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Can truth of a single predicate be defined by another language that only shares that predicate?
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I have been looking at Tarski's work and asked a question here:
Can truth be defined by a 'same-level' language according to Tarski's undefinability theorem?
I am trying to refine my question as follows: if a 'meta-language' L* is meta- only as far as one predicate is concerned, can the truth of that predicate in the lower language L be proven? Or is it essential that L* contains all of L (plus more), to prove the truth of something in L?
logic first-order-logic predicate-logic
asked Jul 18 at 12:43
matteoeoeo
365
 |Â
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0
down vote
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I have been looking at Tarski's work and asked a question here:
Can truth be defined by a 'same-level' language according to Tarski's undefinability theorem?
I am trying to refine my question as follows: if a 'meta-language' L* is meta- only as far as one predicate is concerned, can the truth of that predicate in the lower language L be proven? Or is it essential that L* contains all of L (plus more), to prove the truth of something in L?
logic first-order-logic predicate-logic
asked Jul 18 at 12:43
matteoeoeo
365
Not clear... you say "the truth of that predicate"; I think it must be "the truth-predicate".
– Mauro ALLEGRANZA
Jul 18 at 12:50
I guess I mean the truth of one single statement, without having to prove the truth of the entire language L.
– matteoeoeo
Jul 18 at 13:49
1
Truth of a single statement $S$ can be expressed without any met-language; just say $S$.
– Andreas Blass
Jul 18 at 16:30
But then I’m missing something important. When would a meta-language be needed? Can’t every truth be expressed within L then?
– matteoeoeo
Jul 18 at 19:50
1
Truth of any single statement of $L$ can (trivially) be expressed in $L$. The general notion of truth for $L$-sentences cannot be expressed in $L$ (Tarski's theorem). The liar paradox is essentially the proof of Tarski's theorem. A (fairly) general notion of truth is needed in order to produce the liar sentence ("This sentence is false"), and that's why such a general notion of truth can't exist.
– Andreas Blass
Jul 19 at 11:27
 |Â
show 1 more comment
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I have been looking at Tarski's work and asked a question here:
Can truth be defined by a 'same-level' language according to Tarski's undefinability theorem?
I am trying to refine my question as follows: if a 'meta-language' L* is meta- only as far as one predicate is concerned, can the truth of that predicate in the lower language L be proven? Or is it essential that L* contains all of L (plus more), to prove the truth of something in L?
logic first-order-logic predicate-logic
asked Jul 18 at 12:43
matteoeoeo
365
I have been looking at Tarski's work and asked a question here:
Can truth be defined by a 'same-level' language according to Tarski's undefinability theorem?
I am trying to refine my question as follows: if a 'meta-language' L* is meta- only as far as one predicate is concerned, can the truth of that predicate in the lower language L be proven? Or is it essential that L* contains all of L (plus more), to prove the truth of something in L?
logic first-order-logic predicate-logic
asked Jul 18 at 12:43
matteoeoeo
365
asked Jul 18 at 12:43
matteoeoeo
365
asked Jul 18 at 12:43
matteoeoeo
365
asked Jul 18 at 12:43
matteoeoeo
365
365
Not clear... you say "the truth of that predicate"; I think it must be "the truth-predicate".
– Mauro ALLEGRANZA
Jul 18 at 12:50
I guess I mean the truth of one single statement, without having to prove the truth of the entire language L.
– matteoeoeo
Jul 18 at 13:49
1
Truth of a single statement $S$ can be expressed without any met-language; just say $S$.
– Andreas Blass
Jul 18 at 16:30
But then I’m missing something important. When would a meta-language be needed? Can’t every truth be expressed within L then?
– matteoeoeo
Jul 18 at 19:50
1
Truth of any single statement of $L$ can (trivially) be expressed in $L$. The general notion of truth for $L$-sentences cannot be expressed in $L$ (Tarski's theorem). The liar paradox is essentially the proof of Tarski's theorem. A (fairly) general notion of truth is needed in order to produce the liar sentence ("This sentence is false"), and that's why such a general notion of truth can't exist.
– Andreas Blass
Jul 19 at 11:27
 |Â
show 1 more comment
Not clear... you say "the truth of that predicate"; I think it must be "the truth-predicate".
– Mauro ALLEGRANZA
Jul 18 at 12:50
I guess I mean the truth of one single statement, without having to prove the truth of the entire language L.
– matteoeoeo
Jul 18 at 13:49
1
Truth of a single statement $S$ can be expressed without any met-language; just say $S$.
– Andreas Blass
Jul 18 at 16:30
But then I’m missing something important. When would a meta-language be needed? Can’t every truth be expressed within L then?
– matteoeoeo
Jul 18 at 19:50
1
Truth of any single statement of $L$ can (trivially) be expressed in $L$. The general notion of truth for $L$-sentences cannot be expressed in $L$ (Tarski's theorem). The liar paradox is essentially the proof of Tarski's theorem. A (fairly) general notion of truth is needed in order to produce the liar sentence ("This sentence is false"), and that's why such a general notion of truth can't exist.
– Andreas Blass
Jul 19 at 11:27
Not clear... you say "the truth of that predicate"; I think it must be "the truth-predicate".
– Mauro ALLEGRANZA
Jul 18 at 12:50
Not clear... you say "the truth of that predicate"; I think it must be "the truth-predicate".
– Mauro ALLEGRANZA
Jul 18 at 12:50
I guess I mean the truth of one single statement, without having to prove the truth of the entire language L.
– matteoeoeo
Jul 18 at 13:49
I guess I mean the truth of one single statement, without having to prove the truth of the entire language L.
– matteoeoeo
Jul 18 at 13:49
1
1
Truth of a single statement $S$ can be expressed without any met-language; just say $S$.
– Andreas Blass
Jul 18 at 16:30
Truth of a single statement $S$ can be expressed without any met-language; just say $S$.
– Andreas Blass
Jul 18 at 16:30
But then I’m missing something important. When would a meta-language be needed? Can’t every truth be expressed within L then?
– matteoeoeo
Jul 18 at 19:50
But then I’m missing something important. When would a meta-language be needed? Can’t every truth be expressed within L then?
– matteoeoeo
Jul 18 at 19:50
1
1
Truth of any single statement of $L$ can (trivially) be expressed in $L$. The general notion of truth for $L$-sentences cannot be expressed in $L$ (Tarski's theorem). The liar paradox is essentially the proof of Tarski's theorem. A (fairly) general notion of truth is needed in order to produce the liar sentence ("This sentence is false"), and that's why such a general notion of truth can't exist.
– Andreas Blass
Jul 19 at 11:27
Truth of any single statement of $L$ can (trivially) be expressed in $L$. The general notion of truth for $L$-sentences cannot be expressed in $L$ (Tarski's theorem). The liar paradox is essentially the proof of Tarski's theorem. A (fairly) general notion of truth is needed in order to produce the liar sentence ("This sentence is false"), and that's why such a general notion of truth can't exist.
– Andreas Blass
Jul 19 at 11:27
 |Â
show 1 more comment
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Not clear... you say "the truth of that predicate"; I think it must be "the truth-predicate".
– Mauro ALLEGRANZA
Jul 18 at 12:50
I guess I mean the truth of one single statement, without having to prove the truth of the entire language L.
– matteoeoeo
Jul 18 at 13:49
1
Truth of a single statement $S$ can be expressed without any met-language; just say $S$.
– Andreas Blass
Jul 18 at 16:30
But then I’m missing something important. When would a meta-language be needed? Can’t every truth be expressed within L then?
– matteoeoeo
Jul 18 at 19:50
1
Truth of any single statement of $L$ can (trivially) be expressed in $L$. The general notion of truth for $L$-sentences cannot be expressed in $L$ (Tarski's theorem). The liar paradox is essentially the proof of Tarski's theorem. A (fairly) general notion of truth is needed in order to produce the liar sentence ("This sentence is false"), and that's why such a general notion of truth can't exist.
– Andreas Blass
Jul 19 at 11:27