Maybe the idea of compossibility is relevant to this discussion. Wikipedia describes compossibility as follows:

"According to Leibniz, a complete individual thing (for example a person) is characterized by all its properties, and these determine its relations with other individuals. The existence of one individual may negate the possibility of the existence of another. A possible world is made up of individuals that are compossible—that is, individuals that can exist together."

So it seems the existence of Nixon with a different possible set of properties would be not compossible with the same Nixon with other actual properties in a given possible world; they could not both exist therein. But one of them can exist therein.

Notice that "both" Nixons would be different "individuals" by Leibniz's definition even though each refers to the same person as that person is rigidly defined.

Again, I think the key is that Nixon's other properties are just possible properties and that being the case, there is no contradiction with them being alongside his actual properties. The fact that Metaphysician Undercover talks about them as if they were other actual properties introduces a problem that is not really there.

Maybe the idea of compossibility is relevant to this discussion. — NotAristotle

Interesting.

There's a difference between characterising a thing and referring to it. There was a lively discussion about this in the middle of the last century...

Some philosophers had supposed that there were no individuals, only collections of properties. A name, they supposed, referred only in virtue of those properties - it was called "the description theory of reference". A few good arguments put paid to the - it's now very much a minority opinion.

And example might help here. Supose that all we know of Thales is that he was from Miletus and claimed that every thing was water. Then on the description theory, "Thales" refers to whomever is the philosopher from Miletus who believed all was water.

But supose that in some possible world, Thales went into coopering, making barrels of all sorts, and never gave a thought to ontology. But some other bloke, also from Miletus, happened to think that everything was made of water.

Then, by the description theory, "Thales" would not refer to Thales, but this other bloke. — Banno

Will we say that Thales was a Cooper? I think that a better account than calling some other bloke "Thales" just because he went into doing philosophy.

Again, I think the key is that Nixon's other properties are just possible properties and that being the case, there is no contradiction with them being alongside his actual properties. — NotAristotle

Yep.

The fact that Metaphysician Undercover talks about them as if they were other actual properties introduces a problem that is not really there. — NotAristotle

Yep. I've pointed out elsewhere that Meta confuses metaphysics and logic in this way.

You don't appear to be available for learning at the moment. — frank

Yes, I think I'm more in the mood for calling out bullshit than for learning.

Twaddle. Both sentences are about Nixon. The same Nixon in two different worlds, each of which is evaluated extensionally without contradiction. The basic modal view that you have not understood. — Banno

It appears I went through weeks of discussion with you in the other thread, where we hammered out the difference between referencing the metaphysical world, and referencing the modal world, to no avail. Do you have an extremely short memory? Please remember the distinction we made between what "Nixon" refers to in the real, independent metaphysical world, and what "Nixon" refers to in the modal model. Or were you just pretending to understand in that other thread?

Are you suggesting that a definition of red things that includes all red things is circular? You want a definition that leaves some of them out? — Banno

No, I want an intensional definition, because a purely extensional definition is a free-floating self-referential definition.

Notice that "both" Nixons would be different "individuals" by Leibniz's definition even though each refers to the same person as that person is rigidly defined. — NotAristotle

Yes, I think you could say that, that they must be different individuals, but I prefer to think that they are not even individuals at all. They are just conceptual structures, ideas, descriptions without any real thing being described. This is the issue of Platonism. Is an abstraction an object, or is it something else. In the possible worlds context, we can ask whether the ideas which the symbols refer to are properly called individuals or not. What is referenced is ideas, not physical things or individuals.

Again, I think the key is that Nixon's other properties are just possible properties and that being the case, there is no contradiction with them being alongside his actual properties. The fact that Metaphysician Undercover talks about them as if they were other actual properties introduces a problem that is not really there. — NotAristotle

I don't see what you are saying. Having possible properties along side actual properties is a problem. That's why in modalism they must all be modeled as possible. As Banno argued in the other thread, we can stipulate that some properties are actual and give them special status in this way, by stipulation, but that does not mean that we are talking about a real independent individual named "Nixon". It is all still modal, conceptual, and we must maintain the separation between having the words reference ideas, and having words that reference physical individuals.

It appears I went through weeks of discussion with you in the other thread, where we hammered out the difference between referencing the metaphysical world, and referencing the modal world, to no avail. — Metaphysician Undercover

The one were you repeatedly conflated metaphysics and semantics? I remember it well. You are making the same mistake here. We can plainly talk about what the world would be like were Nixon not re-elected, without thereby committing ourselves to supposing that he had indeed in the actual world not been re-elected.

It's such a simple point. You astonish me.

We can plainly talk about what the world would be like were Nixon not re-elected, without thereby committing ourselves to supposing that he had indeed in the actual world not been re-elected. — Banno

Obviously, and I agreed.

The one were you repeatedly conflated metaphysics and semantics? I remember it well. You are making the same mistake here. — Banno

You have a very strange form of straw manning, in which you project your own errors on to someone else. You equivocate, and blame the interpreter for not being able to distinguish the different meanings you give to the same word. Interesting psychology.

The basic point of extensionality is substitutivity. — frank

As says, not according to the SEP article that is being discussed. Indeed the article literally implies that extensional logic need not have substitutivity principles at all (my bolding):

An extensional logic will thus typically feature a variety of valid substitutivity principles. — Menzel

@Metaphysician Undercover's complaint that little attention is being paid to the SEP article is understandable.

You have a very strange form of straw manning — Metaphysician Undercover

Do I?

Please remember the distinction we made between what "Nixon" refers to in the real, independent metaphysical world, and what "Nixon" refers to in the modal model. — Metaphysician Undercover

These are both Nixon. The Nixon who did not get elected is not a different Nixon to the one who was. They are the very same fellow, but under different circumstances.

When asks what things might be like were Nixon not elected, he is not asking about some other fellow. Not one, who happens to be actual, and another, who is imagined.

That, so far as I can make out, is your mistake.

These are both Nixon. The Nixon who did not get elected is not a different Nixon to the one who was. They are the very same fellow, but under different circumstances. — Banno

Uh-hu, tell me another one bro. Can you tell me how I can get myself into some of these different circumstances? I want the one where I'm the same fellow who won the lottery.

Is it difficult for you to understand that we're not talking about a fellow at all, we're talking about a complex concept?

Let's try for clarrity, again.

As I explained previously, in the SEP article, extension has a narrowly defined, technical meaning:

• The extension of a predicate is the set of objects that satisfy it.
• The extension of a name is the object it refers to.
• The truth of a formula is defined purely in terms of these extensions.
• This is the sense in which Tarski’s semantics is extensional:
• truth is a matter of extensions only.

Importantly, this definition makes no reference to substitution.

In logic, extensionality is standardly understood syntactically:
If two expressions have the same extension, then one may be substituted for the other in any sentence without changing its truth value.
This yields:

• co-referring names are intersubstitutable
• co-extensional predicates are intersubstitutable
• logically equivalent formulas are intersubstitutable

This is the working notion of extensionality in FOL and classical semantics and in the working within the article.

These two ways of understanding extensionality are not at odds.

This is exactly the problem which the extensionality of "possible worlds" produces. It creates the illusion that we are talking about a bunch of different worlds, similar to the world which we actually live in, full of fellows and other things with describable properties. This might mislead the naive. In reality we are not talking about any worlds, or fellows, or things like that, we are talking about conceptual possibilities

This is exactly the problem which the extensionality of "possible worlds" produces. It creates the illusion that we are talking about a bunch of different worlds, similar to the world which we actually live in, full of fellows and other things with describable properties. This might mislead the naive. In reality we are not talking about any worlds, or fellows, or things like that, we are talking about conceptual possibilities — Metaphysician Undercover

What we can do is note this warning and proceed with the article. Is that ok with you?

I want the one where I'm the same fellow who won the lottery. — Metaphysician Undercover

Odd. Who is "...the one where I'm the same fellow who won the lottery" about, if not you??

Basic grammar.

Yep. Will do.

So to sum up to this point,

1. The ideas of necessity and possibility intuitively conjure possible worlds.

2. The obstacle to addressing modality in predicate logic was that there was no recognized path to extensionality. In other words, Tarskian semantics could handle:

BASIC is one of John's pets.

but not:

BASIC is necessarily one of John's pets.

By defining necessity as f(x) being true in all possible worlds, we have extensionality within possible worlds, but not across them. In this case, necessity is a quantifier. It's telling us how many.

So that gives us a brief history. This is the SEP's summary:

Summary: Intensionality and Possible Worlds. Analyzed in terms of possible world semantics, then, the general failure of classical substitutivity principles in modal logic is due, not to an irreducibly intensional element in the meanings of the modal operators, but rather to a sort of mismatch between the surface syntax of those operators and their semantics: syntactically, they are unary sentence operators like negation; but semantically, they are, quite literally, quantifiers. Their syntactic similarity to negation suggests that, like negation, the truth values of ⌈□φ⌉ and ⌈◇φ⌉, insofar as they are determinable at all, must be determined by the truth value of φ. That they are not (in general) so determined leads to the distinctive substitutivity failures noted above. The possible worlds analysis of the modal operators as quantifiers over worlds reveals that the unary syntactic form of the modal operators obscures a semantically relevant parameter. When the modal operators are interpreted as quantifiers, this parameter becomes explicit and the reason underlying the failure of extensionality in modal logic becomes clear: That the truth values of ⌈□φ⌉ and ⌈◇φ⌉ are not in general determined by the truth value of φ at the world of evaluation is, semantically speaking, nothing more than the fact that the truth values of ‘∀xFx’ and ‘∃xFx’ are not in general determined by the truth value of ‘Fx’, for any particular value of ‘x’. Possible world semantics, therefore, explains the intensionality of modal logic by revealing that the syntax of the modal operators prevents an adequate expression of the meanings of the sentences in which they occur. Spelled out as possible world truth conditions, those meanings can be expressed in a wholly extensional fashion. (For a more formal exposition of this point, see the supplemental article The Extensionality of Possible World Semantics.) — ibid

Fascinating stuff. @Banno Do you agree and do you have anything to add?

:up: :wink:

It's hard to grasp the counterarguments here, but perhaps they do think in terms of "an irreducibly intensional element in the meanings of the modal operators". But that wrinkle has been smoothed over by a bit of brilliance form Kripke and others.

The supplement adds a bit of detail. It also gives a neat sumamtion fo the structure here:

• Worlds (World(w)),
• Truth at a world (T(φ, w)),
• Domains of worlds (dom(w)),
• Extensions of predicates at worlds (ext(π, w)),
• Denotations of terms (den(τ)),
• And a designated actual world (@).

It's a bit of a triumph.

To be sure, possible world semantics doesn’t make the modal object language extensional (modal substitutivity still fails), but the semantic theory that defines the truth conditions of the modal language is extensional because it is written in a fully extensional first‑order logic.

And this stuff is not easy, so if you have followed so far, give yourself some credit.

It's a bit of a triumph. — Banno

It's pretty cool. I think it does capture some of what's going on when we talk about possibility For instance, I want to be able to say that if Nixon lost, x and y would be true. That would part is a modal auxiliary verb (modifying be). Not all languages have a separate word that serve that function. For them modality can be conveyed by borrowing from other linguistic functions like capability and desire. Also in English, there's a close connection between would and should. English modal auxiliary verbs

I can understand objections related to ontology, but it's better to get the basics straight before moving onto that.

English modal auxiliary verbs — frank

Yep. English and Germanic language might lend themselves to these formalisations, perhaps, which is not a surprise since the formalities were mostly done by German and English speakers. Not sure if this is structural or cultural.

And in a similar way to English, there are variants of modal logic that apply possible world semantics quite broadly - deontic and temporal logics for a start, and indexicals.

The next section is quite interesting. It gives the formal definition of intension.

The next section is quite interesting. It gives the formal definition of intension. — Banno

:up:

What we can do is note this warning and proceed with the article. Is that ok with you? — frank

OK, so here's the warning from the SEP

Possible world semantics, therefore, explains the intensionality of modal logic by revealing that the syntax of the modal operators prevents an adequate expression of the meanings of the sentences in which they occur. Spelled out as possible world truth conditions, those meanings can be expressed in a wholly extensional fashion. — ibid

And as I explained, extensional definitions have the fundamental problem of being self-referential.

Odd. Who is "...the one where I'm the same fellow who won the lottery" about, if not you??

Basic grammar. — Banno

Jesus Banno! Did you not take English in school? The subject of that phrase is "the one", and this refers to the "circumstances". The phrase is about that set of circumstances, not about me.

See what I mean about your unusual straw man habits? You take your own error (faulty grammar in this case), and project it onto the other person in a false representation, as if it is the other person's error.

1.3 Two Applications: The Analysis of Intensions and the De Re / De Dicto Distinction

@Banno
I was wondering if you could give an example of what they're talking about here?:

More specifically, as described above, possible world semantics assigns to each n-place predicate π a certain function Iπ — π's intension — that, for each possible world w, returns the extension Iπ(w) of π at w. We can define an intension per se, independent of any language, to be any such function on worlds. More specifically:

A proposition is any function from worlds to truth values.
A property is any function from worlds to sets of individuals.
An n-place relation (n > 1) is any function from worlds to sets of n-tuples of individuals. — ibid

And as I explained, extensional definitions have the fundamental problem of being self-referential. — Metaphysician Undercover

At this point, if I were to try to summarize your view back to you, I wouldn't know what to say. I have no idea what you're trying to express.

The concept of an Infinite extension can only be circurarly defined, because in that case all that we actually have is an intensional definition of a sequence in the form of a self-looping algorithm whose output is a finite extension of of random length decided by the user of the algorithm.

Possible world semantics, while used for denoting possibilia, (i.e. state relative actualia via an overspill expansion of the domain of the quantifiers), has no notion of dynamics or interaction, that is necessary for understanding the language-game of possibility.

As a static set-theoretic model. possible world semantics can describe a game tree, but not the execution path of a given game, or the prior processes of interaction by which a game tree emerges.

By defining necessity as f(x) being true in all possible worlds, we have extensionality within possible worlds, but not across them. In this case, necessity is a quantifier. It's telling us how many. — frank

Does the following make sense:

In possible world 5 - a chess set = {64 squares, made of stone}
In possible world 6 - a chess set = {64 squares, made of ivory}

But some of these properties may be necessary and some may be contingent.

But chess has to be defined.
Therefore, ☐ ∃x(B(x)), where x is the subject “a chess set”, and where B is the predicate “has 64 squares”
Then, it is necessarily the case that the proposition “a chess set has 64 squares” is true.
Therefore, having 64 squares is necessary.
Therefore, the proposition “a chess set has 64 squares” is true in all possible worlds.

But this definition says nothing about material.

Therefore ◊∃x(B(x))
Then, it is possible that a chess set is made of ivory.
Therefore, being made of ivory is contingent
Therefore, being made of ivory is possibly true in some possible world.

As you say, i) defining necessity as being true in all possible worlds and where ii) necessity is a quantifier (meaning “all”).

But is it not the case that:
1 - We have intentionality across all possible worlds (because necessary meaning is an intension and the necessary meaning is the same across all possible worlds)
2 - We have extensionality within each possible world (because contingent properties are an extension and contingent properties are particular to each possible world).