Logic of truth
Im re-starting the thread that was locked by the mods because it was thought to overlap with another thread.
Im doing this in good faith, after due consideration and in recognition that the mods may well decide to lock it again. I maintain that the content here is parallel to but distinct from the material in the other thread, that it is substantial enough to constitute a complete thread, that it is better positioned here in the logic sub-forum in the hope of attracting attention from those around here who have a more substantial understanding of logic, so they can point out where I go wrong.
Basically this thread is exegesis, while the other thread is a general discussion. So if you want to add your own theory of truth, do it on the other thread.
And besides, it will keep my posts together in a way that allows me to track my own reasoning.
In my enthusiasm, which might well evaporate, I anticipate discussing Tarski's T-sentences, Kripke's three-valued logic and revision theory.
Tarski's strategy
There's a strategy that Tarski used in his original paper, and that Davidson later mirrored. I'll have a go at paraphrasing it.
We want a theory of truth.
We can ask what such a theory might look like. If it is adequate to its task, it will deliver, for every sentence, something that tells us if that sentence is true.
So it will have the following form, which is not yet a T-sentence:
Further, to avoid circularity, the notion of truth cannot occur in ?.
And finally, this will not work for a language strong enough to talk about its own sentences, because directly it will be able to generate a sentence of the form
Putting these together, if we have as one of our sentences
then our theory will produce a sentence in the metalanguage that looks like
where s is a sentence in the metalanguage.
You should be able to see where this is going. All we need to do now is work out what s might be. So far Tarski is setting out what is needed for any theory of truth. A bare minimum is that it generate for each sentence in our language something that is true exactly when that sentence is true.
Since this assumes that these sentences are either true or false, it assumes realism. That is, so far, nothing Is needed along the lines of proof, knowledge, belief, justification or whatever for our sentences to be true.
Kripkes theory doesnt make this presumption.
Im doing this in good faith, after due consideration and in recognition that the mods may well decide to lock it again. I maintain that the content here is parallel to but distinct from the material in the other thread, that it is substantial enough to constitute a complete thread, that it is better positioned here in the logic sub-forum in the hope of attracting attention from those around here who have a more substantial understanding of logic, so they can point out where I go wrong.
Basically this thread is exegesis, while the other thread is a general discussion. So if you want to add your own theory of truth, do it on the other thread.
And besides, it will keep my posts together in a way that allows me to track my own reasoning.
In my enthusiasm, which might well evaporate, I anticipate discussing Tarski's T-sentences, Kripke's three-valued logic and revision theory.
Tarski's strategy
There's a strategy that Tarski used in his original paper, and that Davidson later mirrored. I'll have a go at paraphrasing it.
We want a theory of truth.
We can ask what such a theory might look like. If it is adequate to its task, it will deliver, for every sentence, something that tells us if that sentence is true.
So it will have the following form, which is not yet a T-sentence:
For any sentence p, p is true if and only if ?
Further, to avoid circularity, the notion of truth cannot occur in ?.
And finally, this will not work for 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
Putting these together, if we have as one of our sentences
Snow is white
then our theory will produce a sentence in the metalanguage that looks like
"snow is white " is true iff s
where s is a sentence in the metalanguage.
You should be able to see where this is going. All we need to do now is work out what s might be. So far Tarski is setting out what is needed for any theory of truth. A bare minimum is that it generate for each sentence in our language something that is true exactly when that sentence is true.
Since this assumes that these sentences are either true or false, it assumes realism. That is, so far, nothing Is needed along the lines of proof, knowledge, belief, justification or whatever for our sentences to be true.
Kripkes theory doesnt make this presumption.
Comments (229)
So we have, as a general form for any theory of truth, what Tarski called "Material adequacy",
And we want to understand what ? is.
And we have that in order to avoid the Liar Paradox, we avoid having a language that can talk about itself. Instead, we employ a second language, and use it to talk about the truth of our sentences. We call this the metalanguage, and it talks about the object language. Our sentence "For any sentence p, p is true if and only if ?" is a part of the metalanguage, referring to any sentence p of the object language and ? is a sentence in the metalanguage
So what is ??
The obvious solution is that ? and p are the same. ?=p.
But the problem here is that ? and p are in different languages. In the metalanguage, p is effectively a name for a sentence in the object language.
Tarski worked around this by introducing terms in his metalanguage that refer to the same thing as terms in the object language; the notion of designation; and then using this to define truth in terms of satisfaction.
Suppose we restrict the object language to being about a group of people, Adam, Bob and Carol...
And in the metalanguage we can have a definition of "designates":
Doubtless this looks cumbersome, despite my having skipped several steps, but it gives us
a metalanguage and and object language both talking about the same objects, Adam, Bob and Carol..., and a way to use the same name in both languages.
We want to add predication. To do this, Tarski developed satisfaction. Suppose we have two nationalities in our object language, English and French. We need a way of talking aobut those nationalities in the metalanguage. We can define "satisfaction":
And so, in a cumbersome way, we have the object language and the metalanguage talking about the same predicates and objects.
Here I've used finite lists, but it is possible to construct similar definitions for designation and satisfaction for infinite objects and predicates, and for n-tuple predicates. I'm just not going to do it here.
So now we have the material adequacy condition for a theory of truth, together with definitions of designation and satisfaction that serve to allow us to talk about the object language using the metalanguage.
Putting these together we get, in the metalanguage, a definition of the truth of any sentence in the object language.
We have
and given designation and satisfaction, we can take any sentence p in the object language and develop a sentence in the metalanguage that both designates the same things and is satisfied in the same conditions.
So we take any sentence p of the object language, apply to it our definitions of designation and satisfaction, and produce a new sentence in the metalanguage that means the very same. We might call this new sentence in the metalanguage S, and write
Tarski started with a general requirement for any theory of truth, then used satisfaction to show that what was needed for ? is another sentence.
The more casual way of setting this out is to treat p as a sentence in the metalanguage that has the same conditions of satisfaction as a sentence in the object language, and then to name that sentence in the object language "p". p is some sentence in the metalanguage, "p" a sentence in the object language that satisfies the same conditions, and so
Stepping up on level, what Tarski has done is to set out material adequacy as a condition for any theory of truth, then analyses meaning in terms of satisfaction, and use satisfaction to define truth. He was able to tie meaning down using the notion of satisfaction, and use it to define truth.
We have material adequacy:
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.
...and hey, presto, we have a definition of truth.
A diversion, which I am allowed to do because no one has read this far. In Truth and Meaning and elsewhere, Davidson flips this around. Where Tarski held down meaning and defined truth, Davidson holds down truth to get at meaning.
So Davidson takes
and points out that if we take a sentence p and produce anther sentence ? that satisfies the condition of material adequacy, then ? gives the meaning of p.
Brilliant!
This stuff:
Quoting Banno
...gives an interpretation to the terms of our metalanguage by setting out which objects those terms designate. Doing this is constructing a model, and here is perhaps the main contribution of Tarski to logic: Model Theory.
Another side issue, but with relevance. And again I will jump all over it without too much formal consistency, but if you want more detail, use google.
We know from Gödel that any logic sufficiently powerful to include arithmetic will be incomplete, or inconsistent.
So suppose we have a language, with a set of axioms, from which we can deduce bits and pieces of arithmetic. Gödel showed that if it is consistent, then there will aways be some bits of arithmetic that remain outside of that deductive sequence. Things that are unprovable, but nevertheless true.
Tarski took that notion and applied it to truth, and showed that, just as there are always theorems that cannot be proved, there cannot be a definition of truth within that language. Another language is needed, or at least an extension of the language.
The proof takes a first-order language with "+" and "=", and assigns a Gödel number to every deduction, as in the incompleteness proofs. It then finds a Gödel number for a definition of truth, and shows that it is not amongst the list of Gödel numbers of the deductions. Hence, that definition is not amongst the deductions of the language.
In plain language, an arithmetic system cannot define arithmetic truth, for itself.
Hence it was apparent to Tarski that in order to talk about truth, one needed an object language and a metalanguage. This is what he developed in his definition of truth.
Tarski's ideas lead to a hierarchy of languages that, like Russian Dolls, each give the truth of the language that they enclose.
Can a language contain its own truth predicate? Various theories do manage this trick. The one I'd like to bowdlerise next derives from a paper by Kripke. The trick, as mentioned earlier, is avoiding the liar paradox: "This sentence is false".
Again, suppose we restrict the language to being about a group of people, Adam, Bob and Carol... and their respective nationalities, English, French... We can construct any number of sentences from these: Adam is English", "Bob is English", "Adam and Bob are french"...
We start by adopting three truth values instead of two. So as well as assigning "true" and "false" to the statements of our language, we add a third value, pictured as sitting in between - not true and not false. (a Kleen evaluation)
Let's call this third value "meh"
We assign "meh" to all the statements of our language.
Then we can give an interpretation to the language, and assign "true" or "false" to these as appropriate; so "Adam is English" is true, and "Adam is French" is false, and so on.
Notice that so far any sentence that contains the term "true" will still have the truth value "meh". So "'Adam is English' is true" is neither truth nor false.
We then start to permit sentences that contain "true" or "false" to be assigned values other than "meh", but under strict conditions. So:
If "Adam is English" is true, then we allow that "'Adam is English' is true" is also true.
If "Adam is French" is false, then we allow that "'Adam is French"' is false" is true.
And so on. Generally, if p is true, then "p is true" is true, and '"p is true" is true' is true, and so on; if p is false, then "p is false" is true, and '"p is false" is true' is true, and so on.
But notice that in this construction, we never get to assigning a truth value to the sentence "this sentence is false". So it remains with the truth value "meh" - neither true nor false.
So suppose our language were the whole of mathematics, and we adopted a constructivist position, such that a mathematical theorem is true only if there is a proof that it is true. We can adopt the antirealist position that the Goldberg Conjecture, since it is unproven, has the truth value "meh" - is neither truth nor false.
Something not quite right there. Did you mean (the Goldbach conjecture is) true XOR false? Any proposition is either true or false (principle of bivalence).
In the constructive case, the truth value of an arithmetic proposition is considered a 'Win' or 'True' if there exists a proof of the proposition, and is considered a 'Loss' or 'False' if there is a proof of it's refutation. But introducing a truth value for the status of undecided arithmetic formulas is tantamount to calling a failure to prove or refute them a 'Draw', which distorts the concept of mathematical truth by muddying the distinction between a mathematician's abilities and his subject matter.
IMO, in constructive logic it is better to resist assigning a truth value to undecided propositions so that truth values always refer to what has been proved, rather than to what hasn't been proved. Draws should only be considered a third truth value in cases where there is a constructive definition of drawn games such as in Chess, unlike arithmetic that doesn't possess a natural concept of a draw
As for the classical case, the Law of Excluded Middle suffices to denote the truth value of undecided propositions; unlike in the constructive case, the classical meaning of A OR B doesn't entail either a proof of A or a proof of B, therefore A OR ~A interpreted as meaning TRUE OR FALSE suffices as the truth 'value' for undecided propositions of classical arithmetic.
I'm not sure, but: you mean object language? The interpretation is that fragment of the metalanguage that interprets terms of the object language?
What if our premisses are wrong when we try to make a theory of truth as we try to do it?
That we just assume it's a straight forward logic "if...then" as we ordinarily do. But self-reference, or generalizing to all possibilities and we end up in problems where we need a metalanguage or some hierarchical system to avoid a paradox.
And the wrong premiss is that when we add self-reference, or the infinite, our finite logic simply cut it? And this is what the paradoxes tell us.
Accepting a third truth value basically rejects the principle of bivalence.
A good read @Banno - you simplified it enough that I think I followed along :)
Something I'm not following -- if we designate our meta-language to refer to the same objects, does it still, at the same time, function as a meta-language? Sort of like having L1 and L2, with the same strings, but slightly different meanings?
I have a hard time thinking in terms of a meta-language. Like, clearly formal and defined -- but I'm not sure I understand how the meta-language performs both the meta-language's function of talking about L1 and the function of the object language which talks about the objects. At this part:
Quoting Banno
I might just have to crack open the paper again to follow these steps, and that's fine, but I thought I'd note something I'm not following.
Doubtless you are right. But that's how I understand Kleen evaluations; albeit using "meh" for "undefined" - it's easier to type.
So the truth table for conjunction is
[math]\begin{array} {|r|r|}\hline \&= & true & meh & false \\ \hline true & true & meh & false \\ \hline meh & meh & meh & meh \\ \hline false & false & meh & false \\ \hline \end{array}[/math]
(What's the simplest way to construct a table here? Neither HTML nor MathJax seem to work... doubtless my coding skills are not up to it.)
Ah... Thanks, Whichever mod that was. IS there a better way? Can we do HTML tables?
No and yes. In Kripke's system the truth value of an unproven conjecture would be neither true nor false. Hence, meh. It's a non-classical logic, so the principle of bivalence is dropped.
No, although perhaps those sentences are in a metametalanguage.
I'm not real fussed, since the point is not to set out the exact formal logic - What I've written is a long way from that, but to explain roughly what is going on, mostly to myself.
Thanks - that's a bonus, since the aim was to simplify it enough that I seem to understand it.
Quoting Moliere
Yep, because the object languagecan talk about Adam and Bob, but can't talk about itself, however the metalanguage can talk about Adam and Bob, and about the sentences of the object language.
So we have Adam, Bob, Carol,...
And in the object language we can write about them: (Adam is English).
And in the metalanguage we can write about them : (Adam is English), and add sentences from the object language: ("Adam is English" is true)
Doesn't it make more sense to say the truth value of unproven statements is unknown (it is true/false, we just don't know which) instead of neither true nor false.
I see but wouldn't it be better to say we don't know if p or ~p instead of we know neither p nor ~p. There's a difference, no?
By logic, do you mean first order logic?
To me it seems like one takes the natural numbers and then assumes you can deduce from the natural numbers things like infinity or irrational/transcendental numbers. Doesn't go like that.
But perhaps first a question that someone hopefully could answer:
Tarski's indefinability theorem and the Incompleteness theorems of Gödel are in some literature called as incompleteness results. Are they equivalent or just how they link to each other? For me it seems like talking about the same issue just from a bit different viewpoint. Am I wrong?
Quick question.
I perceive something in the world that is cold, white and frozen, and I name it "snow".
Therefore, "snow" means something in the world that is cold, white and frozen.
For Tarski, Convention T is "p" is true IFF p. Therefore, "snow is white" is true IFF snow is white.
Tarski's T convention assumes that in the object language the subject is satisfied by its predicate, in other words, the subject "snow" has the property "is white".
I perceive something in the world that is the ground and name it "the ground".
Therefore, Tarski's Convention T may be written as "snow is on the ground" is true IFF snow is on the ground.
But Tarski's T convention assumes that in the object language the subject is satisfied by its predicate. This would mean that "snow" has the property "is on the ground".
But is it true that "snow" has the property "is on the ground", as for snow, being on the ground is a contingent rather than a necessary fact ?
What am I missing ?
OK, that helps me understand "object language" a lot better. It's a literal moniker - a language for objects and objects only, and especially not its own sentences.
So a thought -- I balked at the meta-language because of its artificiality, however this makes me wonder -- could the meta-language just be a natural language? Like, the meta-language is for our object language, but it can have other functions too. So really it's just its role and relationship to the object language that makes it the meta-language.
Or no?
There's no such think in philosophy.
Quoting RussellA
That's pretty much the theory of descriptions. Although intuitively appealing, it's fraught with issues and generally held to be incorrect. It's outside the scope of this thread.
Similarly, arguably, neither being on the ground nor being white are necessary properties of snow, since in some possible world snow is black and floats in a layer at head height. That is, there is no logical contradiction in snow behaving in this way. But again, modality is outside the scope of this thread.
But
are all true.
The image I have of the place of natural language in the metalanguages is a bit like the place of [math]\omega[/math] in Cantor's ordinal numbers. So we can number the object language as language 1. We can number the metalanguage which allows us to talk about language 1, as language 2. We can number the meta-metalanguage, which allows us to talk about both language 1 and language 2, as language 3, and so on. Each language allows us to talk about truth in the languages with a lower cardinality. So every metalanguage has a cardinality.
But a natural language allows us to talk about the truth values of the whole sequence of metalanguages, more or less as [math]\omega[/math] is larger than any whole number.
This is just my own musing, not a part of this exegesis.
More the first order logic with the additional items that make the stuff discussed here possible.
So you asked:Quoting ssu
Premises might have a few different meaning here. So there are a bunch of rules that set out the game of first order logic - I found this neat summary. There are also axiomatisations, systems in which a specified set of tautologies is assumed. See rules 1 through 8 in this axiomatisation.. This system is both consistent and complete. Only and every true tautology can be deduced from the axioms. The proof of this is called "Gödel's completeness theorem", mostly in order to cause utter confusion.
There's a potted history on SEP.
Tarski and Kripke proceed by adding stuff to this system.
Not true in the same sense that any of the left side utterances are though.
Interesting.
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 ?
I think, for me at least, the next step would be -- if you accept that a natural language can be a meta-language -- to actually say that that's the end of the infinite regress.
The object-language kind of does function along the lines of conversations about objects. We just accept the object language as its being used, and even if people are actually using English they do use it in such a way that "passes over" the liar's paradox
As implied here , this is a fine example of equivocation. The final phrase, "are all true", uses "true" in a different way from the other three.
So, which form of "true" are you talking about in this thread? Or is it the case that "the logic of truth" is itself just deceptive sophistry?
I'm not sure if it's an equivocation of "true" or a way to show that both correspondence and coherence are exhausted. The latter would be very unique.
I had a read an ignored that, since I couldn't make sense of "sense" there - it's truth-functional Boolean algebra all the way down.
Now you have Meta agreeing with you. A sure sign of a problem.
Better. Neither correspondence nor coherence are at work here. It's a formal language with truth defined in terms of satisfaction.
Quoting Banno
The problem with your approach is that logic is the means for justification. So when you define truth in terms of satisfaction you simply reduce "true" to justified in a special way. Some of us like to maintain a separation between true and justified, such that they are distinct properties. Therefore we can see that defining truth in this way is simply a way of avoiding what it really means to be true. So you propose a purposeless exercise.
See the truth table for ?
and, p?q is by definition (p?q)&(q?p)
hence
Notice the last line. The equivalence is true when the terms on either side have the same truth value, false otherwise. Quoting RussellA
To be sure, "S" is constructed using designation and satisfaction.
An analogue would be finding the person designated by "Jean" in French and designating them "John" in English, and taking the predicate "...est anglais" and finding it refers to the very same individuals as the English phrase "...is English", and writing
This is how Tarski avoids the problem to which drew attention, which i summarised here:
Quoting Banno
S is not just any sentence, it's the one that has the same meaning as "S".
Yes, I understand that using logic as a justification is a problem for you.
So we looked at Tarski and at Kripke's theories fo truth. Tarski seems to be the theory in which folk are most interested, from the replies. I suppose Kripke's use of a trinary logic is daunting.
Each of those theories has as a central problem, the liar paradox:
The liar is perhaps the simplest case of a series of sentences that cause all sorts of grief.
Consider the Knights and knaves problem:
The solution is found by assigning different truth values to the guardian's statements, and working out the consequences.
"Would the other guard tell me that your door leads to the castle?"
We consider the potential case.
Here's the first option: Guard one says the other knight would say "yes"
if guard one is a knight, then guard two is a knave, and "yes' is a lie, so guard one's door does not lead to the castle.
If Guard one is a knave, then Guard two is a knight, and "Yes" is a lie, so again guard one's door does not lead to the castle.
In either case Guard two's door leads to the castle.
Here's the second option, a revision of the first: Guard one says "no"
If guard one is a knight, then guard two is a knave, and "no" is the a lie, so guard one's door leads to the castle.
If guard one is a knave, then guard two is a knight, and "no" is a lie, so guard one's door leads to the castle.
So which ever guard one asks, if they answer "yes" their door does not lead to the castle, and if they answer "no" is does.
With this question, the knight will tell the truth about a lie, while the knave will tell a lie about the truth. Therefore, the given answer will always be the opposite of the correct answer to the question of whether the door leads to the castle.
The problem was described by considering one potential response, then revising that response. In this case, the solution is closed, and an answer is found.
Now lets' follow the same process for the Liar.
First we suppose that "this sentence is false" is true. We deduce that the sentence is therefore false.
Given this result, we revise our supposition, supposing instead that "this sentence is false" is false. We deduce that therefore the sentence is true.
And so on. Unlike the Knights and Knaves example, the answer flip flops between true and false.
It seems the revision theory does not seek to resolve the liar paradox, but instead to classify it as one outcome amongst a range of outcomes, closed or open or undecided, and hence to develop a theoretical basis for the examination of such sentences. It is somewhat sympathetic to the prosentential version of deflation, treating truth as a function rather than as a property.
Doubtless this is not a very clear exposition, but it's newish stuff. it's included here as indicating a different direction for the logical of truth.
Obviously you don't understand. It appears you didn't read the post, yet claimed to understand me. That's a mistake. You ought to read the post, or not bother to make a reply, which is what you usually do.
Using logic is the means for justification, as I said. No problem. The problem is with your approach to truth, as I also said. Truth is not a special case of being justified, as you seem to think. Do you have a coherent reply to this, or will you stick to your usual?
It's not my approach. It's formal logic over the last hundred and fifty years. You are behaving here in much the same eccentric way as when you discuss [math] 0.\dot9 \neq 1[/math].
When you accept, adopt, and preach that approach, then it is yours. All of us here are rational human beings with the capacity to think for ourselves, and freely make our own decisions. That it is the approach of formal logic over the last hundred and fifty years, to portray "true" as a special form of justified, is no reason to accept this approach yourself. That's a fallacy known as appeal to authority.
So forgive me if I don't follow through on your comments.
Sorry, you must be talking about someone else.
It's much better conversation if you actually read a person's posts.
Quoting Banno
Quoting Metaphysician Undercover
Quoting Metaphysician Undercover
So there's that. Cheers.
So, where's this use of = and ? you are talking about? I only see "Iff" here, which I clearly did not confuse with "=", nor did I confuse it with "?". In case you are unfamiliar with language, different symbols have different uses, and also different meanings. Please do not misquote me anymore.
Again, it would be beneficial to the conversation if you would actually read what a person writes.
So you can't even see where this is wrong.
Now we're getting somewhere. Perhaps you might explain why you see it as wrong.
No.
So you imported that post for no reason? You have no desire to discuss it?
:up:
Look Banno, the judgement that "p" is true requires the fulfillment of a very special and unique set of circumstances, a particular set of circumstances. Saying "p" is true IFF p, is to say that there is a unique and special relation between "p" and p. If you also say "p" is true IFF q, then you say that both p and q have the very same unique and special relation with "p". Therefore both p and q must refer to the very same particular set of circumstances. Both p and q must refer to the very same thing.
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.
So "a bachelor" is p, and "an unmarried man" is q. You can see that p and q refer to the very same thing.
If you make any changes to what p refers to (a bachelor), or to what q refers to (an unmarried man), such that they no longer both refer to the very same thing, you can no longer make the same "is true iff" statements. Therefore p and q necessarily refer to the very same thing.
Quoting Michael
No it does not, that's the point I made in the other thread. It just shows that "bachelor" and "unmarried man" refer to the very same thing, but it shows nothing about the meaning of those terms. For that, we'd have to look at the meaning of "unmarried", and of "man". The attempt to avoid the infinite regress of definitions is an illusion, and really a farce because it's so obviously simple sophistry.
"Small moves, Banno, small moves"
Quoting 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.
Better, Tarski looks to those things to which "Schnee" points in the object langauge and chooses new words in the metalangauge to point to the same things. SO "Snow" has the same extension as "Schnee"
It is not clear at all because the language used in logic is very complex. I can explain it to myself but how can I know that I am not wrong? This is why I don't tend to take part in threads about logic. The Language used like
Is difficult to follow and understand if you do not have basic skills inside logic.
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.
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.
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:
Quoting 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.
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!
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.
Quoting 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.
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
A theory and an axiomatization is a pair
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.
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.
A fair point. I doubt that I will be able to adopt it, force of habit and all.
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.
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"
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 ?)
Quoting Banno
A one-to-one translation from object language to another language then gets us nowhere, truth value remains unaffected, and a truth maker ? 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?
This:
Quoting Banno
If we have a one-to-one translation we have a definition of truth.
And it's a distinction that Quine shoed the weakness of.
Quoting 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 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.
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.
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.
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.
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.
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.
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:
Yep.
Quoting 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.
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...
[math]\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}[/math]
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...
[math]\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}[/math]
[math]\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}[/math]
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.
Also https://www.jstor.org/stable/4545102?seq=1
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.
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. Quoting RussellA
Relations don't exist. Individuals, {a,b,c...} are what exist.
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.
Quoting 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.
Quoting RussellA
??
[Revised post:]
Consider the sentence:
'Snow is white' is true if and only if snow is white.
That sentence is written in a metalanguage.
The left side of the biconditional is:
'Snow is white' is true.
There 'Snow is white' is a quotation of the sentence:
Snow is white.
which is a sentence of the object language.
The whole sentence is of the form:
'P' is true if and only if P.
/
But we can make this even more precise, using Tarski's method of models:
Consider a formal sentence such as:
0+2 = 2
'0+2 = 2' is true if and only if the denotation of '0+2' is the same as the denotation of '2'.
If, with the interpretation of the language we are using, the denotation of '0' is the number zero, and the denotation of '2' is the number two, and the denotation of '+' is the addition operation, and the denotation of '=' is the identity relation, then:
'0+2 = 2' is true in this interpretation if and only if zero plus two is identical with two.
/
With the original example:
If, with the interpretation of the language we are using, the denotation of 'snow' is:
precipitation in the form of small white ice crystals formed directly from the water vapor of the air at a temperature of less than 32°F (0°C)
and the denotation of 'white' is:
has the achromatic object color of greatest lightness characteristically perceived to belong to objects that reflect diffusely nearly all incident energy throughout the visible spectrum
then:
'Snow is white' is true in this interpretation if and only if precipitation in the form of small white ice crystals formed directly from the water vapor of the air at a temperature of less than 32°F (0°C) has the achromatic object color of greatest lightness characteristically perceived to belong to objects that reflect diffusely nearly all incident energy throughout the visible spectrum.
It seems to be a reply to your
Quoting RussellA
I should have made clear that I'm not opining about RussellA's posts, but rather I meant my own post as a rendering of my own explanation, not necessarily as agreeing with or disagreeing with RusselA's perspectives.
Quoting 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.
Quoting 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
Quoting TonesInDeepFreeze
Summarising @TonesInDeepFreeze (I hope correctly)
1) In the world is the achromatic object color of greatest lightness characteristically perceived to belong to objects that reflect diffusely nearly all incident energy throughout the visible spectrum. Designate this "white"
2) In the world is precipitation in the form of small white ice crystals formed directly from the water vapor of the air at a temperature of less than 32°F (0°C). Designate this as "snow"
3) "Snow is white" is true IFF what has been designated "snow" has what has been designated "white"
Designating
Names are designated in Institutional Performative Acts and written up in the annals (metaphorically). For example, "apples" have been Institutionally named, but the object part my pen and part the Eiffel Tower hasn't (yet) been Institutionally named.
Tarski's T-sentence
I observe the world and see something cold, white and frozen and a relation between them, the relation snow.
If cold, white and frozen didn't exist in the world, then neither would the relation snow.
Let white be designated "white" and snow be designated "snow". It is also possible that white had been designated "green" and snow designated "apple". The world of the ML is not our world, and, in the world of the ML, anything is possible.
The T-sentence is a biconditional, meaning that the truth of the proposition "snow is white" is conditional on something.
But "snow" being "white" is not conditional on snow being white, as snow is of necessity white. Snow only exists as a mereological object having the parts cold, white and frozen. Snow doesn't exist independently of its parts, cold, white and frozen.
"Snow is white" is conditional on i) the existence in the world of cold, white and frozen and a relation between them, snow ii) snow being named "snow" and white being named "white"
Simplifying, it seems to me that the T-sentence becomes: "snow is white" is true IFF snow is white, snow has been named "snow" and white has been named "white".
Your report is quite confused about these notions. I'm afraid that if one wants to properly understand this subject then one has to read a good textbook on it.
Quoting RussellA
The sentence is not the metalanguage. The sentence is written in the metalanguage. (I think that's what you meant.)
Quoting RussellA
No, the sentence is a biconditional in the metalanguage. Both sides of the biconditional are in the metalanguage.
Quoting RussellA
No, we don't specify a domain in the object language. In the metalanguage we specify an interpretation of the object language. Part of that interpretation is specification of a domain.
Quoting RussellA
Right.
Quoting RussellA
No, 'cooking', 'cleaning', etc. are vocabulary of the object language; they are not in the domain.
Quoting RussellA
No, in the metalanguage we specify interpretations for the object language. Part of an interpretation is specification of a domain. And cooking, cleaning, etc. are predicates over members of the domain.
Quoting RussellA
No, as explained above.
Quoting RussellA
Right.
Quoting RussellA
Hmm, not sure that's the way to put it. 'Snow is white' is true iff what 'snow' stands for has the property that 'white' stands for.
Quoting RussellA
In Tarski's sentence, 'snow' is the noun. So 'snow' stands for an object, which, per an interpretation is a member of the domain. So snow is not a relation or even adjective ('is snow') in this particular case. The adjective is 'white' ('is white' or 'has the property of whiteness')
Quoting RussellA
Hmm, okay. A biconditional is just the conjunction of a conditional with the converse of that conditional. The Tarski sentence says:
If 'snow is white' is true, then snow is white. And if snow is white, then 'snow is white' is true.
So, yes, the condition for 'snow is white' being true is that snow is white. And the condition for snow being white is that 'snow is white' is true.
Quoting RussellA
That doesn't enter into it at all. Of course the word 'snow' is not the word 'white'. And of course the word 'white' is not an adjective regarding the word 'snow'.
.
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).
Would you (or Tarskian model theory) accept
Quoting TonesInDeepFreeze
?
Correct that here 'is' is not for equality but for indicating a predicate.
Quoting RussellA
I didn't say anything like that.
I don't see any improvement in your revision.
But no problem, either? Talk of properties when glossing use of a logical predicate is eliminable?
Even in model theory?
I don't know what that means.
Quoting Goodman, p49
I don't know what that means. I'd have to read the rest of the context.
Whatever is meant by 'predicate' and 'property' there, you asked about model theory.
Predicate symbols map to relations on the domain.
So, yes, if I were to render the T-sentence more formally, not so much as an example of a philsophical principle but as a recap of a formal model theoretic formulation, then I wouldn't need to mention 'property'.
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).
'Snow is white' is true if and only if snow is white.
I merely unpacked, pedantically really, the right side.
Nothing is missing.
Quoting TonesInDeepFreeze
I assumed that by 'predicate' was meant primarily linguistic predicate or adjective, but that this corresponds roughly to a unary predicate in FOL? And properties are the corresponding unary relations?
But you wouldn't need to mention those? I mean, you did mention them, as is usual, and my proposed revision was clearly eccentric, in context. So I was pleased when you didn't immediately reject it. I'm ready to hear that model theory would require reference to corresponding properties, contra Goodman.
So I was interested (and still am) in what you have to say.
n-ary for any natural number n
Quoting bongo fury
n-ary for any natural number n
Predicate symbols can be n-place for any natural number n.
Relations can be n-place for any natural number n.
Of course, in the case of the 'snow is white' example, 'is white' is 1-place.
But it's only an example of a principle. Presumably, the principle applies to n-place for any natural number n, as indeed it does in model theory.
He would eliminate n-ary relations (n>1) from the method of models?
You can reduce n-ary predicate symbols (n>2) to 2-ary predicate symbols. But you have to have at least 2-ary if you want more than the monadic predicate calculus.
Don't want more!
Quoting bongo fury
I don't know what it means to not "want" more than monadic logic. You can't do much mathematics with just monadic logic.
We didn't really get started here, did we? Never mind, I may try again tomorrow, with your forbearance.
Got started with what? I don't know where you're going with this.
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" ?
Denotations are stipulated. Though it is not as clear cut in natural languages as with semantics for formal languages.
Quoting RussellA
Do you mean the denotation of 'fire' is x and the denotation of 'hot' is y?
With 'snow' and 'white' I just looked in a dictionary.
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 ?
'snow' is not denoted as snow, and 'white' is not denoted as white.
'snow' denotes snow, and 'white' denotes white.
Hmm.
Why denotation and not extension?
The denotation attempts to set out the necessary and sufficient characteristics of snow and white. The extension just is the things that are snow and white.
The denotation of a word is the thing the word refers to. In formal semantics, the model maps n-place (n any natural number) predicate symbols of the language to n-place relations on the domain, and maps n-place (n any natural number) function symbols to n-place functions on the domain.
That is semantical.
I think we could say that the extension of a predicate or function symbol is the relation or function the symbol maps to. (?)
/
Where biconditionals (necessity and sufficiency) enter are in definitions:
Definitional Axiom Schema - predicate symbol:
Px1...xn <-> Q
where P is a new n-place predicate symbol; x1,..., xn are distinct variables; and Q is a formula (of the language of the source theory) in which P does not occur and in which the only free variables are among x1,..., xn.
(If n = 0, then P is just a propositional symbol and there are no free variables in Q.)
Definitional Axiom Schema - function symbol:
fx1...xn = y <-> Q
where f is a new n-place function symbol; and x1,..., xn, y are distinct variables; and Q is a formula (of the language of the source theory) in which f does not occur and in which the only free variables are among x1,..., xn, y; and all closures of E!yQ are theorems of the source theory.
(If n = 0, then f is just a 0-place function symbol and there are no free variables in Q other than y.)
Those are syntactical.
/
The extension of a property is the set of all things that have the property.
That is philosophical.
Then what is the extension of a word?
I just now added:
I think we could say that the extension of a predicate or function symbol is the relation or function the symbol maps to. (?)
Then have:
The model maps n-place (n any natural number) relation symbols of the language to n-place relations on the domain, and maps n-place (n any natural number) function symbols to n-place functions on the domain.
I think we could say that the extension of a relation or function symbol is the relation or function the symbol maps to.
So we have
Quoting TonesInDeepFreeze
and
Quoting TonesInDeepFreeze
I'm not seeing a great difference.
Indeed, the second is a formal way of saying the first.
But necessity and sufficiency (the biconditional) does not enter there, in the semantics, but rather in the definitions, which are syntactical.
(Note that Tarski's definition of 'is true' is also in canonical biconditional form.)
and
have the same extension. Is it true that they have the same denotation? And further, is it really the case that the former is the denotation of the latter? Or do they just happen to denote the very same things, the denotation being those very things?
The problem is that we don't want it to be the. case that one doesn't know what snow is until one knows it is precipitation in the form of small white ice crystals formed directly from the water vapour of the air at a temperature of less than (0°C).
They have the same denotation and extension (but not the same intension).
There's perhaps a slight problem with the choice of 'snow' for the Tarski example.
'snow' in the sentence is a noun, not an adjective.
But 'snow' is a mass noun, so it's not as easy to work with here.
Better might have been:
The snow on the lawn is white.
Or maybe 'cueball':
'The cueball on the table is white' is true if and only if the cueball on the table is white.
/
Quoting Banno
No.
the phrase 'precipitation in the form of small white ice crystals formed directly from the water vapor of the air at a temperature of less than 32°F (0°C)' does not denote the word 'snow'
and
the word 'snow' does not denote the phrase 'precipitation in the form of small white ice crystals formed directly from the water vapor of the air at a temperature of less than 32°F (0°C)'.
Quoting Banno
They denote the same thing.
Quoting Banno
That's a separate epistemic question. But I don't see how it's a problem for Tarski's definition or my remarks about it.
Hah! Replace the "real object" with the abstract object and thus reveal the semiotic game being played. If you can't see a difference, there never was a difference. The claims about reality were always indirect and mediated. Conditioned on some abstract definition. :up:
Quoting Banno
But that's not snow on the lawn. It's sleet! Etc, etc. :grin:
I don't know where you're headed with this, but in case my hunch is right, I would say:
Tarski is not saying how we know that 'snow is white' is true. He's only saying what it is for 'snow is white' to be true. The latter may help for the former. But the definition itself is only of the latter.
I unpacked 'snow is white' with the longer phrases. That works okay, because the context is extensional, not intensional. If we go to an intensional context:
Bob knows that ('snow is white' is true if and only if precipitation in the form of small white ice crystals formed directly from the water vapor of the air at a temperature of less than 32°F has the achromatic object color of greatest lightness characteristically perceived to belong to objects that reflect diffusely nearly all incident energy throughout the visible spectrum)
then of course, that could be false.
The motivation for the unpacking was just to show that a careless reading of Tarski's formulation as being vacuous for being tautological would be not only prima facie incorrect but illustrated as incorrect.
Tarski's definitIon itself:
For any sentence 'P':
'P' is true if and only if P.
An instance:
Let 'P' be 'snow is white'.
'snow is white' is true if and only if snow is white.
Quoting TonesInDeepFreeze
More than slight.
Quoting TonesInDeepFreeze
The issue is more to do with developments from Tarski's definition. I'm not disagreeing with you but thinking out loud.
What caught my eye was
Quoting TonesInDeepFreeze
It is apparent that this is a co-extensional expression for the items that are snow, but also that that the denotation is not the expression but the items themselves.
Consider
The last strikes me as most closely resembling what Tarski does.
I don't know why you regard it as the most close. All three seem reasonable to me. Though the first is Tarski's own form.
The advantage of Tarki's form is that is is general. It applies to all sentences.
'P' is true if and only if P.
Then we plug we plug in any sentence for 'P'.
Again, I have no particular direction in mind, just attempting to sort out a few of the issues around the wider use of Tarski's schema.
Quoting TonesInDeepFreeze
Simply because the item and the list is closer to the strategy of designation and satisfaction Tarski adopts.
Of course the denotation is not the expression. The denotation is, formally, as given by the method of models.
That's an excellent point.
I like the second because it's illustrative. And I like the third because, as you observe, it is getting closer to the actual formal method (and also hews closer to Tarski's "plugging in" strategy).
Cheers. I keep responding to your previous point...
The thing about the first is, is it incontrovertible. Hence, it can serve as a definition of "...is true...", if the difficulties of opacity and circularity can be overcome.
Which is what, apparently, Gupta claims to have accomplished with his revision theory of truth.
That is the basic idea.
For a 0-place function symbol, the denotation is a member of the domain.
For an n-place (n>0) function symbol, the denotation is an n-place total function on the domain.
For a 0-place relation symbol, the denotation is a truth value.
For an n-place (n>0) relation symbol, the denotation is an n-place relation on the domain.
Quoting Banno
I prefer 'set' rather than 'list', since 'list' could be taken as a sequence of the things in the set, or even suggesting a countable sequence.
Right.
But recall that my unpacking was a conditional:
If 'snow' stands for blahblahblah and 'white' stands for 'bleepbleepbleep', then
'snow is white' is true if and only if blahblahblah is bleepbleepbleep.
That's a consequence of the Tarski formulation but not an equivalent (since the Tarski formulation doesn't have antecedents like that).
Sure. I chose "list" by way of avoiding using a technical term, hence checking the applicability of the notions involved to natural languages. The list is not in any particular order and might be innumerable.
What difficulties of opacity and circulaity?
I don't know what sense of 'opacity' you have in mind.
And Tarski's formulation is not circular. Indeed he stated a requirement that a definition not be circular, and he gave one that is not circular.
Yeah; I didn't notice that until a second reading, so at first I misunderstood you as being categorical.
Quoting TonesInDeepFreeze
@Michael made a point of these in another thread. Tarski has
"S" is true IFF X
And we can't just substitute any sentence p for S and X.
Gupta seems to have a strategy that allows us to conditionally substitute p for both S and X and by considering all the possibilities shows that no inconsistency ever results.
'set' is no more technical than 'list'
Quoting Banno
I usually take 'list' as 'sequence' or 'series' or 'enumeration'.
And since we're interested in hewing to Tarski, the term to use is 'set'.
A 1-place relation symbol maps to a 1-place relation on the domain. (A 1-place relation on the domain is a subset of the domain.)
Not in the way I used it. Your background read it that way... :wink: Set it aside as a moot point.
Thank you very much for that, and for saying it. Refreshing to read something like that in this forum.
Quoting 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.
Starting at "But a certain reservation" and ending where?
Gupta takes ? (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.
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.
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:
The standard method is the method of models
Quoting 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.
Quoting 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.
---
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.
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
(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
(4) The pumpkin market has been seasonably slow. Farmers have been turning to other crops.
You say.
Typo?
Quoting TonesInDeepFreeze
Quoting TonesInDeepFreeze
Why/how? Is it that you aren't sure whether by "property" you (or others) mean
?
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.
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.
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.
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 Tarskis 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 isnt 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.
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
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).
Quoting 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.
Quoting 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.
Quoting 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.
Quoting 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.
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.
Quoting 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.
Quoting 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?
Quoting 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.
Quoting RussellA
Of course.
Quoting RussellA
Tarski's schema is a definition not a claim of a logical truth.
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.
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.
"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.
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.
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.
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.
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.
===============================================================================
Quoting 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.)
===============================================================================
Quoting 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.
===============================================================================
Quoting 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
===============================================================================
Quoting 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.
===============================================================================
Quoting 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
===============================================================================
Quoting 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.
===============================================================================
Quoting 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
===============================================================================
Quoting 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.
===============================================================================
Quoting 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.
It doesn't have to conform to ordinary linguistic usage in this narrower technical one. And this technical one itself isn't universally agreed. Or rather, @TonesInDeepFreeze will be in the know as to how far Tarski's usage (as clarified above) is agreed, in modern logic and maths related discourse. I'm only vaguely aware that many people diverge from this usage, and follow such examples as Russell's classic (On Denoting) which appears to imply singular reference.
The reason I'm aware of that divergence is that I'm a fan of Goodman's (and others') deliberate indulgence of the multiple (along with singular) interpretation. According to which, denotation is any pointing of a word (or in Goodman's case even a picture) at one or more things. And thus not necessarily singular (as for Russell). Specifically, naming or definite description when singular (e.g. "snow" of snow), or description when general (e.g. "white" of white things). But generally, any pointing (of a word), or labelling (with or by a word).
Quoting RussellA
@TonesInDeepFreeze - is it right to say it's both, for Tarski (as for Goodman and Quine)? Naming is a species of denotation?
Quoting RussellA
Yes! (Passing over infers for implies.) And no! It's all pointing!... just how is "snow" doing more than pointing to snow??
Would you please tell me what textbook(s) in mathematical logic is the source of your understanding of the basics of this subject? Then, if I have the book, I can see better how we can at least agree on the basics.
Quoting RussellA
'x' denotes y.
That is grammatically correct.
That usage is well established in everyday language and in logic.
Quoting RussellA
That is not correct.
The word 'denotes' doesn't infer. People infer; words don't infer.
Quoting RussellA
I'd like to see somewhere an example of
'x' is denoted as y.
If I saw someone write that, then I wouldn't know what it is supposed to mean:
(1) 'x' is denoted by y.
Which is backwards.
or
(2) y is denoted by 'x'.
Which is correct.
Your Cambridge examples are about relations between things: fetal heart rate and percentage of time. We're talking about the relation between a word and a thing.
Quoting RussellA
Two separate things:
(1) An inconsistent theory has theorems that are contradictions. That is syntactical.
(2) There are no true contradictions. That is semantical.
Quoting RussellA
In the kind of formal theories most pertinent to this subject, contradictions are stated in a language, but the concern is whether they occur in theories. Tarski is not worried about whether languages are consistent, since 'consistent language' doesn't even make sense. What are consistent or not are theories.
I don't know where Tarski is supposed to have couched anything he said in a way similar to what you wrote. I don't know where Tarski is supposed to have said anything fairly paraphrased as "the world is logical". What are logical or not are inferences and arguments.
Quoting RussellA
Right. And the way that is achieved is by adhering to certain definitional forms with certain restrictions.
Quoting RussellA
No, you are adding that, from your own notions. Unless you show me exactly what Tarski wrote that can be reasonably paraphrased as "ensuring that the language corresponds with a world that is logical". Again, where does Tarski say "the world is logical"? What would he mean by it when what is logical or not is not a world but rather an inference or argument? And where does Tarski say a language corresponds with the world. What corresponds or not with a world is a sentence or theory.
Quoting RussellA
Does Tarski even say that snow is white? In the "standard interpretation" of 'white' and 'snow', we get that 'snow is white' is true. But it is not precluded that the words can be interpreted otherwise. I went into detail about that in my previous post.
Quoting RussellA
Please say where Tarski says 'snow is white' is analytic.
I really do not want to get bogged down into an endless debate about analytic/synthetic, but you are taking a leap with it here regarding Tarski, unless you show us where he said that 'snow is white' is analytic.
Quoting RussellA
He could have used that also.
We say for any statement P,
'P' is true iff P.
Whether P is synthetic, analytic, or supercalifragilistic:
'P' is true iff P.
'Bob's car is white' is true iff Bob's car is white.
That's another instance of the schema.
The schema does not at all require that its instances be analytic sentences.
Quoting RussellA
That's a good point. I overlooked that I chose a definition that happened to include 'white' in the definiens of 'snow'. That was a mistake. I don't know whether Tarski even had a scientific definition of 'snow' in mind, and especially one that has 'white' in the definiens. So, I don't know whether Tarski thought of a particular definition so that he regarded 'snow is white' as analytic. I highly doubt that he did.
I should have chosen one such as this:
"precipitation in the form of ice crystals, mainly of intricately branched, hexagonal form and often agglomerated into [s]snow[/s]flakes, formed directly from the freezing of the water vapor in the air"
The point of the Tarski schema is not to define 'is true' for just analytic sentences.
"grass: vegetation consisting of typically short plants with long, narrow leaves, growing wild or cultivated on lawns and pasture, and as a fodder crop"
So 'grass is green' is not analytic.
'grass is green' iff grass is green.
Quoting RussellA
The interpretation of the words takes place in the metalanguage.
What is your definition of 'external'?
Quoting RussellA
Correct. To define 'is true' for sentences in a metatheory, requires a meta-metatheory, ad infinitum.
But that doesn't entail that for the object language we can't both interpret it and state the definition of 'is true' in the metatheory. In ordinary mathematical logic we do give interpretations of an object language and also state the definition of 'is true' per each interpretation - all in one metatheory. (Or is there something I'm overlooking?)
Quoting RussellA
Let L be the object language.
Let x(y) be the metalanguage for y.
m(L) interprets L
mm(L) interprets m(L)
ad infinitum
There are not clashes of interpretation.
Quoting RussellA
An axiom is written in a language but occurs in a theory.
For example, the axiom
Ax(x+0 = x)
is written in the language of first order PA and it occurs in first order PA.
The language of a theory and the theory itself are different things.
I know some of the basics of mathematical logic pretty well, but I'm not even within a telescope's distance of being an expert on it or anything other than jazz, and even on that subject I'm outclassed by a number of greater experts.
Quoting bongo fury
I am using 'denote' in the everyday sense:
"stand as a name or symbol for"
'denotes', 'names', 'stands for', 'symbolizes'. All good.
'Chicago' denotes Chicago.
Chicago is denoted by 'Chicago'.
The denotation of 'Chicago' is Chicago.
In the most ordinary context of mathematical logic:
An interpretation (model) is a certain kind of function from the set of symbols. The function maps a symbol to an element of the domain, an n-ary total function on the set of n-tuples of the domain or an n-ary relation on the domain, depending on the kind of symbol.
What about 'describes'? If that's too different, then probably I jumped to conclusions, and 'denotes' is singular for Tarski, as for Russell and many others?
Quoting TonesInDeepFreeze
And in the case of an adjective or one-place relation symbol, perhaps the denotation is, for Tarski, singular, even though a set?
1) The type of truth object (sentences vs propositions)
2) The nature of the equivalence relation (analytic necessity vs material necessity vs modal necessity)
3) Whether the schema is used prescriptively to exhaustively define the meaning of "truth" e.g as in deflationary truth, or whether the schema is used to non-exhaustively describe truth but not explain the truth predicate, as in inflationary truth.
I don't have answers to those questions.
1) Tarski was dealing with sentences.
2) The biconditional is for material equivalence.
2) It's a definition.
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.
You are quibbling over details and things I never said.
Not really. In maths and logic it covers at least "designate", "name", "refer to", "map to", "point to", "apply to", "be satisfied by", and "be true of"; and for Goodman and others also "describe" and "depict".
It goes from word to thing, not from thing to word, or from word to word.*
Quoting RussellA
Maybe. It gets interesting. But hopeless if you misunderstand "denote".
Quoting RussellA
Don't edit when quoting.
* Except where the object denoted is itself a word. But not from word to co-referring word.
It's just the ordinary sense here.
And that is supported by the fact that it was Tarski himself who specified how symbols of the object language map with an interpretation mapping.
Look it up. Read up on Tarksi and model theory.
Quoting RussellA
When you just completely skip what I wrote about that, you are in bad faith as a discussant. Here it is:
Quoting TonesInDeepFreeze
They are critical details. And I also posted about matters that are not mere details. And you went right past the very critical substantive reply to you, as instead you restated your own contention again, ignoring that I had just clearly rebutted it. That is bad faith posting and bad faith to put the onus on your interlocuter in that instance.
And if you think I've misrepresented what you've said, or strawmamned you, then you can say specifically where.
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.
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.
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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Quoting 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".
===============================================================================
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.
Are you suggesting you quoted from a different edition of this article? https://www.google.com/url?sa=t&source=web&rct=j&url=https://sites.ualberta.ca/~francisp/Phil426/TarskiTruth1944.pdf&ved=2ahUKEwjWm9HtsZ76AhX8S0EAHZ8tANMQFnoECA8QAQ&usg=AOvVaw2wFEunA8J8bXe0NoxT516Z
If so, grateful for a link. Otherwise, how on earth are your four (three??) quotes Quoting RussellA
They are presented as a single quote; they come from 3 different pages; and your number 3 is a half sentence seemingly about "denote" and completed by number 4 that turns out to be about "true".
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 as separate quotations, no they aren't. I had to go to the article to discover the editing.
Quoting RussellA
Oh that's ok then? No, it isn't. Don't edit when quoting.
Quoting RussellA
Trying to make Tarski look confused isn't helping you.
At no time have I ever suggested that Tarski was confused.
Are you making this up? Bye.
That's good. You started out in this thread pretty confused. Hopefully some of the comments from posters have helped you.
Quoting RussellA.
The details I have mentioned are not unimportant.
What misunderstandings? Your own? Or posters misunderstanding one another? In either case, misunderstandings deserve to be remedied. I have misunderstood things in this thread. I corrected myself.
Quoting RussellA
You seem not to know what you've written, even immediately upon quoting yourself: "Denotes infers [...]". That which infers is doing the inferring.
Quoting RussellA
No, you just protested that you don't like all the detail. I don't know what you mean is included in that. I didn't claim what you think is the unneeded detail. But an important point that you kept not responding to is that the schema is not concerned with only analytic sentences. So I reiterated that point, stressing it's not a mere detail, whether you think it is or not.
Quoting RussellA And now have skipped twice the very exact response I gave about that.
Quoting RussellA
Granted. And to be exact, I did not say you had. Rather, you kept saying that you think 'snow is white is analytic' (while skipping my replies about that). I wouldn't know why you think it being analytic is important, unless perhaps you think that Tarski chose it for the reason that it is analytic. So, without claiming that you had said that the schema disallows synthetic instances, I stressed that indeed it does not.
To recap:
'Snow is white' is analytic or not depending on the definition of 'snow'. With a definition of 'snow' that includes 'white' then I'd say 'snow is white' is analytic. But I showed you another definition in which 'white' is not included.
If the analyticity of 'snow is white' does not imply that Tarski chose it because it is analytic and thus that the schema is meant to convey that its only instances should be analytic, then what is the point of going on about the purported analyticity of 'snow is white'?
But now I see that you finally state your agreement that the instances of the schema don't have to be analytic.
So what's next? Most importantly, what's for lunch?
And that is misleading.
First, at least it is misleading formatting. The items are from very different parts of the paper, and not a list he made, as it looks from your formatting, of considerations about 'denote'.
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.
(Thanks to bongo fury for catching all that and causing me to notice it too.)
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.
It's ludicrous that you bring Umberto Eco into this. Tarski in 1931 and 1944 is concerned with mathematical logic, not literary criticism. He uses 'denotes' in a very ordinary sense, not anticipating what Umberto Eco might say decades later.
And to drive the point home about what Tarski means by 'denote', he gives the formulation of the method of models in which symbols are mapped to individuals, functions and relations. That is a formalization of the idea of symbols denoting those individuals, functions, relations.
It's the very simple idea:
'Chicago' maps to Chicago.
'Carl Sagan' maps to Carl Sagan.
'Cats' map to cats.
'maps to in an interpretation', 'denotes', 'names', 'refers to'. Different words doing the same job here.
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.
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" ?
===============================================================================
Quoting 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" ?
===============================================================================
Quoting 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. ?
===============================================================================
Quoting 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.
It is remarkable that you say that. It is quite an example of willfully ignoring a mass of context. 'truth' or 'true' are mentioned 15 times in the section in which that paragraph is in the middle.
[begin quote; bold and italics added]
14. IS THE SEMANTIC CONCEPTION OF TRUTH THE "RIGHT" ONE?
***The subject of this section is whether the semantic conception of truth is the only right one.
I should like to begin the polemical part of the paper with some general remarks.
I hope nothing which is said here will be interpreted as a claim that the semantic conception of truth is the "right" or indeed the "only possible" one. I do not have the slightest intention to contribute in any way to those endless, often violent discussions on the subject: "What is the right conception of truth?" I must confess I do not understand what is at stake in such disputes; for the problem itself is so vague that no definite solution is possible. In fact, it seems to me that the sense in which the phrase "the right conception" is used has never been made clear. In most cases one gets the impression that the phrase is used in an almost mystical sense based upon the belief that every word has only one "real" meaning (a kind of Platonic or Aristotelian idea), and that all the competing conceptions really attempt to catch hold of this one meaning; since, however, they contradict each other, only one attempt can be successful, and hence only one conception is the "right" one.
*** We're not getting in arguments premised on the view that there is only one right concept of truth.
Disputes of this type are by no means restricted to the notion of truth. They occur in all domains where - instead of an exact, scientific terminology - common language with its vagueness and ambiguity is used; and they are always meaningless, and therefore in vain.
*** The concept of truth is not the only subject over which there are such arguments. When the subject matter is not exact enough, there are such arguments.
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.
*** In situations where a word is used to denote more than one concept, we should agree to use different words.
Referring specifically to the notion of truth, it is undoubtedly the case that in philosophical discussions - and perhaps also in everyday usage - some incipient conceptions of this notion can be found that differ essentially from the classical one (of which the semantic conception is but a modernized form). In fact, various conceptions of this sort have been discussed in the literature, for instance, the pragmatic conception, the coherence theory, etc.
*** There have been a lot of different concepts of truth.
It seems to me that none of these conceptions have been put so far in an intelligible and unequivocal form. This may change, however; a time may come when we find ourselves confronted with several incompatible, but equally clear and precise, conceptions of truth. It will then become necessary to abandon the ambiguous usage of the word "true," and to introduce several terms instead, each to denote a different notion. Personally, I should not feel hurt if a future world congress of the "theoreticians of truth" should decide - by a majority of votes - to reserve the word "true" for one of the non-classical conceptions, and should suggest another word, say, "frue," for the conception considered here. But I cannot imagine that anybody could present cogent arguments to the effect that the semantic conception is "wrong" and should be entirely abandoned.
*** There could be competing concepts of truth that are all just as rigorous. In that case, we would oblige by using a word other than 'truth'.
[end quote]
Yes, the concept of denote itself is one of those that has differing views. But that's not Tarski's point. Rather his point is that, among concepts having differing views, the concept of truth is in particular one of them. The section is not about what we mean by 'denote' but about the concept of truth and the word 'truth' and the fact that the word 'truth' denotes different things for different people. Especially, Tarski is not at all saying he uses the word 'denote' in different ways.
/
Quoting RussellA
Yes, and I didn't say that you did. It's my point that he wasn't discussing literary criticism; and dragging Umberto Eco into this is quite aside understanding Tarski.
Quoting RussellA
Yes, and 'denotes' there is in the sense you've been told about in this thread.
Quoting RussellA
Wow. You miss the very central point of his articles. Tarski is concerned with providing a rigorous mathematical formulation of the adjective 'is true'. Read the articles.
Quoting RussellA
When Tarski is using everyday examples, he uses the everyday sense of 'denote'. Then he goes on to make it even more rigorous mathematically.
Quoting RussellA
Tarski talks about the fact that in everyday situations we don't have precision. He says that he does not claim to provide an explication of the concept of truth that can withstand all the vagaries of natural language. Nor does he claim to explicate the notion of 'denote' that can withstand whatever disagreements there may be among different settings.
Quoting RussellA
Of course it can be as complex as one wants it to be. But Tarski starts with an ordinary intuitive sense and then goes on to pin it down for more rigorous contexts. Read his articles.
/
In sum:
(1) Tarski's overriding concern is with defining 'is true' in context of formal languages for mathematics and the sciences.
(2) 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.
(3) Whether 'snow is white' is analytic or not is not part of Tarski's concern in the two articles. Moreover, as pointed out: Whether 'snow is white' is analytic depends on which definition of 'snow' we're looking at. There are common enough definitions in with 'white' is not in the differentia.
The statement P is true IFF P.
An example:
"Snow is white" is true IFF Snow is white.
Argument A
1. Snow is white [math]\to[/math] "Snow is white" is true.
2. Snow is white.
Ergo
3. "Snow is white" is true. [1, 2 MP]
Argument B
1. "Snow is white" is true [math]\to[/math] Snow is white.
2. "Snow is white" is true.
Ergo
3. Snow is white. [1, 2 MP]
---
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.
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.
Quoting 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.
Quoting TonesInDeepFreeze
I doubt there are many definitions of "snow" whereby being white isn't included as a property.
First, my point stands that you said the word 'true' is not in the paragraph you quoted, but you intentionally ignored that the words 'truth' or 'true' are mentioned 15 times in the surrounding paragraphs. My point stands: He was talking about 'truth' having different meanings to different people, not about 'denotation' having different meanings. One might think that, for purpose of clear communication, you would admit this now. Instead, where I gave you the actual quoted context, with 15 mentions of 'truth' or 'true', you just go on to post as if it doesn't exist.
About formal languages, mathematics, and the sciences, read the papers! And read the IEP and SEP articles about them. It is a plain fact that Tarski's overriding concern is with languages for mathematics and the sciences. It's throughout the papers.
Quoting RussellA
No, it is overwhelmingly clear that his primary concern is formal languages for mathematics and the sciences. Tarski takes pains to point out that we can't expect to nail the definition of 'true' for informal contexts and that his proposal is directed at mathematics and the sciences. Moreover, a great amount of the paper goes on to actual mathematical formulations of the definition of 'true'. The papers are steeped in it. The primary area of concern is model theory, which is a subject of mathematical logic. Man, you need to read the papers, and for even more much needed background, get a textbook in mathematical logic so that you can appreciate the fruit of Tarski's work.
Quoting RussellA
But he did! He showed the method of models, mathematically. Read the papers!
My points stand: (1) Tarski's overriding concern is with formal languages for mathematics and the sciences. (2) In the passage you quoted, Tarski is not about the fact that 'denotation' has different senses but about the fact that 'truth' ('true' also) has different senses. Either merely blithely or dishonestly, you misrepresented Tarski with your original 1-4 decoction, and then again by choosing just one paragraph from the middle of the other paragraphs. (3) Whether or sentence is analytic or not is not part of Tarski's concern in the particular regard of his schema.
But you will keep replying with your confused, uninformed and dogmatic misinterpretations. I'm running out of time to keep correcting you.
What does this have to do with Tarski?
Where did Tarski write that?
The quote marks are crucial.
'P' is true iff P.
Quoting Agent Smith
It's a biconditional. Formal definitions are biconditionals. He's not saying how we know that 'snow is white' is true. He's only giving a definition of 'is true'.
Moreover, if you want to know that 'snow is white' is true, then you can look at snow to find out.
The mere fact that a formulation is a biconditional doesn't entail that we can't move from right to left (or from left to right) to make our inferences.
You really really need to read a book on the subject of formal logic. Otherwise, you will continue to incessantly spin out in your own confusions.
If I understand what you're after, because the meaning of denoting (designating) is central to Tarski's Semantic Theory of Truth.
In the sense that words have to mean something for his theory to have any relevance, sure. But Tarski doesn't need to give a comprehensive account of meaning to make his point.
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".
Tarski mentioned no such pointless tautology.
But if we don't know what Tarski means by the words he uses, then how do we know what he means ?
That's not how it works in the context of Tarski. Rather, 'is white' is an undivided unit.
In simple terms his argument is just that "Schnee ist weiß" is true if and only if snow is white, where "snow is white" is a translation in the meta-language (which in this case is English) of the object-language (which in this case is German) sentence "Schnee ist weiß".
You seem to want to know how it is that the English sentence "snow is white" is a translation of the German sentence "Schnee ist weiß". That's irrelevant to Tarski's point. He just argues that assuming that it is the translation the T-schema follows.
Whether or not any given meta-language sentence is a translation of any given object-language sentence is a separate matter entirely.
I quite agree that Tarski never said ""snow" denotes snow because "snow" denotes snow".
But I imagine you are inferring that I said that Tarski made this tautological comment.
I have never said Tarski made this tautological comment.
I made the tautological comment to illustrate, as Umberto Eco and others have done, the complexity of meaning in the word "denote".
:zip:
I'm not inferring that you are claiming that Tarski made that comment. I'm pointing out that the comment is irrelevant to understanding Tarski.
Meanwhile, it would be helpful if you'd recognize the 15 instances of 'truth' or 'true'. And take note that 'is' is not a standalone denoter in this context.
I asked previously, what textbook in mathematical logic is your main reference for the subject of mathematical logic. The context of Tarski here is overwhelmingly mathematical logic.
I'm pretty confident that 100,000 years ago people weren't going around saying "snow is white".
That would be a translation issue, oui monsieur? They just used different words to convey the meaning snow is white.
Anyway, as I said in my last post, how do we determine "snow is white" is true? We first look at the individual words and see if they match the denoted objects and then if there's a correspondence, we conclude "snow is white" is true. That makes sense to me and I suppose that's how we break the vicious cycle.
One poster coemmented that P is true IFF P is a definition, but a definition is basically a list of sufficient and necessary conditions which means my argument stands that this is a chicken-and-egg situation.
Susan Haack Philosophy of Logics
"Tarski emphasises that the (T) schema is not a definition of truth though in spite of his insistence he has been misunderstood on this point. It is a material adequacy condition: all instances of it must be entailed by any definition of truth which is to count as 'materially adequate'. The point of the (T) schema is that, if it is accepted, it fixes not the intension or meaning but the extension of the term 'true'".
Given the T-schema - "snow is white" is true IFF snow is white
From my reading, Tarski's T-schema doesn't give the meaning, ie intension, of "snow is white", but if "snow is white" is true, then the T-schema does give the extension of "snow is white", ie snow is white. The T-schema, in not giving an intension for "snow is white", is using the concept of "satisfaction" to allow for a recursive definition of truth.
As the T-schema doesn't give the intension of "snow is white", then it doesn't allow translation between "snow is white" and "schnee ist weiss".
Yes, whether "snow is white" is a translation of "schnee ist weiss" is a separate matter to Tarski's T-schema.
No, I did not.
I said:
'P' is true iff P
is a definition of 'is true'.
Leaving off the quote marks is ruinous.
As far as I can tell, you are unfamiliar with use-mention.
Quoting Agent Smith
Saying "''P' is true iff P" does not preclude that one cannot investigate P without first establishing that 'P' is true: (1) Establish P (such as looking at snow), then (2) infer 'P' is true. That is not circular.
If you actually READ Tarski's papers, you would see how he goes on to explicate that.
You're putting the cart before the horse. Tarski is saying that if "snow is white" is the translation of "schnee ist weiss" then "schnee ist weiss" is true iff snow is white.
Yes, since Tarski's context is formal, he is not opining on the intensional senses.
Back to the point:
Quoting RussellA
That unedifying tautology has nothing to do with understanding Tarski.
Here is his actual proof of that theorem:
https://liarparadox.org/Tarski_275_276.pdf
You seem to have the same passion for this subject that I do.
It is unknown whether the Goldbach conjecture is true or false because there is currently no known shortcut to the infinite proof of simply testing every natural number.