What is truth?
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From: Aatu Koskensilta on 6 May 2010 09:45 rods <rodpinto(a)gmail.com> writes: > OK. So I think that we at least agree on our disagreement. There's nothing to agree or disagree about here -- this is purely a matter of technical terminology. Just look up the relevant definitions. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 6 May 2010 09:46 bert <herbertglazier79(a)msn.com> writes: > Truth is reality Bible is not reality. Gods are not > reality.Humankind's quest is to find reality. Possibly, possibly. But what does this have to do with the incompleteness theorems? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: rods on 6 May 2010 09:48 > He's saying that the claim "The model is consistent" is literally > nonsense. Theories are consistent or not. Consistency is not a term > that applies to models. It's really that simple. So just change the word model by the word theory. My base argument don't change. This is just a semantic change. That's what I was saying about Tarski semantic concept of truth. Rodrigo
From: Aatu Koskensilta on 6 May 2010 10:34 rods <rodpinto(a)gmail.com> writes: >> He's saying that the claim "The model is consistent" is literally >> nonsense. �Theories are consistent or not. �Consistency is not a term >> that applies to models. �It's really that simple. > > So just change the word model by the word theory. My base argument > don't change. Well, let's have a look at your argument with this change: Let's say a theory is consistent. And let's say in this theory we have something called "empirical truth". My theory cannot include a "statement of its own consistency" because if it does so I can use Godel's Second Incompleteness Theorem to show that theory is inconsistent. With some good-will we can make sense of this. We have a consistent formal theory, containing enough arithmetic so that the incompleteness theorems apply, some formulas of which are designated "empirical". For example, we may consider a theory T obtained by adding to first-order Peano arithmetic a distinct sort of variable, taken to range over all sheep on Earth at some given time, with a predicate B true of black sheep, and with a function F taking a sheep to the number of its young, possibly with some facts about blackness of sheep and their young as axioms. In this theory we can formulate assertions such as "There is a black sheep with 500 + 500 young", which we may naturally regard "empirical", in that their truth or falsity depends on contingent facts about sheep at some given time. And by the second incompleteness theorem this theory doesn't prove its own consistency. The way Tarski deals with this is to use a semantical approach. So instead of saying that there is an "experimental truth" that would lead to an inconsistent theory we can just use "Truth". And it is not required to have a "statement of its own consistency" to prove that my "defined" is really "truth". Alas, at this point your argument becomes utterly incomprehensible. What is it that you take Tarski to be dealing with? What is an "experimental truth" that would lead -- how? -- to an inconsistent theory? What does it mean to use "Truth"? And so on and so forth. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: rods on 6 May 2010 11:29
On 6 maio, 11:34, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > rods <rodpi...(a)gmail.com> writes: > >> He's saying that the claim "The model is consistent" is literally > >> nonsense. Theories are consistent or not. Consistency is not a term > >> that applies to models. It's really that simple. > > > So just change the word model by the word theory. My base argument > > don't change. > > Well, let's have a look at your argument with this change: > > Let's say a theory is consistent. And let's say in this theory we have > something called "empirical truth". My theory cannot include a > "statement of its own consistency" because if it does so I can use > Godel's Second Incompleteness Theorem to show that theory is > inconsistent. > > With some good-will we can make sense of this. We have a consistent > formal theory, containing enough arithmetic so that the incompleteness > theorems apply, some formulas of which are designated "empirical". For > example, we may consider a theory T obtained by adding to first-order > Peano arithmetic a distinct sort of variable, taken to range over all > sheep on Earth at some given time, with a predicate B true of black > sheep, and with a function F taking a sheep to the number of its young, > possibly with some facts about blackness of sheep and their young as > axioms. In this theory we can formulate assertions such as "There is a > black sheep with 500 + 500 young", which we may naturally regard > "empirical", in that their truth or falsity depends on contingent facts > about sheep at some given time. And by the second incompleteness theorem > this theory doesn't prove its own consistency. > > The way Tarski deals with this is to use a semantical approach. So > instead of saying that there is an "experimental truth" that would > lead to an inconsistent theory we can just use "Truth". And it is not > required to have a "statement of its own consistency" to prove that my > "defined" is really "truth". > > Alas, at this point your argument becomes utterly incomprehensible. What > is it that you take Tarski to be dealing with? What is an "experimental > truth" that would lead -- how? -- to an inconsistent theory? What does > it mean to use "Truth"? And so on and so forth. > OK. Now I think we can start talking about something useful again. If we want to talk about something (anything) we must agree about some basic definitions. We can start an endless argument here about the definition of truth and we can end up in a disagreement. That's why I brought the uncertainty principle up on one of my previous posts. You probably knows better than me that Hilbert was looking for a complete formalization of math, and Godels theorems showed that this was not possible. Going back to the definition of truth. If we want to discuss about an empirical truth we should at least agree on what is truth and what is empirical truth. So let's say we agree on the definition of Truth. This agreement is not necessarily true. I can propose a truth definition that you don't like and vice-versa. This is were I bring Tarski to this discussion. If we talk about "pure" mathmatics we can arrive at stuff like Russell's paradox or things like this. The way Tarski works this out is to use a pure semantic definition of truth. Let's say that we agree on the definition of truth in a pure semantic approach. We both recognize the word "truth" and we both agree on its meaning. We don't need necessarily agree that something is or is not true. So now we can discuss what is experimental truth. For me, an experimental truth is something that can be expected as a result of an experiment. But then we have to agree on the definition of experiment. To avoid all this, we can just "assume" that an experimental truth is a tautology. Again, you don't have to agree on this definition, but then we should go back and try to find out a definition of experiment. Godel's incompleteness theorem shows that if there is a theory, you can have something that is true in this theory and not provable. But something that cannot be proved is not an experimental truth. So if you say that there is something that can be defined as an experimental truth you are saying that there is at least the possibility to have another kind of truth that is not experimental. I think it is easier if we just assume that truth is something that cannot be provable and a experimental truth is a tautology. But again, you may or may not agree with me. Rodrigo |