'set' is no more technical than 'list' — TonesInDeepFreeze

Not in the way I used it. Your background read it that way... :wink: Set it aside as a moot point.

second reading — Banno

Thank you very much for that, and for saying it. Refreshing to read something like that in this forum.

"S" is true IFF X

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

'S' is true iff X

is the general definitional form for a predicate symbol, whether 'is true' or other.

Then Tarski wanted to specify what X should be for the definition of 'is true'.

He came up with S.

I don't see opacity or circularity.

Merriam online has definientia with 'series' and 'enumeration'. (There's another definiens closer to your sense, but I don't think it's common, and especially I've never seen 'list' to mean 'set' in this kind of mathematics or logic.) Not a nit; 'set' (acutually 'subset of the domain') is the word to use.

Wider context, again. I'm referring to the difficulties Tarski discusses from bottom of p. 157 here. These must be addressed in order to broaden the context.

Starting at "But a certain reservation" and ending where?

say mid p159.

Gupta takes β (p.158) as a canonical expression in his examination of pathological definitions.

(Edit) I think... (1) in https://www.jstor.org/stable/4545102 , p.228.

@TonesInDeepFreeze my apologies; I'm packing much too much into each post. There is no specific question here for you, rather I am using you to sound off about a series of interrelated issues that have my attention. The T-sentence is intuitively appealing as a definition of truth for natural languages; it's certainly not wrong. Kripke has a solution but at the cost of adopting antirealsim. Gupta has an approach that might capture the intuition but which is too convolute fro me to clearly understand.

The T-sentence is intuitively appealing as a definition of truth for natural languages — Banno

If I'm not mistaken, the 'snow is white' example was mainy to illustrate, while the main application of the Tarskian schema is for formal theories.

Logic of truth

How does a system of logic handle truth/falsity?

1. Consistency: The law of noncontradiction (LNC). A truth may not entail a contradiction (p & ~p) for if ut does, it can't be a truth.

Contradictions (p & ~p) can't be true, they're always false.

2. Some compound statements are tautologies, true always, not semantically, but solely due to logical form e.g. (p v ~p) [the law of the excluded middle]

3. Fuzzy logic: Degrees, on a continuum, of truth/falsity. The statement it'll rain tomorrow is (say) 90% true[/i].

4. Polyvalent logic: True/False/Unkown (trivalent system), an easy-to-grok variant.

5. :confused:

How does a system of logic handle truth/falsity? — Agent Smith

The standard method is the method of models

Consistency: The law of noncontradiction (LNC). A truth may not entail a contradiction (p & ~p) for if ut does, it can't be a truth. — Agent Smith

The notion of 'consistency' is purely syntactical, it does not mention 'truth'. You've added past the definition (I don't know why you do that; why you wouldn't just take a standard definition as it is written without imposing other stuff on it). The definition is:

A set of sentences S is consistent <-> S does not prove a contradiction

An equivalent definition (since first order logic has explosion):

A set of sentences S is consistent <-> there is sentence not derivable from S

So the definition of 'consistent' is syntactic not semantic.

The one sense in which you're close to something (though not part of the definition of 'consistency') is that if a sentence is true in any model then the sentence does not entail a contradiction. The reason is that entailment, by definition, preserves truth, and a contradiction is false in every model.

Some compound statements are tautologies, true always, not semantically, but solely due to logical form e.g. (p v ~p) [the law of the excluded middle — Agent Smith

The more general notion is of sentences being validities:

A sentence P is a valid <-> P is true in every model.

So the definition of 'valid' is semantic, not syntactic.

(Note: 'the sentence is valid' can also be said as 'the sentence is a validity')

Some authors use 'tautology' for 'validity', but other authors (most?) say the tautologies are the validities that are valid by virtue of evaluation of their connectives alone. (And since sentential logic is decidable, it can be semantic evaluation or syntactical evaluation. In other words, we can look at the syntactic structure or we can look at the truth tables.)

But (the context here is first order logic), due to the soundness and completeness theorems, a sentence is a valid iff it is provable from logical axioms alone. So the set of validities is the set of theorems of the pure first order predicate calculus.

However, for predicate logic that is at least dyadic, there is no algorithm to test for validity, which is to say there is no algorithm for checking the form of sentences to see whether they are valid. However, in sentential logic and in monadic predicate logic there are such algorithms.

:up:

---

The point to logic seems to be to come up with, to use a mathematical analogy, functions (argument forms) such that if the inputs are truths (the premises), the output is a ... further ... truth (the conclusion). The objective here is to grow knowledge as a farmer would pumpkins.

functions (argument forms) such that if the inputs are truths (the premises), the output is a ... further ... truth (the conclusion) — Agent Smith

I might see what you're getting at, but you've not put it in a way that is clear to me.

We should take it in steps:

(1) Df. An argument is an ordered pair <G C> where G is a set of sentences (the set of premises) and C is a sentence (the conclusion).

(2) Df. A set of sentences G entails a sentence C iff for every model M, if all the sentences in G are true in M, then C is true in M.

(3) Df. An argument <G C> is valid iff G entails C.

(4) The pumpkin market has been seasonably slow. Farmers have been turning to other crops.

Danke! I have a lot to work on.

You say.

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

Typo?

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

That is semantical. — TonesInDeepFreeze

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

That is philosophical. — TonesInDeepFreeze

Why/how? Is it that you aren't sure whether by "property" you (or others) mean

• a one-place relation symbol
• the subset of the domain the symbol maps to, or
• something more exotic?

?

The first is suggested by the notion of its having an extension.
The second is what I referred to by the ungainly "unary relation".
The third is suggested by the notion of things 'having' it.

Is the best course just to drop the term, then, as the nominalist recommends? Was my point originally.

Typo? — bongo fury

Yes. I fixed it now. Thanks.

Church says, "A property, as ordinarily understood, differs from a class only or chiefly in that two properties may be different though the classes determined by them may be the same (where the class determined by a property is the class whose members are the things that have that property). Therefore we identify a property with a class concept, or concept of a class in the sense [mentioned earlier]. And two properties are said to coincide in extension if they determine the same class." [italics original]

I take that to be philosophical explication.

Is the best course just to drop the term, then, as the nominalist recommends? Was my point originally. — bongo fury

With no comment on nominalism, I think you're right that it is cleaner not to drag in 'property'. But my original use of 'property' was not meant in a philosophically technical sense, but in an everyday sense to emphasize how the right side of the biconditional is substantively different from the left side (i.e. to highlight that the definition is not "circular"), and to do that without invoking set theoretic notions that are not everyday notions. But your suggestion is better for rigor and crispness.

@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.

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

I need to read the rest of your post carefully, but I am not familiar with people saying:

[word] denoted as [thing]

I guess you mean:

[thing] denoted by [word]

or

[word] denotes [thing]

I am not raising this as a mere grammar nit, but rather that we can get lost if we're not very careful to be clear what is denoting and what is denoted.

Anyway, I suggest not saying:

'snow' is denoted as snow

But instead:

snow is denoted by 'snow'

or

'snow' denotes snow

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 ?" — RussellA

Do you mean this?:

Where in Tarski's example is snow denoted by 'snow' and white denoted by 'white'? If not in the example, then how can the schema be formally correct?

It's formally correct because it meets the criteria for formal correctness that Tarski specified, and which also are the usual criteria in mathematical logic.

Just to review:

Tarski is defining the adjective 'is true'. (More explicitly, for a given interpretation of a language, a definition of 'is true', or a definition of 'is true per the interpretation'.)

A definition of that adjective will be of the form (let M be an interpretation of the language):

'P' is true iff X

or

'P' is true per M iff X

where what X meets certain criteria (the criteria of formal correctness).

Tarski then says, 'P' itself will be X, so

'P' is true iff P

or

'P' is true per M iff P

And that is formally correct since it meets the criteria, and we show that it does

If one claims that it is not formally correct, then one needs to show that one of the criteria is not met. Saying, "How can it be formally correct if [whatever]?" doesn't have culpatory weight, any more in form than "How can an airplane fly if ducks have feet?"

Then the question you asked: Where is snow denoted by 'snow' and white denoted by 'white'? The answer, for formal languages, is in the interpretation of the language (the model for the language). The answer, for natural languages, is in the semantical assignments for words (usually per a dictionary or per the referential habits of speakers).

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". — RussellA

I don't see why they can't both be in the same metatheory. Or is there a liar paradox problem that comes up? If so, I'd like to see a proof:

Show if the metatheory gives an interpretation of the object theory and also a definition of truth, per that interpretation, of the language for the object theory, then that metatheory is inconsistent.

That doesn't seem right to me.

(Of course we know Tarski's theorem [simplified and roughly stated:] If if a theory has its own truth predicate, then the theory is inconsistent.

Therefore, white is a necessary condition for snow — RussellA

Tarski doesn't say that. It's your claim, I guess.

[There's redundancy in the rest of my post, because I want these points to come across in different phrasings:]

Indeed, Tarski doesn't even say that 'snow is white' is true. Rather, he is merely giving an instance from a definition of the adjective 'is true'. The example can work even with a false statement:

'Snow is black' is true iff snow is black.

The schema follows by form alone, and does not depend on what happens to be true or false or even necessarily true or necessarily false.

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". — RussellA

I don't see that the statement of the interpretation and the T-sentence can't be given in the same meta-theory.

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

Yes, truth depends on interpretation of the language. A sentence can be true in one interpretation of the language and false in another interpretation of the language.

So, when we talk about the truth value of 'snow is white', we take it implicitly that some particular interpretation has been given.

Or we can make it explicit in this manner:

Let M and Q be interpretations for a language:

Examples:

Let the language have the constant symbols 'c' and 'd' and a 1-place predicate symbol 'F'.

Let M specify:

domain = {0 1}

'c' for 0

'd' for 1

'F' for {0}

So:

'Fc' is true per M iff Fc.

So:

'Fc' is true per M iff 0 is an element of {0}.

So:

'Fc' is true per M.


Now let Q specify:

domain = {0 1}

'c' for 1

'd' for 0

'F' for {1}

So:

'Fc' is true per M iff Fc.

So:

'Fc' is true per M iff 1 is an element of {1}.

So:

So 'Fc' is true per Q.


It is not assumed that we can only use an interpretation in which snow has been named 'snow' and white has been named 'white'. Rather, whatever 'snow' and 'white' name, the schema holds per that naming. If 'snow' named fire and 'white' named black, the schema would still hold. The schema does not dictate what 'snow' and 'white' should name. Indeed, what they name is interpretation-dependent. The schema works, per each interpretation. If a certain interpretation says 'snow' names fire and 'white' names black, then the schema still holds. My example of precipitation and chromaticity was conditional, and we may take that conditional as tantamount to an interpretation stipulating denotations. We could have stated the antecedent of the conditional so that 'snow' denotes fire and 'white' denotes black, thus tantamount to an interpretation stipulating different denotations from the usual ones, and the schema would still work.

It just happens that the "standard" interpretation (i.e. semantic assignments in English) has 'snow' standing for snow (precipitation ...) and 'white' standing for white (the chromaticity ...), so that's the most intuitive interpretation to use as an example. The schema though does not depend on any particular interpretation; we could use some other set of semantic assignments for English words, and the schema still would apply.

We could even say hypothetically that there's a natural language in which 'snow' stands for the thing we regard as fire and 'white' stands for the color we regard as black. The schema would still hold with that natural language taken as the standard one.

Therefore, "snow is white" is true because i) snow is white, ii) snow is named "snow" and white is named "white" — RussellA

No, (ii) is not included in the schema. The same point I just made The truth or falsehood of 'snow is white' is not dependent on 'snow' naming snow (precipitation...) and 'white' naming white (the chromaticity...). No matter what you say 'snow' denotes and no matter you say 'white' denotes, 'snow is white' is true iff the thing you set as the denotation of 'snow' has the property [extensional sense] that you set as the denotation of 'white'.

/

I hope to take time to carefully read your remarks about Putnam.

More on Putnam and Tarski - Panu Raatikainen — RussellA

I haven't carefully read that article, but are your own remarks dependent on the article? If so, should we take it that Raatikainen's summary of Putnam is correct? And do you agree with Putnam as he is summarized and reject Raatikainen's rebuttals? Or is it just certain parts of Putnam you think need to be answered?

In other words, I don't know what specifically you would like me to agree with in Putnam.

In the meantime, I'll respond to your own remarks, not necessarily vis-a-vis Putnam himself.

the Criterion of Adequacy, and being a test of a definition, is formulated only in the metametalanguage (MML). — RussellA

I'm not sure. At first glance, I don't see that in a metatheory we can't state the criteria and prove that a certain schema upholds that criteria.

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. — RussellA

Of course.

And that allows that a formal language can have different interpretations.

If the interpretation has 'snow' denoting the frosty stuff you see on the ground in winter, and 'white' denoting the color of a surrender flag, then 'snow is white' is true per that interpretation.

If the interpretation has 'snow' denoting the rising thing you see when you light a match, and 'white' denoting the color you see when you look at a matador's cape, then 'snow is white' is true in that interpretation.

If the interpretation has 'snow' denoting the frosty stuff you see on the ground in winter, and 'white' denoting the color you see when you look at a matador's cape, then 'snow is white' is false in that interpretation.

Etc.

But I understand that it might be tricky. I'm not sure, 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?

Is that what you're driving at?

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 — RussellA

'green' isn't in the particular instance 'snow is white', so I think you mean.

If 'white' denotes green, then

'snow is white' is true iff snow is white

is not true.

But it is still true. Made explicit

Let M interpret 'snow' as the frosty stuff, and 'white' as the color of a St. Patrick's day T-shirt, and the frosty stuff as not in the set of things having the color of a St. Patrick's day T-shirt.

'Snow is white' is true per M iff the frosty stuff is the color of a St. Patrick's day T-shirt.

Both sides of the biconditional are false. So the biconditional is true.

it isn’t a logical truth that the (German) word ‘Schnee’ refers to the substance snow — RussellA

Of course.

nor is it a logical truth that the sentence ‘Schnee ist weiss’ is true in German if and only if snow is white. — RussellA

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

...the main application of the Tarskian schema is for formal theories. — TonesInDeepFreeze

Tarski's schema is the most influential idea in philosophical analysis of truth. See the SEP article, which lists the classical theories, then talks about Tarski, then spends the remainder of the article discussing were Tarski is used or rejected by the various alternatives.

So while your interest may be in formal systems, others see formal systems as ways of clarifying the use of the notion of truth in a broader philosophical context.

There is no doubt that the schema has wide and pervasive application and interest throughout philosophy.

But in one of the SEP articles it also mentions that Tarski's main [or whatever word was used] focus for it was for formal theories.

Sure, Tarski likes formal theories. But it's the use of his ideas in wider philosophical discussions that made him his name.

It's why I'm interested in him.

On that topic, , you might enjoy Truth and Meaning, in which Davidson plays with Tarski's strategy by using truth to illuminate meaning. You have noticed the direct connection between meaning and truth that the T-schema displays. Tarski used meaning to explicate truth. Davidson uses truth to explicate meaning.

But it's the use of his ideas in wider philosophical discussions that made him his name. — Banno

"Made his name" is not definite enough for me know whether that's true or false. But, of course, Tarski is a giant in mathematics and philosophy, and his mathematics leads to great philosophical interest. He is one of my real heroes. A mind of deep of beautiful wisdom and breathtaking creativity.

Sure, an interesting life. My interest in Tarski comes from his use in Davidson and thereafter. If his ideas had no application outside of formal systems, I doubt I would have paid him much attention at all. And I suspect that's true of the wider philosophical community. Pure logic has a certain aesthetic appeal, however the point is to put it to use.

About a meta-metalanguage:

What is wrong with this?:


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.

the point — Banno

I don't take it that there is "the" point, but rather many forms of application, engagement and appreciation. Some of them not necessarily that of "putting to use" except in the broad sense that virtually any attention, even if purely aesthetic, is a form of "putting to use". Each person may find the appeal of mathematical logic on their own terms, which of course includes using it in the sense you mention, but also may be primarily enjoyment of seeing concepts of reasoning so ingeniously, rigorously and elegantly articulated even irrespective of the use in philosophy and the sciences. Then the usefulness in philosophy and the sciences is a huge added bonus.