Doesn't this lead to a chicken-and-egg situation?

We can't know "Snow is white" is true unless we know Snow is white and/but we can't know Snow is white unless we know "Snow is white" is true. — Agent Smith

We can know snow is white before we know "theluji ni nyeupe".

There was a world pre-language
In the world 100,000 years ago, there was something that was snow having the property white. In this world, the English language had not yet been invented, and therefore not only did the words "snow", "is" and "white" not exist, but neither did the proposition "snow is white".

IE, even if there was a life-form that knew snow is white, it couldn't have known that "snow is white".

The world post-language
At some point in the past, in an Institutional Performative act in the English-speaking world (metaphorically speaking), snow was named "snow", is was named "is" and white was named "white". Subsequently, these words were then accepted by society as a whole as the proper names of these things.

Today, an individual within society can decide whether "snow is white" is true or false in two ways.

First, even without observing white snow in the world, but knowing from the dictionary that "snow is white". IE, I know that Aristotle was a Greek philosopher and polymath during the Classical period in Ancient Greece even though I have never met him.

Note: I know Aristotle was Greek not be acquaintance but by description. IE, I don't know the person, but I do know the description. My knowledge is of the description, not the person.

Second, by observing white snow in the world, knowing "snow" names snow, "is" names is and "white" names white, they know "snow is white". IE, I know from my knowledge of language and the world that "snow is white"

IE, in answer to your question, the situation is not circular as we can know snow is white by observing the world even if we don't know that "snow is white". IE, I know snow is white by observing the world even if I don't know that "theluji ni nyeupe".

What does this have to do with Tarski? — Michael

If I understand what you're after, because the meaning of denoting (designating) is central to Tarski's Semantic Theory of Truth.

Tarski's overriding concern is with defining 'is true' in context of formal languages for mathematics and the sciences. — TonesInDeepFreeze

Tarski's approach is certainly rigorous. I would say logical rather than mathematical or scientific.

He wrote in The Semantic Conception of Truth and the Foundations of Semantic: "The predicate "true" is sometimes used to refer to psychological phenomena such as judgments or beliefs, sometimes to certain physical objects, namely, linguistic expressions and specifically sentences, and sometimes to certain ideal entities called "propositions." By "sentence" we understand here what is usually meant in grammar by "declarative sentence"; as regards the term "proposition," its meaning is notoriously a subject of lengthy disputations by various philosophers and logicians, and it seems never to have been made quite clear and unambiguous. For several reasons it appears most convenient to apply the term "true" to sentences, and we shall follow this course".

As formal languages include logic, mathematics, the sciences and linguistics, it is clear from his article that his definition of "true" is more relevant to the formal language of linguistics than either mathematics or science.

He uses an ordinary sense of 'denote' (or cognates of 'denote), but then moves on to instead specify the method of formal modals, where 'denote' is subsumed by certain kinds of functions from linguistic objects to model theoretic objects. This is the movement from informal semantics to formal semantics that Tarski provides. — TonesInDeepFreeze

I agree that Tarski was concerned with formal rather than informal language.

As he wrote in The Semantic Conception of Truth and the Foundations of Semantic: "While the words "designates," "satisfies," and "defines" express relations (between certain expressions and the objects "referred to" by these expressions)." IE, "designates" ("denotes") is about relations.

The word "denote" may be used in different ways, but as there is no substantial difference in meaning between the "ordinary" sense of the word "denote" and a formal sense of the word "denote", he cannot have moved from an "ordinary" sense to a formal sense.

Whether 'snow is white' is analytic depends on which definition of 'snow' we're looking at. — TonesInDeepFreeze

I doubt there are many definitions of "snow" whereby being white isn't included as a property.

Second, (4) is not about the meaning of the word 'denote' but rather it's about the meaning of 'true'. That 'true' has different conceptions. — TonesInDeepFreeze

The complete paragraph containing item 4) is:
It seems to me obvious that the only rational approach to such problems would be the
following: We should reconcile ourselves with the fact that we are confronted, not with one
concept, but with several different concepts which are denoted by one word; we should try to make these concepts as clear as possible (by means of definition, or of an axiomatic procedure, or in some other way); to avoid further confusions, we should agree to use different terms for different concepts; and then we may proceed to a quiet and systematic study of all concepts involved, which will exhibit their main properties and mutual relations.

How is this paragraph about the meaning of true. The word "true" isn't mentioned. ?

Tarski wrote that we are confronted with several concepts denoted by one word, ie, one word may denote several concepts.

How is this not about the meaning of the word "denote" ?
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Tarski in 1931 and 1944 is concerned with mathematical logic, not literary criticism. — TonesInDeepFreeze

I have never said that Tarski was concerned with literary criticism.

Tarski's article The Semantic Conception of Truth and the Foundations of Semantics was published in 1944.

Within the article he wrote:
Semantics is a discipline which, speaking loosely, deals with certain relations between
expressions of a language and the objects (or "states of affairs") "referred to" by those expressions. As typical examples of semantic concepts we may mention the concepts of
designation, satisfaction, and definition as these occur in the following examples:
the expression "the father of his country" designates (denotes) George Washington; snow satisfies the sentential function (the condition) "2 is white"; the equation "2 . x = 1" defines (uniquely determines) the number 1/2.

The Cambridge Dictionary defines semantics as the study of meanings in a language.

I haven't said that Tarski was not concerned with mathematical logic. I pointed out that Tarski had a concern with the semantic conception of truth, and the semantic conception of truth is not the same as the mathematical conception of truth.

You write that Tarski is using "denote" in the ordinary sense of the word.

@TonesInDeepFreeze: "However, Tarski does mention elsewhere that there are different senses of 'denote', but it's a highly technical matter he's addressing. Usually, he uses 'denotes' or 'names' in the very ordinary sense of the words."

Are you saying that the ordinary sense of the word "denote" is the mathematical sense of the word "denote" ?
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It's the very simple idea: 'Chicago' maps to Chicago. 'Carl Sagan' maps to Carl Sagan. 'Cats' map to cats. — TonesInDeepFreeze

It is a simple idea until one considers how "a unicorn" maps to a unicorn, or "beauty" maps to beauty.

How exactly does "beauty" map to beauty. ?
===============================================================================

If you want to understand Tarski and not be bogged down in misunderstandings, then you'd do well to start there, and to refrain from dripping goop all over by ridiculously dragging Umberto Eco into it. — TonesInDeepFreeze

We are specifically discussing the meaning of the word "denote".

You wrote that "He uses "denotes" in a very ordinary sense"

I am pointing out, as Umberto Eco pointed out, that the meaning of "denote" is far more complex than as used in the ordinary sense of "a cat" denotes a cat.

If you reject Umberto's Eco's later contribution, then perhaps consider earlier contributions by Frege and his theory of sense and denotation 1892, Russell's On Denoting 1905 or Saussure's Course in General Linguistics based on notes of lectures 1906 to 1911.

How exactly does "snow" denote snow ?

In the ordinary sense, "snow" denotes snow because "snow" denotes snow.

If the answer to that was agreed, then many of the problems in the philosophy of language would be well on the way to a solution.

Trying to make Tarski look confused isn't helping you. — bongo fury

At no time have I ever suggested that Tarski was confused.

Are you making this up? Bye.

They are presented as a single quote; they come from 3 different pages — bongo fury

They are quite clearly not presented as a single quote, because the four quotations are individually numbered 1), 2), 3) and 4).

You have the document so obviously know they aren't a single quote.

The important knowledge to be gained from these quotations is that Tarski can use one expression to denote one or more objects, concepts or expressions.

Not really — bongo fury

I had written: "the meaning of "denote" is much debated".

I agree that the word "denote" can mean from a word to thing or things. Yet there is more to it than this. For example, people agree that beauty is a combination of qualities, such as shape, colour, or form, that pleases the aesthetic senses, yet millions of words have been written about the meaning of beauty.

For example, Umberto Eco in Meaning and Denotation 1987 wrote: "Today denotation (along with its counterpart, connotation ) is alternatively considered as a Property or function of (i) single terms,(ii) declarative sentences (iii) noun phrases and definite descriptions. In each case one has to decide whether this term has to be taken intensionally or extensionally: is denotation tied to meaning or to referents? Does one mean by denotation what is meant by the term or the named thing and, in the case of sentences, what is the case ?"
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But hopeless if you misunderstand "denote". — bongo fury

I had written: "when I hear the word "unicorn", the word "unicorn" is doing more than pointing to a unicorn".

The sentence is about "pointing", not "denoting".
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Don't edit when quoting. — bongo fury

When an article is edited, the article is changed. My three quotes were neither edited nor paraphrased, they were verbatim and in context.

I'll say it again (as this is certainly not a mere "detail"): The schema says that for any sentence P, we have: 'P' is true iff P. He does not say that 'P' has to be analytic. Look it up. Anywhere. — TonesInDeepFreeze

My goal is to understand Tarksi's Semantic Theory of Truth, not get bogged down in unimportant detail and misunderstandings.

I wrote "Denotes infers points to, and "snow" is doing more than pointing to snow."
You wrote "That is not correct. The word 'denotes' doesn't infer. People infer; words don't infer."
Of course I am not suggesting that the word "denote" is doing the inferring.

Of course the T-sentence "P" is true IFF P is not a detail. It is extremely important. I never said it was a detail.
I said that in my opinion "snow is white" is an analytic proposition.
I never said that in my opinion "P" is an analytic proposition.
I never said that Tarski said that "P" has to be analytic.

@TonesInDeepFreeze

You are quibbling over details and things I never said.

And no! It's all pointing!... just how is "snow" doing more than pointing to snow?? — bongo fury

The meaning of denote
The exact meaning of "denote" is debated, whether in linguistics or mathematics, and books have been written about the topic, e.g., Umberto Eco Meaning and Denotation, John Lyons Language and Linguistics, Bertrand Russell On Denoting.

"Snow" does more than point to snow
Starting with @Tones whereby 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). Remove the expression "formed directly from the water vapor of the air at a temperature of less than 32°F (0°C)", as this describes how "snow" formed rather than what "snow" is. As "snow" is precipitation, can remove the expression precipitation. Therefore, can simplify the denotation of "snow" as small white ice crystals.

In the world we observe small, white, ice crystals, which we name "snow". The unstated and reasonable assumption is that small, white, ice crystals is snow. Therefore, we have named snow as "snow", ie, "snow" is the name for snow and "snow" refers to snow.

Note that the intension of "snow" is white, such that white is necessary but not sufficient for snow.

But note also that snow does not exist independently of its properties. Snow is small, white ice crystals in the sense that A is A. If the properties small, white and ice crystals were removed, there would be nothing left. IE, it is not the case that first we observe small, white ice crystals and then we observe snow, rather, we observe them contemporaneously as they are the same thing.

When I hear the word "snow", there are two aspects. On the one hand, "snow" denotes snow, in that "snow" is actively pointing out something in the world, namely snow. On the other hand, "snow" is passively being denoted by small, white ice crystals.

Similarly, when I hear the word "unicorn", the word "unicorn" is doing more than pointing to a unicorn. When I hear the word "beauty", the word "beauty" is doing more than pointing to beauty.

IE, words have meaning even when there are no unicorns or beauty in the world for them to point to.

Tarski's usage of denote
Tarski in The Semantic Conception of Truth and the Foundations of Semantics uses denote for one or more items.

For example, he wrote:
1) The expression "the father of his country" designates (denotes) George Washington.
2) We have seen that this conception essentially consists in regarding the sentence "X is true" as equivalent to the sentence denoted by 'X' (where 'X' stands for a name of a sentence of the object language).
3) While the words "designates," "satisfies," and "defines" express relations (between certain expressions and the objects "referred to" by these expressions)
4) We should reconcile ourselves with the fact that we are confronted, not with one concept, but with several different concepts which are denoted by one word

In summary, the meaning of "denote" is much debated, and words do more than pointing to snow and unicorns in the world.

Anyway, I suggest not saying: 'snow' is denoted as snow
But instead: snow is denoted by 'snow' or 'snow' denotes snow — TonesInDeepFreeze

Tarski used "denote", but I don't think this term is strictly grammatically correct, but that is the word he used. I think snow is named "snow" would be better, rather than "snow" denotes snow. Denotes infers points to, and "snow" is doing more than pointing to snow.

I agree with "snow" denotes snow and snow is denoted by "snow", but I still believe that "snow" is denoted as snow is grammatically correct.

Within a sentence, "as" points forwards, and "by" points backwards. The Cambridge Dictionary supports this, giving the examples of i) Fetal heart rate is denoted as the percentage of time in fetal tachycardia per 12-hour period ii) a marking is graphically denoted by a distribution of tokens on the places of the net.
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liar paradox problem — TonesInDeepFreeze

In L, "this proposition is false" is a paradox.
In the world there are no paradoxes. An apple is an apple, if an apple is to the right of an orange then the orange is to the left of the apple, an apple can never be a non-apple.
To avoid paradox in language we need to ensure that language corresponds with the world, because the world is logical.
Tarski is aiming at the same goal.
From the IEP - The Semantic Theory of Truth - "To be satisfactory SDT must conform to so-called conditions of adequacy. More specifically, this definition must be (a) formally correct, and (b) materially correct Condition (a) means that the definition does not lead to paradoxes and it is not circular."

IE, paradoxes in language may be avoided by ensuring that language corresponds with a world that is logical.

(As an aside, correspondence works not when a concept in the mind corresponds with an object in the world, but rather when a concept in the mind corresponds with a public word that has been established during an Institutional Performative Act. The word can be concrete as in "apple" or abstract as in "beauty". Conversation then becomes about the public word, which in its turn corresponds with concepts in the minds of all those taking part in the conversation.)
===============================================================================

Tarski doesn't even say that 'snow is white' is true — TonesInDeepFreeze

You wrote - 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)
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

It is true that Tarski does not say that white is a necessary condition for snow.
However, this is part of the problem that Tarski uses the analytic proposition "snow is white" rather than a synthetic proposition such as "snow is always welcome" .

You wrote - "snow" is precipitation ..............white...............
You didn't write "snow" is precipitation.........which may or may not be white.........

This infers that white is an intension of "snow", meaning that white is a necessary condition for "snow".

Ask anyone in the street whether snow is white or purple, and I am sure nearly all would say white. People know "snow" is white, in an analytic sense.
===============================================================================

Tarski says, "Let us suppose we have a fixed language L whose sentences are fully interpreted." — TonesInDeepFreeze

As an example of interpretation, "snow" is frosty stuff and "white" is the colour of St Patrick's Day T-shirt are external

From the IEP - The Semantic Theory of Truth
"A standard objection against STT points out that it stratified the concept of truth. It is because we have the entire hierarchy of languages Lo (the object language), L1 ( = MLo), L2 (= ML1), L3 (M L2), …. Denote this hierarchy by the symbol HL. It is infinite and, moreover, there is no universal metalanguage allowing a truth-definition for the entire HL."

IE, for each MML there is a language L, and for each language L there is a ML.
Where L = "snow is white"
MML = "snow" is snow and "white" is white
ML = "snow is white" is true IFF snow is white
===============================================================================

The truth or falsehood of 'snow is white' is not dependent on 'snow' naming snow (precipitation...) and 'white' naming white (the chromaticity...). — TonesInDeepFreeze

Given snow is white

If in MML One, "snow" denotes snow - and "white" denotes green
Then in the ML "snow is white" is false

If in MML Two, "snow" denotes snow - "white" denotes white
Then in the ML "snow is white" is true

IE, the truth or falsehood of "snow is white" is dependent on naming in the MML.
===============================================================================

should we take it that Raatikainen's summary of Putnam is correct? — TonesInDeepFreeze

Raatikainen argues against Putnam's objections to Tarski's theory.
However, for me, Raatikainen doesn't make his case, and Putnam's objections to Tarski's theory of truth make sense to me.

Panu Raatikainen, More on Putnam and Tarski
===============================================================================

but maybe Tarski is conceding that we can't have a truth definition that covers all interpretations, but only, for each interpretation, its own truth definition? — TonesInDeepFreeze

You wrote: "Let M interpret 'snow' as the frosty stuff, and 'white' as the color of a St. Patrick's day T-shirt"

Yes, within a particular MML, there is only one interpretation. Between different MML's there are different interpretations.
===============================================================================

If 'white' denotes green, then 'snow is white' is true iff snow is white is not true. But it is still true. Made explicit — TonesInDeepFreeze

Yes, but each new denotation requires a new MML.

In MML One, "snow" denotes snow and "white" denotes green.
Therefore the T-sentence "snow is white" is true IFF snow is white is valid

In MML Two, "snow" denotes snow and "white" denotes white
Therefore the T-sentence "snow is white" is true IFF snow is white is valid
===============================================================================

Tarski's schema is a definition not a claim of a logical truth — TonesInDeepFreeze

IEP - The Semantic Theory of Truth
"To be satisfactory SDT must conform to so-called conditions of adequacy. More specifically, this definition must be (a) formally correct, and (b) materially correct Condition (a) means that the definition does not lead to paradoxes and it is not circular."

Yes, but is founded on logic in order to avoid paradox and circularity.
===============================================================================

Given a language L, and an interpretation M of L, and a sentence P of L: A sentence 'P' is true per M iff P. That's just like any textbook in mathematical logic. No meta-metalanguage. — TonesInDeepFreeze

From Wikipedia - Mathematical Logic - Concerns that mathematics had not been built on a proper foundation led to the development of axiomatic systems for fundamental areas of mathematics such as arithmetic, analysis, and geometry.

In a language L there could be "1 + 1 = 2", "1 + 1 = 5", "1 + 1 = 3"
These may be true or false
The axiom 1 + 1 = 2 exists within a Metametalanguage (MML)
This allows in the Metalanguage (ML) the T-sentence: "1 + 1 = 2" is true IFF 1 + 1 = 2
Note that the axioms are not in the ML, and the ML cannot question the axioms that it has been given.

Truth and Meaning — Banno

Thanks, I have downloaded it. I understand more this week about Tarski's STT than last week, and hopefully more next week than this week, but I think that there is coherent light at the end of the tunnel, unless I'm mistaken.

@TonesInDeepFreeze @Banno

I'm answering my own (grammatically correct) question: "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 ?"

Definitions
"Snow" denotes 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
'White' denotes 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.

Let "snow" denote snow and "white" denote white. Tarski used the word "denote", and so I will continue to use the same word, even if not strictly grammatically correct.

I observe precipitation, etc and name it "snow". The mereological object precipitation, etc is snow. In the sense that A is A, snow is precipitation, etc. To say that snow has the properties precipitation, etc is metaphorical.

I observe achromatic, etc and name it "white". The mereological object achromatic, etc is white. In the sense that A is A, white is achromatic, etc. To say that white has the properties achromatic, etc is metaphorical.

The proposition "snow is white"
1) Snow is precipitation + in the form of ice crystals + that are small + and white + formed directly from water vapour + of the air + at a temperature of less than 32F.
2) Therefore, white is a necessary condition for snow
3) Snow is white in the sense that the intension of snow includes white
4) Even though the T-sentence "snow is white" is true IFF snow is white is given in a Metalanguage (ML), it is assumed that in a Metametalanguage (MML) snow has been named "snow" and white has been named "white".
5) "Snow is white" in the sense that the intension of "snow" includes "white"
6) Therefore, "snow is white" is true because i) snow is white, ii) snow is named "snow" and white is named "white"
7) IE, "snow is white" is not dependent upon a biconditional, as it is an analytic proposition.

Putnam's argument against Tarski's Theory of Truth
Taken from More on Putnam and Tarski - Panu Raatikainen, Tampere University.

Hilary Putnam argued against Tarski's Theory of Truth. He had two basic objections, ’the unsoundness objection’ and ‘the modal objection’.

I doubted that the T-sentence could be formally correct, if snow had not been named "snow" and white had not been named "white" within the ML.

The answer to my own question is that the notion of naming does not occur in Tarski’s definition of truth, but only in the Criterion of Adequacy, and being a test of a definition, is formulated only in the metametalanguage (MML).

Tarski always said that truth can only be defined for a particular formalized language, a language that had already been interpreted, where the meaning of the object language was fixed and constant. Truth is relativized for a particular object language

In the event that the object language was reinterpreted, for example defining "green" as white, the language changes to a different language, requiring a different T-Sentence

IE, precipitation, etc has been denoted as "snow", and achromatic, etc has been denoted as "white" in a MML.

This raises the problem that truth in the ML depends on arbitrary decisions in the MLL, ie, naming white as "white" rather than as "green". Putnam complained that it isn’t a logical truth that the (German) word ‘Schnee’ refers to the substance snow, nor is it a logical truth that the sentence ‘Schnee ist weiss’ is true in German if and only if snow is white.

Putnam made the point that the truth in the ML now becomes dependent on a truth in a MLL, saying "And, pray, what semantical concepts will you use to state these ‘semantical rules’? And how will those concepts be defined?” (Putnam 1988)

In summary, the truth of Tarski's T-sentence in a ML has been pushed back to a MML.

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 ?

Before the truth value of any proposition is known, metaphysical or otherwise, the meaning of the words must be known. For example, is the proposition "mbeya majipu" true or false ?

First, do the words have a meaning in the first place, and if they do, who or what determined their meaning ? And if their meaning has been determined, where is this meaning to be found ?

IE, the truth value of a synthetic proposition cannot be known empirically until the meaning of the words within it are known analytically.

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" ?

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).

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).

@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".

The T-sentence is in the metalanguage, while the quotes name a sentence in the object language. — Andrew M

Considering Tarski's T-Sentence "snow is white" is true IFF snow is white, you are right that the T-sentence is the metalanguage, not the right-hand side of the biconditional.

Yes, but naming it doesn't affect what it is. 200,000 years ago, snow wasn't named "snow", and the color white wasn't named "white", yet snow was still white. — Andrew M

Certainly if the word "white" were used to denote the color green then the sentence, "snow is white" would be false (since snow is not green). — Andrew M

We agree that snow is white, and we agree that "white" could have been used to denote the colour green.

Assume that "green" was used to denote the colour white.
Given that snow is white, "snow" denotes snow, "white" denotes green and "green" denotes white, then "snow is white" denotes snow is green.
Also, "snow is green" denotes snow is white.
If snow is white, then "snow is green" is true.
IE, "snow is green" is true IFF snow is white.

In that event, doesn't this mean that Tarski's T-sentence would be false ?

Yes, but naming it doesn't affect what it is. 200,000 years ago, snow wasn't named "snow", and the color white wasn't named "white", yet snow was still white. — Andrew M

The truth of Tarski's T-Sentence depends on how snow and white have been named

Today, we have Tarski's T-Sentence "snow is white" is true IFF snow is white. The left hand side is the object language, the right hand side is the metalanguage.

In the metalanguage, snow is white, meaning that, in the domain of the metalanguage, snow is white, ie, in the world of the metalanguage, snow is white. The world of the metalanguage may or may not correspond with our world.

I agree that in our world snow is white. However, in the world of the metalanguage, snow may or may not be white.

Possibility One - in the world of the metalanguage, snow has the property white and apples have the property green

1) Let "snow" name snow, "white" name white
Then "snow is white" is true
2) Let "snow" name snow, let "white" name green
Then "snow is white" is false

1) Let "snow" name apple, "white" name white
Then "snow is white" is false
2) Let "snow" name apple, let "white" name green
Then "snow is white" is true

Possibility Two - in the world of the metalanguage, snow has the property green and apples have the property purple.

1) Let "snow" name snow, "white" name white
Then "snow is white" is false
2) Let "snow" name snow, let "white" name green
Then "snow is white" is true
3) Let "snow" name snow, let "white" name purple
Then "snow is white" is false

1) Let "snow" name apple, "white" name white
Then "snow is white" is false
2) Let "snow" name apple, let "white" name green
Then "snow is white" is false
3) Let "snow" name apple, let "white" name purple
Then "snow is white" is true

Summary

If snow is white in a metalanguage, "snow is white" may or may not be true dependant upon how snow and white have been named. Therefore, it is not necessarily true that "snow is white" is true IFF snow is white.

IE, the truth of Tarski's T-Sentence depends on how snow and white have been named, or as Kripke said, "baptised".

Tarski didn't think of the T-sentence as being a definition of truth and, I'd add, neither was his actual definition of truth circular. — Andrew M

I am curious why naming plays no part in Tarski's T-sentence, as naming seems to affect the truth or falsity of the T-sentence itself. Am I missing something ?

The problem of naming

Tarski proposed:

The T-sentence - "snow is white" is true IFF snow is white.
A definition of truth can be obtained in a very simple way from that of another semantic notion, namely, of the notion of satisfaction.
Satisfaction is a relation between arbitrary objects and certain expressions called "sentential functions." These are expressions like "x is white,"

A sentence such as "snow is white" is true if in the sentential sentence "x is white", x is satisfied by snow.

200,000 years ago snow had not been named. Today, snow has been named, whether "white" in English or "schnee" in German. Therefore, there must have been a point in time when snow was named "snow", ie, what Kripke calls "baptised".

Although the right hand side of Tarski's biconditional is a metalanguage, as he uses the example of the object snow and the property white, for the moment consider a world whereby snow is white. In a world whereby snow is not white, or we consider the general T-sentence "P is Q" is true IFF R is S, the same problem of naming occurs.

Before naming snow as "snow" and white as "white"
As "white" didn't exist, in the sentential function "x is white", there is no x that satisfies "white", therefore "snow is white" can never be true.

After naming snow as "snow" and white as "white"
As snow has been named "snow" and white has been named "white", in the sentential function "x is white", x is always satisfied by snow. Therefore, "snow is white" is always true.

In summary, the T-sentence is false before snow had been named "snow" and white named "white". The T-sentence is always true after snow had been named "snow" and white named "white". IE, the T-sentence itself may be either true or false dependant upon how its parts have been named.

If the parts exist, their collection necessarily exists too.........Collections in a spacetime can have causal relations between them — litewave

I don't see much difference between a galaxy posited as an abstract entity and me as an actual entity — universeness

Convention of quotation marks
Using the convention of Davidson's T-sentence "snow is white" is true IFF snow is white, where with quotation marks refer to language and the mind and without quotation marks refers to a world.

Sets (to my understanding)
A set is a collection of elements. A set with no elements is "empty", a set with a single element is a "singleton", elements can be numbers, symbols, variables, objects, people and even other sets. A set is an abstract, such that its elements don't have to be physically connected for them to constitute a set. An object is not a set, though it can be a set of objects.

Platonists vs Nominalists
A Platonist would argue that "galaxies" exist in a mind-independent world, whereas a Nominalist would argue that they don't. For the Nominalist, an apple in the world is a projection of the concept "apple" in the mind onto the world.

See SEP - Abstract Objects - https://plato.stanford.edu/entries/abstract-objects

I agree that galaxies exist in a mind-independent world, I agree that "galaxies" exist in the mind, but I don't agree that "galaxies" exist in a mind-independent world.

Argument Three against Platonism
A Platonist would argue that "apples" exist in a mind-independent world, a Nominalist would argue that apples exist in a mind-independent world.

It is argued that if two people observe the same world, and both independently perceive an "apple" then an apple exists in the world. However, I may observe the world and perceive a "duck", whilst someone else perceives a "rabbit".

IE, it does not necessarily follow that because we both perceive the same "object", then that object exists in the world.

Sets and Galaxies
@Kuro started the thread by asking about a set of all that exists. The word "exist" needs to be defined.

A Nominalist would argue that as sets are abstract, and as abstracts don't exist in a mind-independent world, neither do sets. Therefore, sets can only exist in the mind. A set of stars exists in the mind as a "galaxy". Galaxies exist in a mind-independent world.

A Platonist would argue that although abstracts exist in a mind-independent world, they are independent of any physical world. As abstracts exist in a mind-independent world, and as sets are abstract, then sets can exist in a mind-independent world. Sets can also exist in the mind. Therefore, "galaxies" exist both in the mind and in a mind-independent world. Galaxies also exist in a mind-independent world.

A galaxy is a gravitationally bound system of stars, stellar remnants, interstellar gas, dust, and dark matter. These physical parts are connected by physical forces, such as gravity.

A "galaxy" is an abstract entity of physical parts. These physical parts are connected, but not physically. If an object, such as a "galaxy", is a collection of parts, such as stars, there must be some kind of connection between the parts, otherwise it wouldn't be an object.

For the nature of connections see SEP - Relations - https://plato.stanford.edu/entries/relations

In summary, "galaxies" must be distinguished from galaxies.

I would say that all combinations exist regardless of the mind..........But who says that parts of an object need to be in causal contact? — litewave

In Mathematical Platonism, sets exist in the world as abstract entities. The parts don't need to be in causal contact. Yet the parts must be connected in some way in order for the set to exist. How exactly ? How are things in the world abstractly connected ? By what mechanism ?

Infinite number of objects doesn't seem sensible? — litewave

For a world to start off with 3 objects and end up with an infinite number of objects because of the ontological existence of sets doesn't seem sensible.

Tree's rocks and stars can exist as composites without the labels tree, rock, or star. — universeness

You write that trees, rocks, stars, solar systems, etc are combinatory systems that can exist
independently of a lifeform and can exist without the labels tree, rock, star, solar system, etc.
@Kuro wrote "Suppose that all that exists forms a set."

Taking the Milky Way Galaxy as an example, The Milky Way Galaxy is an object, a combinatory system, a collection of things. As an object it is a set of parts.

The Mathematical Platonist would argue that the Milky Way Galaxy as a set of parts exists as an abstract entity, independent of any mind. A Nominalist would disagree.

If the Milky Way Galaxy exists as an abstract entity, by what mechanism do you propose that the parts are connected, parts that could be 87,000 light years apart ?

Taking sets to exist is the most natural interpretation of the existential quantifier in set theory without awkward paraphrases — Kuro

I am sure that both Platonists and Nominalists agree that sets exist. The question is where, in the mind or mind-independent.

Frege argued for mathematical Platonism as the only tenable view of mathematics, yet objectors include Psychologists, Physicalists and Nominalists.

Quine-Putnam's Indispensability Argument argued for the existence of abstract mathematical objects, such as numbers and sets, yet persuasive objectors include Harty Field.

I agree that the Existential Quantifier, having the meaning "there exists", "there is at least one", "for some", is invaluable in logic. For me, however, the most natural interpretation of "exist" means within the mind.

IE, the most natural interpretation for one person may be different to another person's.

What doesn't exist only in the mind then? Non-composite objects? — litewave

Yes, a tree is a combination as is a grain of sand, a rock or a star. They need no lifeform to exist as combinations of fundamentals. — universeness

If combinations don't ontologically exist in a mind-independent world (aka relations) but do exist in the mind, then:
i) what exists in the mind-independent world are fundamental forces and fundamental particles. These fundamental particles may be called "objects", and are non-composite.
ii) a tree, which is a combination of parts, can only exist in the mind.

Argument One against sets as combinations existing in the world
From before, if only 3 things were introduced into a world, and if sets as combinations did exist, then an infinite number of other things would automatically be created. This doesn't seem sensible.

Argument Two against sets as combinations existing in the world
If combinations exist in the world, then an object such as an apple would exist as a set of parts. It would follow that one part 8cm distant from another part would be in combination.

The Earth would exist as an object, meaning that one part 12,000 km from another part would be in combination.

The Milky Way Galaxy would exist as an object, meaning that one part 87,000 lights years from another part would be in combination.

If being in combination was instantaneous, then the combination between two parts of the Milky Way Galaxy 87,000 light years apart would be instantaneous. But this would break the physical laws of nature as we know them, and would need to be justified.

If being in combination followed the physical laws of nature as we know them, then two parts could only be in combination once information had travelled between them at the speed of light. This raises a further problem.

If, during the 87,000 years it took for the two parts to become in combination, one or both of the parts ceased to exist, then a combination would come into existence without any parts. This doesn't seem sensible.

IE, Platonic Sets existing in a mind-independent world sounds fine until one considers the real world implications.

The apple exists as a set of parts in the mind. When the mind believes that it is observing an apple in the world, for the apple to also exist in this observed world as the same set of the same parts would be an example of overdetermination.

IE, an apple does not need to exist in the world in order for the mind to believe that it is observing an apple in the world.

Some people think of a set as being some abstract, Platonic entity that "exists" in some sense, distinct from its members? — Michael

The 'extra things' come from combination of the fundamentals — universeness

A set is a combination of things. I would accept that "combinations" exist in the mind. I would accept that the mind observes "combinations" in the world when observing the world. I would accept that there are forces between things in a mind-independent world, but the concept of force is different to the concept of "combination". A set of things does not require that there are forces between these things.

Q1 - do "combinations" exist in the world when the world is not being observed ?

Is there any persuasive argument that "combinations" do exist in a mind-independent world ? I have yet to come across one.

Suppose that all that exists forms a set — Kuro

In the world exists x, y and z.
Suppose x,y and z exist in the world.

This gives us 6 sets - (x) - ( y) - (z) - (x,y) - (x, z) - (y,z).

If sets exist in the world
You say that sets "exist".

Q1 - If sets exist in the world, we start off with 3 things that exist and end up with 6 things that exist. Where did the extra 3 things come from ?

But if sets do exist in the world it gets worse.
Let set F be (x), set G be ( y), set H be (z), set J be (x,y), set K be (x,z) and set L be (y,z).
This gives us the additional sets (F), (G), (H), (J), (K), ( L), (F,G), (F, H), (F, J), ((F,K), (F,L), (F,G,H), (F,G,J), (F,G,K), (F,G,L), (F,H,J), (F,H,K), (F,H,L), (F,J,K) - etc - a lot.
We can continue the same process and end up with the existence of an infinite number of possible sets.

Q2 - if sets do exist in the world, we start off with 3 things that exist and end up with an infinite number of things that exist. Where did the extra things come from ?

If sets don't exist in the word
If sets don't exist in the world, life is a lot simpler, and the only things that exist in the world are x,y,z.

The implication is, that as an object such as an apple is only a set of parts, as sets don't exist in the world, then apples don't exist in the world, which is my belief.

IE, set E (x,y,z) is the set of all that exists in the world.

That in another possible world snow is green, but 7+5 is 12 in all possible worlds................And is not obviously related to T-sentences. — Banno

T-Sentences
Consider the T-sentence "snow is white" is true IFF snow is white.

Note that "snow is white" is being used in the sense that white is one of the properties of the object snow, not that white is the only property of snow.

The right hand side of the biconditional
For Tarski, the right hand side is a Metalanguage, which is not the world.
For Davidson, the right hand side is the world, in that for Davidson, T-Sentences are laws of empirical theory. For Davidson, I can hear someone saying "schnee ist weiss" and see them pointing to white snow. A similar approach to Wittgenstein's Tractatus, in that the understanding of language is founded on what is shown rather than what is said.

Naming
In the world one million years ago, snow existed but the word "snow" didn't. Today, the word "snow" exists. Therefore, there must have been a moment in the past whereby snow was named "snow". This may be called a Performative Act.

Does snow exist in the world?
Yes, if relations ontologically exist in the world. No, if relations don't ontologically exist in the world.

Note that if relations don't ontologically exist in the world, then neither can the equation 7+5=12 exist in the world. In this event, the equation 7+5=12 cannot exist in all possible worlds, and therefore cannot be necessary.

As I have not come across any persuasive argument that relations do ontologically exist in the world, my belief is that they don't.

Assuming for the sake of argument that relations do ontologically exist in the world

Situation One - in the world, the properties cold, white and frozen exist

The properties cold, white and frozen exist as the mereological object snow.

Therefore, snow is white is true.

Let the property cold be named "cold", the property white be named "white" and the property frozen be named "frozen".

Let the properties cold, white and frozen be named the object "snow".

"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"

"Snow is white" is false IFF not only i) snow is white but also ii) snow has not been named "snow" and white has not been named "white".

Situation Two - in the world, the properties cold, white and frozen don't exist

Then the mereological object snow doesn't exist.

Snow is white is false because the properties cold, white and frozen don't exist.

"Snow is white" is false because snow is white is false

Summary

"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".

Anil Gupta says that Tarski's biconditionals are central to the concept of truth, yet introduce circularity, such that i) from "p is true" can infer p ii) from p can infer "p is true" iii) such that "p is true" is equivalent to p.

However, the biconditional given above is not circular, as the truth of "snow is white" depends on a contingency, namely, that of the Performative Naming of properties and objects observed in the world.

"Snow is white" is not analytic...Keep working on it. — Banno

Random searches on the internet agree that "snow is white" is not analytic. For example, from www.oxfordbibliographies.com: "The existence of analytic truths is controversial. Sceptics have sometimes argued that the idea of an analytic truth is incoherent".

I am still not convinced. The problem is one of logic. In what fundamental way is "snow is white" different to "seven plus five is twelve". I hope next time I will have a deeper understanding and a more persuasive argument. :smile:

The analytic.synthetic distinction makes not difference to the T-sentence; in works for both.............Your use of "designation" is nothing like Tarski's. It's closer to Austin's discussion of performative utterances — Banno

Hopefully, I'm not repeating myself too much.

Designation has at least two senses, one as used by Tarski, and one as used by Austin. Both are relevant to the T-Sentence.

Austin and designation
"I name this ship the Queen Elizabeth" is a Performative act, whereby the ship has been christened the "Queen Elizabeth". The performative utterance gives an unnamed object a name, a designation, by which it is henceforth known. There is a free choice as to what objects may be named. For example, snow may equally be named "white" or "black". If snow is named "white", then "snow is white" is true.

Tarski and designation, satisfaction and definition
Tarski sets out certain definitions in The Semantic Conception of Truth and the Foundations of Semantics

The expression "the father of his country" designates (denotes) George Washington.
Snow satisfies the sentential function (the condition) "x is white".
The equation "2*x = 1" defines (uniquely determines) the number 1/2.
Where the words "designates", "satisfies" and "defines" express relations between certain
expressions and the objects "referred to" by these expressions.
While the words "designates," "satisfies," and "defines" express relations (between certain
expressions and the objects "referred to" by these expressions), the word "true" is of a different
logical nature: it expresses a property (or denotes a class) of certain expressions, viz., of
sentences.

"All notions mentioned in this section can be defined in terms of satisfaction. We can say, e.g., that
a given term designates a given object if this object satisfies the sentential function "x is identical
with T" where 'T' stands for the given term.
Similarly, a sentential function is said to define a given object if the latter is the only object which
satisfies this function."

In other words:

Designation and satisfaction
As regards analytic propositions:
i) If snow satisfies "x is identical with white" then "is white" designates snow.
ii) If snow satisfies "x is identical with black" then "is black" designates snow.
Snow may be identical to "white" in the sense that snow has the property of being "white".

As regards synthetic propositions:
iii) If snow satisfies "x is identical with being on the ground" then "being on the ground" designates snow.
iv) If snow satisfies "x is identical with not being on the ground" then "not being on the ground" designates snow.
Snow may be identical to "being on the ground" in the sense that snow may be observed "being on the ground".

Definition and satisfaction
i) If snow is the only object that satisfies "x is white" then "x is white" defines snow. As many objects can be white, "x is white" doesn't define snow.
ii) If snow is the only object that satisfies "x is on the ground" then "x is on the ground" defines snow. As many objects can be on the ground, "x is on the ground" doesn't define snow.

Analytic proposition "snow is white"
During a Performative Utterance, a previously unnamed property is designated "white". Subsequently, a previously unnamed object with the property "white" is designated "snow"
As "snow is white" is always true, then "snow is white" is true.

During a Performative Utterance, a previously unnamed property is designated "black". Subsequently, a previously unnamed object with the property "black" is designated "snow"
As "snow is black" is always true, then "snow is black" is true.

Synthetic proposition "snow is on the ground"
Subsequently, during a Performative Utterance, a previously unnamed object is named "ground"

The object named "snow" may or may not be on the object named "ground"

If snow is on the ground:
"Snow is on the ground" is true IFF snow is on the ground
"Snow is on the ground" is false IFF snow is not on the ground

If snow is not on the ground:
"Snow is not on the ground" is true IFF snow is not on the ground
"Snow is not on the ground" is false IFF snow is on the ground

Summary
To my understanding, whether a proposition is analytic or synthetic makes a difference to the T-Sentence, because the truth of an analytic proposition is determined by a Performative Utterance, which is not the case for a synthetic proposition.

white is not a property but just the most commonly seen appearance of snow...Unfortunately this leads away from the OP topic which presumes truth for T-sentences. — magritte

In the dictionary, snow is defined as atmospheric water vapour frozen into ice crystals and falling in light white flakes or lying on the ground as a white layer. A property is defined as an attribute, quality, or characteristic of something. For example, the dictionary does not define snow as "as atmospheric water vapour frozen into ice crystals and falling in light flakes of various colours or lying on the ground as a layer of various colours".

To this reading, white is a property of snow.

It is also true that FH Bradley noted that the nature of an object's properties is problematic.

However, I do think that the difference between analytic and synthetic propositions is central to the nature of T-Sentences.

Snow is black shocks because it is contradictory to white and thus supposedly logically impossible. — magritte

Snow can appear black at night, can appear white in sunlight, can appear red at sunset and can appear grey at dusk.

All these are contradictory yet logically possible.

It seems unlikely that the fundamental nature of snow changes with the light.

Adding white to snow is a synthetic addition to my more modern understanding because on a dark night snow could be black instead. — magritte

The word "is" has many meanings. For example - i) "snow is black on a dark night", where "is" means "appears to be" - ii) "snow is white", where "is" means "has the property" - iii) "snow is angry", where "is" is being used metaphorically - iv) "snow is welcome", where "is" is being used ironically, etc.

Tarski in "snow is white" is using "is" to mean "has the property", in which case "snow is white" is analytic.

To say "snow is black on a dark night" is a synthetic proposition, as it can be expanded to "snow which has the property of being white appears black on a dark night"

So yes, the T-sentences are not a theory of truth, at least in that they do not tell us which sentences are true and which false, but which sentences have the same truth value. — Banno

Propositions may be either analytic or synthetic
I would move on to Davidson if it weren't for my confusion with Tarski's Semantic Theory of Truth (STT), in that it does not differentiate between analytic and synthetic propositions. For example, the proposition "snow is white" is analytic, whereas "snow is on the ground" is synthetic. Note that the word "is" does not mean "is a synonym of" but rather "has the properties of", thereby avoiding Quine's Two Dogmas of Empiricism problem.

The matter is complicated by the fact that Tarski himself used an analytic proposition "snow is white" to illustrate the T-Schema "p" is true IFF p which is dependent on synthetic propositions.

Tarski wrote in The Semantic Conception of Truth: and the Foundations of Semantics 1944 - "Consider the sentence ‘snow is white.’ We ask the question under what conditions this sentence is true or false. It seems clear that if we base ourselves on the classical conception of truth, we shall say that the sentence is true if snow is white, and that it is false if snow is not white. Thus, if the definition of truth is to conform to our conception, it must imply the following equivalence: The sentence ‘snow is white’ is true if, and only if, snow is white."

Analytic Propositions
Consider something in an Object Language (OL) that is "green, circular and distant". Still within the Object Language, I designate this something as "snow".

As long as "snow" has been designated "green, circular and distant", then it follows that not only is "snow" satisfied by the predicate "green, circular and distant" but also that "snow is green, circular and distant" is true.

"Snow" is therefore independent of anything that may or may not exist in the Metalanguage (ML). Similarly with all analytic propositions.

For example, as long as "snow" has been designated as "white" in the object language, then it follows that not only is "snow" satisfied by the predicate "white" but also that "snow is white" is true.

For example, as long as "unicorn" has been designated as "a horse with a single horn", then it follows that not only is "unicorn" satisfied by the predicate "a horse with a single horn" but also that "a unicorn is a horse with a single horn" is true.

IE, analytic propositions don't require a Metalanguage in order to be true. Analytic propositions are Theories of Truth using the Performative Speech Act.

"Designation" is a Theory of Truth
Designation is a Performative Speech Act, in that "I name this ship Queen Elizabeth" means the same as "I designate the name of this ship the Queen Elizabeth".

Designation as a Performative Speech Act is a Theory of Truth, in that "designation" establishes what is true. Once one knows what is true, it follows that one knows the conditions of satisfaction.

IE, designating something "green, circular and distant" as "snow" establishes that "snow is green, circular and distant" is true. It follows that the predicate "green, circular and distant" then must satisfy the subject "snow".

Synthetic Propositions
Consider the synthetic proposition "snow is on the ground" in the Object Language.

In the Metalanguage, either snow is on the ground or snow is not on the ground.

Let "snow" in the OL be designated snow in the ML, and let "ground" in the OL be designated ground in the ML.

Situation A) - in the ML, snow is on the ground.
i) The predicate "is on the ground" in the OL is satisfied by the predicate is on the ground in the ML
ii) "The snow is on the ground" is true IFF the snow is on the ground.
iii) "The snow is on the ground" is false IFF the snow is not on the ground.

Situation B) - in the ML, snow is not on the ground.
i) The predicate "is not on the ground" in the OL is satisfied by the predicate is not on the ground in the ML
ii) "The snow is not on the ground" is true IFF the snow is not on the ground.
iii) "The snow is not on the ground" is false IFF the snow is on the ground.

IE, as the T-Schema does not tell us whether snow is or isn't on the ground, Tarski's SST is not a Theory of Truth.

The STT is not a Theory of Truth
The IEP article "The Semantic Theory of Truth" notes that "STT as a formal construction is explicated via set theory and the concept of satisfaction. The prevailing philosophical interpretation of STT considers it to be a version of the correspondence theory of truth that goes back to Aristotle"

As to my understanding, the STT is not a Theory of Truth, including the Classical Correspondence Theory of Truth, it seems to me that the quote above from the IEP is incorrect.

"This sentence is false"
There are many possible Theories of Truth - Correspondence Theory of Truth, Evidence Theory of Truth, Performative Theory of Truth, Coherence Theory of Truth, Common Agreement Theory of Truth, Utilitarian Theory of Truth, etc. Tarski requires a Theory of Truth to be formally correct, ie to avoid paradox.

If a Particular Theory of Truth leads to paradox, the conclusion is that this particular Theory of Truth is not valid, not that there isn't a Theory of Truth that doesn't lead to paradox.

Summary
To my understanding, 1) Tarski's T-Schema "p" is true IFF p is not a Theory of Truth, but establishes the conditions necessary for a Theory of Truth for synthetic propositions.

2) Designation is a Performative Act which is a Theory of Truth for analytic propositions.

To my understanding: Tarski's Semantic Theory of Truth is not a Theory of Truth

Tarski in his Semantic Theory of Truth (STT) requires any Theory of Truth to be formally correct and materially correct. Formally correct means it does not lead to a paradox. Materially correct is formulated as Convention T, whereby the truth of the proposition "schnee ist weiss" in an Object Language is given in a Metalanguage as snow is white

In the Object Language are names of objects, such as "snow", "house", "government", etc, and names of properties, such as "red", "distant", "large", etc. In the Metalanguage are the same names, ie, snow, house, government, red, distant, large, etc.

Any name in the Object Language can be designated any set of names in the Metalanguage. For example, "snow" may be designated green, circular and distant.

But who designates "snow" as green, circular and distant? Either an individual or an Institution can designate a name, although generally this is done by Institutions.

And on what basis does an Institution designate a name? It could be designated in either a performative act, such that "truth is what I say it is", or by correspondence with the world, such that "snow" corresponds with snow.

Tarski's Semantic Theory of Truth is not a Theory of Truth, in that it doesn't specify which Theory of Truth should be used, only that a Theory of Truth must be used. The Semantic Theory of Truth is establishing the conditions under which a Theory of Truth may be used.

For example, if the Theory of Truth to be used is the Performative Theory of Truth, let "snow" designate distant, green, circular. As "snow" is satisfied by circular, then "snow is circular" is true. The T-Schema may be written "snow is circular" is true IFF snow is circular.

If the Theory of Truth to be used is the Correspondence Theory of Truth, let "snow" designate cold, white, frozen, As "snow" is satisfied by white, then "snow is white" is true. The T-Schema may be written "snow is white" is true IFF snow is white.

Within Tarski's Semantic Theory of truth, both i) "snow is circular" is true IFF snow is circular is true and ii) "snow is white" is true IFF snow is white is true.

Within the Performative Theory of Truth, only "snow is circular" is true IFF snow is circular is true. Within the Correspondence Theory of Truth, only "snow is white" is true IFF snow is white is true.

IE, Tarski's Semantic Theory of Truth is establishing the conditions under which a Theory of Truth may be used.

Tarski seems to be the theory in which folk are most interested — Banno

"Small moves, Banno, small moves"

This is how Tarski avoids the problem to which @Michael drew attention — Banno

My understanding is correct IFF my understanding is correct.

My instinctive belief is as @Michael wrote ""'p' is true iff p" isn't the definition of truth but something which follows from whatever the actual definition is". This leads into @Banno's quote that "Neither correspondence nor coherence are at work here. It's a formal language with truth defined in terms of satisfaction."

Consider "schnee ist weiss" is true IFF snow is white

What is "designation"
I perceive the word "snow" and designate it "schnee", such that "schnee" mean "snow". I perceive the word "white" and designate it "weiss", such that "weiss" means "white".

What is the mechanism of "satisfaction"
From the IEP: "The Semantic Theory of Truth"
Consider the open formula "x is a city", open because it has a free variable.
The formula is satisfied by London, so "London is a city" is true
The formula is not satisfied by The Thames, so "the Thames is a city" is false
Satisfaction turns an open formula into a true sentence, and non-satisfaction turns an open formula into a false sentence

Based on @Banno, an object o satisfies a predicate f IFF either the object o is snow and the predicate f is "is snow" or the object o is schnee and the predicate f is "ist weiss".

What is intensional and extensional
The quote marks around "snow is white" makes it intensional. Intensional means analytic, reasoning from abstract rules, what Quine calls "meaning" and what is necessary to make a concept.

P ↔ Q means biconditional, it means P IFF Q, it also means P implies Q and Q implies P. The meanings of P and Q are extensional. Extensional means synthetic, involves examples from the world, what Quine calls "reference" and what is contingent to a concept.

Replacing the T-Schema "schnee ist weiss" is true IFF snow is white by the equivalent T-Schema "snow is white" is true IFF snow is white

@Banno wrote: "Tarski gets past this for formal languages by developing the mechanism of satisfaction, so that he has extensionally transparent terms on both sides of the equivalence". This means that the extensional meaning of "schnee ist weiss" is the same as the extensional meaning of snow is white, because of the mechanism of satisfaction.

Because "schnee" has been designated as "snow", and as ""weiss" has been designated as "white", the intensional meaning of "schnee is weiss" is the same as the intensional meaning of "snow is white".

Therefore we can replace the T-Schema "schnee ist weiss" is true IFF snow is white by the equivalent T-Schema "snow is white" is true IFF snow is white.

Introducing extensional meaning
Tarski is saying that the extensional meaning of "snow is white" is equivalent to the extensional meaning of snow is white.

In this case, the T-Schema may be written as: the extensional meaning of "snow is white" is true IFF it is equivalent to the extensional meaning of snow is white.

The problem with the extensional meaning
The problem is, why should the extensional meaning of "snow is white" be equivalent to the extensional meaning of snow is white?

The T-Schema has become tautological. It may be written in full as: the extensional meaning of "snow is white" is true IFF it is equivalent to the extensional meaning of snow is white given that the extensional meaning of "snow is white" is equivalent to the extensional meaning of snow.

The core problem with the T-Schema as a definition of truth without circularity is that it is founded on a conditional, the conditional IFF, which is saying no more than x is true IFF x is true. The T-Schema is a tautology, it is analytic.

The truth tables reinforce the conditionality of the T-Schema
The truth table for material conditional uses the conditional, and the truth table for material biconditional uses the conditional.

A valid definition of truth cannot be founded on a conditional
A definition of truth cannot be founded on a conditional, as this leads to a tautology. A valid definition of truth must avoid the conditional. For example, i) "truth is what I say it is" is a valid definition of truth, ii) the performative act "I name this ship Queen Elizabeth" means that it is true that this ship is named Queen Elizabeth and iii) I perceive something in the world and name it "snow" means that it is true that "snow" is snow.

As an aside, my perception of something in the world that is cold, white and frozen and name it "snow" means that there is a correspondence between "snow" and snow. However such correspondence is not purely cognitive, but is founded on something visceral, thereby avoiding the problem of belief as a truth bearer.

Summary
The T-Schema is based on the conditional IFF, which is fixed by the mechanism of satisfaction. Yet the mechanism of satisfaction is itself based on another conditional, again leading to circularity.

It seems to me that a valid definition of truth cannot rely on a conditional, which Tarski's Semantic Theory of Truth does.

"Snow is turquoise with purple polkadots" is true IFF snow is turquoise with purple polkadots — Banno

I am attempting to understand Tarski's logic of of truth.

Tarski's T-Schema states "S" is true IFF S

Tarski's Semantic Definition of Truth establishes the T-Schema, whereby "S" is true IFF S, where "S" is in an Object Language, and S is in a Metalanguage.

The following T-Schema are all true:
"Snow is turquoise with purple polkadots" is true IFF snow is turquoise with purple polkadots.
"Snow is white" is true IFF snow is white
"Snow is a volcano" is true IFF snow is a volcano.

If S was not limited in some way, the T-Schema would be "S" is true

For every possible statement "S" in an object language, an S may be found in a metalanguage. For example, given the proposition "snow is a volcano", there is a true T-Schema such that "snow is a volcano" is true IFF snow is a volcano.

It follows that for every possible "S", a true T-Schema may be found, meaning that every possible "S" will be true.

If every possible "S" is true, the term IFF becomes redundant, and the T-Schema may be reduced to "S" is true.

What limits possible values of S ?

However, there is a term IFF in the T-schema, meaning that not all propositions "S" in an object language are true. It follows that there are limitations as to what S can be in the metalanguage.

My belief is that the S in the metalanguage is limited by correspondence with the world, in that I perceive something in the world that is cold, white and frozen, but I don't perceive something in the world that is cold, a volcano and frozen.

However, if S is not limited by correspondence with the world, yet S must be limited by something (otherwise the T-Schema would be "S" is true), then what does limit S ?

What prevents some values of S from being a possibility in the metalanguage ?