We are verging on some interesting recent stuff here. There's an argument from David Chalmers that the sort of account given above is problematic in that it would have "water is H₂O" and "water is water" have the same intension. He and others have proposed what they call a 2-dimensional semantics in order to overcome this. It looks like modal logic together with with Kaplan's treatment of indexicals.

Others have proposed hyperintensionality, in which finer levels are found inside possibility and necessity. Belief is an example of a hyperintensional context, in virtue of how it exhibits degrees.

"Nixon might not have one the election" is about Nixon, not some other non-physical...whatever — Banno

Sure, but "Nixon might not have won the election" is obviously a blatant falsity.

And, if you set up a modal model, possible worlds, within which Nixon might not have won the election then this is "some other non-physical...whatever". It's nothing other than a conceptual structure.

What baffles me is that you and I spent weeks hammering out the fact that there is a real difference, and significant separation, between the "actual world" of the conceptual modal model, and the real independent "actual world". And, when the difference was finally made clear, and agreed upon by both of us, you repeatedly accused me of not respecting that difference. Now, you are firmly in that position of refusing to respect the difference.

How can you repeatedly accuse me of making the error of ignoring this difference, and now you insist that there is no difference? In the other thread you insisted that "actual world" could refer to the metaphysically independent world, and also that "actual world" could refer to a conceptual model modal, and it is a significant error to confuse these two meanings. Now you claim the exact opposite, that there is no such duality of meaning for "Nixon". What's going on?

Sure, but "Nixon might not have won the election" is obviously a blatant falsity. — Metaphysician Undercover

Given that, you are not even in the game, Met.

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). — RussellA

I think all of this is correct.

Possible world semantics preserves Tarski’s notion of extension, but lifts it to a function from worlds to extensions.

This function is the intension. Speaking roughly, the intension of π is the rule that tells you what π’s truth-value would be in every possible world. If you prefer you can treat this as a term of art, as being quite different to the other intensions mentioned in my previous post. But the issue of whether and to what extent this clearly defined notion of intension is the same as the others is alive in the literature. — Banno

So are intension and interpretation the same thing?

So are intension and interpretation the same thing? — frank

Not really. The interpretation is the link between all the things and predicates at a world, while the intension kinda goes in the other direction, as well as across worlds. So in a world the interpretation tells us which thing "Algol" picks out and that it is a pet - that it is true that Algol is a pet. The intension goes the other way, telling us that "Algol is a pet" is true.

So
The interpretation tells us: "Algol" denotes this particular dog, "is a pet" denotes {Algol, BASIC, ...}

The intension tells us: "Algol is a pet" maps w₁ to TRUE (and also what it maps w₂, w₃, ... to)

Got it, thanks.

I think all of this is correct. — frank

Continuing the quest for truth.

The SEP article Possible Worlds writes:
1 - Modal logic, by contrast, is intensional.
2 - 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.

This initially makes sense.

Consider the sentence “swans are white”. The truth value of this sentence cannot be known just from the sentence itself

Consider the extension of “swan”. For example {that swan in Hyde Park, this swan on Ullswater}. Knowing what a “swan” is is impossible from a finite number of examples.

As the article writes, the truth value of the sentence “swans are white” can only be determined by something over and above the form of the sentence “swans are white” and over and above the extension of its component “swan”.

Suppose I observe something X having the properties {waterfowl, flighted, white}.

I obviously cannot know whether being white is internal (necessary) or external (contingent) to X without knowing how X has been defined, and that it has been named "swan".

Suppose a “swan” has been defined as having the properties {waterfowl, flighted, white}. Then I know I am seeing a swan, and being white is an internal, necessary property. Given this definition of “swan”, swans are necessarily white in all possible worlds

From Wikipedia Extensional and intensional Definitions
Intensional definition = gives meaning to a term by specifying necessary and sufficient conditions for when the term should be used.
Extensional definition = listing everything that falls under that definition

However, it is not necessarily the case that the same definition obtains in all possible worlds.

For example:
In possible world 5, “swan” may have been defined as {waterfowl, flighted, white}
In possible world 6, “swan” may have been defined as {waterfowl, flighted, black}

The SEP article suggests that the truth value of the sentence “all swans are white” must be determined over and above its form and over and above its extension.

From the Wikipedia article Modal Logic, ☐ P is true at a world if P is true at every accessible possible world. In other words, necessarily “swans are white” is true at a world if “swans are white” is true at every accessible possible world.

However, in modal logic, this something over and above cannot be a definition, so what could it be?

How does modal logic determine truth values?

How does modal logic determine truth values? — RussellA

I think the simple answer is that it doesn't. You have to provide that. You build a model with a domain and predicates. You have to know what the things you're filling the domain with are and how the predicates relate. It's like you're building a little world. Truth is defined in a certain way.

So the point is more about rigorously handling an expression like "All swans are white" as opposed to determining if it's true.

Here's a block of text from the SEP :grin:

A model consists of a collection of states, some determination of which states are relevant to which, and also some specification of which propositional letters hold at which of these states. States could be states of the real world at different times, or states of knowledge, or of belief, or of the real world as it might have been had circumstances been different. We have a mathematical abstraction here. We are not trying to define what all these states might ‘mean,’ we simply assume we have them. Then more complex formulas are evaluated as true or false, relative to a state. At each state the propositional connectives have their customary classical behavior. For the modal operators. □X, that is, necessarily X, is true at a state if X itself is true at every state that is relevant to that state (at all accessible states). Likewise ◊X, possibly X, is true at a state if X is true at some accessible state. If we think of things epistemically, accessibility represents compatibility, and so X is known in a state if X is the case in all states that are compatible with that state. If we think of things alethically, an accessible state can be considered an alternate reality, and so X is necessary in a state if X is the case in all possible alternative states. These are, by now, very familiar ideas. — Intensional Logic from the SEP

Given that, you are not even in the game, Met. — Banno

So here's a summary of the progress which you and I have made, in our discussion of modal logic.

We both agree that there is a very clear and significant difference between "the actual world" in a modal model, and "the actual world" as a real, independent metaphysical object. However, you persistently refuse to apply this principle in you interpretation of modal logic. And, when I insist on applying this principle in our interpretation of modal logic, you reject me as erroneous, and refuse to include me in your "game".

The SEP article suggests that the truth value of the sentence “all swans are white” must be determined over and above its form and over and above its extension.

From the Wikipedia article Modal Logic, ☐ P is true at a world if P is true at every accessible possible world. In other words, necessarily “swans are white” is true at a world if “swans are white” is true at every accessible possible world.

However, in modal logic, this something over and above cannot be a definition, so what could it be?

How does modal logic determine truth values? — RussellA

That is the problem of extensional definition which I pointed to, calling it "self-referential". Banno called it "circular", but refused to acknowledge it as a problem. If the definition is purely extensional, then what makes something what it is, is being categorized as such. What makes a swan a swan is being in the set of swans. You can see the problem of having no intensional criteria. There is nothing to state what it means to be a swan, which justifies classifying something that way. Extensional understanding produces meaningless statements like "it's true that the cat is on the mat if the cat is on the mat". You can see that there is no principle by which we might judge the truth of a proposition.

The something "over and above" referred to by the SEP is much more nuanced than a definition. Truth is determined by the modal operators, necessity, etc.. The application may be based in intuition, empirical principles, or pragmatic reasons, but as indicates it's fundamentally arbitrary.

I think the simple answer is that it doesn't. You have to provide that. You build a model with a domain and predicates. You have to know what the things you're filling the domain with are and how the predicates relate. It's like you're building a little world. Truth is defined in a certain way. — frank

I am sure that truth in possible world semantics is not down to personal preference, though I am definitely no expert in modal logic.

From SEP Possible Worlds

Consequently, there was no rigorous account of what it means for a sentence in those languages to be true and, hence, no account of the critical semantic notions of validity and logical consequence to underwrite the corresponding deductive notions of theoremhood and provability. A concomitant philosophical consequence of this void in modal logic was a deep skepticism, voiced most prominently by Quine, toward any appeal to modal notions in metaphysics generally, notably, the notion of an essential property. (See Quine 1953 and 1956, and the appendix to Plantinga 1974.)

The purpose of the following two subsections is to provide a simple and largely ahistorical overview of how possible world semantics fills this void.

In my words, the article is saying that in traditional modal logic there was no rigorous account of what it meant for a sentence to be true. This led to a deep scepticism. However, this void of what it means for a sentence in modal logic to be true was filled by possible world semantics.

IE, possible world semantics do somehow give a rigorous account of what it means for a sentence in modal language to be true.

In other words, in modal logic, truth is not a personal thing, in that I think that it is true that "swans are white” whilst you may think that it is true that “swans are black”.

The section you posted from the SEP mentions the concept “true”, but does not specify “true for whom”.

For the modal operators. □X, that is, necessarily X, is true at a state if X itself is true at every state that is relevant to that state (at all accessible states).

But what is the foundation for truth in modal logic?

IE, possible world semantics do somehow give a rigorous account of what it means for a sentence in modal language to be true. — RussellA

Yes, a definition of "true" is provided.

In other words, in modal logic, truth is not a personal thing, in that I think that it is true that "swans are white” whilst you may think that it is true that “swans are black”. — RussellA

I didn't say it was a personal thing. I said the matching of members of the domain and predicates is part of the model or interpretation.

But what is the foundation for truth in modal logic? — RussellA

Foundation?

Foundation? — frank

I didn't say it was a personal thing. I said the matching of members of the domain and predicates is part of the model or interpretation. — frank

In my model, “swan” = {waterfowl, flighted, white}
In John’s model, “swan” = {waterfowl, flighted, black}

I agree that the matching of members of the domain and predicates is part of the model or interpretation.

But what judges whether white or black is part of the domain of a “swan”?

If any judgement is not a personal thing, what is the source of any impersonal judgement?

If mathematical truths are necessary truths (e.g., there is no possible world where Pi isn't 3.14...), then aren't mathematical truths also logically true? Or at least carry the same weight?

But what judges whether white or black is part of the domain of a “swan”? — RussellA

I'll ask @Banno to weigh in on that because I thought I already answered it. :smile:

then aren't mathematical truths also logically true? — RogueAI

What do you mean by "logically true"?

You can see the problem of having no intensional criteria. — Metaphysician Undercover

On the other hand.

Suppose you are given the extensional definition of the foreign word “livro”, where “livro” = {Pride and Prejudice, The Terminal Man, The Great Gatsby, In Cold Blood}

I am sure you could make a good guess as to the meaning of “livro” just from its extensional definition.

Once you have the concept of “livro” in your mind, you could then apply your concept to include other objects, such as {Harry Potter and the Chamber of Secrets}

IE, we can only ever observe extensional definitions, as intensional definitions only exist within our minds.

then aren't mathematical truths also logically true?
— RogueAI

What do you mean by "logically true"? — frank

P or not P; If P then Q, P therefore Q; all bachelors are unmarried men, etc.

I think Kripke would say the value of pi is necessarily 3.14..., but this is known a posteriori.

What about something simple, like 2+2=4? Isn't that discoverable through pure reason?

What about something simple, like 2+2=4? Isn't that discoverable through pure reason? — RogueAI

I think so.

If I recall correctly, Quine's position relates to the following observation:

Suppose that the modal operators merely refer to the quantifiers of First Order Logic (FOL), which is the case if every set of possible worlds is describable by a first-order predicate. In which case, we can eliminate the modal operators from any modal formula f to produce an equi-satisfiable formula f' without the modal operators, e.g by using skolemization. This means that although we obviously cannot represent modal truth as a formula in modal logic unless the modal logic is trivial (as per Tarski's undefinability theorem), we do at least have an automatable procedure for verifying the logical falsity of a modal formula f , i.e by substituting it for an equi-satisfiable formula f' without quantifiers, and then checking whether ~ f' is satisfiable. If it is, then f isn't satisfiable in all models, meaning that f cannot be logically true and hence isn't provable, as per Godel's completeness thorem.

On the other hand, suppose that the modal operators refer to second-order quantification over sets of possible worlds that cannot be described in terms of first-order predicates. In which case we "really have" modal operators above and beyond first-order quantification, since modal formulas are now assumed to not be reducible to FOL. But the price we pay is to lose Godelian completeness, and hence we can no longer use skolemization to determine the truth of the modal formula.

Essentially modal logic is just a game of let's pretend. Underneath the philosphical posturing one either has FOL with syntactically defined quantifiers equipped with a verifiable notion of external truth (skolemisation applied to a model), else one has second-order modal formulas without an unbiased means of deciding a truth value.

Would you want to dumb that down a tad?

Yes, that's it. except perhaps for some expressions that are true in every possible world.


It's not exactly personal preference, more agreed background. See if I can make this work.

The paragraph you quote is saying that in the early part of last century there was no Tarski-style way of treating truth for modal logic. That's what Kripke provided. So First-order logic had satisfaction as an extensional path to truth, but given that modal logic is intensional, it seemed impossible to use satisfaction there. Kripke did just that,

In first order logic, Algol satisfies {Algol, BASIC}, which is the extension of 'John's Pets". In modal logic, Algol satisfies {Algol, BASIC} in some world w, which is the extension of 'John's Pets" in that world, w.

Or changing examples, in England "Swans are white" is true just in the case that every instance of "swan" satisfies "...is white". How do we check this out? The logic doesn't say. That's not what it is for. In that possible world, Australia, "Swans are white" is also true just in the case that every instance of "swan" satisfies "...is white". But here, each instance of a swan is black, so the extension of "Swans are white" is empty, and "Swans are white" is false.

What we have is a rigorous account of what it is for "Swans are white" to be true. But it doesn't tell us if swans are white.

To work that out we are gong to have to go out and take a look.

This seems to be pretty much what you were saying.

Sot he "foundation for truth" in modal logic, as for first order logic, is satisfaction, and is extensional, but in modal logic we have truth-at-a-world, since what is true can vary from world to world.

This is good:

In my model, “swan” = {waterfowl, flighted, white}
In John’s model, “swan” = {waterfowl, flighted, black} — RussellA

If you both insist on this definition - this stipulation, if you will - then you and John will not agree as to what is a swan and what isn't. And this amounts to you and John using the word differently. You will say that there are swans in England, but not in Australia, while John may say that there are swans in Australia, but not in England.

Notice that it is an intensional definition: it does not list the very things that are swans, but gives a rule for deciding of something is a swan.

But what judges whether white or black is part of the domain of a “swan”? — RussellA

Well, it might be worth pointing out the relativity of the relation between you and John. You are well aware that John thinks all swans are black, and he is perhaps aware you think all swans are white. You can get together and have a chat about the use of these words, and either come to an accommodation or go to war... It's perhaps a "personal" thing as to which definition you choose, but it is not "private".

And further, since you understand each other, you know that when John talks about swans, he means the black birds in Australia, not your white English ones. So you know that when John says "Swans are Black", what he is saying is true in his peculiar language. You know that "Swans are Black" as spoken by John is true if the individuals John picks out with his word "Swan" in Australia satisfy "...is black"; and that they do.

But that's the principle of charity at work, rather than anything to do with modal logic. It's you charitably recognising that John is making use of a different interpretation

to your own, one that you can translate.

So back to

But what judges whether white or black is part of the domain of a “swan”? — RussellA

It's built in to the interpretation.

If mathematical truths are necessary truths (e.g., there is no possible world where Pi isn't 3.14...), then aren't mathematical truths also logically true? Or at least carry the same weight? — RogueAI

Not sure what "logical truth is here - but the value of pi is presumably the same in all possible worlds, and so a necessary truth. And the case is similar for 2+2=4, P or not P; If P then Q, P therefore Q; and perhaps "all bachelors are unmarried men", given certain precautions.

If what you are saying is that mathematical and logical truths are true in all possible worlds and hence necessarily true, then yep.

Is there a problem here?

On the other hand.

Suppose you are given the extensional definition of the foreign word “livro”, where “livro” = {Pride and Prejudice, The Terminal Man, The Great Gatsby, In Cold Blood}

I am sure you could make a good guess as to the meaning of “livro” just from its extensional definition.

Once you have the concept of “livro” in your mind, you could then apply your concept to include other objects, such as {Harry Potter and the Chamber of Secrets}

IE, we can only ever observe extensional definitions, as intensional definitions only exist within our minds. — RussellA

I don't really think so. Strictly speaking, there is no further "meaning" to an extensional definition, only the set of items. If we switch to a meaning, and extend the set on that principle, then we've used an intensional definition to do that. That's why logic always consists of both aspects. the intensional must be grounded in substance (extensionality), but the extensionality cannot force necessary limits on the intensional, to free us to go beyond the limited capacity of human observation.

It may be the case, that intensional definitions only truly exist in minds (denying Platonism which allows for independent ideas), but the extensional definition is also only the product of minds. Even though the extensional utilizes empirical observations, it actual becomes trapped by that dependence, unless we allow for arbitrariness to infiltrate. In my other post, I explained how Pythagoras used the theory of participation to escape that trap, in the development of what is now known as Platonism.

By the theory of participation, which Plato explains very well and attributes to Pythagoras, the set of things which compose the extensional definition, are members of that set because they partake in the Idea, which is the defining meaning. This is an independently existing Idea (Platonism), and so it is an objective intensional definition, in a stronger sense than inter-subjective objectivity, because there is supposed to be a real independent idea which provides the meaning.

Platonism is common in mathematical interpretations. The Idea of "two" for example, is supposed to have real meaning, independent from human minds, so the symbol stands for that intensional package of meaning, as an object. Then these mathematical objects can provide the substance for extensionality. Intensionality and extensionality are separated in analysis, theory, but in practise they're all wrapped up in each other.

We both agree that there is a very clear and significant difference between "the actual world" in a modal model, and "the actual world" as a real, independent metaphysical object. However, you persistently refuse to apply this principle in you interpretation of modal logic. — Metaphysician Undercover

No. In modal logic there is a difference between the actual world and other possible words. It's that the actual world is w₀ and the world at which accessibility relations begin. In metaphysics the actual world is a bit different, and no where near so clearly explained. But, for some conversations, we can use modal logic and take the metaphysically actual world as the modally actual world, and look that the accessibility relations that originate in the metaphysically actual world.

This constipated way of talking is a result of your convolute form of expression. It just says that we can consider how things might have been different.

...but refused to acknowledge it as a problem. — Metaphysician Undercover

That'd be 'casue it's only a problem of you misunderstand modal logic in your peculiar fashion.

...we can only ever observe extensional definitions, as intensional definitions only exist within our minds. — RussellA

Pretty much. Another way to think of a intension is the rule we apply in order to decide, say, if that bird is a swan or not. But the truth of "That bird is a swan" is completely determined by the extension of "That bird" and the extension of "...is a swan": it will be true if and only if "That bird" satisfies "...is a swan"

But it's worth noting that we agree on most "intensional definitions". They are not private. And extensions are not decided only by taking a look. That 3 is a prime is not an empirical fact.

If what you are saying is that mathematical and logical truths are true in all possible worlds and hence necessarily true, then yep.

Is there a problem here? — Banno

Mathematical truths are distinct from logical truths. It seems strange that mathematics cannot be reduced to logic when both logical and mathematical truths have the force of necessity.

Mathematical truths are distinct from logical truths. — RogueAI

Well, maybe. More often formal logic is treated as a branch of maths, seen as grounding set theory and so the whole edifice. Whether this is correct remains contentious - logicism vs formalism vs structuralism.

DO we know maths cannot be reduced to logic? We know that specific logicist programmes such as Frege–Russell and Principia, failed in specific ways. Do we know that maths cannot be reduced to logic? We know that in any sufficiently strong formal system (including arithmetic), there are true but unprovable statements in that system. But we know this because of logic... So its going to depend on wha that reduction is.

It might b simplest to treat logic and maths as much the same sort of thing.