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. — Metaphysician Undercover

The claim that individuals in possible worlds might lose identity is false in standard semantics. The formal system already handles non-existence cleanly by having the individual absent from a predicate’s extension. That is, if it does not exist in w, then it is not int he domain of w.

(SEP) says that while extension establishes relations with things, intension provides the semantics which determines the extension. — Metaphysician Undercover

Modal logic is intensional: truth cannot be determined by reference in the actual world alone. But Tarski-style extensional semantics can be applied within each world. The intension of a term or predicate is a function from worlds to extensions, and this intension determines the extension in each world. Extensions still define truth inside a world, while intensions describe how extensions vary across worlds. Modal operators (□, ◇) are intensional because they quantify over extensions in multiple worlds. This is the account given in the SEP article.

the extensionality inside any world is fixed by intensionality — Metaphysician Undercover

In Kripke-style possible-world semantics, each world w has a domain of individuals, D(w),and extensions of each predicate: $\text{Ext}_w(P)$ Within that world, extensional truth is evaluated directly, exactly like Tarski semantics: $P(a) \text{ is true at } w \iff a \in \text{Ext}_w(P)$

Nothing "semantic" or "intensional" is needed inside the world. The evaluation is purely extensional.

So I'm afraid you are incorrect here, too.

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. — Metaphysician Undercover

Well, no. In formal Kripke semantics, extensionality inside a world is real and exact. Nothing “artificial” or “intentionally produced” is involved inside the world. Possible-world semantics does not care whether the individuals are “real” or “fictional", since the extension of a predicate in a world is always a well-defined set of individuals in that world. Intensions tell us how the extension changes across worlds, but inside each world, extensionality is fully Tarskian, such that the truth of a sentence depends only on the domain and the extension in that world. Intension is a tool for cross-world reasoning, not a replacement for extensional truth inside a world.

Yep. Rigid designation isn't mentioned in the article, but it 'drops out' of the explanation of domains. Very roughly there is a domain for each world, and we can add these together to form a domain of all the possible individuals. And what this means is that Algol is Algol in any possible world in which it exists. The same Nixon in multiple worlds.

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

John asked if Frosty the Snowman is a Christmas themed character.

The extension of "is a Christmas-themed character" is

{Santa Claus, Mrs. Claus, Reindeer (especially Rudolph), Snowmen (like Frosty), Elves, Belsnickel & Befana, The Grinch, Jack Skellington, Ebenezer Scrooge}

C(x) = "Is a Christmas-themed character."

C(Frosty the Snowman) is true.

It doesn't matter that Frosty the Snowman isn't real.

Yep.

:up:
I was poring over your example trying to get that right.

:grin: Cheers.

It's worth looking at the difference between the definitions of truth (satisfaction) for atomic sentences, negation, material conditional and universal quantification, in the Tarski account and in the possibel world accounts.

The difference is the same in each case.Consider negation. in Tarski:

A negation ⌈¬ψ⌉ is true-in-I if and only if ψ is not true-in-I.

...where "I" is the interpretation.

And for negation in possible worlds:

A negation ⌈¬ψ⌉ is true-in-M at w if and only ψ is not true-in-M in w.

...were w is some world and M is a possible world interpretation.

The "true-in-M at w if and only if" makes explicit that each is true at a world.

It's perhaps worth pointing out that while the list includes only atomic sentences, negation, material conditional and universal quantification, the whole of first-order logic can be defined therefrom.

And to this we can now add

A necessitation ⌈◻ψ⌉ is trueM at w if and only if, for all possible worlds u of M, ψ is trueM at u.

Which is just that a proposition is necessarily true exactly when it is true in all possible worlds. ◇ is then defined as ~☐~, in the same relative way as ∃(x) and U(x).


Neat stuff.

:up:

This is excellent:

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.

In syntax, modal operators (□, ◇) block substitution and fail to behave like extensional connectives. But semantically, if we treat each world as a Tarskian interpretation, then modal truth conditions are entirely extensional within each world. Intensionality arises from the syntax, not from some deep semantic mystery.

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. — NotAristotle

Sure, but as I said, with possible worlds we are talking about different properties at the same time. That is what prevents the name from referring to the same thing.

The claim that individuals in possible worlds might lose identity is false in standard semantics. — Banno

The truth or falsity of this statement depends on how one would define "identity". By the law of identity, identity is a relation between a thing and itself, stating that the thing is the same as itself. Mathematics, specifically set theory, has produced a distinct form of identity, which is based in the concept of equality, rather than the empirical observations of "a thing".

Standard possible worlds semantics appears to borrow this form of "identity", from mathematics, allowing that individuals in possible worlds have the same identity through an equality relation. This form of "identity" is in violation of the law of identity. And if the equivalent individuals, in distinct possible worlds, have contradictory properties, at what is said to be the same time, and are also said to be the same individual (have the same identity), this would violate the law of non-contradiction. Therefore it is best for proper understanding, to recognize this violation of the law of identity, and that the individuals within distinct worlds who bear the same name, have an equality relation rather than an identity relation, so that the law of non-contradiction is not violated.

Modal logic is intensional: truth cannot be determined by reference in the actual world alone. But Tarski-style extensional semantics can be applied within each world. The intension of a term or predicate is a function from worlds to extensions, and this intension determines the extension in each world. Extensions still define truth inside a world, while intensions describe how extensions vary across worlds. Modal operators (□, ◇) are intensional because they quantify over extensions in multiple worlds. This is the account given in the SEP article. — Banno

The way I see it, and as described by the SEP, any logic has intensional and extensional aspects. There are very good reasons why logic could not exist as just one of these.

Nothing "semantic" or "intensional" is needed inside the world. The evaluation is purely extensional. — Banno

You are not paying close attention to what the SEP is saying:

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. — SEP

Rules of extension are intensional. So the rules of Tarskian semantics which you stated, are intensional, and they apply specifically "inside the world".

But it's not the case that extentionality produces good logic, and intensionality produces bad logic, or anything like that, as they are both necessary aspects of logic. The way I see it is that intensionality provides the creative aspect required for what the SEP calls "rigorous definitions of truth", while extensionality provides the demonstrative aspect, to show, or prove to others, the usefulness of those intensional definitions. If you are interested in reading further, my perspective on this, check my reply to frank below.

Well, no. In formal Kripke semantics, extensionality inside a world is real and exact. Nothing “artificial” or “intentionally produced” is involved inside the world. Possible-world semantics does not care whether the individuals are “real” or “fictional", since the extension of a predicate in a world is always a well-defined set of individuals in that world. Intensions tell us how the extension changes across worlds, but inside each world, extensionality is fully Tarskian, such that the truth of a sentence depends only on the domain and the extension in that world. Intension is a tool for cross-world reasoning, not a replacement for extensional truth inside a world. — Banno

I don't think you are understanding what I meant. Being a "possible world", the entire world is intentionally produced, and it is imaginary in the sense that it is a description which does not necessarily describe anything "real", as in independent, in the physical world. This is why the semantics are such that it doesn't matter if things are real or fictional, because everything is treated as fictional. That's the same as pure mathematics, the axioms are assumed to be fictionalbecause this provides for the required freedom.

So the extensions within a world are produced intensionally, through a set of rules, Tarskian in this case. They are not "real" extensions in the sense of being demonstrated or proven through reference to "real" empirical objects in the physical world, they are proven through reference to the rules, which you say in Kripke semantics are "real and exact".

John asked if Frosty the Snowman is a Christmas themed character.

The extension of "is a Christmas-themed character" is

{Santa Claus, Mrs. Claus, Reindeer (especially Rudolph), Snowmen (like Frosty), Elves, Belsnickel & Befana, The Grinch, Jack Skellington, Ebenezer Scrooge}

C(x) = "Is a Christmas-themed character."

C(Frosty the Snowman) is true.

It doesn't matter that Frosty the Snowman isn't real. — frank

I think you are missing out on the foundation, or basic point of "extension". Notice, all your examples of "Christmas-themed characters" are intensional concepts. Not one is a physical "thing" which you can point to, and say that is an example of a Christmas-themed character. Even "snowmen" is a concept, and you would need to point to individual snowmen, as an extensional demonstration of what a snowman is.

Let's take an example, the concept "red", and I'll try to draw this out threw some historical references.

Suppose we say that the meaning of the concept "red" is demonstrated by all the things in the world that are red, that is the extension. So we might be inclined to define "red" that way. If it's the colour of any of these things, then its red. There would be a problem with this definition because it self-referential, and lacks objectivity. And, even if we have agreement from the majority of people which things are red, the things referred to as "red" could shift over time, and we could be adding gold things, orange things, whatever. So conventional agreement on extensionality does not suffice for objectivity. And extension is therefore not a good base or foundation for logic.

Pythagoras got around this problem with the theory of participation, which we now know as Platonism. Every red thing is correctly called "red", or "is red", because it partakes in the Idea of red. Notice that this inverts the situation, giving priority to intension, meaning, rather than empirical observations. From this perspective it is not the case that the idea of "red" is derived from the extension (seeing, and calling things red), but the idea of what it means to be red is prior to there being red things, and we call things "red" because they fulfil the criteria of this intension.

Giving priority to the semantic idea, intension, opens the door to the very productive ideas of the empty set, zero, and possibility in general. Notice that if "red" is defined extensionally, through reference to red things, there cannot be a "red" if there is no red thing. Giving priority to the idea, intensionality, allows that "red" may be a defined concept, without having any red things. This principle allows for "zero", and "possibility" in general. We can say that we have found zero red things, while maintaining the possibility that we may find some red things.

So logic is fundamentally intensional. Logicians produce axioms, definitions and rules for logical proceedings, and these are intensional. However, philosophers are by nature skeptical, and they will doubt these logical principles, requesting demonstrations. This forces the logicians to produce extensions to demonstrate the usefulness of the principles. The philosopher says to the logician, you have an idea of red, and an empty set of red things, prove to me that this is a valid idea. So the logician must formulate extensions, ways in which "red" is useful. Aristotle for example, was very strict in his demands, insisting that the extensions must ultimately refer to substance.

I think you are missing out on the foundation, or basic point of "extension". Notice, all your examples of "Christmas-themed characters" are intensional concepts. — Metaphysician Undercover

The basic point of extensionality is substitutivity. Extension and intension are ways to define an expression.

What were you thinking it was?

Thanks for your thorough comments earlier. I am afraid I cannot do it just with a similarly thoroughgoing reply.

I do not really have any reason to argue against what you have said. However, I do want to phone-in on this assertion:

When we talk about what is possible, we are not talking about a set, namely the set of all possible things. — Leontiskos

I am unsure whether a possible world semantics interpretation of modal logic can still be extensional if it refers to, not only currently existing things, but in addition, "possible things."

To Metaphysician Undercover's point, we might wonder whether a "black frog" refers to anything if its existence is limited to something like possible worlds. And yet, if we consider all the existing frogs, that is what I take us to be referring to if we were to list all the things that fit into the domain for the predication "black frog." It is the property "black" that is "possible" not the referent, which is all extant frogs, now existing, and all of which could be black. On the other hand, perhaps imaginary things like "Frosty the Snowman" can be referents too; but of that I am less certain. In terms of just intension, it is clear that "Frosty the Snowman is a holiday character, but I am less certain whether Frosty is extensional in the sense of having a referent. All that said, I think possible world semantics definitely works extensionally, at least when the referents are well-defined in the actual world.

Sure, but as I said, with possible worlds we are talking about different properties at the same time. That is what prevents the name from referring to the same thing. — Metaphysician Undercover

We are talking about possible "properties" of a thing, the referent, in this case "Nixon." Insofar as those properties are merely "possible" I don't see why they can't be attributed to Nixon, even at the same time, as Nixon's actual properties.

Frank, Kripke's use of rigid designators is not discussed in the SEP article according to Banno. Would you object if we hear from Richard B's critique of rigid designators in this thread anyways?

Frank, Kripke's use of rigid designators is not discussed in the SEP article according to Banno. Would you object if we hear from Richard B's critique of rigid designators in this thread anyways? — NotAristotle

That's fine.

Thanks.

Frank, Kripke's use of rigid designators is not discussed in the SEP article according to Banno. Would you object if we hear from Richard B's critique of rigid designators in this thread anyways?
— NotAristotle

That's fine. — frank

I think now is a good time to hear your critiques whenever you are ready Richard B.