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.

How, in your mind, does possible worlds semantics establish extensionality for modal logic? — Metaphysician Undercover

Step by step, Meta. Step by step. The aim here is to see what standard modal theory says before critiquing it.

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. — Metaphysician Undercover

Yep.

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

But this is an equivocation between natural language and a specific formal language. When @Metaphysician Undercover spoke about "possible worlds semantics," he surely was not talking about natural language that uses the words that possible worlds semantics attempts to formalize. Yet your construal of his claim is mistakenly committed to the idea that this is precisely what he is doing. Indeed, if @Metaphysician Undercover were incorrect then your post would make no sense, for it is bound up with the fact he pointed to.

Possible worlds semantics reifies "possible worlds" into an (extensional) set, in order to make it conform to extensional presuppositions. But this is confused given that modal terms in natural language are not extensional in that manner. The words "necessarily" and "possibly" do not denote extensional sets. It is this confusion that lies at the bottom of so many of the problems that come up in this area. The modal logician basically says, "Modal terms are not extensional, but we're going to pretend they are." Modal terms are made into a nail by those who have only a hammer, and those who pay attention to this sleight of hand are not surprised by the strange outcomes. The various "puzzles" that inevitably come up in the latter parts of the SEP articles are largely reducible to this pretense.

Step by step, Meta. Step by step. The aim here is to see what standard modal theory says before critiquing it. — Banno

Extensionality is the very first step. We ought to understand what it means before proceeding. It appeared like my interpretation was not consistent with frank's so I asked frank to clarify what he was saying.

See the Open Logic text, Appendix A1.

Make up your own definition is counterproductive here.

Definition A.1 (Extensionality). If A and B are sets, then A=B iff every element of A is also an element of B , and vice versa.

I'm not interested in your attempt to change the subject.

:rofl:

Heaven forbid we talk about the definition of "extension" in modal logic.

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.

Since the middle ages at least, philosophers have recognized a semantical distinction between extension and intension. The extension of a denoting expression, or term, such as a name or a definite description is its referent, the thing that it refers to; the extension of a predicate is the set of things it applies to; and the extension of a sentence is its truth value. 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.[2] A logic is extensional if the truth value of every sentence of the logic is determined entirely by its form and the extensions of its component sentences, predicates, and terms. An extensional logic will thus typically feature a variety of valid substitutivity principles. A substitutivity principle says that, if two expressions are coextensional, that is, if they have the same extension, then (subject perhaps to some reasonable conditions) either can be substituted for the other in any sentence salva veritate, that is, without altering the original sentence's truth value. 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

It’s the same. That’s the definition of extensionality used in logic and maths.

Tarskian Semantics
The next section looks pretty fearsome. Its formality belies a fairly simple and direct way to deal with truth, which was developed by Tarski. It's not his T-sentences, although it comes from the same body of work.

We are already almost there with the following:

The extension of a denoting expression, or term, such as a name or a definite description is its referent, the thing that it refers to; the extension of a predicate is the set of things it applies to; and the extension of a sentence is its truth value. — 1.1 Extensionality Lost

In predicate logic, every predicate symbol has an arity, the number of arguments it takes. A proposition, such as p, is 0-ary, it takes zero arguments; a 1-ary predicate such as f(x) takes one argument - the "x"; a 2-ary predicate such as f(x,y) takes two arguments - the "x" and the "y". Generally, an n-ary predicate takes n arguments

What's added is the definition of "...is true" as follows:

n = 0 (i.e., π is a sentence letter) and the extension of π is the truth value TRUE; or
n = 1 and aτ1 is in the extension of π; or
n > 1 and ⟨aτ1, ..., aτn⟩ is in the extension of π. — 1.2 Extensionality Regained

These give the meaning of "...is true" for each of the n-ary predicate symbols.

So what is being said is that a proposition, p, will be true in the case that its extension is the truth value TRUE. This might seem odd at first, but it's standard, so take it as it stands for now.

A 1-ary predicate such as f(x) will be true in the case that the referent of x is one of the things that is in the extension of f.

A 2-ary predicate such as f(x,y) will be true in the case that the referent of x and y are among the things in the extension of f.

Going back to John's two dogs, The sentence "John has two dogs" has as its extension "TRUE", and so is a true sentence. The predicate "John's dogs" has the extension {Algol, BASIC}; and "Algol is one of John's dogs" will be true precisely if Algol is in that extension; which it is.

We can add a bit of terminology. We say that "TRUE" satisfies "John has two dogs", and that Algol satisfies "One of John's dogs".

We might add the predicate "Loved by", with the extension {(John, Algol), (John, BASIC)} - "John loves Algol" and "John Loves BASIC" are both true. We get such the 2-ary predicates as "Loved by (John, Algol)" which will be true exactly if (John, Algol) is a member fo the extension of "Loved by" - that (John, Algol) satisfies "Loved by"

What Tarski did here was to provide a way to evaluate the truth of any formula, using satisfaction, and hence purely in terms of extensionality.

:up:

I'm not overly happy with that. I might try a different approach.

We have a language - roughly, first order calculus.We give it the following interpretation...

We have a domain consisting of three things: John, Algol and BASIC (Who names their dogs after extinct computer languages?)

We have a few predicates, "Is John's pet", with the extension {Algol, BASIC}; "Is a dog" with the extension {Algol, BASIC}; "Is loved by" with the extension {(John, Algol), (John, BASIC)}.

We can note immediately that "Is John's pet" is co-extensional with "Is a dog" - all John's pets are dogs.

We then set out satisfaction; An individual satisfies a predicate exactly if it is a member of the extension of that predicate. So Algol satisfies "Is a dog", and the pair (John, Algol) satisfies "Is loved by".

And then we can define being true for any sentence in our interpretation in terms of satisfaction. A proposition is true if the individuals involved satisfy the predicates involved.

This approach might make it easier the follow the next section.

Again, we've defined truth in our language using only extensions.


This does the work in the section of Tarski's semantics, with

Playing with MathJax...

The equivalences between my last post and the section on Tarski's semantics.

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. — frank

Yep. The U and the ∃ quantify within a world, the ☐ and the ◇ across worlds.

That's the next step.

How is the domain set?

the individuals are the domain. So it’s whatever you would include. In our case,

Domain: D = { John, Algol, BASIC }

But potentially anything.

:up: