Statements are true?
What does it mean to say that a statement is true?
Is it, as some assert, that a statement has the property of truth or is it that a statement is merely labelled as true?
The trouble with asserting that truth is a property of a statement is in finding a logical process by which the property of truth can be identified.(Tarski's artificial meta-system fails to answer this question.)
On the other side, if statements can only be labelled as 'true' by someone who considers that the statement conforms with the facts of the world as they see them or that it conforms with the axioms of a formal abstract system then the above-mentioned problem evaporates.
Then statements like' this statement is true' make no more sense than 'this statement is blue'. and statements like ' this statement is false' would no longer be problematic.
Is it, as some assert, that a statement has the property of truth or is it that a statement is merely labelled as true?
The trouble with asserting that truth is a property of a statement is in finding a logical process by which the property of truth can be identified.(Tarski's artificial meta-system fails to answer this question.)
On the other side, if statements can only be labelled as 'true' by someone who considers that the statement conforms with the facts of the world as they see them or that it conforms with the axioms of a formal abstract system then the above-mentioned problem evaporates.
Then statements like' this statement is true' make no more sense than 'this statement is blue'. and statements like ' this statement is false' would no longer be problematic.
Comments (43)
That if people believe that statement and use it to inform their actions, they will be more likely to make useful decisions related to what the statement refers to.
Most generally, it is labelled as true.
When you feed a logic sentence to a theory, you need to provide the two-tuple ([math]\ulcorner s\urcorner[/math],s) to the formal system. Example:
There are exceptions. Some but not all sentences happen to be provable in the theory T from existing tuples ([math]T \vdash s[/math]). According to Gödel's semantic completeness, a statement provable from true two-tuples, is also true (in all models for T) ([math]T \vDash s[/math]):
[math]T \vdash s \Rightarrow T \vDash s[/math]
Meaning: if sentence s is provable from T then sentence S is also true in all models for T.
Quoting A Seagull
This is exactly what is NOT possible.
Tarski's theorem concerning the undefinability of the truth predicate makes this impossible. A theory is not allowed to evaluate the truth of a sentence. So, this does not exist:
[math]s \leftrightarrow isTrue(\ulcorner s \urcorner)[/math]
Such isTrue predicate cannot possibly exist. In terms of computability, the following cannot be done:
Such [math]T.eval(\ulcorner s \urcorner)[/math] function that can evaluate the truth of any logic sentence s in a first-order (number) theory T cannot exist, and is therefore uncomputable. The fact that some -- just some -- logic sentences will be provable and can therefore be computed to be true, is the exception. It is not the rule.
Quoting A Seagull
Well, given the theorem about the undefinability of the truth in the object theory, Tarski proposes a work around. He injects the truth via an encompassing meta-theory. The meta-theory is an extension of the object theory that can evaluate the truth of sentences in the object theory but not in itself.
If T1 is the object theory and T2 is the meta theory, then T2 feeds two-tuples to T1:
([math]T1.accept(\ulcorner s\urcorner, T2.eval(\ulcorner s\urcorner))[/math])
This convoluted protocol seeks to bypass the limitations imposed be the fact that T1.eval(([math]\ulcorner s\urcorner[/math]) is undefinable.
Quoting A Seagull
The truth predicate is part of the meta theory, T2.eval([math]\ulcorner s\urcorner[/math]), but the provability predicate is part of the object theory, T1.isProvable([math]\ulcorner s\urcorner[/math]).
So, if you do not need to prove anything in the object theory T1, then yes, the truth of its statements does not matter. If you want to prove sentences in T1, then the truth of its statements is an essential extra piece of mostly external information.
The T1.isProvable([math]\ulcorner s\urcorner[/math]) predicate is rather weak. It cannot handle all true sentences s. The following is Gödel's incompleteness theorem, expressed in Tarski's convention T:
[math]\exists \ulcorner s\urcorner (T2.eval(\ulcorner s\urcorner) \leftrightarrow \neg T1.isProvable(\ulcorner s\urcorner) )[/math]
Which means: T1 is inconsistent (It will prove false statements) or there are true statements that can be expressed in the language of T1 that are not provable from T1. You can use the expression above to derive Gödel's second incompleteness theorem: If T1 is capable of proving its own consistency, then T1 is provably inconsistent.
All of this kicks in, for a first-order theory that embodies enough arithmetic to reach the level of complexity of Q (=Robinson's arithmetic).
Yes, this is exactly right. Without this link words, statements and even philosophy are meaningless.
Who is 'feeding a logic sentence to a theory'?
What is a 'logic sentence'? - as opposed to a 'non-logic sentence'?
What theory are you referring to?
And why does it 'need' a two-tuple?
The thing about formal systems, which are necessarily abstract, is that any proof or truth they generate is only applicable within that particular formal system.
In order to apply the theorems of a formal system to some other system, a mapping is required between the elements or symbols of one system with those of the other. This mapping process is not a logical deductive process and so when the mapping is complete the deductive proof and certainty provided by the formal system cannot be carried over onto the other system.
All too often formal systems and semantic systems are conflated, this is unjustifiable.
The boring wittgensteinian in me would just relate truth to language games: that thing outside is a "tree" because of its use in our language, just like it would be "derevo" if we were in russia (which I believe means wood as well as tree). it would be absurd to point to it and ask which one it "really" is - tree or derevo?
going outside of language though - if we're able to do this - it does seem that certain languages are superior to others. if i were to go down this route and explain this it would be a different type of truth than the simple one explained above.
even the word "truth" has different meanings across culture and languages. russians distinguish between "pravda" and "instina": that which is apparently true and that which is an unshakeable fact of the universe.
The user of the theory.
Quoting A Seagull
A logic sentence evaluates to true or false. A non-logic sentence may evaluate to something else or to nothing at all. Example:
5+3 --> non-logic sentence, because it evaluates to a number.
It is raining now --> logic sentence, because it evaluates to a boolean value.
Quoting A Seagull
The truth of a sentence can generally not be determined by the theory. The general case is that it must be externally supplied. Other properties of such sentence could be decidable by the theory. For example, the number of characters in a logic sentence is decidable in a sufficiently strong theory. In the general case, the truth value of the sentence is not.
Quoting A Seagull
The term "semantic" itself is already a quite confusing and annoying one in model theory. The term "semantic" is used to indicate truth value while the term "syntactic" is used to indicate provability. This does not mean such semantics have a real-world interpretation or other such meaning. In fact, everything is synctactic in mathematics. Therefore, the meaning of the term "semantic" in mathematics is quite different from what you could otherwise expect. I do not really like that situation but there is nothing that I can do about it! ;-)
Since we have agreed that the truth of a statement is a label and not a property, how can a statement be 'evaluated' to a truth or falsity?
Just because one can apply a statement to a formal system, this does not mean that it is expedient or useful to do so. It seems to me to be a pointless exercise.
Actually, you are right. The term 'evaluated' is a bit ambiguous here.
It is a syntactic thing. For example, you can legitimately write:
s1 [math]\leftrightarrow[/math] s2
That is legitimate. It means that (s1,s2) is (true,true) or (false,false). The tuple of variables (s1,s2) could, for example, "evaluate" to (true,true). You could also say that it then "resolves" to (true,true).
This is ambiguous because s1 looks like "true" or "false" while [math]\ulcorner s1 \urcorner[/math] could look like "Socrates is mortal". The term "evaluate" does not mean in that case that the theory can determine its truth. It just means that it can replace the symbol "s1" by "true".
Properties, on the other hand, are implemented through predicates:
Quoting Wikipedia on predicates
Even though it would mean exactly the same as the sentence above, the following is not allowed:
truth([math]\ulcorner s1 \urcorner[/math]) [math]\leftrightarrow[/math] truth([math]\ulcorner s2 \urcorner[/math])
because truth cannot be defined as a predicate.
In another case, sometimes a sentence is actually even provable in the theory, and then through the computational steps of some proof procedure, the sentence "evaluates" to true.
The term to "evaluate" can be ambiguous, actually; especially in this context ...
You are referring to a specific formal system. And you can have many theorems and analyses within that formal system. You can even specify an interpretation of 'truth' within that system. But you cannot apply that system to statements about the real world without going outside of that system to a domain where the rules, axioms and 'truths' of the system do not apply.
The way that I see it there are two sources of analytical truth:
(a) Some expressions are stipulated to have the semantic property of Boolean true.
We must do this when we assign meaning to otherwise totally meaningless finite strings.
Meaning is represented as relations between finite strings. The source of these relations is
the model of the world. These relations are encoded in the various language of the world.
(b) Some expressions are deduced from the first set using sound deductive inference.
We see a patch of colour in the world. I label it as red and you label it as blue. A third person makes the statement "this patch of colour is red".
For me, the statement "this patch of colour is red" is true, whilst for you, the statement "this patch of colour is red" is false.
Whether the statement is true or false depends on what the patch of colour has been labelled.
Who or what determines what a patch of colour is labelled? It cannot be the world, as the world has no language. It could be a god if one believed in a god. It could be a Public Institution, but then one Public Institution could name it red and another name it rouge. Labelling could be up to each individual, but then the truth of a statement would be relative to each individual.
Whether the statement "this patch of colour is red" is true or not depends on what colour the patch of colour has been labelled as.
The problem is that there can be no mechanism of naming that is independent of the users of the language, meaning that naming can only be done by the users of the language. This results in the situation that a word only has meaning to the users of a language because the users of the language have given a meaning to that word.
So what does it mean to say that a statement is true. If the users of a language say "this patch of colour is the colour X" then it is true that "this patch of colour is the colour X".
IE, truth is what the users of a language say it is.
Thanks for your very clear exposition of the logic.
Applying this to ordinary language, which does not separate T1 and T2 and is therefore liable to get into a bit of a mess when talking about itself and 'truth', one finds that the language itself cannot decide whether "The cat is on the mat." is true or false. This is exactly what most of us want, because we do not want to be talking about the language, but about the cat. Generally, the cat determines where the cat is, and thereby the truth value of the sentence. This is only a problem for folks that want to live entirely in their own heads and have their inner monologue determine everything.
Shame they've been banned these last three years.
No, this is exactly wrong. Plenty of people believe in statements that inform their actions and these beliefs are not true.
Truth is reality. Our ideas can be concurrent with reality, or contradicted by reality. While we can never know if our concurrent beliefs with reality are just because we have not stumbled onto a point of reality that will contradict it, we can know when reality contradicts our beliefs.
I observe a patch of colour and name the colour red. The statement "this patch of colour is red" is then true. Naming is a logical process, in that if I name a patch of colour red then it is true that the patch of colour is red.
Tarski proposed an object language and metalanguage, whereby the truth of a statement cannot be found in the object language but only externally in a metalanguage.
Given the statement "this patch of colour is red" in an object language, it is not possible to know whether the statement is true or false within the object language itself. The truth or falsity of the statement can only be found external to the object language, as Tarski proposed, in a metalanguage .
In an object language there may be a set of words "this patch of colour is red". This set of words is neither true nor false until the words have been given meaning by naming, where naming is external to the object language. The object language cannot name itself, it cannot talk about itself. Once I have named this patch of colour as red, then the statement "this patch of colour is red" becomes true.
Naming is external to the object language, and once the words in an object language have been given meaning by naming, then statements within the object language can then be known to be either true or false by reference to a naming external to the object language itself, ie, in a metalanguage.
Generally speaking, I could say that a statement can be considered to be if it can be proved to be in accordance to specific rules, conditions, facts, etc. within a certain frame of reference. More strictly, it must not leave any doubt about its truthfulness --in the same frame of reference-- and it can be proved to be so.
Note however that this can be hardly applied to Philosophy! :smile:
Of course, since proofs can only be offered and accepted in Science and in Law.
To evaluate the truthfulness of a statement, 1) the statement must make sense, 2) we must know the context or frame of reference in which it is made or applied and 3) assume that all its elements are well defined/understood, so that there's no doubt or ambiguity about the semantics involved. In other words, the statement must not raise plausible questions about any of the elements it is composed of. E.g. saying "This man is a stranger" has no meaning as a standalone statement. It raises the basic question "'Stranger' in what way?" Which has a lot or ramifications:
1) "Is he a newcomer to some place?" and "in what place?" (One meaning of the word "stranger")
2) "Is he not known, with no personal acquaintances" and "not known by whom? (A second meaning of the word "stranger")
3) "Has he visited a place without being invited"? (A third meaning of the word "stranger")
And so on.
From the moment we know the answers to these questions, we can then judge whether this statement is true or false. E.g. This man is a Dutchman and is a stranger in my country, which is Greece. Or this man is not talking with anyone in the party. Or he has managed to sneak into a golf club without being a member. In all these cases, the statement "This man is a stranger" can be considered to be true.
We have to assume of course that the facts involved in each case can be verified to be also true. So, there's a second level in testing the truthfulness of a statement: The verification of the facts offered as evidence! And this may seem to go ad infinitum. So there must be some agreement --between all parts involved-- at some point, where we must conclude definitely about the truthfulness of the statement! This is how court decisions are made about the innocence or guiltiness of the accused when a jury is involved.
As you say, the verification of the truth will go on ad infinitum until someone makes a determination as to the truth of a statement or situation.
I made the metaphorical comment that "truth is what the users of a language say it is."
As your example of a court decision shows, a jury made up of resident citizens and registered electors after deliberation determines guilt or innocence, upon which the judge passes sentence.
In this case, it is the jury after consideration of the evidence that makes the judgement as to the truth. What the jury says is the truth of the matter is then accepted as the truth by the wider society.
Right. The larger the degree or volume of acceptance or agreement, the more "solid" the truth of a statement --or a matter in general-- is.
Which makes us see that truth is relative. There is always a margin for doubt or error, however small this may be.
In other words, there is no such a thing as an absolute truth. Not that we can think of and talk about. To establish such a truth would require an infinite power of perception and undestanding, i.e. our perception and undestanding should be absolute too.
So, we can say that truth is what is agreed to be true. That is, in accordance with facts, actual states or conditions.
Three Classic Objections and Responses
All 3 cases described in the page of the mentioned link refer to beliefs and beliefs are opinions that can be countered, contradicted, etc., but not facts that can be refuted, i.e. proven to be false.
I cannot even say, to you e.g. "Your belief in the existence of ghosts is false". It has no meaning. It is your right to believe in ghosts. And you do believe in ghosts. For you, ghosts are real. So, except if one is disillusioned, one's beliefs are always "true".
Then, beliefs show a certain amount of certitude about something. We say, e.g. "I believe that this is what most probably have happend.". Ans, when we are totally certain about something, we say e.g. "I don't just believe that this is so; I know this is so". But still though, that would not be a fact that can be proven to be true or false[/b].
So, lets stick to the facts, please! :smile:
:up:
5.2 of SEP Pragmatic Theory of Truth concludes that pragmatic theories of truth do make a difference in shaping inquiry and assertoric discourse, and unlike other accounts of truth "do not block the way of inquiry".
[i]It is also possible to ask this question of the pragmatic theory of truth itself: what difference does this theory make? Or to put it in James’ terms, what is its “cash value”? One answer is that, by focusing on the practical function of the concept of truth, pragmatic theories highlight how this concept makes certain kinds of inquiry and discourse possible. In contrast, as Lynch (2009) notes, some accounts of truth make it difficult to see how certain claims are truth-apt:
consider propositions like two and two are four or torture is wrong. Under the assumption that truth is always and everywhere causal correspondence, it is a vexing question how these true thoughts can be true. (Lynch 2009)
If that is so, then pragmatic theories have the advantage of preserving the possibility and importance of various types of inquiry and discourse. While this does not guarantee that inquiry will always reach a satisfying or definite conclusion, this does suggest that pragmatic theories of truth do make a difference: in the spirit of Peirce’s “first rule of reason”, they “do not block the way of inquiry”[/i]
Taking the statements "two and two are four", "torture is wrong", "grass" is green", "force equals mass times acceleration", "DNA is the code of life" or life evolved by natural selection" as examples, what Theory of Truth other than the Pragmatic Theory of Truth is able to give a useful answer as to the truth or falsity of these statements ?
Where is there a realist and metaphysical Theory of Truth that is capable of defining the truth of a statement ?
Obviously Av~A is true and A&~A is false. Other statments are contingents, as in are not a tautology nor contradictions.
That's all there's to it really... :-)
Well that depends.Take A&~A is false. How so? There is a symbol A and there is a symbol ~A thus it is true. Or substitute for A “I am in pain”, in the case Av~A is false because ~A is not sensical so the “or” cannot function if utter in the situation of expressing pain.
The objections refer to beliefs, and not to truths? Then can you give an account, in your pragmatic terms, of the difference between something's being true, and something's being only believed true? It seems that pragmatism ignores truth in favour of collective belief.
Supposing that one could provide a definition of truth is supposing that somehow truth is analysable, that it has components that might be identified. I think that rather, truth is something that has to be taken as granted in order for such analysis to occur. If you like, being true is what we do with felicitous statements; or "P" is true IFF P. There is more that can be said, and has been by Kripke and Tarski and Davidson and many others, discussions that are closed off if one simply drops truth in favour of communal belief.
"I believe in God" is different than "I assert that God exists". The first is not a posit that requires a proof. It cannot be agreed or disagreed upon, argued and refuted. The second is an assertion that requires a proof and it can be agreed upon, argued and refuted.
Truth has to be proven. So beliefs are exempted from it.
Seems to me that both of these express a belief - a relation between a person and a proposition. If someone asserts that god exists, and yet there is no god, then their belief was wrong.
Truth, on the other hand, need not be a relation between a person and a proposition. "God exists" says nothing about what me or you or anyone else think.
Quoting Alkis Piskas
Why? What do you mean by this? Seems to me that there can be unproven truths.
OK, Banno.
And you don't want any critique of your views?
Ok, Alkis. That's one way to for you to have certainty.
Fitch’s Paradox of Knowability
Yes, been there before, but it is relevant here.
Socrates in Disneyland...
If you believe that something is true, it doesn't mean that it is true in general, or also true for me or someone else ... (But I believe that it is true for you.)
:smile:
Given the expression "the bird is blue" is true IFF the bird is blue, "the bird is blue" exists in language, and the bird is blue exists in the world.
The question is, where does this world exist, in the mind or independent of the mind. Wittgenstein in the Tractatus, para 1 "The world is everything that is the case", carefully avoided giving his understanding of where this world existed.
The answer to the question whether is it true that in the world the bird is blue first depends on deciding where this world exists.
Is it true that in a world independent of the mind there is a bird that is blue, or is it true that in the world in the mind there is a bird that is blue.
If there is a realist, metaphysical truth existing in a world independent of the mind, how can we discover this truth, given the inherent problem of trying to understand a world that is independent of the mind when all we have to understand this mind-independent world is the mind.
I observe a colour in the world and name it blue. The statement "the colour is blue" is then true because I have named the colour blue. Regardless of whether the world I have observed exists independently of my mind or exists in my mind, the statement "the colour is blue" is true. IE, "The colour is blue" is true IFF I have named the colour blue.
The problem remains, what mechanism will enable the mind to know what is true independent of the mind?
I would say a statement has the property of truth if it refers to/conforms to the world (specifically, the way things are at some time).
Related: Crispin Sartwell wrote a recent article, arguing one commonality between analytic and continental philosophers was their emphasis on language- the linguistic turn. He goes on to conclude that emphasis seems off, and even destructive, in a world where truth matters (of course, it has always mattered). His example is climate change. In general, I have noticed a shift. It used to be sexy to say things like, "Your truth is whatever you want it to be." I don't see much celebration of the malleability of truth anymore. I think the political landscape, at least in the US, has shown us how reckless those attitudes can be. At any rate, Sartwell:
"Any philosophy that seemed to undermine the reality of the natural world, or make it a malleable human artefact, has come to feel potentially destructive."
The post-linguistic turn
Besides of course tautologies and contradictions, and statements like "I am a Liar" which are by definition always true or always false.
As for Socrates I am not even sure he ever existed, perhaps he is/was a fictional character...'
What's the difference between fiction and reality in that case? it depends on what do you believe of course...
Of course it doesn't matter. But please complete the sentence "As long as you or I or him believe it to be true ..." (Then what?)
Quoting A Realist
But this statement is neither true nor false! :smile: It's a known "paradox", or better, a self-contradictory statement. If I am a liar then this statement is a lie (false). But then, if this statement is a lie (false), it means that I'm not a liar. See the self-contradiction involved?
Quoting A Realist
What have I to do with Socrates in this thread? :smile: Have you confound me with someone else?
Anyways, the Liar paradox can also be regarded as both true and false.
As long as you or I or him believe something to be true, then it's true.
Tarski said that the truth of a statement cannot be found in an object language, but only in a metalanguage
Naming is extra-linguistic. For example, suppose I observe a colour in the world and name it "chekundu". As naming is extra-linguistic, naming is, in Tarski's terms, in a metalanguage. Once this colour has been named "chekundu", then the statement "this colour is chekundu" is true. The truth of the statement cannot be found in the object language itself, but in the act of naming, which is extra-linguistic, and part of a meta-language.