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.

How, in your mind, does possible worlds semantics establish extensionality for modal logic? — Metaphysician Undercover

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

Banno, are you bored? — Philosophim

Not really. Although this topic is not of any particular interest to me, beyond the misuse of philosphy of language I've been pointing out.

A word is ambiguous when it has two or more possible meanings, and it is unclear which meaning is intended in a given context.

A word is polysemous when it has multiple related meanings that are all legitimate and established, and the word’s meaning shifts depending on context.

It's not that hard.

Woman is polysemous, not ambiguous.

And, in the gender-social sense, “trans women are women” is true.

Again, Meta, what I have been espousing here is not "mine" in the way that what you have been saying belongs so specifically to you. The account I have been using is standard, accepted modal logic; and now, because of your extended eccentricities, standard accepted mathematics.

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. You treats the representational construct inside modal logic (the “actual world” symbol in a Kripke frame) as if it is the metaphysical actual world. The model’s “actual world” is a description; it is not the metaphysical actual world.

I've been pointing this put for pages. Quite literally.

Numbers are extensional objects - you can substitute them in equations, which is the very definition of extensionality. Modal logic uses an intensional syntax, modelling it extensionally. If you continue with the other discussion, instead of seeking to pervert it, you might actually see how this happens.

You know I never stated an essential meaning for woman — Philosophim

you did say:

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

And that's specifically what I addressed. Again,

Insisting on only the biological sense is a misunderstanding of how language works, not a logical or empirical requirement. — Banno

...my conclusion was that the phrase is ambiguous — Philosophim

And I pointed out that it is polysemous rather than ambiguous. You conflate the two.

Is this kind of like how "sick" "means" "impressive" and "hot" "means" "attractive" and/or "stolen", etc.? — Outlander

What do you think?

I'm claiming the context of 'woman/man' unmodified is most rationally interpreted to mean 'sex of the person', not 'gender of the person.' That's what the 'trans' and 'cis' modifiers are for. — Philosophim

Not at all. We went through this. There is no "context of 'woman/man' unmodified", no "true" meaning for such terms, beyond your preference for choose a "true" meaning in order to justify your claims concerning trans folk.

So are transwomen women? Are transwomen men? No. The terms man and woman indicate a person's age and sex, not gender. Are transwomen men who act with a female gender? Yes. Are transmen women who act with a male gender? Yes. — Philosophim

But "woman" is a polysemous term; one established meaning is biological, and another established meaning is gender-social. Contrary to the OP, in the gender-social sense, “trans women are women” is true. Insisting on only the biological sense is a misunderstanding of how language works, not a logical or empirical requirement.

cheers. Very pleasing.

Good questions. There is a use of "intension" that is the same as "meaning" or "sense" or "the concept of...". And there is a use of extension that amounts to "that very thing".

Some gross oversimplification follows. I'm concerned about getting the overall picture in place rather than the detail.

Go back to John's pets. The extension of "John's pets" is {Algol, BASIC}. It is exactly the set of things, taken as a whole. The extension of "John's pets" = {Algol, BASIC} is the same as saying the extension is "that very thing" - the extension is those specific dogs.

The intension is much less specific. The intension of 'John's dogs" is it's meaning or sense, whatever that is, or the concept of a dog owned by John.

Its much easier to work with extension. Intensionaly speaking, to check if "Algol is one of Joh's dogs" is true might require us to check the sense of "John's dogs", what that concept means or how it is used, then to do the same with "algol", and bring the two together.

Extensionally speaking, to check if "Algol is one of Joh's dogs" is true we look to see if "Algol" is in {algol, BASIC}.

the important bit is to notice that in the intensional way of checking, the truth of the sentence depends on concepts and meaning and such. But in the extensional approach, what's involved is a relative y simple process of checking if the referent of the term is an element of the extension of the predicate.

There are formal definitions of intension, used in formalising intensional logic. These pretty much consist in relations between terms and their extensions. But this is not central to the article we are considering.

So when we say modal logic wasn't extensional, it's that the items mentioned in modal expressions didn't pick out anything in the world. — frank

Not quite. It's not that "possibly, Algol might not have been one of John's dogs" does not refer to anything - it clearly does. It's that substitution, the very core of extensionality, might not preserve the truth of such sentences. In modal contexts, knowing what something ‘actually is’ is not enough to determine truth; you have to consider how it might be in other possible worlds.

Sure. Acknowledged.

The social model of disability started in the seventies, as a change in perspective that involved listening to the voices of the disabled, to wha tit was that they needed rather than what others were willing to do for them.

The improvements to how we deal with folk with disabilities is in a very large part down to this move in emphasis. Accessibility standards, anti-discrimination laws, independent-living movements, deinstitutionalisation, personal-assistance schemes, and now programs like the NDIS—owe their momentum to this change. The driving force was disabled activists insisting that disability is not a deviation from the normal human body, but the consequence of social design.

That we now do more to help disabled people thrive is because the social model reframed disability as a matter of rights, participation, and capability, not charity or medical adjustment. Listening to disabled people, not than assuming the abled know what they need, was the hinge on which that change turned.

To say historically implies that it is a practice put in place. — L'éléphant

That exactly what the social model suggests: that disability is "a practice put in place" as much as it is a feature of a body. Is disability a property of a body, or a relation between that body and it's environment, including it's social context?

Saying that a particular body has a deficit is making an evaluation, that it ought be otherwise.

But that ought is embedded in an historical and cultural context.

The issue here is perhaps one of presumption, that in the first instance we look at what a disabled body cannot do, when we could look at what the disabled body might require in order to achieve it's full capabilities. to ask, “Given the person’s body and circumstances, what supports or adjustments would allow them to exercise their capabilities?”

So instead of checklists of what a person cannot do, ask which valued activities is the person currently unable to achieve? What barriers (physical, social, environmental) prevent capability realisation? What supports would remove those barriers?

It's about shifting the narrative from “what’s wrong with you?” to “what do you need to thrive?”

Simplifying a bit, we have that all John's pets are dogs. His pets are the same as his dogs.

We can substitute in some sentences; so that since all john's dogs are mammals, by substitution we have that all John's pets are mammals. All good - truth is preserved, the context is extensional.

And we have

(5) Necessarily, all John's dogs are mammals: □∀x(Dx → Mx),

Of course this is true since all dogs are mammals. In no possible world does is there a dog that is nto a mammal.
but by substitution that gives

(6) Necessarily, all John's pets are mammals: □∀x(Px → Mx)

But he might have had a pet lizard.

Substitution fails in the modal sentence. And another name for such a failure is that the context is not extensional. Modal sentences are not extensional.

:wink:

One strategy in that culture war has been the denigration of the term "liberal". It's odd, since if we scratch most folk, outside of religious traditions, their core values will be classically liberal: Individual freedom, the rule of law, equality before that law, protection of rights and liberties and so on.

These are what lead to tolerance, and to acceptance, as much as vice versa.

So we might accept that others live lives quite divergent from our own, on the condition that they do not obligate us to do as they do. Acceptance of divergent lives does not imply agreement or obligation. This maintains moral consistency: one can uphold their own values while ethically recognising the legitimacy of other ways of living.

A quick note that model and modal are not the same, but that we are using both. Modal is to do with necessity and possibility. A model is an assignment of truths to a set of sentences or propositions.

I am still confused about why modal logic itself is not extensional — NotAristotle

Simply, substitution fails.

Here's an example fo the sort of thing that threw Quine:

• Necessarily, eight is greater than seven
• The number of planets =eight

Note that the first sentence is modal - the modal operation "Necessarily" wraps around the whole of "eight is greater than seven". Now extensionality is simply the substitution of equal expressions. And "The number of planets =eight" expresses an equality. So we shoudl be able to substitute "The number of planets" for "eight". But that gives
[*] Necessarily, the number of planets is greater than seven

But that does not seem right - it might have been the case that there were only five planets, and the ancients thought.

So substitution fails, and the modal context is not extensional.

But possible world semantics gets around all this.

See this thread on Quine if you need more.

By this I understand you to be saying that the symbols need to refer to something (or predicate something) in the world (or in a possible world if we are using possible world semantics). — NotAristotle

Roughly, yes. But it's freer than that. It's fine in a formal system to say things like "a" stands for a, perhaps in explaining what the " does in separating mention form use. The symbols do not have to denote actual-world entities.

So they can, but need not.

I was trying to draw a broad sharp line between those who support institutions even if they often suck and those who want to shake the Etch a Sketch upside down. I am not aware of any of the former kind who subscribe to the purely emotional view you propose to be a significant factor in political discourse. — Paine

Somewhere in between we have Popper's ad hoc social engineering, piecemeal improvement. Small, testable reforms, improving society step by step while avoiding catastrophic overreach.

But is that enough?

:wink:

So we have the supposed paradox of tolerance; that the left, in advocating "tolerance", is hypocritical in not tolerating the right - in not tolerating intolerance.

One way to view this is as confusing tolerance with acceptance. In this usage, to tolerate is roughly to refrain from using coercion, while to accept is to place the account in the domain of public discussion.

The left can coherently tolerate the more extreme views of those on the right without accepting them.

Why not accept them? Popper's response is well-known, even if the attribution might be lost. To accept intolerance is to undermine the broader ethic of tolerance. It's not hypocrisy but consistency. On this account intolerance might be tolerated, but certainly not accepted.

I think Kant was muddle din his talk of such things, his confidence misplaced. But that's a side issue here.

The choices between what is acceptable or not is worked out each day wherever we are. — Paine

...is much better.

It's not only the result of obeying a series of rules, although rules may have their place; it's not algorithmic — Banno

Kant was pretty confident he was up to speed about the correct rules. — Paine

Always with the Kant. Oh, well.

Being consistent is all very well, but it doesn't tell us what to do in every case. The central problem with rules is that they are incomplete; there is always something they do not cover.