The words "necessarily" and "possibly" do not denote extensional sets. — Leontiskos

Can you say what you mean by this?

Do you mean a sentence with these terms cannot have a truth value, or do you mean they fail at substitutivity always?

...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.

So with Nixon winning the election, the domain is

{Nixon}

W(x) = won the election
L(x) = lost the election

W(Nixon) is true.

But in the possible world(s) in which he lost the election,

L(Nixon) is true.

Is that right?

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.

- So substitution fails across worlds, not because modal logic is intensional,
but because predicate extensions vary from world to world. — Banno

:up: :up: :up:

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. — Banno

That is contrary to what the SEP article states. Modal logic is intensional. And, it is only the expression of it, the interpretation of separate "possible worlds", which produces extensionality. There is no extensionality between possibilities because possibilities are inherently imaginary. It is only by assigning distinct "worlds", ("domains" or whatever you wish to call them), each with its own rules of extensionality, that the illusion of extensionality is artificially created.

However, the rules of extensionality cannot extend from one supposed "world" to another, to provide for the semantic modality of "possible". Therefore only intensionality relates the distinct "worlds" because the fact is that modal logic which relates possibilities is inherently intensional. This intentionality is described by you as "exactly what necessity and possibility require". Notice that the structure is ultimately designed to accommodate the intensional meaning, what necessity and possibility "require", rather than an extensional reality.

Your objection against possible worlds makes sense to me. Extensionality seems to require a referent of some sort; and I am not sure the article has defended any such referent up to now.

Kripke postulates "rigid designators," I think. So if Nixon is the referent of the term "Nixon" in any given possible world, maybe that alone solves extensionality without having to worry about the existence of "possible worlds." What do you think?

That is contrary to what the SEP article states. Modal logic is intensional. And, it is only the expression of it, the interpretation of separate "possible worlds", which produces extensionality. — Metaphysician Undercover

Could you quote the passage you're referring to here?

predicate extensions vary from world to world. — Banno

And consequently sentence extensions; that is, truth value, also varies across worlds.

1.1 -> "Modal logic, by contrast, is intensional."

But i think Banno was not making the claim that modal logic is not intensional.

His point was that the intensionality of modal logic is irrelevant to the fact that possible world semantics establishes extensionality by predicates having different individuals in their domains depending on the possible world, and that it is this difference that defeats substitutivity for modal logic. At least I think that is correct.

I reject your definition as completely different from the one in the article we are supposed to be reading, which I quoted above. Taking a definition from a different context is not helpful, only a distraction or a deliberate attempt at equivocation. — Metaphysician Undercover

Yep. :up:

---

The words "necessarily" and "possibly" do not denote extensional sets. — Leontiskos

Can you say what you mean by this? — NotAristotle

Modern logic is built on the foundation of set theory (cf. SEP - Emergence of First-Order Logic), and therefore its ability to translate and track natural reasoning depends on how closely the meaning of any given natural reasoning coheres with set theory.

So we begin with something as simple as set inclusion, with things like this:

x ∈ B (x is an element of the set B)
x ⊆ B (x is a subset of the set B)

A set such as B involves quantitative extension. For example, B might be the set of three things: {x, y, z}.

Quantificational logic builds on this foundation by envisioning everything, at bottom, as a form of set membership. So if we have a natural language statement such as, "There is a green frog," then quantificational logic will try to translate it as follows:

∃x(G(x) ∧ (F(x))

...Which, commonly expressed, means, "There exists an x such that x is green and x is a frog." But everything here is based on set theory. A predicate such as G(x) will be true precisely when x is an element in the set defined by the predicate G. Similarly, the quantifiers (∃ and ∀) are also founded on set theory, where the quantifiers are intended to "quantify over" all "things" (or elements of the domain one is speaking to). They are statements about the domain conceived as a set.

Modal logic builds on this foundation by thinking of natural language modal terms ("necessarily" and "possibly") basically as extensional sets (cf. SEP - Modern Origins of Modal Logic). So if we have a natural language statement such as, "It is possible that there is a black frog," then modal logic will try to translate it as follows:

◊∃x(B(x) ∧ F(x))

The meaning here is similar to the first-order sentence above, except that the possibility operator is included (◊). Possibility and necessity operators in modal logic (◊ and □) are conceived along the lines of quantifiers, and are built on the same foundation of set theory that quantifiers are built on. So, simplifying a bit, the formal modal logic sentence about the black frog envisions a set which contains all possible things, and it affirms that the "existence" of a black frog belongs to that set of all possible things (i.e. possible "worlds"). So everything is reducible to set theory: the functions (B and F), the existential quantifier (∃), and the modal quantifier (◊).

Now this is a simplification, for set theory, first-order logic, and modal logic all took a long time to develop, and this time was used to iron out all sorts of wrinkles. But the gist remains true, namely that set theory is the foundation for all of these forms of modern logic. In a sense logicians did this because once set theory was developed, if one could make other forms of discourse conform to set theory then the developed power of set theory could be applied to boolean logic, or predicates, or modal notions, etc. It's a bit like an engineer who engineers an engine for a train, and then tries to simply modify that engine for all other purposes rather than creating a new kind of engine for other things. It is also closely related to the desire to extend the precision of mathematics to all human reasoning (which is an old error that Aristotle points out explicitly in his Metaphysics). But let's come back to your question.

When someone says, "It is possible that there is a black frog," they are actually not saying the equivalent of ◊∃x(B(x) ∧ F(x)). When we talk about what is possible, we are not talking about a set, namely the set of all possible things. And when we talk about what is necessary, we are not talking about a set, namely the set of all necessary things. In natural language there are a million different shades of possibility and necessity, and the binary logic of set theory simply cannot capture the semantic nuance involved. Pigeonholing modal language into set theory can result in half-truths and partial representations, but to pretend it is semantically equivalent is a serious error. Now good logicians actually know this, but there are lots of bad logicians who either do not know it or else are not consistent in applying it. Those bad logicians have no understanding of the historical situatedness of modern forms of logic, and they will tend to try to interpret someone's claim as ◊∃x(B(x) ∧ F(x)) and pretend that there is no difference at all, even browbeating the person if they protest that they are not talking about sets, or that the set machine is inadequate to represent their claim.

Menzel is fairly clear about the equivocation that occurs when trying to shoehorn modal language into a quantificational apparatus:

The idea of possible worlds raised the prospect of extensional respectability for modal logic, not by rendering modal logic itself extensional, but by endowing it with an extensional semantic theory — one whose own logical foundation is that of classical predicate logic and, hence, one on which possibility and necessity can ultimately be understood along classical Tarskian lines. Specifically, in possible world semantics, the modal operators are interpreted as quantifiers over possible worlds... — Menzel, Possible Worlds (SEP)

The point is clear enough, "Modal logic is not extensional, but modern logicians endow it with an extensional semantic theory." Or as I said earlier, modern logicians pretend that modal terms are extensional because they have a pre-made extensional engine, and that engine can't power non-extensional reasoning.

The historical background for all of modern logic is late Medieval nominalism (cf. SEP - The Medieval Problem of Universals, by Gyula Klima). Basing logic on set theory is a quintessentially nominalistic move, and was already being anticipated by figures like Peter Abelard in the 12th century. Yet the Medievals always understood something that Moderns consistently fail to understand, namely that repurposing a formalization engine for logic has intrinsic limitations and problems (link). The reason the modern does not see this is because the modern period has to do with control over nature, following Francis Bacon, and a set-theoretic engine aligns well with that telos.

(Edit: As alludes to, embedded sub-forms of extensionality were introduced into modal logic as it evolved, primarily by Saul Kripke. There are questions about whether sub-extensionalities indexed to possible worlds are adequate to represent modal language, but there is also the fact that the super-structure and paradigm is all extensional. Given that extensionality is the water that the fish of the modern logician swims in, it is no coincidence that "possible worlds" are inherently conceived along the lines of quantitative, set-theoretic entities.)

Kripke postulates "rigid designators," I think. So if Nixon is the referent of the term "Nixon" in any given possible world, maybe that alone solves extensionality without having to worry about the existence of "possible worlds." What do you think? — NotAristotle

That doesn't really make sense. Since the properties of the thing named "Nixon" in this case, are different in the different possible worlds, we cannot say that there is a single referent, the subject is different in each different world. The "Nixon" in one world would not be the same person as the "Nixon" in another. There could be some semantic rules about the use of the name, making it a "rigid designator", but that does not constitute a referent.

Could you quote the passage you're referring to here? — frank

I have already, here:

I think your EDIT is the proper interpretation. It makes modal logic the subject of an extensional logic. Here's a quote from the referenced supplement at the end of 1.2:

"As noted, possible world semantics does not make modal logic itself extensional; the substitutivity principles all remain invalid for modal languages under (basic) possible worlds semantics. Rather, it is the semantic theory itself — more exactly, the logic in which the theory is expressed — that is extensional." — Metaphysician Undercover

Could you quote the passage you're referring to here? — frank

Modal logic, therefore, is intensional: in general, the truth value of a sentence is determined by something over and above its form and the extensions of its components. — Menzel, 1.1

Your quote of Menzel follows directly on this premise.

Note:

the central motivation for possible world semantics was to deliver an extensional semantics for modal languages — Menzel

For Menzel the whole issue has to do with attempting to turn something that is prima facie intensional into something that is extensional. This is also one of the basic lines of demarcation between "concretism" and "abstractionism." He presents possible world semantics as an attempt to "extensionalize" modal logic, or as an attempt to argue that the intensionality of modal logic is only apparent.

Since the properties of the thing named "Nixon" in this case, are different in the different possible worlds — Metaphysician Undercover

The same thing cannot have different properties at different times?

So we have multiple domains and interpretations. That gives us extension within worlds, but not across worlds.

I wonder, what do you make of the heading "1.2 Extensionality Regained"?

What do you think is going on in that section?

That is contrary to what the SEP article states. — Metaphysician Undercover

No, Meta. Let's go through it step by step.

1. What “extensional” means here A logic is extensional when:
To know whether a sentence is true, you only need to know the extensions (the things the predicates apply to).


Example in ordinary Tarskian semantics:

“...is John’s pet” has the extension {Algol, BASIC}.

So “Algol is John’s pet” is true just because Algol ∈ that set. Nothing else matters. That’s extensionality.

2. Why modal logic is intensionalModal logic contains operators like □ “necessarily” and ◇ “possibly.”
Now the truth of “□φ” does not depend only on what is true in the actual world. It depends on what happens in other worlds (other interpretations).
That is why modal logic is intensional.
We need more information than just the extension in the actual world.
This is exactly what the SEP says.

3. Kripke’s move: extensionality inside each world
Here is the key point Meta missed:
Even though modal logic is intensional globally, each individual world is fully extensional in the plain Tarskian sense.
Inside any world w

• The domain is fixed
• Predicate extensions are fixed
• Truth is evaluated purely extensionally, just like ordinary first-order logic

Example:
In world w₀: Ext(“John’s pet”) = {Algol, BASIC}, so at w₀: “Algol is John’s pet” is true extensionally.
In world w₁: Ext(“John’s pet”) = {BASIC} so at w₁: “Algol is John’s pet” is false extensionally.

But in both cases the evaluation rule is exactly the same.
This is why Frank is correct.

4. Why substitution fails across worlds
Meta insists that failure of substitution “proves” intensionality between worlds. But that is exactly the point of possible-world semantics:
Each world has its own extensions. Therefore substituting co-referential terms across worlds need not preserve truth.
That is not a problem — it is the definition of intensionality.
There is no “illusion” here.

Just different Tarskian interpretations, one per world.



So, while modal contexts are intensional overall, because the truth of □φ and ◇φ depends on more than what happens in the actual world, each possible world is internally extensional in the plain Tarskian way: a fixed domain and fixed extensions for predicates. The intensionality appears only when you compare worlds, because extensions can vary from one world to another.

Again, what must you make of the heading "1.2 Extensionality Regained"?

And consequently sentence extensions; that is, truth value, also varies across worlds. — NotAristotle

Yes.

extensionality lost = no referent for modal logic claims. "Necessarily all John's pets are mammals" is false, but because there is no extension to corroborate the falsity; that is, there is only Algo and Basic in the set satisfying the predicate "all John's pets" it appears that maybe the statement should be true. At least then substitution would be preserved.

However, I think "all John's pets" in the statement "necessarily all John's pets are mammals" means all John's possible pets. Thus the semantic interpretation of the modal claim. Extensionality regained = all John's possible pets are the referents of the expression "John's pets" in the statement "Necessarily all John's pets are mammals."

His point was that the intensionality of modal logic is irrelevant to the fact that possible world semantics establishes extensionality by predicates having different individuals in their domains depending on the possible world, and that it is this difference that defeats substitutivity for modal logic. At least I think that is correct. — NotAristotle

Yep. Modal logic uses the extensional definition of truth as satisfaction within a world. Strictly, it is the interpretation that varies form world to world, as that includes the different individuals. So if we compare w₀, in which we have {Algol, BASIC}, and with w₁ in which we have {Algol, BASIC, COBOL}, the difference in the domain shows itself in a difference in the interpretation of the predicate.

As we go on and fill the logic out we will find things that remain true across possible worlds.

Can I ask, how are you going with the jargon and use of letters in what I've had to say? Too much?

Yes - the ☐ quantifies over multiple worlds, including those in which John has other pets and the interpretation of "All john's pets" includes non-mammals.

Exactly right.

SO the logic restores extensionality in deciding truth.

negatory Banno, not too much, the lingo is :up: :up:

Cheers.

The same thing cannot have different properties at different times? — NotAristotle

We're talking about at the same time, in different possible worlds. If you start trying to describe the difference between one possible world and another as a difference in time (i.e. same object at a different time), you'll open a real can of worms.

That gives us extension within worlds, but not across worlds. — frank

Yes, but even the extension within worlds is artificial, because the worlds (possibilities) are imaginary.

1. What “extensional” means hereA logic is extensional when:
To know whether a sentence is true, you only need to know the extensions (the things the predicates apply to). — Banno

Right, now you're on board with the SEP definition. Notice "the things" which the predicates apply to. Traditionally these would be objects with an identity by the law of identity.

So “Algol is John’s pet” is true just because Algol ∈ that set. Nothing else matters. That’s extensionality. — Banno

If there is a thing called Algol, and it is John's pet, then it fulfils that extension. In the case of possible worlds, Algol can be an imaginary thing, a thing which does not have an identity by the law of identity. then the supposed "thing" is not even a thing. I suggest to you that this is a very significant matter.

. Why modal logic is intensionalModal logic contains operators like □ “necessarily” and ◇ “possibly.”
Now the truth of “□φ” does not depend only on what is true in the actual world. It depends on what happens in other worlds (other interpretations).
That is why modal logic is intensional.
We need more information than just the extension in the actual world.
This is exactly what the SEP says. — Banno

No it is not exactly what the SEP says about intension. It says that while extension establishes relations with things, intension provides the semantics which determines the extension. Please look again:

By contrast, the intension of an expression is something rather less definite — its sense, or meaning, the semantical aspect of the expression that determines its extension. For purposes here, let us say that a logic is a formal language together with a semantic theory for the language, that is, a theory that provides rigorous definitions of truth, validity, and logical consequence for the language.
...
In an intensional logic, the truth values of some sentences are determined by something over and above their forms and the extensions of their components and, as a consequence, at least one classical substitutivity principle is typically rendered invalid. — SEP

Here is the key point Meta missed:
Even though modal logic is intensional globally, each individual world is fully extensional in the plain Tarskian sense.
Inside any world w
The domain is fixed
Predicate extensions are fixed
Truth is evaluated purely extensionally, just like ordinary first-order logic — Banno

Like I explain above, the extensionality inside any world is fixed by intensionality. This is because a possible world may contain fictional, imaginary things. Therefore the extensionality is not fixed through reference to real things, it is fixed by semantics.

Meta insists that failure of substitution “proves” intensionality between worlds. But that is exactly the point of possible-world semantics: — Banno

Please don't misquote me. I have said nothing about substitution. You keep insisting that extensionality is about, or defined by substitution. In reality substitution is a logical consequence, relying also of intension.

Each world has its own extensions. Therefore substituting co-referential terms across worlds need not preserve truth.
That is not a problem — it is the definition of intensionality.
There is no “illusion” here. — Banno

Really? This is the definition of intension? You really need to pay closer attention to the reading instead of just assuming your preconceptions.

Again, what must you make of the heading "1.2 Extensionality Regained"? — Banno

As I explained, the extensionality regained is an artificial extensionality, produced intensionallly, rather than through reference to real physical things with an identity. That is required, because we need to allow that a possible world has imaginary, fictional things. Since we cannot rely on true extensions ("things the predicates apply to") in the imaginary world, the referents are really a semantical (intensional) recreation of extensionality.

Surprisingly good. Most of what you have said about first order logic is correct.

A few things. While it's true that historically, set theory proceeds first order logic stands independently of set theory, it would be more accurate to say that logically, set theory uses first-order language. Zermelo–Fraenkel set theory (ZF/ZFC), for example, is formulated in a first-order language with the single primitive symbol ∈, and uses first-order logic to express its axioms. Hence it depends on FOL for its syntax and proof system.

Hence

...therefore its ability to translate and track natural reasoning depends on how closely the meaning of any given natural reasoning coheres with set theory. — Leontiskos

is an bit of an over-reach. Even if a logic's semantics uses sets the meaning of natural language does not thereby become extensional. Indeed, we ought keep the intentional aspect of natural languages not found in extensional logics.

Modal logic is not built on set theory, and as we've been reading, it does not treat possibility and necessity as extensional sets. Possible-world semantics interprets □ and ◊ using relations between worlds, not by forming extensional sets whose members are propositions. But that's jumping ahead in the article.

Logicians understand that formal languages approximate modalities, but do not claim semantic equivalence with natural language.

The point is clear enough, "Modal logic is not extensional, but modern logicians endow it with an extensional semantic theory." Or as I said earlier, modern logicians pretend that modal terms are extensional because they have a pre-made extensional engine, and that engine can't power non-extensional reasoning. — Leontiskos

Not quite. Menzel states that the semantics is extensional, meaning it is a Tarskian model-theoretic semantics. This does not mean that modal operators are extensional, nor that modal language is reducible to sets, nor that modal reasoning becomes extensional. It simply means the model theory uses standard tools (sets, functions, relations). Logicians are not pretending that modal terms are extensional.

We might do well to keep in mind that what Menzel is presenting is standard, accepted logic and has been so for many years.

The same thing cannot have different properties at different times? — NotAristotle

Of course it can. Indeed, there are temporal logics that build on the framework of possible world semantics. See the semantics of the system TL

So we have multiple domains and interpretations. That gives us extension within worlds, but not across worlds. — frank

Yep.

Since the properties of the thing named "Nixon" in this case, are different in the different possible worlds, we cannot say that there is a single referent, — Metaphysician Undercover

This is what you said. But you presumably also agree that the same thing can have different properties over time. If the same thing can have different properties over time, then the same thing can have different properties and still be the same thing. Therefore, different possible attributes of Nixon can refer to the very same Nixon, as would be the case whether Nixon was actually fat or actually skinny.

EDIT: Or put another way, the fact that different possible Nixons have different properties does not render them different Nixons.