[Revised post:]

Consider the sentence:

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

That sentence is written in a metalanguage.

The left side of the biconditional is:

'Snow is white' is true.

There 'Snow is white' is a quotation of the sentence:

Snow is white.

which is a sentence of the object language.

The whole sentence is of the form:

'P' is true if and only if P.

/

But we can make this even more precise, using Tarski's method of models:

Consider a formal sentence such as:

0+2 = 2

'0+2 = 2' is true if and only if the denotation of '0+2' is the same as the denotation of '2'.

If, with the interpretation of the language we are using, the denotation of '0' is the number zero, and the denotation of '2' is the number two, and the denotation of '+' is the addition operation, and the denotation of '=' is the identity relation, then:

'0+2 = 2' is true in this interpretation if and only if zero plus two is identical with two.

/

With the original example:

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)

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

then:

'Snow is white' is true in this interpretation 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 (0°C) 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.

So, , is helpful?

It seems to be a reply to your

For Tarski, the right hand side is a Metalanguage, which is not the world. — RussellA

I should have made clear that I'm not opining about RussellA's posts, but rather I meant my own post as a rendering of my own explanation, not necessarily as agreeing with or disagreeing with RusselA's perspectives.

@Banno

What do "the domain of the metalanguage" and "the world of that metalanguage" refer to? — TonesInDeepFreeze

Metalanguage
I used to think that "For Tarski, the right hand side is a Metalanguage, which is not the world", however, @Andrew M made me rethink. I now believe Tarski's T-sentence is the Metalanguage (ML). As the truth cannot be found in either the RHS or LHS by themselves, but only in a combination, the T-sentence must be the ML.

The LHS is the Object Language (OL).

It seems sensible that the RHS is the world of the ML, where world is a synonym for domain, where the world of the ML is snow, apples, houses, white, mountains, etc. However, the world of the ML is not necessarily our world, though it could be.

A language itself doesn't have a domain nor a world. Rather, an interpretation of a language has a domain of discourse — TonesInDeepFreeze

In the OL, we can say that the domain of the wife is cooking, cleaning and housekeeping, where the set wife = {cooking, cleaning, housekeeping}

In the OL, "Terry left the bar and walked through a thick fog". In the ML we can say that the writer used the expression "thick fog" to symbolize Terry's state of mind. The OL is interpreted in the ML.

The domain of the OL on the LHS of the biconditional is "cooking", "cleaning", "bar", "fog", etc
The domain of the ML on the RHS of the biconditional is cooking, cleaning, bar, fog, etc

IE, the T-sentence relates the domain of the OL with the domain of the ML

Consider the sentence: 'Snow is white' is true if and only if snow is white. — TonesInDeepFreeze

Summarising @TonesInDeepFreeze (I hope correctly)
1) In the world is the achromatic object color of greatest lightness characteristically perceived to belong to objects that reflect diffusely nearly all incident energy throughout the visible spectrum. Designate this "white"
2) In the world 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). Designate this as "snow"
3) "Snow is white" is true IFF what has been designated "snow" has what has been designated "white"

Designating
Names are designated in Institutional Performative Acts and written up in the annals (metaphorically). For example, "apples" have been Institutionally named, but the object part my pen and part the Eiffel Tower hasn't (yet) been Institutionally named.

Tarski's T-sentence
I observe the world and see something cold, white and frozen and a relation between them, the relation snow.

If cold, white and frozen didn't exist in the world, then neither would the relation snow.

Let white be designated "white" and snow be designated "snow". It is also possible that white had been designated "green" and snow designated "apple". The world of the ML is not our world, and, in the world of the ML, anything is possible.

The T-sentence is a biconditional, meaning that the truth of the proposition "snow is white" is conditional on something.

But "snow" being "white" is not conditional on snow being white, as snow is of necessity white. Snow only exists as a mereological object having the parts cold, white and frozen. Snow doesn't exist independently of its parts, cold, white and frozen.

"Snow is white" is conditional on i) the existence in the world of cold, white and frozen and a relation between them, snow ii) snow being named "snow" and white being named "white"

Simplifying, it seems to me that the T-sentence becomes: "snow is white" is true IFF snow is white, snow has been named "snow" and white has been named "white".

Your report is quite confused about these notions. I'm afraid that if one wants to properly understand this subject then one has to read a good textbook on it.

Tarski's T-sentence is the Metalanguage — RussellA

The sentence is not the metalanguage. The sentence is written in the metalanguage. (I think that's what you meant.)

The LHS is the Object Language (OL). — RussellA

No, the sentence is a biconditional in the metalanguage. Both sides of the biconditional are in the metalanguage.

In the OL, we can say that the domain — RussellA

No, we don't specify a domain in the object language. In the metalanguage we specify an interpretation of the object language. Part of that interpretation is specification of a domain.

The OL is interpreted in the ML. — RussellA

Right.

The domain of the OL on the LHS of the biconditional is "cooking", "cleaning", "bar", "fog", etc — RussellA

No, 'cooking', 'cleaning', etc. are vocabulary of the object language; they are not in the domain.

The domain of the ML on the RHS of the biconditional is cooking, cleaning, bar, fog, etc — RussellA

No, in the metalanguage we specify interpretations for the object language. Part of an interpretation is specification of a domain. And cooking, cleaning, etc. are predicates over members of the domain.

the T-sentence relates the domain of the OL with the domain of the ML — RussellA

No, as explained above.

1) In the world is the achromatic object color of greatest lightness characteristically perceived to belong to objects that reflect diffusely nearly all incident energy throughout the visible spectrum. Designate this "white"
2) In the world 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). Designate this as "snow" — RussellA

Right.

3) "Snow is white" is true IFF what has been designated "snow" has what has been designated "white" — RussellA

Hmm, not sure that's the way to put it. 'Snow is white' is true iff what 'snow' stands for has the property that 'white' stands for.

I observe the world and see something cold, white and frozen and a relation between them, the relation snow. — RussellA

In Tarski's sentence, 'snow' is the noun. So 'snow' stands for an object, which, per an interpretation is a member of the domain. So snow is not a relation or even adjective ('is snow') in this particular case. The adjective is 'white' ('is white' or 'has the property of whiteness')

The T-sentence is a biconditional, meaning that the truth of the proposition "snow is white" is conditional on something. — RussellA

Hmm, okay. A biconditional is just the conjunction of a conditional with the converse of that conditional. The Tarski sentence says:

If 'snow is white' is true, then snow is white. And if snow is white, then 'snow is white' is true.

So, yes, the condition for 'snow is white' being true is that snow is white. And the condition for snow being white is that 'snow is white' is true.

"snow" being "white" — RussellA

That doesn't enter into it at all. Of course the word 'snow' is not the word 'white'. And of course the word 'white' is not an adjective regarding the word 'snow'.



.

Of course the word 'snow' is not the word 'white'. And of course the word 'white' is not an adjective regarding the word 'snow'. — TonesInDeepFreeze

The meaning of "is"
It seems that most of our disagreement relates to the meaning of certain words that have multiple meanings.

For example, I wrote : But "snow" being "white" is not conditional on snow being white, as snow is of necessity white.

You wrote: That doesn't enter into it at all. Of course the word 'snow' is not the word 'white'. And of course the word 'white' is not an adjective regarding the word 'snow'.

When Tarski wrote "snow is white", this is obviously not intended literally, in that A is A, but rather that "snow has the property white". Similarly, when I wrote "Snow" being "white", my intended meaning was that of "snow" having the property "white".

Language is problematic when key words have multiple meanings.

Tarski's T-sentence
You wrote:
1) 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)
2) 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
3) 'Snow is white' is true in this interpretation 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 (0°C) 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.

From 1) Let S = 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)
From 2) Let W = the achromatic object color of greatest lightness characteristically perceived to belong to objects that reflect diffusely nearly all incident energy throughout the visible spectrum
From 1,2,3) "snow is white" is true IFF i) S is W ii) where the denotation of "snow" is S and the denotation of "white" is W

I wrote: "Snow is white" is true IFF not only i) snow is white but also ii) snow has been named "snow" and white has been named "white"

Tarski's T-sentence is "snow is white" is true IFF snow is white

It seems that we both agree that the T-sentence is missing a necessary condition on the RHS of the biconditional.

(As an aside, I am of the opinion that i) snow is white is the condition of satisfaction, and ii) snow has been named "snow" and white has been named "white" is the condition of designation).

'Snow is white' is true iff what 'snow' stands for has the property that 'white' stands for. — TonesInDeepFreeze

Would you (or Tarskian model theory) accept

'Snow is white' is true iff what 'snow' stands for has the property that 'white' stands for [it, among other things]. — TonesInDeepFreeze

?

Correct that here 'is' is not for equality but for indicating a predicate.

It seems that we both agree that the T-sentence is missing a necessary condition on the RHS of the biconditional. — RussellA

I didn't say anything like that.

I don't see any improvement in your revision.

But no problem, either? Talk of properties when glossing use of a logical predicate is eliminable?

Even in model theory?

Talk of properties when glossing use of a logical predicate is eliminable? — bongo fury

I don't know what that means.

E.g.

The nominalist cancels out the property and treats the predicate as bearing a one-many relation directly to the several things it applies to or denotes. — Goodman, p49

I don't know what that means. I'd have to read the rest of the context.

I can't find a pdf, but here's the paragraph.

A second thread of Hochberg's article comes to something like this: a common predicate applies to several different things in virtue of a common property they possess. Now I doubt very much that Hochberg intends to deny that any two or more things have some property in common; thus for him as for the nominalist there are no two or more things such that application of a common predicate is precluded. Advocates of properties usually hold that sometimes more than one property may be common to exactly the same things; but Hochberg does not seem to be arging this point either. Rather, he seems to hold that a predicate applies initially to a property as its name, and then only derivatively to the things that have that property. The nominalist cancels out the property and treats the predicate as bearing a one-many relation directly to the several things it applies to or denotes. I cannot see that anything Hochberg says in any way discredits such a treatment or shows the need for positing properties as intervening entities.

Whatever is meant by 'predicate' and 'property' there, you asked about model theory.

Predicate symbols map to relations on the domain.

So, yes, if I were to render the T-sentence more formally, not so much as an example of a philsophical principle but as a recap of a formal model theoretic formulation, then I wouldn't need to mention 'property'.

I didn't say anything like that. — TonesInDeepFreeze

1) 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)
2) 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
3) 'Snow is white' is true in this interpretation 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 (0°C) 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.

The only conclusion that can be drawn from what you wrote is that the T-sentence "snow is white" is true IFF snow is white is missing a necessary condition on the RHS of the biconditional, otherwise you wouldn't have included items 1) and 2).

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.

Whatever is meant by 'predicate' and 'property' there, you asked about model theory. — TonesInDeepFreeze

I assumed that by 'predicate' was meant primarily linguistic predicate or adjective, but that this corresponds roughly to a unary predicate in FOL? And properties are the corresponding unary relations?

But you wouldn't need to mention those? I mean, you did mention them, as is usual, and my proposed revision was clearly eccentric, in context. So I was pleased when you didn't immediately reject it. I'm ready to hear that model theory would require reference to corresponding properties, contra Goodman.

So I was interested (and still am) in what you have to say.

unary predicate — bongo fury

n-ary for any natural number n

unary relations — bongo fury

n-ary for any natural number n

I mean one-place? (Sure I've seen "unary" quite a lot but I doubtless have it wrong.)

I mean one-place? — bongo fury

Predicate symbols can be n-place for any natural number n.

Relations can be n-place for any natural number n.

Of course, in the case of the 'snow is white' example, 'is white' is 1-place.

Exactly.

But it's only an example of a principle. Presumably, the principle applies to n-place for any natural number n, as indeed it does in model theory.

Absolutely. And Goodman goes on in the next paragraph (and the link in the first quote will take you there via page 49) to explain how he would eliminate talk of corresponding relations for n > 1. But let's stick to n = 1?

He would eliminate n-ary relations (n>1) from the method of models?

He doesn't say models. I said models. That's what I'm asking about.

Presumably he is talking about interpreting FOL sentences in general, though.

You can reduce n-ary predicate symbols (n>2) to 2-ary predicate symbols. But you have to have at least 2-ary if you want more than the monadic predicate calculus.

monadic — TonesInDeepFreeze

Don't want more!

let's stick to n = 1? — bongo fury

Don't want more! — bongo fury

I don't know what it means to not "want" more than monadic logic. You can't do much mathematics with just monadic logic.