The composition may change in terms of NaCl, etc., but if you do not have H2O then you do not have water. Your response?

I don't think I really understand the question here. — Metaphysician Undercover

Why am I not surprised?

Wouldn't we have to check every snowman, and make sure that it is not Frosty before we conclude that Frosty the Snowman does not exist. — Metaphysician Undercover

Yes Metaphysician, check every snowman in the whole world and double check that none of them is Frosty; that would be an excellent use of your reason.

This leaves "truth" as either completely arbitrary, or rescued from arbitrariness by subjectivity. — Metaphysician Undercover

The truth tables are important in Tarski’s First Order Logic. For example, the material implication truth table, whereby:

P.....Q.....if P then Q
==================
T.....T..........T
T.....F..........F
F.....T..........T
F.....F..........T

Kripke extended First Order Logic into Modal Logic K adding necessity and possibility, where the truth table shown above remains applicable to each accessible world.

Therefore truth, as expressed by truth tables, cannot be said to be arbitrary and is in this sense objective rather than subjective.

:up:

Yep. Truth tables for propositions and logical operators. Tarski also added satisfaction - f(a) is true IFF a satisfies f...

There's nothing arbitrary here. It's determined by the formal structure. The modal operators ◇ and ☐ are defined in relation to that formal structure by the introduction of possible worlds. The rules of logic and the structure of models fix truth independently of anyone’s opinion, so truth is objective in the formal sense.

Meta hasn't been able to follow this. But it is how it works.

Can I also at this stage express my appreciation to you, and @Frank for putting in the effort to understand what is happening here before launching into a critique. And thanks for the opportunity presented by this thread. paraphrasing is an excellent way to improve my comprehension.

Fundamentally, I think it is a problem to try and establish identity between two distinct ideas. There is always nuanced differences which makes such an identity incorrect. Some people would say that it's a difference which doesn't make a difference, but that is contradictory because if it is noticed as a difference it has already made a difference.

Mathematicians are often inclined to do this with equality (=). They will say that "2+2" represents the same idea as "4". But this is clearly false because there is an operator "+" within "2+2", so obviously it cannot be the same idea as "4". This is why it is best for good philosophy, to maintain a very clear distinction between identity and equality. Equality is a relation between two individuals within a category (kind). You and I as human beings are equal. But identity is unique to an individual.

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

My proposed compromise was for you to recognize that what you call "the standard account" is Platonist. That shouldn't be difficult. Modern "standard" interpretations of mathematics are clearly Platonist. The rule of consistency would suggest that modal logic would be interpreted in a Platonic way as well. Surely there is "space" for that unless you have some good reason not to.

Also, your supposed "standard account" is not the only account. That's why we're reading the SEP to find out about all the alternative interpretations. That's what good philosophy is all about, understanding the difference between the different possibilities.

Frodo" refers to Frodo, a fictional character in LOTR. It does not refer to the idea of Frodo. — Banno

A fictional character is an idea, not a thing. That's pretty obvious. Why would you deny it?

We have two different things - Frodo, who carried the one ring, and the idea of Frodo, which never carried anything. "Frodo" is the name of Frodo, not the name of the-idea-of-Frodo. — Banno

What is this nonsense? We have the idea of Frodo carrying a ring, and the idea of Frodo not carrying a ring. Two distinct ideas.. Why do you attempt to make ideas which are very simple and easy to understand, extremely complex and difficult?

Kripke extended First Order Logic into Modal Logic K adding necessity and possibility, where the truth table shown above remains applicable to each accessible world. — RussellA

It is those additions which introduce subjectivity. The subjectivity being the intentional products of the mind which enter due to the variance in purpose, and are allowed to contaminate judgement, rendering "truth" as fundamentally subjective.

On the assumption that there is a (nonempty) set of all possible worlds and a set of all possible individuals, we can define “objective” notions of truth at a world and of truth simpliciter, that is, notions that are not simply relative to formal, mathematical interpretations but, rather, correspond to objective reality in all its modal glory. Let ℒ be a modal language whose names and predicates represent those in some fragment of ordinary language (as in our examples (5) and (6) above). Say that M is the “intended” interpretation of ℒ if (i) its set W of “possible worlds” is in fact the set of all possible worlds, (ii) its designated “actual world” is in fact the actual world, (iii) its set D of “possible individuals” is in fact the set of all possible individuals, and (iv) the referents assigned to the names of ℒ and the intensions assigned to the predicates of ℒ are the ones they in fact have. Then, where M is the intended interpretation of ℒ, we can say that a sentence φ of ℒ is true at a possible world w just in case φ is trueM at w, and that φ is true just in case it is trueM at the actual world. (Falsity at w and falsity, simpliciter, are defined accordingly.) Under the assumption in question, then, the modal clause above takes on pretty much the exact form of our informal principle Nec. — SEP

Notice, necessity is not based in the set of all possible worlds, it is based in the assumption that there is a set of all possible worlds. @Banno, this is inherently Platonist. It assumes an idea "all possible worlds" which is unknown to us, independent. Then, (i) the interpretation M, is dependent on W being "in fact the set of all possible worlds". Of course, one could never, in fact, know the set of all possible worlds, so the judgement of "in fact the set of all possible worlds" is purely subjective.

Further, (ii), "its designated 'actual world' is in fact the actual world" is something which is truly impossible. This is the ongoing discussion I've had with Banno. It is a problem which Banno seems to acknowledge but refuses to respect. So what happens here is that a subjective representation of "the actual world" is assumed to be "in fact the actual world", as this is a requirement.

Then (iii) repeats the subjectivity of (i), and (iv) repeats the problem of (ii).

Platonist. It assumes an idea "all possible worlds" which is unknown to us, independent. — Metaphysician Undercover

No, it doesn't.

That's a useless and baseless assertion if I've ever seen one.
Thank you for your opinion nonetheless.

IF logic did not apply to Middle Earth, the books would be unreasonable. Our logic ought apply in such cases. And indeed it does. — Banno

I feel that there is some truth in the following, but cannot clearly see it. Hopefully it adds something.

JL Austin’s performative and constative utterances is relevant to Wittgenstein’s Language Games

Suppose in Possible World 5 there is a form of life and a language game.

Before any performative utterances by an authority

JL Austin discussed performative and constative utterances.

Suppose in this world people see a family resemblance between the elements of the set {this swan in Hyde Park, that swan on the Thames, those swans on the Serpentine}

We can then say that there is something X that the elements of this set have in common. In other words, the elements of this set are part of the domain of X

As regards X = {this swan in Hyde Park, that swan on the Thames, those swans on the Serpentine}
1 - This is not an extensional definition, as the set does not include every element that falls under the definition.
2 - This is not an intensional definition, as the set does not include necessary and sufficient elements to be analytically valid.

Suppose in this world people also see a family resemblance between the elements of the set {waterfowl, flighted, white}

We can then say that there is something Y that the elements of this set have in common.

As regards Y = {waterfowl, flighted, white}
1 - This is not an extensional definition, as the set does not include every element that falls under the definition.
2 - This is not an intensional definition, as the set does not include necessary and sufficient elements to be analytically valid.

We now have two sets. Set X whose elements are concrete things and set Y whose elements are abstract properties.

But people also observe the following:
“This swan in Hyde Park” = {waterfowl, flighted, white}
“That swan on the Thames” = {waterfowl, flighted, white}
“Those swans on the Serpentine” = {waterfowl, flighted, white}

As regards “This swan in Hyde Park” = {waterfowl, flighted, white}
1 - This is not an extensional definition, as the set does not include every element that falls under its definition
2 - This is not an intensional definition, as the set does not include necessary and sufficient elements to be analytically valid.

From this we can say that there is a concrete something X that has the properties Y.

After performative utterances by an authority

What X is is unknown, but for linguistic convenience it can be given a name, and in a performative act someone in authority names it “swan”.

As regards “swan” = {this swan in Hyde Park, that swan on the Thames, those swans on the Serpentine}.
1 - This is not an extensional definition, as the set does not include every object that falls under the definition.
2 - This is not an intensional definition, as the set does not include necessary and sufficient elements to be analytically valid.

Both the intensional and extensional definition of “swan” are still unknown, but what is known is that the elements of the set have a family resemblance. This means that “swan” is the name of a family resemblance between the elements of the set.

What Y is is unknown, but for linguistic convenience it can be given a name, and in a performative act someone in authority names it “swanness”.

As regards “swanness ” = {waterfowl, flighted, white}.
1 - This is now an extensional definition, because a performative utterance by an authority, and as the set does include every object that falls under the definition
2 - This is now an intensional definition, because a performative utterance by an authority, and as the set does include necessary and sufficient elements to be analytically valid.

Therefore, if something is observed that is {waterfowl, flighted, black} then by definition it has no "swanness".

As regards swan = {swanness}
1 - This is an extensional definition, because swanness is an extensional definition
2 - This is an intensional definition, because swanness is an intensional definition.

In summary, in a language game before performative utterances, sets of concrete and abstract elements can be neither extensional nor intensional definitions, but within a language game, performative utterances can create extensional and intensional definitions

Possible world 8, Tolkein's Middle Earth

“Creatures who walked into Mordor” = {Frodo, Samwise} was a performative rather than constative utterance by Tolkein.

Therefore, it is not an extensional definition, because the set does not include every element that falls under its definition. I am sure other creatures than Frodo and Samwise walked into Mordor.

Neither is it an intensional definition, because although Tolkein tells us that Frodo and Samwise necessarily walked into Mordor, that Frodo and Samwise walked into Mordor is not sufficient to the truth of the expression “creatures who walked into Mordor”.

Question

Before any performative utterance by an authority, X = {this swan in Hyde Park, that swan on the Thames, those swans on the Serpentine}.

Does X refer to the set of elements or does it refer to the family resemblance between the elements, ie, does it refer to the elements as part of the domain of X?

That's a useless and baseless assertion if I've ever seen one. — Metaphysician Undercover

Dude. I could resurrect Frege and transport him to your house to explain to you what an abstract object is and you still would maintain some other baloney you made up.

It is those additions which introduce subjectivity. The subjectivity being the intentional products of the mind which enter due to the variance in purpose, and are allowed to contaminate judgement, rendering "truth" as fundamentally subjective. — Metaphysician Undercover

In our world, the proposition “pigs cannot fly” is true. This is an objective fact. My judgement that “pigs cannot fly” is not a subjective judgement.

Modal logic K developed by Kripke introduced the concepts necessary and possible. He introduced possible world semantics, not just any possible world but accessible possible worlds.

What are accessible possible worlds?

Intuitively, an unknown world cannot be an accessible possible world.

Not “all” possible worlds are accessible, because some worlds will be unknown to us.

I could say that possible world 5 is accessible because it follows the logic of our world, such that it is possible in world 5 that “pigs can fly” is true. Or I could say that possible world 5 is accessible because it follows the natural laws of our world, such that “pigs can fly” is false but “pigs can vote” is true.

Suppose I use the model that a possible world is accessible because it follows the logic of our world. Then in possible world 5, pigs can fly.

Then in possible world 5 the proposition “pigs can fly” is true is not a subjective judgement, because in possible world 5 pigs can fly, which is an objective fact within possible world 5.

(I am willing to be corrected about my knowledge of modal logic).

I think the answer is: extensionally, yes; intensionally no, not until an utterance is performed.

"....a Tarskian interpretation I for ℒ specifies a set D for the quantifiers of ℒ to range over (typically, some set of things that ℒ has been designed to describe) and assigns, to each term (constant or variable) τ of ℒ, a referent aτ ∈ D and, to each n-place predicate π of ℒ, an appropriate extension Eπ — a truth value (TRUE or FALSE) if n = 0, a subset of D if n = 1" - SEP

My understanding of the above text is that predicating "swan" will refer to some subset in the domain (of all swans; that is, of all the things that conform to the predication).

Mathematicians are often inclined to do this with equality (=). They will say that "2+2" represents the same idea as "4". But this is clearly false because there is an operator "+" within "2+2", so obviously it cannot be the same idea as "4". This is why it is best for good philosophy, to maintain a very clear distinction between identity and equality. Equality is a relation between two individuals within a category (kind). You and I as human beings are equal. But identity is unique to an individual. — Metaphysician Undercover

This is the confusion that underpins Meta previously not accepting that 0.9̈ = 1, and rejecting instantaneous velocity; indeed, in his not understanding limits, generally. He confuses what is represented with the representation.

2+2 and 4 are different expressions for the same number. The "=" is used to express this. Hence we can write

• 2+2=4
• Hesperus=Phosphorus
• 0.9̈ = 1
• Superman=Clark Kent

The claim that equality is only a relation “within a kind” (like moral or political equality) equivocates between normative or comparative equality (you and I are equal as citizens), and mathematical identity (2 + 2 = 4). Put simply, folk do differentiate normative equity and identity. We recognise a difference between two citizens being equal and two numbers being equal.

How does this relate to Meta's misunderstanding of modal logic? We can have different descriptions of the very same object. Meta seems to think that if we have different descriptions, we must thereby have different objects. Hence his insistence that when we consider what it might have been like if Nixon had not won the 1972 election, we cannot be talking about Nixon. Hence his rejection of cross-world identity.

Now there are philosophical issues here, to be sure. But while Meta insists that we cannot have different descriptions of the same thing, he cannot address these other issues.