the possible worlds is the actual world", or "the actual world is not possible". Since it violates the rules of the model for one of the possible worlds to be the actual world, then within the modal model, the actual world is not possible. Therefore model logic is not consistent with realism.

I'm glad you are at least starting to consider this as a possibility.

That's why ontologies like model-dependent realism are not true realism. Here's an analogy which might help you to understand. Under relativity theory, any rest frame is a valid rest frame, and each frame is made to be consistent with each other, as a valid "possibility". Each is a "possible" frame of reference, but none provides a true rest frame, which would be the "actual rest frame". In order that all the frames of reference may equally be valid rest frames, it is imperative that none is the "actual rest frame". Likewise, in modal logic it is imperative that none of the possible worlds is the "actual world", or the possibility of the others is invalidated.

The person who flipped the coin knew it. — EricH

You are changing to a different definition of "know", a subjective one, claiming that one person knows what others do not. We have been discussing this issue under the premise that knowing is a property of "we", not the property of one individual subject. I cautioned against equivocating between these two senses of "know" a few days back, because the conditions are completely different.

So in your example, there is a number of people, and discrepancy between what one person thinks that they know, and what others think that they know. Therefore it does not qualify as "we know", and is not a valid example. Anytime that it is proposed that there is inconsistency between what one knows and another knows, or between what some know and others do not, it is not a case of "we know", and not a valid example for the purpose of this discussion.

Any of the possible worlds could be the actual world - hence, "there is no modal difference between the actual world and the other possible worlds". Modal theory does not tell us which possible world is actual. — Banno

That's an incorrect interpretation, for obvious reasons. None of the possible worlds could be the actual world, as that would constitute an invalid difference, within the collection of possible worlds, one would be the actual world. Therefore it is clearly not the case that one of the possible worlds is the actual world because that would invalidate the model. Therefore we must interpret that none of the possible worlds is the actual world.

So for some p, the possibility of p ends when p occurs and for other p it doesn't. — Ludwig V

Yes, this acknowledges the difference between a particular and a universal. So you use "p" in an equivocal way. "When p occurs" refers to a particular, while "other P" refers to a type. It is clear that "when p occurs" must refer to a particular, because if it was a type, an instance of p occurring would not preclude the occurrence of another instance of p. But, the occurrence of a particular precludes the possibility of that same particular occurring again. So, we must clear up the equivocation in that statement, where "p" refers to a particular, and also to a type.

So do you accept that there are some unknown things? — Ludwig V

I believe there is a lot which is unknown. Strictly speaking it would not be correct to call the unknown "things", because that implies some sort of knowledge of the unknown, knowledge that the unknown consists of things. In other words, saying that there is "unknown things" wrongly projects knowledge onto the unknown. This is similar to, but more subtle, than saying that there is unknown statements. "Unknown statements" is obviously a problem, even to those without metaphysical education. But "unknown things" requires metaphysical understanding to recognize as fundamentally incorrect.

Someone in the next room flips a coin. We cannot see the result. Now we ask the question - is it possible that the coin is showing heads? The answer is of course yes. Then we walk into the next room and indeed the coin is showing heads.

So while it is not necessarily so (the coin could be tails), something can be both possible and also be real/actual at the same time. — EricH

I went through this all, way back. When we know that the coin is showing heads, it is incorrect to saying it is possible it is heads. When we do not know that the coin is showing heads it s correct to saying that that it is possible the coin is showing heads. Your example refers to two different times, before walking into the room, and after, so your conclusion of "at the same time" is incorrect. Before walking into the room we say it is possible, and after, we say it is actually showing heads, and we can no longer say it is possible. There is no "at the same time" indicated.

I'll give you this - I cannot win the 2025 Kentucky Derby twice. But that's not because I won it, but because it has happened that the result - win or lose - is settled. But if whatever the result of the 2025 race, it remains possible for me to win the 2026 race. So the possibility of my winning the Kentucky Derby does not cease when I win it. — Ludwig V

How's that relevant? You change from a specific possibility to a more general, so it is a different referent.

That's true. But the fact that the existence of the statement that Mount Everest is 29,000 ft high depends on human beings, does not show that the existence of Mount Everest depends on human beings at all. — Ludwig V

Again, I don't see the relevance. What I was responding to was unknown true statements, not unknown things.

If we look at EricH's example of the coin, there is implied an unknown real thing, the coin before looking at it. But that is not a statement, it is simply something unknown.

As if we could not talk about the actual world. — Banno

Why does what I say to you indicate that we cannot talk about the actual world? How can you make such a conclusion from what I wrote? What I said, is that if we are realist, we cannot put "the actual" into a semantic model in which it is a possible world. That would contradict our realist principles. In no way does it imply that we cannot talk about the actual world. It only implies that we cannot talk about the actual world in that specific context, because that would contradict our realist principles.

Because, as explained many times, it's not the semantic model that shows which possible world is actual. — Banno

That doesn't matter. By the principle upon which the semantic model is produced, we cannot conclude that any of the possible worlds is the actual world without contradicting realist principles. If you take a set of possible worlds, and apply some realist principles to deduce "the actual world", then you must relinquish the claim that the others are possible. You are then not within the constraints of the semantic model, so the other worlds are no longer "possible". "Possible" is a word applied in a very specific way, within that model, and you have moved outside that model, so the application is not valid.

Look what you said already yesterday:

"There is no modal difference between the actual world and the other possible worlds... ".

That is the very issue with modal models, by what principles do we produce an actual world. When you apply some principles to designate "the actual world", you violate the modal model by assigning special status to one of the worlds, placing yourself outside the model, and no longer correct in referring those other worlds as "possible". We only have those "possible worlds" within that semantic model which denies any such special status to any world. Assigning special status violates the model.

Are you serious? — frank

Sure, you start the thread, I'll follow.

Why should a semantic model commit us to the existence of the things quantified over? — Banno

The semantic model does not commit us to the existence of its content. But if the actual world is affirmed to be a part of that semantic model, as you and others here continue to insist, then this is contrary to realism which assumes that the actual world is independent from any semantic model. Why is that difficult to understand? You can't have it both ways, assert that the actual world is a part of a semantic model, with no claims to existence, and also assert that there is a real independent, existing actual world.

Your whole edifice still depends on an equivocation between what is and what is said. — Banno

The point is, that when you claim that the actual world is one of the possible worlds, you deny that there is a distinction between what is said and what is. If you assert that there is another "actual world" which is independent from what is said (realism), as well as the "actual world" which is part of the semantic model, then you equivocate. The equivocation in my "edifice" is just a reflection of the equivocation in what you are telling me, which my edifice is built upon. You are telling me that there is a metaphysical "actual world" and an epistemic "actual world", and the use of these contradict each other.

The solution to this problem is to maintain the distinction between the actual world and the possible worlds, i.e. the actual world is not a possible world. But as soon as we accept the proposition "if it is actual, it is possible" we negate that distinction and we are left with either a denial of realism, or a contradictory equivocation.

Here's an example of the contradiction you fed to me in that link:

They [possible worlds] are semantic or metaphysical constructs used to interpret modal statements.

They exist (or are defined) independently of human imagination. — CHAT GPT

"Constructs" which are independent from human imagination. Who constructs these metaphysical constructs if not human beings?

From the fact that humans think about alternative possibility-structures, it does not follow that those possibility-structures depend on human thought. — CHATGPT

Duh, the possibility structures are the thought ("constructs"), produced by the minds that think them. Are you arguing Platonism now? 'From the fact that human beings think ideas, it doesn't follow that ideas are dependent on human thought''. What about the "construct" part boss?

Come on Banno, quit the bullshit and say something real.

I've answered already. Several times. Here's the best I am willing to do. — Banno

That's not very good. Where's the answer?

I've explained what possible worlds are and how the actual wold is a possible world. If there is a problem set it out. The view I've set out it quite standard. If you see it as problematic, set out how. — Banno

I did set it out. This will be the third time I post the very same paragraph. Please, could you read it and reply accordingly. This is it:

So to reply to your SEP article, human beings think that "things might have been different in countless ways". These different ways that human beings think that things might have been different, are thought up by human beings, and so they are not independent from us. Therefore, "possible worlds" are worlds which are not independent from us, they are dependent on us. If, "the actual world" is said to be one of the possible worlds, then the actual world is not independent from us. Possible worlds are not independent. — Metaphysician Undercover

The possibility that it will rain tomorrow does not depend on whether we recognize it. — Ludwig V

Yes it does, very explicitly, it is something very specific, that human beings draw up in words, "the possibility that it will rain tomorrow".

The trouble is that we cannot know what they are. — Ludwig V

Of course we cannot know what they are because they cannot exist. The existence of statements is dependent on human beings. How could there be statements which we cannot know what they are, when a human being must have made the statement?

If the actual world was not a possible world, then it could not exist. — Ludwig V

That's nonsense. The possibility for something, precedes in time the actual existence of that thing. Once it is actualized, it is not longer a possibility, but an actuality. Suppose president Trump is actually assassinated. At this time, Trump has been assassinated, is a true statement of the actual world. At this point it is not possible for him to be assassinated, because he already has been.. So it's nonsense to think that if, in the actual world, Trump has been assassinated, it must be possible that Trump could be assassinated, because it is actually impossible, as he has already been assassinated. The same must be true of every actuality, and "the actual world" in general. Once it is actual, it is false to claim that it is possible.

There is no modal difference between the actual world and the other possible worlds... That'll confuse Meta no end. — Banno

It doesn't confuse me, I fully understand this, and it is the base of my argument. If you understand this, then it is undeniable that realism is incompatible with modal logic. Obviously, realism requires a difference between possible worlds and the actual world.

ecause the difference is not modal. It's metaphysical.

And Meta, as I've pointed out, has failed to see this distinction. — Banno

As I've told you, to have inconsistency between your metaphysics and epistemology is to have contradictory philosophy. To say that in my epistemology "the actual world is the same as any other possible world", but in my metaphysics "the actual world is completely independent and different from possible worlds", is nothing but contradiction within your concept of "actual world".

The philosophy behind actual versus possible is lengthy and complex. If you want to walk through two SEP articles on it we can examine the views of all the interested parties. There's even a tie-in to negative dialectics!!! — frank

Whatever you wish, I'm willing to follow.

You have not addressed the issue. That is to define "the actual world" in a way which is consistent with "a possible world", and also realism. I explained it to frank here:

So to reply to your SEP article, human beings think that "things might have been different in countless ways". These different ways that human beings think that things might have been different, are thought up by human beings, and so they are not independent from us. Therefore, "possible worlds" are worlds which are not independent from us, they are dependent on us. If, "the actual world" is said to be one of the possible worlds, then the actual world is not independent from us. Possible worlds are not independent. — Metaphysician Undercover

You have not addressed this question:

Care to explain what you're talking about? As far as I'm aware of, only human beings make statements, and only human beings make judgements of true and false. That is why "a true statement that's beyond our knowledge" makes no sense to me. It has nothing to do with whether I'm realist or not, it's a matter of how I understand the terms you are using. You appear to be using these terms in a way which I am not familiar with. Maybe you could define "statement" and "true"? — Metaphysician Undercover

Again, a world does not consist of a set of statements. — Banno

What does a possible world consist ofthen?

You said:
" A possible world does not consist of stipulations, so much as a complete description of a state of affairs - which statements are true and which are false."

Doesn't that mean to you, that a possible world consists of a set of statements? No wonder I'm confused, you keep contradicting yourself.

Real? They're both abstract objects. :lol: — frank

Abstractions aren't real for you, frank?

It makes sense to realists. Apparently you aren't one. — frank

I think I'm realist, that's why I have difficult making "possible worlds" (worlds which are not real), consistent with "the actual world" (a world which is real).

Care to explain what you're talking about? As far as I'm aware of, only human beings make statements, and only human beings make judgements of true and false. That is why "a true statement that's beyond our knowledge" makes no sense to me. It has nothing to do with whether I'm realist or not, it's a matter of how I understand the terms you are using. You appear to be using these terms in a way which I am not familiar with. Maybe you could define "statement" and "true"?

All I was trying to say is, even a child can come across "true knowledge"—he or she simply might be "incapable of knowing (processing it?)" at the time (but might, given enough time, thus illustrating the concept of the unknowable becoming knowable, at least in one valid manner of thinking). — Outlander

Knowledge is the property of knowers. Is this knowledge which no human beings possess supposed to be possessed by God?

If so, how does that make the "actual world", as known by God, consistent with "possible worlds" which are statements made by human beings?

An important distinction is, much like a child, we assume either we—or someone we know—knows all there is to know (that is to say, can simply be "exposed" to such knowledge, such as walking into a room where it's written and automatically understand it in full depth and detail as others do; this is merely the ego at work, the driving force and cause of all human suffering). — Outlander

Speak for yourself. I don't see why anyone would ever assume that there is someone who knows all there is to know. Since knowledge varies from one person to another, it's very counterintuitive to think that there would be one person who knows everything. Since much knowledge is context dependent wouldn't this require someone who is everywhere, all the time?

Since knowledge is the property of knowers, are you proposing God to support the idea of knowledge which is unknowable to current human beings?

A realist says the actual world contains true statements that are beyond our knowledge. — frank

Can you clarify this? What is a true statement that's beyond our knowledge? It doesn't make any sense to me.