In which language can one EXPRESS the fact that two sentences are contradictory ? is meta-language required? Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Why do other logical symbols except $to$ and $leftrightarrow$ not have corresponding meta symbols?Can the ongoing need for a meta language be stopped by a loop?Can second order logic express each (computable) infinitary logic sentence?Translating sentences with causality into logical propositionsWhy aren't valid higher order logic sentences recursively enumerable in full semantics?What are the L-sentences that are true in an empty structure?Do all logics have formula syntax specified by context-free grammar?Completeness theorem for second-order logic in the language $$Regular languages not expressible in first-order logic with modular quantifiersHow to translate a first-order formula to natural languageis first-order logic with constants equally expressive as first-order logic without constants?
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In which language can one EXPRESS the fact that two sentences are contradictory ? is meta-language required?
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Why do other logical symbols except $to$ and $leftrightarrow$ not have corresponding meta symbols?Can the ongoing need for a meta language be stopped by a loop?Can second order logic express each (computable) infinitary logic sentence?Translating sentences with causality into logical propositionsWhy aren't valid higher order logic sentences recursively enumerable in full semantics?What are the L-sentences that are true in an empty structure?Do all logics have formula syntax specified by context-free grammar?Completeness theorem for second-order logic in the language $$Regular languages not expressible in first-order logic with modular quantifiersHow to translate a first-order formula to natural languageis first-order logic with constants equally expressive as first-order logic without constants?
$begingroup$
[ Edited: title modified]
Remark.- My question is not : is it possible to produce a contradictory formula in the language of first order logic. ( I actually can produce such a formula: for ex. the formula P & ~ P ) . My question is rather : " is it possible to say, in the language itself , that a formula is contradictory , or to say that the contradiction relation holds between two formulas.
To express the idea that sentence A and sentence B contradict each another, is it sufficient to say that :
(A w B) [ with ' w ' meaning : exclusive OR] ?
or does one have to translate it as :
" necessarily (A w B) " ,
or, maybe as,
" (A w B) is valid, true in all possible interpretations"
in which case the " contradiction " relation would not be expressible in the language of first order logic.
logic
asked Apr 1 at 16:37
Ray LittleRockRay LittleRock
10110
$endgroup$
|
show 3 more comments
$begingroup$
[ Edited: title modified]
Remark.- My question is not : is it possible to produce a contradictory formula in the language of first order logic. ( I actually can produce such a formula: for ex. the formula P & ~ P ) . My question is rather : " is it possible to say, in the language itself , that a formula is contradictory , or to say that the contradiction relation holds between two formulas.
To express the idea that sentence A and sentence B contradict each another, is it sufficient to say that :
(A w B) [ with ' w ' meaning : exclusive OR] ?
or does one have to translate it as :
" necessarily (A w B) " ,
or, maybe as,
" (A w B) is valid, true in all possible interpretations"
in which case the " contradiction " relation would not be expressible in the language of first order logic.
logic
asked Apr 1 at 16:37
Ray LittleRockRay LittleRock
10110
$endgroup$
$begingroup$
Frankly speaking, I cannot understand your concern... $A$ and $lnot A$ obviously contradict each other. Thus, what do you mean with "To express the idea that sentences A and B contradict each another" ? A set of sentences (formulas) is contradictory or unsatisfiable when it is not possible to satisfy all them together. Thus for two sentences $A$ and $B$, we may simply say that $ A, B $ is unsatisfiable. A simple way to express this is to use the falsum constant $bot$ writing : $ A, B vDash bot$.
$endgroup$
– Mauro ALLEGRANZA
Apr 1 at 18:52
$begingroup$
@ Mauro Allegranza My question was related to a previous one concerning the principle of non-contradiction: does ithis principle belong to language or to meta-language? My aim was also at knowing whether there was a modal notion involved in the principle. If I understand correctly your answer, it means that the idea " A contradicts B" has to be expressed in the metalanguage ( via the concept of logical consequence).
$endgroup$
– Ray LittleRock
Apr 1 at 19:00
$begingroup$
I'll have to edit my question. The answer given by Bram 28 made me realize that there is a difference between " A contradicts B" and " A and B are contradictory".
$endgroup$
– Ray LittleRock
Apr 1 at 19:02
$begingroup$
Maybe... in logic a contradiction is a sentence/ formula that is always FALSE : $P land lnot P$. Obviously, $P$ and $lnot P$ contradict each other (in the "usual" sense of the word).
$endgroup$
– Mauro ALLEGRANZA
Apr 1 at 19:10
$begingroup$
I totally agree that (P& ~P) is a contradiction. My question was simply whether there was a way to say " this sentence is a contradiction" in the language itself, or whether it has to be said at the meta-level. What I understand from your answer and Bram28's is that it has to be expressed in the meta-language, or in the language of modal logic.
$endgroup$
– Ray LittleRock
Apr 1 at 19:19
|
show 3 more comments
$begingroup$
[ Edited: title modified]
Remark.- My question is not : is it possible to produce a contradictory formula in the language of first order logic. ( I actually can produce such a formula: for ex. the formula P & ~ P ) . My question is rather : " is it possible to say, in the language itself , that a formula is contradictory , or to say that the contradiction relation holds between two formulas.
To express the idea that sentence A and sentence B contradict each another, is it sufficient to say that :
(A w B) [ with ' w ' meaning : exclusive OR] ?
or does one have to translate it as :
" necessarily (A w B) " ,
or, maybe as,
" (A w B) is valid, true in all possible interpretations"
in which case the " contradiction " relation would not be expressible in the language of first order logic.
logic
asked Apr 1 at 16:37
Ray LittleRockRay LittleRock
10110
$endgroup$
[ Edited: title modified]
Remark.- My question is not : is it possible to produce a contradictory formula in the language of first order logic. ( I actually can produce such a formula: for ex. the formula P & ~ P ) . My question is rather : " is it possible to say, in the language itself , that a formula is contradictory , or to say that the contradiction relation holds between two formulas.
To express the idea that sentence A and sentence B contradict each another, is it sufficient to say that :
(A w B) [ with ' w ' meaning : exclusive OR] ?
or does one have to translate it as :
" necessarily (A w B) " ,
or, maybe as,
" (A w B) is valid, true in all possible interpretations"
in which case the " contradiction " relation would not be expressible in the language of first order logic.
logic
logic
asked Apr 1 at 16:37
Ray LittleRockRay LittleRock
10110
asked Apr 1 at 16:37
Ray LittleRockRay LittleRock
10110
edited Apr 3 at 11:19
Ray LittleRock
asked Apr 1 at 16:37
Ray LittleRockRay LittleRock
10110
asked Apr 1 at 16:37
Ray LittleRockRay LittleRock
10110
asked Apr 1 at 16:37
Ray LittleRockRay LittleRock
10110
10110
$begingroup$
Frankly speaking, I cannot understand your concern... $A$ and $lnot A$ obviously contradict each other. Thus, what do you mean with "To express the idea that sentences A and B contradict each another" ? A set of sentences (formulas) is contradictory or unsatisfiable when it is not possible to satisfy all them together. Thus for two sentences $A$ and $B$, we may simply say that $ A, B $ is unsatisfiable. A simple way to express this is to use the falsum constant $bot$ writing : $ A, B vDash bot$.
$endgroup$
– Mauro ALLEGRANZA
Apr 1 at 18:52
$begingroup$
@ Mauro Allegranza My question was related to a previous one concerning the principle of non-contradiction: does ithis principle belong to language or to meta-language? My aim was also at knowing whether there was a modal notion involved in the principle. If I understand correctly your answer, it means that the idea " A contradicts B" has to be expressed in the metalanguage ( via the concept of logical consequence).
$endgroup$
– Ray LittleRock
Apr 1 at 19:00
$begingroup$
I'll have to edit my question. The answer given by Bram 28 made me realize that there is a difference between " A contradicts B" and " A and B are contradictory".
$endgroup$
– Ray LittleRock
Apr 1 at 19:02
$begingroup$
Maybe... in logic a contradiction is a sentence/ formula that is always FALSE : $P land lnot P$. Obviously, $P$ and $lnot P$ contradict each other (in the "usual" sense of the word).
$endgroup$
– Mauro ALLEGRANZA
Apr 1 at 19:10
$begingroup$
I totally agree that (P& ~P) is a contradiction. My question was simply whether there was a way to say " this sentence is a contradiction" in the language itself, or whether it has to be said at the meta-level. What I understand from your answer and Bram28's is that it has to be expressed in the meta-language, or in the language of modal logic.
$endgroup$
– Ray LittleRock
Apr 1 at 19:19
|
show 3 more comments
$begingroup$
Frankly speaking, I cannot understand your concern... $A$ and $lnot A$ obviously contradict each other. Thus, what do you mean with "To express the idea that sentences A and B contradict each another" ? A set of sentences (formulas) is contradictory or unsatisfiable when it is not possible to satisfy all them together. Thus for two sentences $A$ and $B$, we may simply say that $ A, B $ is unsatisfiable. A simple way to express this is to use the falsum constant $bot$ writing : $ A, B vDash bot$.
$endgroup$
– Mauro ALLEGRANZA
Apr 1 at 18:52
$begingroup$
@ Mauro Allegranza My question was related to a previous one concerning the principle of non-contradiction: does ithis principle belong to language or to meta-language? My aim was also at knowing whether there was a modal notion involved in the principle. If I understand correctly your answer, it means that the idea " A contradicts B" has to be expressed in the metalanguage ( via the concept of logical consequence).
$endgroup$
– Ray LittleRock
Apr 1 at 19:00
$begingroup$
I'll have to edit my question. The answer given by Bram 28 made me realize that there is a difference between " A contradicts B" and " A and B are contradictory".
$endgroup$
– Ray LittleRock
Apr 1 at 19:02
$begingroup$
Maybe... in logic a contradiction is a sentence/ formula that is always FALSE : $P land lnot P$. Obviously, $P$ and $lnot P$ contradict each other (in the "usual" sense of the word).
$endgroup$
– Mauro ALLEGRANZA
Apr 1 at 19:10
$begingroup$
I totally agree that (P& ~P) is a contradiction. My question was simply whether there was a way to say " this sentence is a contradiction" in the language itself, or whether it has to be said at the meta-level. What I understand from your answer and Bram28's is that it has to be expressed in the meta-language, or in the language of modal logic.
$endgroup$
– Ray LittleRock
Apr 1 at 19:19
$begingroup$
Frankly speaking, I cannot understand your concern... $A$ and $lnot A$ obviously contradict each other. Thus, what do you mean with "To express the idea that sentences A and B contradict each another" ? A set of sentences (formulas) is contradictory or unsatisfiable when it is not possible to satisfy all them together. Thus for two sentences $A$ and $B$, we may simply say that $ A, B $ is unsatisfiable. A simple way to express this is to use the falsum constant $bot$ writing : $ A, B vDash bot$.
$endgroup$
– Mauro ALLEGRANZA
Apr 1 at 18:52
$begingroup$
Frankly speaking, I cannot understand your concern... $A$ and $lnot A$ obviously contradict each other. Thus, what do you mean with "To express the idea that sentences A and B contradict each another" ? A set of sentences (formulas) is contradictory or unsatisfiable when it is not possible to satisfy all them together. Thus for two sentences $A$ and $B$, we may simply say that $ A, B $ is unsatisfiable. A simple way to express this is to use the falsum constant $bot$ writing : $ A, B vDash bot$.
$endgroup$
– Mauro ALLEGRANZA
Apr 1 at 18:52
$begingroup$
@ Mauro Allegranza My question was related to a previous one concerning the principle of non-contradiction: does ithis principle belong to language or to meta-language? My aim was also at knowing whether there was a modal notion involved in the principle. If I understand correctly your answer, it means that the idea " A contradicts B" has to be expressed in the metalanguage ( via the concept of logical consequence).
$endgroup$
– Ray LittleRock
Apr 1 at 19:00
$begingroup$
@ Mauro Allegranza My question was related to a previous one concerning the principle of non-contradiction: does ithis principle belong to language or to meta-language? My aim was also at knowing whether there was a modal notion involved in the principle. If I understand correctly your answer, it means that the idea " A contradicts B" has to be expressed in the metalanguage ( via the concept of logical consequence).
$endgroup$
– Ray LittleRock
Apr 1 at 19:00
$begingroup$
I'll have to edit my question. The answer given by Bram 28 made me realize that there is a difference between " A contradicts B" and " A and B are contradictory".
$endgroup$
– Ray LittleRock
Apr 1 at 19:02
$begingroup$
I'll have to edit my question. The answer given by Bram 28 made me realize that there is a difference between " A contradicts B" and " A and B are contradictory".
$endgroup$
– Ray LittleRock
Apr 1 at 19:02
$begingroup$
Maybe... in logic a contradiction is a sentence/ formula that is always FALSE : $P land lnot P$. Obviously, $P$ and $lnot P$ contradict each other (in the "usual" sense of the word).
$endgroup$
– Mauro ALLEGRANZA
Apr 1 at 19:10
$begingroup$
Maybe... in logic a contradiction is a sentence/ formula that is always FALSE : $P land lnot P$. Obviously, $P$ and $lnot P$ contradict each other (in the "usual" sense of the word).
$endgroup$
– Mauro ALLEGRANZA
Apr 1 at 19:10
$begingroup$
I totally agree that (P& ~P) is a contradiction. My question was simply whether there was a way to say " this sentence is a contradiction" in the language itself, or whether it has to be said at the meta-level. What I understand from your answer and Bram28's is that it has to be expressed in the meta-language, or in the language of modal logic.
$endgroup$
– Ray LittleRock
Apr 1 at 19:19
$begingroup$
I totally agree that (P& ~P) is a contradiction. My question was simply whether there was a way to say " this sentence is a contradiction" in the language itself, or whether it has to be said at the meta-level. What I understand from your answer and Bram28's is that it has to be expressed in the meta-language, or in the language of modal logic.
$endgroup$
– Ray LittleRock
Apr 1 at 19:19
|
show 3 more comments
1 Answer
1
active
oldest
votes
$begingroup$
You really need the 'necessary' part in there, i.e. basically what you have in the second half of your post.
To see this, consider:
$A:$ "grass is green"
$B:$ "snow is purple"
Now, in our world, $A$ is true and $B$ is false, and hence $A oplus B$ is true. But, $A$ and $B$ do not logically contradict each other. Indeed, there are other logically possible worlds where both claims are true.
One small point though: I would say that for two sentences to contradict each other it is not necessary that they have opposing truth-values (in which case we call them 'contradictories'), but it is sufficient that they cannot both be true (in which case we call them 'contraries').
For example, I would say that $P land Q$ and $neg P$ contradict each other ... but it would not be the case that in all interpretations, exactly one of them is true. These two sentences are indeed not contradictory, but they are contrary.
So, using modal logic, I would express the claim that $varphi$ and $psi$ contradict each other as:
$square neg (varphi land psi)$
Or, if we use $uparrow$ for the NAND:
$square (varphi uparrow psi)$
answered Apr 1 at 17:14
Bram28Bram28
64.7k44793
$endgroup$
$begingroup$
@ Bram28 - This gives a new life to the antique distinction between " contraries" and " contradictories".! Actually, from " A and B contradict each other" I cannot infer " A and B are contradictories", I had never thought of this in this way.
$endgroup$
– Ray LittleRock
Apr 1 at 17:29
add a comment |
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$begingroup$
You really need the 'necessary' part in there, i.e. basically what you have in the second half of your post.
To see this, consider:
$A:$ "grass is green"
$B:$ "snow is purple"
Now, in our world, $A$ is true and $B$ is false, and hence $A oplus B$ is true. But, $A$ and $B$ do not logically contradict each other. Indeed, there are other logically possible worlds where both claims are true.
One small point though: I would say that for two sentences to contradict each other it is not necessary that they have opposing truth-values (in which case we call them 'contradictories'), but it is sufficient that they cannot both be true (in which case we call them 'contraries').
For example, I would say that $P land Q$ and $neg P$ contradict each other ... but it would not be the case that in all interpretations, exactly one of them is true. These two sentences are indeed not contradictory, but they are contrary.
So, using modal logic, I would express the claim that $varphi$ and $psi$ contradict each other as:
$square neg (varphi land psi)$
Or, if we use $uparrow$ for the NAND:
$square (varphi uparrow psi)$
answered Apr 1 at 17:14
Bram28Bram28
64.7k44793
$endgroup$
$begingroup$
@ Bram28 - This gives a new life to the antique distinction between " contraries" and " contradictories".! Actually, from " A and B contradict each other" I cannot infer " A and B are contradictories", I had never thought of this in this way.
$endgroup$
– Ray LittleRock
Apr 1 at 17:29
add a comment |
$begingroup$
You really need the 'necessary' part in there, i.e. basically what you have in the second half of your post.
To see this, consider:
$A:$ "grass is green"
$B:$ "snow is purple"
Now, in our world, $A$ is true and $B$ is false, and hence $A oplus B$ is true. But, $A$ and $B$ do not logically contradict each other. Indeed, there are other logically possible worlds where both claims are true.
One small point though: I would say that for two sentences to contradict each other it is not necessary that they have opposing truth-values (in which case we call them 'contradictories'), but it is sufficient that they cannot both be true (in which case we call them 'contraries').
For example, I would say that $P land Q$ and $neg P$ contradict each other ... but it would not be the case that in all interpretations, exactly one of them is true. These two sentences are indeed not contradictory, but they are contrary.
So, using modal logic, I would express the claim that $varphi$ and $psi$ contradict each other as:
$square neg (varphi land psi)$
Or, if we use $uparrow$ for the NAND:
$square (varphi uparrow psi)$
answered Apr 1 at 17:14
Bram28Bram28
64.7k44793
$endgroup$
$begingroup$
@ Bram28 - This gives a new life to the antique distinction between " contraries" and " contradictories".! Actually, from " A and B contradict each other" I cannot infer " A and B are contradictories", I had never thought of this in this way.
$endgroup$
– Ray LittleRock
Apr 1 at 17:29
add a comment |
$begingroup$
You really need the 'necessary' part in there, i.e. basically what you have in the second half of your post.
To see this, consider:
$A:$ "grass is green"
$B:$ "snow is purple"
Now, in our world, $A$ is true and $B$ is false, and hence $A oplus B$ is true. But, $A$ and $B$ do not logically contradict each other. Indeed, there are other logically possible worlds where both claims are true.
One small point though: I would say that for two sentences to contradict each other it is not necessary that they have opposing truth-values (in which case we call them 'contradictories'), but it is sufficient that they cannot both be true (in which case we call them 'contraries').
For example, I would say that $P land Q$ and $neg P$ contradict each other ... but it would not be the case that in all interpretations, exactly one of them is true. These two sentences are indeed not contradictory, but they are contrary.
So, using modal logic, I would express the claim that $varphi$ and $psi$ contradict each other as:
$square neg (varphi land psi)$
Or, if we use $uparrow$ for the NAND:
$square (varphi uparrow psi)$
answered Apr 1 at 17:14
Bram28Bram28
64.7k44793
$endgroup$
You really need the 'necessary' part in there, i.e. basically what you have in the second half of your post.
To see this, consider:
$A:$ "grass is green"
$B:$ "snow is purple"
Now, in our world, $A$ is true and $B$ is false, and hence $A oplus B$ is true. But, $A$ and $B$ do not logically contradict each other. Indeed, there are other logically possible worlds where both claims are true.
One small point though: I would say that for two sentences to contradict each other it is not necessary that they have opposing truth-values (in which case we call them 'contradictories'), but it is sufficient that they cannot both be true (in which case we call them 'contraries').
For example, I would say that $P land Q$ and $neg P$ contradict each other ... but it would not be the case that in all interpretations, exactly one of them is true. These two sentences are indeed not contradictory, but they are contrary.
So, using modal logic, I would express the claim that $varphi$ and $psi$ contradict each other as:
$square neg (varphi land psi)$
Or, if we use $uparrow$ for the NAND:
$square (varphi uparrow psi)$
answered Apr 1 at 17:14
Bram28Bram28
64.7k44793
edited Apr 1 at 18:23
answered Apr 1 at 17:14
Bram28Bram28
64.7k44793
answered Apr 1 at 17:14
Bram28Bram28
64.7k44793
answered Apr 1 at 17:14
Bram28Bram28
64.7k44793
64.7k44793
$begingroup$
@ Bram28 - This gives a new life to the antique distinction between " contraries" and " contradictories".! Actually, from " A and B contradict each other" I cannot infer " A and B are contradictories", I had never thought of this in this way.
$endgroup$
– Ray LittleRock
Apr 1 at 17:29
add a comment |
$begingroup$
@ Bram28 - This gives a new life to the antique distinction between " contraries" and " contradictories".! Actually, from " A and B contradict each other" I cannot infer " A and B are contradictories", I had never thought of this in this way.
$endgroup$
– Ray LittleRock
Apr 1 at 17:29
$begingroup$
@ Bram28 - This gives a new life to the antique distinction between " contraries" and " contradictories".! Actually, from " A and B contradict each other" I cannot infer " A and B are contradictories", I had never thought of this in this way.
$endgroup$
– Ray LittleRock
Apr 1 at 17:29
$begingroup$
@ Bram28 - This gives a new life to the antique distinction between " contraries" and " contradictories".! Actually, from " A and B contradict each other" I cannot infer " A and B are contradictories", I had never thought of this in this way.
$endgroup$
– Ray LittleRock
Apr 1 at 17:29
add a comment |
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Required, but never shown
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Frankly speaking, I cannot understand your concern... $A$ and $lnot A$ obviously contradict each other. Thus, what do you mean with "To express the idea that sentences A and B contradict each another" ? A set of sentences (formulas) is contradictory or unsatisfiable when it is not possible to satisfy all them together. Thus for two sentences $A$ and $B$, we may simply say that $ A, B $ is unsatisfiable. A simple way to express this is to use the falsum constant $bot$ writing : $ A, B vDash bot$.
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– Mauro ALLEGRANZA
Apr 1 at 18:52
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@ Mauro Allegranza My question was related to a previous one concerning the principle of non-contradiction: does ithis principle belong to language or to meta-language? My aim was also at knowing whether there was a modal notion involved in the principle. If I understand correctly your answer, it means that the idea " A contradicts B" has to be expressed in the metalanguage ( via the concept of logical consequence).
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– Ray LittleRock
Apr 1 at 19:00
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I'll have to edit my question. The answer given by Bram 28 made me realize that there is a difference between " A contradicts B" and " A and B are contradictory".
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– Ray LittleRock
Apr 1 at 19:02
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Maybe... in logic a contradiction is a sentence/ formula that is always FALSE : $P land lnot P$. Obviously, $P$ and $lnot P$ contradict each other (in the "usual" sense of the word).
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– Mauro ALLEGRANZA
Apr 1 at 19:10
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I totally agree that (P& ~P) is a contradiction. My question was simply whether there was a way to say " this sentence is a contradiction" in the language itself, or whether it has to be said at the meta-level. What I understand from your answer and Bram28's is that it has to be expressed in the meta-language, or in the language of modal logic.
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– Ray LittleRock
Apr 1 at 19:19