Yep. w₀, the actual world, is the one in which Nixon satisfies "Won the election". In some other world, w₁, he does not satisfy "won the election".

An extensional account.

Let:

w₀ = the actual world
w₁ = a counterfactual world

Let the 1-ary predicate:

W(x) = "x won the election"

Tarskian semantics inside each world:

$\mathrm{Ext}_{w_0}(W) = \{ \mathrm{Nixon} \}$
$\mathrm{Ext}_{w_1}(W) = \varnothing$

So:

In w₀, Nixon satisfies "won the election":
$\mathrm{Nixon} \in \mathrm{Ext}_{w_0}(W)$

In w₁, Nixon does NOT satisfy it:
$\mathrm{Nixon} \notin \mathrm{Ext}_{w_1}(W)$

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.

You haven't followed the argument, missing the main point about privileging a sense. I addressed the reasons he gave, you fumbled around. I don't see anything in your post not already addressed.

Ok.

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. — Metaphysician Undercover

Yep. I can do that. The same term is used for two different things. That's not equivocation. It's your error to conflate metaphysics with modality. Think I mentioned that. A few times.

I don't understand the paradox as a paradox. — Metaphysician Undercover

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?

So I suggest that you present it in a way which appears to make sense to you, — Metaphysician Undercover

No. If you would proceed, set it out for us. 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.

Thanks for getting us back on track. The digression with Meta has gone on too long. But it did spawn another thread, thanks to @Frank, which I'm enjoying. More opportunities for me to show off, of course. :halo:

If the question asks is there a possibility that there is an aspect of reality beyond our known reality, how could we rule this out? — Tom Storm

I think we can guarantee that "there a possibility that there is an aspect of reality beyond our known reality". It seems to me that wha this says is "there are things we do not know", and I'm pretty confident that we do not know everything.

But is there something here, some other understanding of "an aspect of reality beyond our known reality" that I'm missing?

if not, then this appears to be a classic case of language leading us astray.

Are you not entertained?! Is this not why you are here?!

:wink:

Links to my posts rather than your own would be preferred, when you are trying to demonstrate a problem with something I said. The bit where I pointed out that responding to your rubbish requires more time than it is worth.

But I tracked down the originals.

We are in the actual world. — Metaphysician Undercover

We are in the actual world.

That's from this:

We are in the actual world. Metaphysics. — Banno

The whole quote makes it clear I am talking metaphysically. See the word "Metaphysics" in the very next sentence? It's kinda a giveaway.

The other is from a different post,

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. — Banno

See how it refers to w₀, and so is clearly modal.

SO your accusation was I

...in the very same argument use "the actual world" to refer to a model-theoretic object, and also to a metaphysical object. — Metaphysician Undercover

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.


A pathetic response, even for you. This is why, if I wasn't chasing posts, I'd have long ago gone back to ignoring you.

are you ready to address the so-called Fitch's paradox — Metaphysician Undercover

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

At the risk of being overly formal, have a think about the difference between what is true and what is known to be true.

To explain the idea, lets' suppose we can list all the facts, every true statement: {f1, f2, f3...} Those facts, taken together, list everything that is the case.

But while you and I know maybe the first few thousand facts, little Jimmy over there only knows the first few hundred.

Will we say that he is living in a different world to us? That he has a different reality? Well, we could, if we restrict facts to only those things that are known, and not toe those things that are true, whether known or not.

So in that way of talking, Little Jimmy's reality is smaller than yours and mine.

Btu notice that this is a different sense to all the facts, taken as a whole.

So we have two different things - what is known, and what is true. On the first, folk can have different realities. On the second, we all share the same reality.

Not an ambiguity, but we should take care as to which sense we are using and be consistent in that use.

You, in the very same argument use "the actual world" to refer to a model-theoretic object, and also to a metaphysical object. — Metaphysician Undercover

Where?

Might be best to quote me. Be precise.

...a Tarskian interpretation fixes the domain of quantification and the extensions of all the predicates. Pretty clearly, however, to capture necessity and possibility, one must be able to consider alternative “possible” domains of quantification and alternative “possible” extensions for predicates as well. — From Tarskian to Possible World Semantics.

The trouble with Tarski's system is that there is but one domain, and one interpretation. Kripke's move was to notice that if we consider multiple domains and interpretations, we can use Tarski's approach to analyse modal statements.

It might have been the case that Algol did not become one of John's pets. That would be a change in the interpretation, but not in the domain. The extension of "Is John's pet" would no longer be { Algol, BASIC }, but just { BASIC }.

And in the previous example the domain was { John, Algol, BASIC }. Now it might have been the case that instead John has a pet canary — COBOL (I'm not choosing these names!). The domain would then be { John, Algol, BASIC, COBOL }. Some of the sentences we used would here keep their truth value - that Algol is one of John's pets would remain true. Others would change - that all of John's pets are dogs would no longer be true.

Notice that this latter instance is also a change in the interpretation. The interpretation is a list of which individuals are assigned to which predicates. Adding an individual to the domain changes the interpretation.

That's all a possible world amounts to. A different interpretation of the symbols in a Tarskian system.

In one possible world, the interpretation has the pets as Algol and BASIC. That's the possible world in which it is true that John's pets are Algol and BASIC. In another, the pets are Algol, BASIC and COBOL. In another, Algol is not one of John's pets.

Notice that extensionality survives within, but not between, these worlds.

Here's were we can explain and overcome the accusations from Quine and others that modal logic cannot be treated extensionally.

That's pretty much in agreement with my view, I think.

Where you talk of welcoming, I used acceptance.

So we both differentiate mere toleration, in which something is thought unacceptable but we put up with it, from welcoming and accepting different ways of living that do not infringe on our own, and a willingness to negotiate when they do.

And both are contrary to the view that one's own way of living is obligatory for others. Such a view cannot be accepted, and ought not be tolerated.

Glad it was of use.

This got complicated. — Tom Storm

Yes, indeed. I'll stand by what I said in my first post:

How do we know there isn’t anything beyond our reality?

Because reality is what there is.

To posit something "beyond reality" is to posit more of what there is. It is to extend reality.

This is why the extent of our language is the extent of our world.

Hopefully, replacing "limit" with "extent" will head off some of the misplaced criticism of that phrase.

The other mistake here is to equate what we experience with what is real, and so to conflate "How do we know there isn’t anything beyond our experience" with "How do we know there isn’t anything beyond our reality".

"Beyond reality" is not a region; it is a grammatical error. — Banno

How's that sit with you?

Philosophim has given you his reasoning for preferring his meaning — AmadeusD

Has he? He claimed that one interpretation was more rational. His reasoning was questionable, and questioned.

He claimed Woman/man unmodified is most rationally interpreted as sex.But he had previously , over the course of days and pages, agreed that there is no one “true” or privileged unmodified meaning for woman/man. Oddly, Philosophim can't bring himself to say he is privileging one sense.

He claimed normal English makes sex the default meaning. But English does not have a single “default” meaning independent of context. Claiming one is simply choosing a preferred meaning for ideological reasons.

He arguers that different uses are marked by modifiers such as cis/trans, and these mark gender, while the unmodified term marks sex. But again, words and sentences are never without context; we do as an issue of fact use "woman" to include both cis- and trans- folk.

He claimed that “trans women are women” is ambiguous without external context, but again, there are no cases that are not in a context. And addition, polysemous is not ambiguous.

I think this represents his position accurately. He can explain if that is incorrect.

When he claims that one interpretation is more rational than the others, he is doing no more than saying that he prefers one interpretation over the others. But his preferences are not a consideration here.

What I have done is to show that there clearly is a sense in which "trans women are women" is true. That undermines his OP.

You seem to be advocating an argument by majority vote. Issues of usage are not decided democratically. If a community uses a word in a particular way, then that usage exists.

, , For better or worse, the term is now embedded in the literature. One needs must learn instead to use it correctly.

But it is possible that he is not doing that. 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

You have not misunderstood.

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. — Metaphysician Undercover

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

Indeed.

An example of Modal realism, David Lewis' ideas were quite sophisticated, and far from Meta's misunderstandings. In brief, Lewis held that since someone in another possible world would consider their world to be the actual world, we should treat it as an actual world; that it was as real as our own. Such a view is annoyingly coherent, leading to a large literature.

Most philosophers will differentiate between treating a possible world as if it were real, and treating it as real.

Meta's idea, so far as I can make out, is that the node in a model (w₀, the designated “actual world”) though it is claimed to be the metaphysical actual world rather than one of many semantic artefacts, w₁, w₂ and so on. He takes these model-theoretic objects as claims about reality then accuses the logician of contradiction because the formal “actual world” differs from the metaphysical actual world.

It all seems perfectly clear. — Ludwig V

Good. Following your analogy, one of the books in your encyclopaedia is about the actual world. You might take it out and read it. In another possible world, another possible you can take out a different book about their world, treating it as their actual world, and read, it perhaps with as much satisfaction as you derive from reading yours.

Can we also write ◇p→(p v ~p)? — Ludwig V

Yep. The consequent is a tautology, hence always true, so the implication as a whole is always true.


Might be more of a surprise that □p→(p∨¬p) is also true.

the individuals are the domain. So it’s whatever you would include. In our case,

Domain: D = { John, Algol, BASIC }

But potentially anything.

A quantifer tells us about the number of items in a domain that have a certain property, like all, or some. So "necessary" will mean that all the items (in every possible world) have the property. Possibly mean at least some of them do. — frank

Yep. The U and the ∃ quantify within a world, the ☐ and the ◇ across worlds.

That's the next step.

Playing with MathJax...

The equivalences between my last post and the section on Tarski's semantics.

I'm not overly happy with that. I might try a different approach.

We have a language - roughly, first order calculus.We give it the following interpretation...

We have a domain consisting of three things: John, Algol and BASIC (Who names their dogs after extinct computer languages?)

We have a few predicates, "Is John's pet", with the extension {Algol, BASIC}; "Is a dog" with the extension {Algol, BASIC}; "Is loved by" with the extension {(John, Algol), (John, BASIC)}.

We can note immediately that "Is John's pet" is co-extensional with "Is a dog" - all John's pets are dogs.

We then set out satisfaction; An individual satisfies a predicate exactly if it is a member of the extension of that predicate. So Algol satisfies "Is a dog", and the pair (John, Algol) satisfies "Is loved by".

And then we can define being true for any sentence in our interpretation in terms of satisfaction. A proposition is true if the individuals involved satisfy the predicates involved.

This approach might make it easier the follow the next section.

Again, we've defined truth in our language using only extensions.


This does the work in the section of Tarski's semantics, with

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

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

I don't necessarily reject this. — Metaphysician Undercover

That's a start. Good.

Tarskian Semantics
The next section looks pretty fearsome. Its formality belies a fairly simple and direct way to deal with truth, which was developed by Tarski. It's not his T-sentences, although it comes from the same body of work.

We are already almost there with the following:

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. — 1.1 Extensionality Lost

In predicate logic, every predicate symbol has an arity, the number of arguments it takes. A proposition, such as p, is 0-ary, it takes zero arguments; a 1-ary predicate such as f(x) takes one argument - the "x"; a 2-ary predicate such as f(x,y) takes two arguments - the "x" and the "y". Generally, an n-ary predicate takes n arguments

What's added is the definition of "...is true" as follows:

n = 0 (i.e., π is a sentence letter) and the extension of π is the truth value TRUE; or
n = 1 and aτ1 is in the extension of π; or
n > 1 and ⟨aτ1, ..., aτn⟩ is in the extension of π. — 1.2 Extensionality Regained

These give the meaning of "...is true" for each of the n-ary predicate symbols.

So what is being said is that a proposition, p, will be true in the case that its extension is the truth value TRUE. This might seem odd at first, but it's standard, so take it as it stands for now.

A 1-ary predicate such as f(x) will be true in the case that the referent of x is one of the things that is in the extension of f.

A 2-ary predicate such as f(x,y) will be true in the case that the referent of x and y are among the things in the extension of f.

Going back to John's two dogs, The sentence "John has two dogs" has as its extension "TRUE", and so is a true sentence. The predicate "John's dogs" has the extension {Algol, BASIC}; and "Algol is one of John's dogs" will be true precisely if Algol is in that extension; which it is.

We can add a bit of terminology. We say that "TRUE" satisfies "John has two dogs", and that Algol satisfies "One of John's dogs".

We might add the predicate "Loved by", with the extension {(John, Algol), (John, BASIC)} - "John loves Algol" and "John Loves BASIC" are both true. We get such the 2-ary predicates as "Loved by (John, Algol)" which will be true exactly if (John, Algol) is a member fo the extension of "Loved by" - that (John, Algol) satisfies "Loved by"

What Tarski did here was to provide a way to evaluate the truth of any formula, using satisfaction, and hence purely in terms of extensionality.

In 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.

A Kripke frame is a representational device, and every world inside it is a representational device.
Even w₀ is just another node in the model.

Calling one of those nodes “the actual world” introduces no metaphysics. It is merely a stipulation in the model: "let this node represent the actual world".

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".

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.

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

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.

Keep in mind that the equation he rejects, p→◇p, is valid in both S4 and S5. In rejecting it he rejects the two most useful systems of modal logic. Meta’s rejection of the principle amounts to rejecting reflexivity, which means rejecting T, and thereby rejecting S4 and S5, which means rejecting every ordinary epistemic, doxastic, and metaphysical modal logic used in philosophy.

I think you saying that this particular world is the only one that corresponds to the facts of reality as we experience them, which is not a strong statement since I don't see how it could be otherwise. — noAxioms

Not really.

The usage comes from "The world is all that is the case", the first lines of the Tractatus Logico-Philosophicus. It's still a pretty good definition. While it says nothing about experience, it remains that it is difficult to see how it could be otherwise.

But not at all idealistic. The world is the facts, experienced or not, known or not.

It’s the same. That’s the definition of extensionality used in logic and maths.

The "counterpoint"?

You mean your attempted restrictive use of "privilege"? It's an obvious dead cat:

Look Over There!! — Philosophim

Get back on the topic.

You privilege one meaning over others.

If you are not doing that, then you cannot maintain that "trans women are women" is false. — Banno

What remains is that the response I've given undermines the OP, so that you now feel the need to change the topic to some feeble argument about the essence of "privilege".

So, again,

You privilege one meaning over others.

If you are not doing that, then you cannot maintain that "trans women are women" is false. — Banno

You privilege one meaning over others.

If you are not doing that, then you cannot maintain that "trans women are women" is false.

Pretty simple stuff.

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.

:rofl:

Heaven forbid we talk about the definition of "extension" in modal logic.

See the Open Logic text, Appendix A1.

Make up your own definition is counterproductive here.

Definition A.1 (Extensionality). If A and B are sets, then A=B iff every element of A is also an element of B , and vice versa.

Are you intent on playing Dictionaries for the remainder of this discussion?

...advantage... — Philosophim

...as, for example, you give the advantage to 'sex of the person' over 'gender of a person' when you say

I'm claiming the context of 'woman/man' unmodified is most rationally interpreted to mean 'sex of the person' — Philosophim

Where have I ever advocated privilege? — Philosophim

Exactly here:

I'm claiming the context of 'woman/man' unmodified is most rationally interpreted to mean 'sex of the person' — Philosophim

You try to privilege one interpretation over all others.

Here, it's only Sky News, and maybe some of it's audience, who are angry. Otherwise the somewhat archaic notion of "a fair go" prevails, and folk just move on.

Ok, I JUST told you I said the term was polysemous, while the phrase was ambiguous. — Philosophim

You can't maintain that while simultaneously maintaining that the One True Meaning is the biological one.

All that stuff about phrases and words is a bit of a furphy. Words and sentences are never without context.

The context of "are transwomen women?" in your OP is just the OP - after all, the purpose of a good OP is to set up a context.

Yours seems a pretty desperate account. The phrase “trans women are women” is meaningful and true in its social-gender sense; claims of ambiguity or fixed biological meaning ignore polysemy and the unavoidable role of context. Your attempt to maintain polysemy while privileging a single biological sense is logically inconsistent.