I'm a bit confused right now. The notion of a definition includes a word which can be any damn thing you want (arbitrary) although etymology-based ones tend to make sense and are more easily recalled + what the definiens lists are, conventionally, essential features (not arbitrary) of that element/set the word is assigned to. I think I'm making a noob mistake; sorry, I'm new to the game (of philosophy).

Yes, you are right and I see your point. But I was referring to the type of language implemented. If you check logic premises they tend to be pretty hard to follow. This is why I wonder if it is more difficult to me to reach "truth" because I can't follow the logic rules.

Yes, you are right and I see your point. But I was referring to the type of language implemented. If you check logic premises they tend to be pretty hard to follow. This is why I wonder if it is more difficult to me to reach "truth" because I can't follow the logic rules. — javi2541997

That's the same problem I face. We're, it seems, in the same boat. Apologies. :smile:

Specialized symbols translate into speed & brevity if you're using pen/pencil & paper. However, have you noticed how cumbersome it is to write logical & mathematical symbols online? LaTex is unweildly & time-consuming. How odd!

And yep I concur, technical symbols tend to be hard to memorize and adds another barrier between us and what's being conveyed - it's like learning an new language and you know how difficult that is. However, once one has the language under one's belt, learning is accelerated. That's what I think anyway; mileage may vary.

I'm a bit confused right now. The notion of a definition includes a word which can be any damn thing you want (arbitrary) although etymology-based ones tend to make sense and are more easily recalled + what the definiens lists are, conventionally, essential features (not arbitrary) of that element/set the word is assigned to. I think I'm making a noob mistake; sorry, I'm new to the game (of philosophy). — Agent Smith

One must be wary of "etymology-based" definitions. The definition employed by the logician will significantly restrict the word's usage in comparison to the common usage. However, the word still has all that baggage within the reader's mind, habitual associations. The dishonest logician (sophist) will employ that baggage (equivocation) to produce the appearance of valid conclusions which are really invalid. The conclusions are invalid because they require making associations outside of what is stipulated by the significantly restricted definition.

Michael provided an example:

“John is a bachelor” is true iff John is a bachelor

“John is a bachelor” is true iff John is an unmarried man

This shows us the meaning of “bachelor”. — Michael

There are no definitions provided here, but we must assume that "true" means the same thing in both instances. Also, "iff" signifies a special relation, and the second phrase on the right side of iff must have the same special relation with the proposition "John is a bachelor" as the first one does. This is stipulated by "is true iff", because the meaning of "is true iff" must remain static.

We can only conclude that "a bachelor" means something different from "an unmarried man" if we allow that the first "true" means something different from the second "true".

This is why the example shows us nothing about the meaning of "bachelor". It does not provide a definition of "bachelor" because a definition is to place the word into a wider context.. Here, the two "bachelor" and "unmarried man" are placed in the exact same context, so we have no definition. It is only if you allow your mind to wander, and think that "man", and "unmarried" have a wider context of meaning, that the illusion is created that something has been said about the meaning of "bachelor". But such a wandering mind is not allowed in logic, because it contaminates the soundness, and produces invalid conclusions by way of equivocation.

One must be wary of "etymology-based" definitions. The definition employed by the logician will significantly restrict the word's usage in comparison to the common usage. However, the word still has all that baggage within the reader's mind, habitual associations. The dishonest logician (sophist) will employ that baggage (equivocation) to produce the appearance of valid conclusions which are really invalid. The conclusions are invalid because they require making associations outside of what is stipulated by the significantly restricted definition. — Metaphysician Undercover

Yeah, it can be misleading I hear - I came across an example or two which I can't recall at the moment (Memory Access Failure). Much obliged for the warning!

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.

Yes, more or less, with a few notes.

Any name in the Object Language can be designated any set of names in the Metalanguage. — RussellA

What designation does is to take each of the things named in the object language and give them another name in the metalanguage. Satisfaction does much the same thing for predicates. It's not that the names in the metalanguage name the names in the object language, but that both languages talk about the same objects.

So if int he object language there was a sentence "snow is green, circular and distant", there would be a sentence in the metalanguage that is about the very same things. "snow* is green*, circular* and distant*". And the T-sentence would be true:

"snow is green, circular and distant" is true IFF snow* is green*, circular* and distant*

...in this case because both left and right sides are false.

Who generates the meta-sentences? Just the process of designation and satisfaction. For every sentence int he object language, that process guarantees a sentence in the metalanguage.

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.

a language strong enough to talk about its own sentences, because directly it will be able to generate a sentence of the form

This sentence is false — Banno

We need to be careful not to conflate 'language' with 'theory', or with 'a theory and an axiomatization' or 'a logic' or 'logistic system'. These are related but different notions.

A theory can have sentences that "talk about" sentences in the language of the theory, without contradiction. However, a consistent theory adequate for "a certain amount" of arithmetic, cannot have a defined truth predicate in the theory.

In general, I see in this thread uses of 'language' that should be 'theory' or other specific notions.

A language is just a set of symbols and a signature that assigns kind (predicate symbol or operation symbol) and arity to the predicate and operation symbols. We also add formation rules for terms and formulas.

A theory is a set of sentences closed under deduction. (Some authors say a theory is just a set of sentences, but I prefer when authors add "closed under deduction".) Every theory has a language, which is the language used to form the sentences of the theory. In this sense, if L is the language and T is the theory, we may write <L T> for the language and theory.

A theory and an axiomatization is a pair <S T> where T is the theory and S is a set of sentences in the language for the theory such that every member of T is deducible from S.

A logic is an entailment relation.

A logistic system (deductive system) is comprised of the logical axioms and rules.

EDIT NOTE: I see that Tarski does talk about languages being inconsistent. However, that is not in accord with the basic mathematical logic regarding languages, models and theories that Tarski spearheaded. I don't know what to make of that situation.

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.

We need to be careful not to conflate 'language' with 'theory', or with 'a theory and an axiomatization' or 'a logic' or 'logistic system'. These are related but different notions. — TonesInDeepFreeze

A fair point. I doubt that I will be able to adopt it, force of habit and all.

Tarski himself used an analytic proposition "snow is white" — RussellA

How is this an analytic proposition? Because if it is taken to be analytic then it is circular at best and shows nothing. However, if it is synthetic then its truth is under-determined because whenever observed by whatever method snow is hardly ever white.

EDIT: I should add that I see the problem of circularity in the IFF. I don't think the arrow can go both ways. It only shows redundancy in the method, schnee is not needed.
EDIT:I notice that if the example was 'snow is colorless' or 'translucent' my objection would fail. Adding white to snow is a synthetic addition to my more modern understanding because on a dark night snow could be black instead.

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"

Good job mate, and a good read. Yes, I read it, including the replies. We have our differences, but I admire the effort.

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" — RussellA

Analytic-synthetic judgment comes with logical difference. Snow is black shocks because it is contradictory to white and thus supposedly logically impossible. Since black is not impossible white cannot be an analytic property of snow. Now if snow is translucent and cannot logically be otherwise then 'snow is translucent' is analytic. Translucent is a real property of snow while all natural appearances of snow color are only contrary within a range and are synthetic. This cuts through the confusion caused by Tarski's example. Tarski's theory might or might not work but this example undermines his intentions and questions his understanding of Kant. (or else I'm blowing bubbles ?)

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

A one-to-one translation from object language to another language then gets us nowhere, truth value remains unaffected, and a truth maker ϕ is still to be sought. What else could constitute a truth maker for any proposition of an object language?

You brought up the metaphor of a Russian doll with each layer being more inclusive thereby more physically powerful. And you mentioned the idea of logical power. I think metaphysical power might give for those T-sentences material adequacy. What do you think?

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

A one-to-one translation from object language to another language then gets us nowhere, truth value remains unaffected, and a truth maker ϕ is still to be sought. What else could constitute a truth maker for any proposition of an object language? — magritte

This:

We have material adequacy:
For any sentence p, p is true if and only if ϕ
and we tie meaning down by sticking to one sentence, so that the meaning cannot be ambiguous. We name the sentence on one side, and use it on the other.

"p" is true if and only if p

...and hey, presto, we have a definition of truth. — Banno

If we have a one-to-one translation we have a definition of truth.

The analytic.synthetic distinction makes not difference to the T-sentence; in works for both.

And it's a distinction that Quine shoed the weakness of.

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. — RussellA

Again, T-sentences work for both analytic and synthetic sentences.

Again, it is not a substantive theory of truth, like coherence or fallibilism. It does not tell us if a sentence is true or false in each case. But it does set out the place of the predicate "...is true" in all cases.

Your use of "designation" is nothing like Tarski's. It's closer to Austin's discussion of performative utterances; Ausitn used the same example, naming a ship. Searle developed this int the notion of institutional facts, which seems to be where you are going. But that's far from a complete theory of truth.

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.

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

Yes. Realism demands that objects have fixed properties just so lack of contradiction can distinguish truth from falsity. This is etched in stone. Appearances can be true or false subjectively and they can change therefore are unreliable, but can be classified and named as are the colors of the rainbow. When there are many rather than just one then the logic of contrariety takes the place of contradiction. This means if not this one then any one of the others without contradiction. This, I think, is a useful logical test and proof for properties.

Based on this thinking, white is not a property but just the most commonly seen appearance of snow. The truth of the alternative colorless or translucent snow is based elsewhere in the stronger language of some applied branch of science. Unfortunately this leads away from the OP topic which presumes truth for T-sentences.

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.

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.

Suppose you have a list of colors {red, blue, green, ...}, and a function that maps an object to a color.

For any color, you could define a predicate based on the function that assigns colors. For example

x is red iff color(x) = red

Is that a definition of red? It is if you mean, narrowly, explaining the use of red as a predicate, given only the use of it as a value. It is a method for turning nouns into adjectives, certainly.

It is not a definition of red that would have been of any use in constructing the color() function. You have to be able to assign color values already. This just shows you how to express your assignment of a color value as predicating.

It's a change in notation. The predicative version is syntactic sugar for the value-assigning version.

And all of this could go the other way, if you start with predicates. You can turn adjectives into nouns by the same method.

If you have neither in hand, this method is no use at all.

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. — RussellA

While I commend your involvement with these ideas, I just find this incomprehensible.

"Snow is white" is not analytic.

T-sentences are as appropriate to analytic as to contingent statements.

I don't see what it is you are trying to do with performative utterances.

Sorry. Keep working on it.

"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 problem is one of logic. — RussellA

Yep.

In what fundamental way is "snow is white" different to "seven plus five is twelve". — RussellA

That in another possible world snow is green, but 7+5 is 12 in all possible worlds.

That's necessity and contingency, of course, not analyticity, but it works.

And is not obviously related to T-sentences.

The Revision Theory — Banno

A bit more... by way of fumbling my way through what I can find on this developing area. What follows is jumbled and incoherent, and hence is my notes on what I've been reading rather than an attempt at an explanation.

Bowdlerising one of the arguments from the SEP article, it seems the process is something like this...

Suppose we have a system in which

A: B is true or C is true
B: A is true
C: ~A is true
D: ~D is true

Then we assign values for a first revision. I think the fist revision is arbitrary... not sure. We assign
A1=f
B1=t
C1=f
D1=f

We then work through the deductions from those values to get our second revision.
For A2, B1 is true hence A2 is true
For B2, A1 is false so B2 is false
For C2, A1 is false so C2 is true
For D2, D1 is false so D2 is true

giving a second revision:
A2=t
B2=f
C2=t
D2=t

We then work through a third revision,

For A3, C2 is true so A3 is true
For B3, A2 is true so B3 is true
For C3, A2 is true so C3 is false
For D3, D2 is true so D3 is false

A3=t
B3=t
C3=f
D4=f

For A4, B3 is true so A4 is true
For B4, A3 is true so B4 is true
For C4, A3 is true so C4 is false
For D4, D3 is false so D4 is true

A4=t
B4=t
C4=f
D4=t

and from there on the pattern repeats...

$\begin{array} {|r|r|}\hline S & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline A & f & t & t & t & t & ... \\ \hline B & t & f & t & t & t & ... \\ \hline C & f & t & f & f & f & ... \\ \hline C & f & t & f & t & f & ... \\ \hline \end{array}$

So what? So the value of A, B, and C settles down to either true or false, but the value of D always cycles.

Same for any first assignment...

$\begin{array} {|r|r|}\hline S & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline A & t & f & t & t & t & ... \\ \hline B & f & t & f & t & t & ... \\ \hline C & f & f & t & f & f & ... \\ \hline C & f & t & f & t & f & ... \\ \hline \end{array}$

$\begin{array} {|r|r|}\hline S & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline A & t & t & t & t & t & ... \\ \hline B & t & t & t & t & t & ... \\ \hline C & t & f & f & f & f & ... \\ \hline C & t & f & t & f & t & ... \\ \hline \end{array}$

The point is that this (when reunited with all the stuff left out in this simplification) gives a way of classifying the sentences of the system as either settling down to true, or settling down to false, or never settling down.

And then it gets complicated, because of course this works for much more complex systems. giving a result at each iteration n as either stably true, stably false or unstable. The system can then be extended for sentences that refer to all the sentences in a given row... I think; it all gets a bit weird. The result is that where a finite series of revisions might settle down to being stable true, its transfinite revisions may be stably false or unstable.

And apparently these transfinite visions allow for consideration of T-sentences, and more generally for circular expressions. The biconditional here becomes, instead of material implication, a definitional equivalence.

And the upshot is something like that T-sentences are essentially circular definitions within a single transfinite language, but in a way that is not vicious.

Yeah, this post is a mess, but if truth has been shown to have a place in a general theory of definitions as a circular concept, revisionist theories of truth might be worth further investigation.

A talk by Gupta, on circular definition of truth.




Also https://www.jstor.org/stable/4545102?seq=1

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.

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

The right hand side is in the metalanguage, and is about the domain of the metalanguage, which is all that the metalanguage can talk about. So it is the world of that metalanguage. Same as for Davidson.

Yes, if relations ontologically exist in the world. No, if relations don't ontologically exist in the world. — RussellA

Relations don't exist. Individuals, {a,b,c...} are what exist.

the domain of the metalanguage — Banno

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

You're talking about Tarski. It was Tarski who invented the now usual method of interpretations of languages. A language itself doesn't have a domain nor a world. Rather, an interpretation of a language has a domain of discourse. Ignoring that very basic and crucial distinction leads to deep confusions.

Relations don't exist. — Banno

Relations on the domain of discourse exist.

EDIT NOTE: I see that Tarski does talk about languages being inconsistent. However, that is not in accord with the basic mathematical logic regarding languages, models and theories that Tarski spearheaded. I don't know what to make of that situation.

Then how would you respond to

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

??