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.

If the claim means that in the actual world, no individual satisfies the description “Frosty the Snowman”, and this holds in every accessible world, then this is a claim about the emptiness of a predicate, not about the non-existence of a named individual. (de dicto reading) — Banno

The de dicto reading makes more sense to me. If I am tracking the terminology correctly, the question I am running into is: if the predicate is an empty set, doesn't that mean there is no extension for that predicate? But if there is no extension, can we refer to anything intensionally? :chin:

To try to reformulate the issue: if there is not something referred to, how can the sentence have a truth value? And yet, it appears to be true.

That some statements about the actual world are objective facts doesn't mean that all are.
From what I see, you've just demonstrated the subjectivity which I referred to.

2+2 and 4 are different expressions for the same number. The "=" is used to express this. — Banno

The axiom of extensionality makes a statement about equality. You can interpret this as a statement of identity if you want. But as I've demonstrated many times in this forum, that is not a very good approach philosophically, as it produces a violation of the law of identity.

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. — Banno

Again, this is your terrible straw man habit. The issue with modal logic we have been discussing, is the notion that the description is the object. "Frodo has a ring" is a description, and you want to interpret it as an object. You said :

We can move on to first order logic. Since Frodo walked in to Mordor, we can conclude that Something walked in to Mordor. This is an instance of the rule of Existential Generalisation. Formally, it's fa → ∃x(fx) — If a is f, then there is an x such that x is f.

Have we proved, by this, that Frodo exists? Not at all. We introduced Frodo when we set up the Domain of Middle Earth. His existence is not a consequence of our deductions, but a presumption or stipulation.

The domain is in a sense a list of the things we are talking about. In first order logic and basic modal logic it is static. (There are variable-domain modal logics.) — Banno

Obviously, when I say "Banno is fool", this does not necessitate the conclusion that there is an existing person called Banno.

No one yet has addressed the quote which I brought this morning, from the SEP article we are reading. On the conditions for truth, it is stated as required, that " (ii) its designated “actual world” is in fact the actual world".

Now you'll have to excuse me, I need to go get ready for Santa Clause, who must be a real existing person because people can describe him.

Now you'll have to excuse me, I need to go get ready for Santa Clause, who must be a real existing person because people can describe him. — Metaphysician Undercover

Nobody tell him; let him have this one

The de dicto reading makes more sense to me. — NotAristotle

Yep. they are generally clearer because they do not involve necessary or possible properties, but propositions.
If the predicate is an empty set in some world then yes, there is no extension for that predicate in that world. Consider "spotted penguin" - there are none in the actual world but they are not impossible. But is it empty in every possible world? If so, then it's necessarily empty, and unlike the spotted penguin there can be no such thing. Consider "Four-sided triangle".

1 - This is not an extensional definition, as the set does not include every element that falls under the definition. — RussellA

Yes, but there's a bit more. It's also intensional as it sets out the conditions under which something is a swan, not a list of the swans. I guess properly we should write x:x is white ∧ x is flighted ∧ x is a waterfowl.

And again, this line of enquiry is about kinds, not individuals. It's slightly different to what is being dealt with in the article. So consider: there are white ducks; and juvenile swans do not fly.

The axiom of extensionality makes a statement about equality. You can interpret this as a statement of identity if you want. But as I've demonstrated many times in this forum, that is not a very good approach philosophically, as it produces a violation of the law of identity. — Metaphysician Undercover

You think you have. You are mistaken.


In standard set theory (ZF, ZFC), the Axiom of Extensionality is that

∀x ∀y (∀z (z ∈ x ↔ z ∈ y) → x = y)

Here, “=” is identity. There is no weaker or alternative relation intended. Sets have no identity conditions other than their members. To deny that “=” here expresses identity is to deny that sets are individuals at all. So Meta’s attempt to treat extensional equality as something other than identity is not merely philosophically optional — it is incompatible with orthodox set theory.

The law of identity is x = x. Interpreting extensional equality as identity cannot violate this law.
On the contrary, if two sets have the same members, they are the same set. This enforces identity, it does not undermine it. Meta seems to think that because two sets can be described differently (or constructed differently), treating them as identical violates identity. That is the same representation/referent confusion diagnosed earlier. So the alleged violation does not exist.

Fair.

To deny that “=” here expresses identity is to deny that sets are individuals at all. — Banno

Sets are not individuals, they are ideas. The unity required for "individual" is simply assumed by the subject, in the case of a set. That's why I criticized your use of "..." in your description of a set. That indicates the lack of a clearly defined boundary, required for a unit. And what about the empty set? What kind of individual is composed of nothing? Sets are not in any way objective individuals. They may amount to objective ideas, in the sense of inter-subjective objectivity, but the unity required for "individual" is simply not there.

That's the reason for Aristotle's law of identity, to hold a separation between real individuals (substance), and ideas, which were held by Platonists to be objects, or individuals. The separation was intended to resist the sophistry which was derived from treating ideas as if they are real existing things, the sophistry which you displayed with your example of Frodo.

I think we've given you enough rope.

The De Re / De Dicto Distinction.

Pretty straightforward. It's a distinction that caused much confusion historically. It dissipates in modal logic, with a small ghost that might be summarised in terms of the scope of the modal operator. perhaps the main issue is that it was unclear exactly what the de dicto/de re distinction was. See The De Re/De Dicto Distinction for more on the history here. Stealing Quine's example from there, "Ralph believes that someone is a spy" is ambiguous between Ralph's believing that there are spies, and Ralph's believing of someone that they are a spy. The first is de dicto, since Ralph's belief is about the sentence "there are spies". The second is de re, since Ralph's belief is about someone.

Very roughly, if the operator has a sentence in its scope, it's de dicto - about the sentence. If it has only a thing or its properties in its scope, its de re. More formally, in quantified modal logic:

• De dicto: □ ∀x φ(x) → the quantifier is inside the modal operator
• De re: ∀x □ φ(x) → the quantifier ranges over individuals outside the modal operator

So in our target article, "Necessarily, Algol is a dog" is de re, being about Algol, while "Necessarily, All dogs are mammals" is de dicto, being about a sentence.

The benefit of formalisation here is that it makes explicit what is going on, an improvement over older approaches to de dicto and de re.

"Necessarily, Algol is a dog" is understood as saying that, in every world in which Algol exists, Algol is a dog. It thereby presumes that Algol exists in multiple possible worlds, that is, it presumes transworld identity. Hence,

(i) permitting world domains to overlap and (ii) assigning intensions to predicates, thereby, in effect, relativizing predicate extensions to worlds. In this way, one and the same individual can be in the extension of a given predicate at all worlds in which they exist, at some such worlds only, or at none at all.

I think we've given you enough rope. — Banno

Sorry Banno, it doesn't work that way. You have to actually hang me.


I will add the following to your description of the de re/de dicto distinction.

It is mentioned in the SEP article, "the truth conditions for sentences exhibiting modality de re involve in addition a commitment to the meaningfulness of transworld identity". This, as I explained above, is supported ontologically by Platonism, and requires a violation of the law of identity. That is why the SEP says:

The subject of transworld identity has been highly contentious, even among philosophers who accept the legitimacy of talk of possible worlds. Opinions range from the view that the notion of an identity that holds between objects in distinct possible worlds is so problematic as to be unacceptable, to the view that the notion is utterly innocuous, and no more problematic than the uncontroversial claim that individuals could have existed with somewhat different properties. Matters are complicated by the fact that an important rival to ‘transworld identity’ has been proposed: David Lewis’s counterpart theory, which replaces the claim that an individual exists in more than one possible world with the claim that although each individual exists in one world only, it has counterparts in other worlds, where the counterpart relation (based on similarity) does not have the logic of identity. Thus much discussion in this area has concerned the comparative merits of the transworld identity and counterpart-theoretic accounts as interpretations, within a possible-worlds framework, of statements of what is possible and necessary for particular individuals. (Similar issues arise concerning the transworld identity of properties.) — SEP

You are repeating the same assertions already shown to be false, and then quoting arguments that are based on the stuff you claim to have disproved...

Yes, but there's a bit more. It's also intensional as it sets out the conditions under which something is a swan, not a list of the swans. I guess properly we should write x:x is white ∧ x is flighted ∧ x is a waterfowl. — Banno

Does my understanding make sense?

Am I right that:
1 - An extensional definition must include everything that falls under the definition.
2 - An intensional definition must include everything that is necessary and sufficient for the definition.

On my walks, I observe the set {this swan in Hyde Park, that swan on the Thames, those swans on the Serpentine}.

I am also thinking about Wittgenstein and JL Austin.

I perceive that the elements of this set have a family resemblance.

I cannot describe this family resemblance, but I can name it X, such that the family resemblance between the elements of the set is X.

If I were the King, I could make it a law of the land such that X was henceforth given the name “swanness”, along the lines of a JL Austin performative utterance. Then “swanness” becomes the official name of the family resemblance between the elements of the set {this swan in Hyde Park, that swan on the Thames, those swans on the Serpentine}

Note that “swanness” does not refer to the elements of the set {this swan in Hyde Park, that swan on the Thames, those swans on the Serpentine}, but refers to the family resemblance between the elements of the set {this swan in Hyde Park, that swan on the Thames, those swans on the Serpentine}.

As regards an intensional definition, the intensional definition of “swanness” is the family resemblance between the elements of the set {this swan in Hyde Park, that swan on the Thames, those swans on the Serpentine}.

Therefore:
1 - On the one hand, I know in my mind that there is a family resemblance. The public word for this can be “swanness”, which I can use in my daily life.
2 - On the other hand, even though I know in my mind that there is a family resemblance, I cannot put my knowledge into words.
3 - Therefore, I can use words such as “swanness” in a public language because I know what “swanness” means, even though I cannot put my knowledge of what “swanness” means into words.

You are repeating the same assertions already shown to be false, and then quoting arguments that are based on the stuff you claim to have disproved... — Banno

It seems to me, that the quote supports what I've been arguing very well.

"The subject of transworld identity has been highly contentious, even among philosophers who accept the legitimacy of talk of possible worlds." — SEP

Possible worlds semantics is a piece of Platonist ontology. And, Platonist ontology violates the law of identity. So, many philosophers do not accept the legitimacy of "possible worlds".

Further, the quote indicates that even among those philosophers who accept the Platonist "possible worlds", many believe that extending the violation of the law of identity from the identity handed to possible worlds , to the "transworld identity" of the individuals within, is so problematic as to be unacceptable.

In other words, once we allow the initial violation of the law of identity, by accepting Platonism, we open a pandora's box of unacceptability. Each step we take within this domain, where imaginary things have identity, takes us further and further from the domain of demonstration. So the things we say become more and more controversial because they cannot be demonstrated. This is very important and significant because the objectivity of logic is derived from convention, agreement, inter-subjectivity. Therefore unacceptability is the most detrimental and destructive thing to logic, leaving it as waste. For a logician to forge ahead with principles which have been declared by philosophers to be unacceptable, is a mistaken venture, a waste of time.