We didn't really get started here, did we? Never mind, I may try again tomorrow, with your forbearance.

We haven't really got started here — bongo fury

Got started with what? I don't know where you're going with this.

No, I'm just unpacking what's already there. 'Snow is white' is true if and only if snow is white. I merely unpacked, pedantically really, the right side. Nothing is missing. — TonesInDeepFreeze

I believe that you are saying that the denotation of "snow" as snow and the denotation of "white" as white are already within the expression snow is white, waiting to be unpacked, waiting to be discovered.

However, if given fire is hot as the expression on the right hand side, this means that the denotation of "x" as fire and the denotation of "y" as hot are already within the expression fire is hot, waiting to be unpacked.

If that is the case, then what are "x" and "y" ?

waiting to be discovered — RussellA

Denotations are stipulated. Though it is not as clear cut in natural languages as with semantics for formal languages.

However, if given fire is hot as the expression on the right hand side, this means that the denotation of "x" as fire and the denotation of "y" as hot are already within the expression fire is hot, waiting to be unpacked.

If that is the case, then what are "x" and "y" ? — RussellA

Do you mean the denotation of 'fire' is x and the denotation of 'hot' is y?

With 'snow' and 'white' I just looked in a dictionary.

Denotations are stipulated — TonesInDeepFreeze

In Tarski's T-sentence, "snow is white" is true IFF snow is white, where exactly is "snow" denoted as snow and "white" denoted as white ?

Because if not included within the T-sentence, then how can the T-sentence be formally correct ?

where exactly is "snow" denoted as snow and "white" denoted as white — RussellA

'snow' is not denoted as snow, and 'white' is not denoted as white.

'snow' denotes snow, and 'white' denotes white.

the denotation of 'snow' is:

precipitation in the form of small white ice crystals formed directly from the water vapor of the air at a temperature of less than 32°F (0°C)

and the denotation of 'white' is:

has the achromatic object color of greatest lightness characteristically perceived to belong to objects that reflect diffusely nearly all incident energy throughout the visible spectrum — TonesInDeepFreeze

Hmm.

Why denotation and not extension?

The denotation attempts to set out the necessary and sufficient characteristics of snow and white. The extension just is the things that are snow and white.

The denotation of a word is the thing the word refers to. In formal semantics, the model maps n-place (n any natural number) predicate symbols of the language to n-place relations on the domain, and maps n-place (n any natural number) function symbols to n-place functions on the domain.

That is semantical.

I think we could say that the extension of a predicate or function symbol is the relation or function the symbol maps to. (?)

/

Where biconditionals (necessity and sufficiency) enter are in definitions:

Definitional Axiom Schema - predicate symbol:
Px1...xn <-> Q
where P is a new n-place predicate symbol; x1,..., xn are distinct variables; and Q is a formula (of the language of the source theory) in which P does not occur and in which the only free variables are among x1,..., xn.
(If n = 0, then P is just a propositional symbol and there are no free variables in Q.)

Definitional Axiom Schema - function symbol:
fx1...xn = y <-> Q
where f is a new n-place function symbol; and x1,..., xn, y are distinct variables; and Q is a formula (of the language of the source theory) in which f does not occur and in which the only free variables are among x1,..., xn, y; and all closures of E!yQ are theorems of the source theory.
(If n = 0, then f is just a 0-place function symbol and there are no free variables in Q other than y.)

Those are syntactical.

/

The extension of a property is the set of all things that have the property.

That is philosophical.

The denotation of a word is the thing the word refers to. — TonesInDeepFreeze

Then what is the extension of a word?

I just now added:

I think we could say that the extension of a predicate or function symbol is the relation or function the symbol maps to. (?)

For uniformity of style we could say 'relation symbol' rather than 'predicate symbol'.

Then have:

The model maps n-place (n any natural number) relation symbols of the language to n-place relations on the domain, and maps n-place (n any natural number) function symbols to n-place functions on the domain.

I think we could say that the extension of a relation or function symbol is the relation or function the symbol maps to.

So we have

The denotation of a word is the thing the word refers to. — TonesInDeepFreeze

and

...the extension of a predicate or function symbol is the relation or function the symbol maps to — TonesInDeepFreeze

I'm not seeing a great difference.

Indeed, the second is a formal way of saying the first.

But necessity and sufficiency (the biconditional) does not enter there, in the semantics, but rather in the definitions, which are syntactical.

(Note that Tarski's definition of 'is true' is also in canonical biconditional form.)

But there is a further issue. We agree

precipitation in the form of small white ice crystals formed directly from the water vapor of the air at a temperature of less than 32°F (0°C)

and

snow

have the same extension. Is it true that they have the same denotation? And further, is it really the case that the former is the denotation of the latter? Or do they just happen to denote the very same things, the denotation being those very things?

The problem is that we don't want it to be the. case that one doesn't know what snow is until one knows it is precipitation in the form of small white ice crystals formed directly from the water vapour of the air at a temperature of less than (0°C).

They have the same denotation and extension (but not the same intension).

There's perhaps a slight problem with the choice of 'snow' for the Tarski example.

'snow' in the sentence is a noun, not an adjective.

But 'snow' is a mass noun, so it's not as easy to work with here.

Better might have been:

The snow on the lawn is white.

Or maybe 'cueball':

'The cueball on the table is white' is true if and only if the cueball on the table is white.

/

is it really the case that the former is the denotation of the latter? — Banno

No.

the phrase 'precipitation in the form of small white ice crystals formed directly from the water vapor of the air at a temperature of less than 32°F (0°C)' does not denote the word 'snow'

and

the word 'snow' does not denote the phrase 'precipitation in the form of small white ice crystals formed directly from the water vapor of the air at a temperature of less than 32°F (0°C)'.

Or do they just happen to denote the very same things, the denotation being those very things? — Banno

They denote the same thing.

we don't want it to be the. case that one doesn't know what snow is until one knows it is precipitation in the form of small white ice crystals formed directly from the water vapour of the air at a temperature of less than (0°C). — Banno

That's a separate epistemic question. But I don't see how it's a problem for Tarski's definition or my remarks about it.

Or maybe 'cueball': 'The cueball on the table is white' is true if and only if the cueball on the table is white. — TonesInDeepFreeze

Hah! Replace the "real object" with the abstract object and thus reveal the semiotic game being played. If you can't see a difference, there never was a difference. The claims about reality were always indirect and mediated. Conditioned on some abstract definition. :up:

The problem is that we don't want it to be the. case that one doesn't know what snow is until one knows it is precipitation in the form of small white ice crystals formed directly from the water vapour of the air at a temperature of less than (0°C). — Banno

But that's not snow on the lawn. It's sleet! Etc, etc. :grin:

I don't know where you're headed with this, but in case my hunch is right, I would say:

Tarski is not saying how we know that 'snow is white' is true. He's only saying what it is for 'snow is white' to be true. The latter may help for the former. But the definition itself is only of the latter.

I unpacked 'snow is white' with the longer phrases. That works okay, because the context is extensional, not intensional. If we go to an intensional context:

Bob knows that ('snow is white' is true if and only if precipitation in the form of small white ice crystals formed directly from the water vapor of the air at a temperature of less than 32°F has the achromatic object color of greatest lightness characteristically perceived to belong to objects that reflect diffusely nearly all incident energy throughout the visible spectrum)

then of course, that could be false.

The motivation for the unpacking was just to show that a careless reading of Tarski's formulation as being vacuous for being tautological would be not only prima facie incorrect but illustrated as incorrect.

Tarski's definitIon itself:

For any sentence 'P':

'P' is true if and only if P.

An instance:

Let 'P' be 'snow is white'.

'snow is white' is true if and only if snow is white.

There's perhaps a slight problem with the choice of 'snow' for the Tarski example. — TonesInDeepFreeze

More than slight.

I don't see how it's a problem for Tarski's definition or my remarks about it. — TonesInDeepFreeze

The issue is more to do with developments from Tarski's definition. I'm not disagreeing with you but thinking out loud.

What caught my eye was

If, with the interpretation of the language we are using, the denotation of 'snow' is:

precipitation in the form of small white ice crystals formed directly from the water vapor of the air at a temperature of less than 32°F (0°C) — TonesInDeepFreeze

It is apparent that this is a co-extensional expression for the items that are snow, but also that that the denotation is not the expression but the items themselves.

Consider

• "The kettle is boiling" is true if the kettle is boiling
• "The kettle is boiling" is true if the water in the kettle has reached the temperature at which its vapour pressure is equal to the pressure of the gas above it.
• "The kettle is boiling" is true if the kettle is one of the items in the list of things which are boiling.

The last strikes me as most closely resembling what Tarski does.

Consider
"The kettle is boiling" is true if [and only if] the kettle is boiling[.]
"The kettle is boiling" is true if [and only if] the water in the kettle has reached the temperature at which its vapour pressure is equal to the pressure of the gas above it.
"The kettle is boiling" is true if [and only if] the kettle is one of the items in the list of things which are boiling.
The last strikes me as most closely resembling what Tarski does. — Banno

I don't know why you regard it as the most close. All three seem reasonable to me. Though the first is Tarski's own form.

The advantage of Tarki's form is that is is general. It applies to all sentences.

'P' is true if and only if P.

Then we plug we plug in any sentence for 'P'.

I don't know where you're headed with this, — TonesInDeepFreeze

Again, I have no particular direction in mind, just attempting to sort out a few of the issues around the wider use of Tarski's schema.

I don't know why you regard it as the most close. — TonesInDeepFreeze

Simply because the item and the list is closer to the strategy of designation and satisfaction Tarski adopts.

the denotation is not the expression but the items themselves. — Banno

Of course the denotation is not the expression. The denotation is, formally, as given by the method of models.

item and the list is closer to the strategy of designation and satisfaction Tarski adopts. — Banno

That's an excellent point.

I like the second because it's illustrative. And I like the third because, as you observe, it is getting closer to the actual formal method (and also hews closer to Tarski's "plugging in" strategy).

So for names the denotation or extension is the individual designated but the name; for predicates the denotation or extension is a list of specific individuals, and for statements the denotation or extension is either true, or false.

That's an excellent point. — TonesInDeepFreeze

Cheers. I keep responding to your previous point...

The thing about the first is, is it incontrovertible. Hence, it can serve as a definition of "...is true...", if the difficulties of opacity and circularity can be overcome.

Which is what, apparently, Gupta claims to have accomplished with his revision theory of truth.

That is the basic idea.

For a 0-place function symbol, the denotation is a member of the domain.

For an n-place (n>0) function symbol, the denotation is an n-place total function on the domain.

For a 0-place relation symbol, the denotation is a truth value.

For an n-place (n>0) relation symbol, the denotation is an n-place relation on the domain.

list — Banno

I prefer 'set' rather than 'list', since 'list' could be taken as a sequence of the things in the set, or even suggesting a countable sequence.

The thing about the first is, is it incontrovertible. — Banno

Right.

But recall that my unpacking was a conditional:

If 'snow' stands for blahblahblah and 'white' stands for 'bleepbleepbleep', then

'snow is white' is true if and only if blahblahblah is bleepbleepbleep.

That's a consequence of the Tarski formulation but not an equivalent (since the Tarski formulation doesn't have antecedents like that).

I prefer 'set' rather than 'list', since 'list' could be taken as a sequence of the things in the set, or even suggesting a countable sequence. — TonesInDeepFreeze

Sure. I chose "list" by way of avoiding using a technical term, hence checking the applicability of the notions involved to natural languages. The list is not in any particular order and might be innumerable.

if the difficulties of opacity and circularity can be overcome. — Banno

What difficulties of opacity and circulaity?

I don't know what sense of 'opacity' you have in mind.

And Tarski's formulation is not circular. Indeed he stated a requirement that a definition not be circular, and he gave one that is not circular.

But recall that my unpacking was a conditional: — TonesInDeepFreeze

Yeah; I didn't notice that until a second reading, so at first I misunderstood you as being categorical.

What difficulties of opacity and circulaity? — TonesInDeepFreeze

@Michael made a point of these in another thread. Tarski has

"S" is true IFF X

And we can't just substitute any sentence p for S and X.

Gupta seems to have a strategy that allows us to conditionally substitute p for both S and X and by considering all the possibilities shows that no inconsistency ever results.

I chose "list" by way of avoiding using a technical term — Banno

'set' is no more technical than 'list'

The list is not in any particular order and might be innumerable. — Banno

I usually take 'list' as 'sequence' or 'series' or 'enumeration'.

And since we're interested in hewing to Tarski, the term to use is 'set'.

A 1-place relation symbol maps to a 1-place relation on the domain. (A 1-place relation on the domain is a subset of the domain.)