I am still confused about why modal logic itself is not extensional — NotAristotle

Simply, substitution fails.

Here's an example fo the sort of thing that threw Quine:

• Necessarily, eight is greater than seven
• The number of planets =eight

Note that the first sentence is modal - the modal operation "Necessarily" wraps around the whole of "eight is greater than seven". Now extensionality is simply the substitution of equal expressions. And "The number of planets =eight" expresses an equality. So we shoudl be able to substitute "The number of planets" for "eight". But that gives
[*] Necessarily, the number of planets is greater than seven

But that does not seem right - it might have been the case that there were only five planets, and the ancients thought.

So substitution fails, and the modal context is not extensional.

But possible world semantics gets around all this.

See this thread on Quine if you need more.

A quick note that model and modal are not the same, but that we are using both. Modal is to do with necessity and possibility. A model is an assignment of truths to a set of sentences or propositions.

:up: Got it.

:wink:

Simplifying a bit, we have that all John's pets are dogs. His pets are the same as his dogs.

We can substitute in some sentences; so that since all john's dogs are mammals, by substitution we have that all John's pets are mammals. All good - truth is preserved, the context is extensional.

And we have

(5) Necessarily, all John's dogs are mammals: □∀x(Dx → Mx),

Of course this is true since all dogs are mammals. In no possible world does is there a dog that is nto a mammal.
but by substitution that gives

(6) Necessarily, all John's pets are mammals: □∀x(Px → Mx)

But he might have had a pet lizard.

Substitution fails in the modal sentence. And another name for such a failure is that the context is not extensional. Modal sentences are not extensional.

I appreciate the offer, but I’m already pretty much a skeptic. That’s not exactly right, it’s more like I don’t see the use of modal logic. Which isn’t to say I don’t think it might not have value for others.

Just a note on extension and intension. When I first learned about those ideas the word "definition" was attached.

An extensional definition of "John's cat" is John's cat, the actual fuzzy creature.

An intensional definition of "John's cat" would be more like a dictionary definition. It's the Siamese feline that belongs to John.

An extensional definition of "purple" is the set of all purple things. In other words, the extension of "purple" is not sense data, or some mental state, it's a set of all the things that can be described as purple.

An intensional definition of "purple" is a color on the high end of the visible spectrum, and so on.

So when we say modal logic wasn't extensional, it's that the items mentioned in modal expressions didn't pick out anything in the world. They had intensional definition, but that's all.

Do you agree with that?

Good questions. There is a use of "intension" that is the same as "meaning" or "sense" or "the concept of...". And there is a use of extension that amounts to "that very thing".

Some gross oversimplification follows. I'm concerned about getting the overall picture in place rather than the detail.

Go back to John's pets. The extension of "John's pets" is {Algol, BASIC}. It is exactly the set of things, taken as a whole. The extension of "John's pets" = {Algol, BASIC} is the same as saying the extension is "that very thing" - the extension is those specific dogs.

The intension is much less specific. The intension of 'John's dogs" is it's meaning or sense, whatever that is, or the concept of a dog owned by John.

Its much easier to work with extension. Intensionaly speaking, to check if "Algol is one of Joh's dogs" is true might require us to check the sense of "John's dogs", what that concept means or how it is used, then to do the same with "algol", and bring the two together.

Extensionally speaking, to check if "Algol is one of Joh's dogs" is true we look to see if "Algol" is in {algol, BASIC}.

the important bit is to notice that in the intensional way of checking, the truth of the sentence depends on concepts and meaning and such. But in the extensional approach, what's involved is a relative y simple process of checking if the referent of the term is an element of the extension of the predicate.

There are formal definitions of intension, used in formalising intensional logic. These pretty much consist in relations between terms and their extensions. But this is not central to the article we are considering.

So when we say modal logic wasn't extensional, it's that the items mentioned in modal expressions didn't pick out anything in the world. — frank

Not quite. It's not that "possibly, Algol might not have been one of John's dogs" does not refer to anything - it clearly does. It's that substitution, the very core of extensionality, might not preserve the truth of such sentences. In modal contexts, knowing what something ‘actually is’ is not enough to determine truth; you have to consider how it might be in other possible worlds.

Ok. I've got it.

cheers. Very pleasing.

Yeah I am still confused about why modal logic itself is not extensional, but possible world semantics is apparently extensional. — NotAristotle

The possible worlds semantics creates the illusion of extensional objects, "worlds" as a referent. This is the same tactic used by mathematical set theory. They use the concept of "mathematical objects" to create the illusion of extensional referents. It's Platonic realism. The problem is that the reality of these "objects" is not well supported ontologically.

The possible worlds semantics creates the illusion of extensional objects, "worlds" as a referent. — Metaphysician Undercover

The kind of expression we're talking about is:

Necessarily, all John's pets are mammals.

There's no mention of possible worlds in this expression. So no, it's not that we give "worlds" a referent by modal logic.

This is the same tactic used by mathematical set theory. They use the concept of "mathematical objects" to create the illusion of extensional referents. It's Platonic realism. — Metaphysician Undercover

This is nonsense because numbers are already abstract objects. We don't need set theory for that. BTW, so are sentences, propositions, words, the whole shebang. We're examining the way we think. A few abstract objects show up. Deal with it. :razz:

1.2 Extensionality Regained

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, as expressed informally in the following two general principles:

Nec A sentence of the form ⌈Necessarily, φ⌉ (⌈◻φ⌉) is true if and only if φ is true in every possible world.[3]
Poss A sentence of the form ⌈Possibly, φ⌉ (⌈◇φ⌉) is true if and only if φ is true in some possible world. — ibid

A quantifer tells us about the number of items in a domain that have a certain property, like all, or some. So "necessary" will mean that all the items (in every possible world) have the property. Possibly mean at least some of them do.

The kind of expression we're talking about is:

Necessarily, all John's pets are mammals.

There's no mention of possible worlds in this expression. So no, it's not that we give "worlds" a referent by modal logic. — frank

I don't understand your argument here frank. How, in your mind, does possible worlds semantics establish extensionality for modal logic? Is it not the case that "necessarily" means true in all possible worlds, and that these "worlds", which are supposed to be the referent objects, provide the foundation for extensionality?

This is nonsense because numbers are already abstract objects. — frank

That abstractions such as numbers are "objects" is a specific ontological claim. That ontology is known as Platonism or Platonic realism. Abstractions are not necessarily understood as objects though. They are considered to be objects if the Platonist perspective is accepted. Set theory stipulates that these abstractions are objects, by axiom. These objects provide the foundation for extensionality. In modal logic "possible worlds" provide the objects for extensionality.