Mixing
Negation of a true proposition
Clash Royale CLAN TAG#URR8PPP
up vote
0
down vote
favorite
It starts by someone asking an exercise question that whether negation of
2 is a rational number
is
2 is an irrational number
Their argument is that they consider it is incorrect if we include complex numbers, because 2 might be complex and not irrational.
My argument is that because first proposition is always true because 2 could not be anything else but rational number, any false sentence could be negation of the first proposition, including the given sentence above. They said what I do is not "negation" in their sense.
So my question is, is it true that every false statements are negation of true statement?
logic
asked Jul 28 at 4:58
tia
1043
add a comment |Â
up vote
0
down vote
favorite
It starts by someone asking an exercise question that whether negation of
2 is a rational number
is
2 is an irrational number
Their argument is that they consider it is incorrect if we include complex numbers, because 2 might be complex and not irrational.
My argument is that because first proposition is always true because 2 could not be anything else but rational number, any false sentence could be negation of the first proposition, including the given sentence above. They said what I do is not "negation" in their sense.
So my question is, is it true that every false statements are negation of true statement?
logic
asked Jul 28 at 4:58
tia
1043
2
The negation of "2 is a rational number" is simply "2 is not a rational number".
– é«˜ç”°èˆª
Jul 28 at 5:27
That's obvious, but the question is about other statements that the obvious one, or even some statements unrelated to the number 2.
– tia
Jul 28 at 6:37
YES: the "negation operation" swaps the truth value of a statement: from TRUE to FALSE and vice evrsa.
– Mauro ALLEGRANZA
Jul 29 at 15:03
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
It starts by someone asking an exercise question that whether negation of
2 is a rational number
is
2 is an irrational number
Their argument is that they consider it is incorrect if we include complex numbers, because 2 might be complex and not irrational.
My argument is that because first proposition is always true because 2 could not be anything else but rational number, any false sentence could be negation of the first proposition, including the given sentence above. They said what I do is not "negation" in their sense.
So my question is, is it true that every false statements are negation of true statement?
logic
asked Jul 28 at 4:58
tia
1043
It starts by someone asking an exercise question that whether negation of
2 is a rational number
is
2 is an irrational number
Their argument is that they consider it is incorrect if we include complex numbers, because 2 might be complex and not irrational.
My argument is that because first proposition is always true because 2 could not be anything else but rational number, any false sentence could be negation of the first proposition, including the given sentence above. They said what I do is not "negation" in their sense.
So my question is, is it true that every false statements are negation of true statement?
logic
asked Jul 28 at 4:58
tia
1043
asked Jul 28 at 4:58
tia
1043
asked Jul 28 at 4:58
tia
1043
asked Jul 28 at 4:58
tia
1043
1043
2
The negation of "2 is a rational number" is simply "2 is not a rational number".
– é«˜ç”°èˆª
Jul 28 at 5:27
That's obvious, but the question is about other statements that the obvious one, or even some statements unrelated to the number 2.
– tia
Jul 28 at 6:37
YES: the "negation operation" swaps the truth value of a statement: from TRUE to FALSE and vice evrsa.
– Mauro ALLEGRANZA
Jul 29 at 15:03
add a comment |Â
2
The negation of "2 is a rational number" is simply "2 is not a rational number".
– é«˜ç”°èˆª
Jul 28 at 5:27
That's obvious, but the question is about other statements that the obvious one, or even some statements unrelated to the number 2.
– tia
Jul 28 at 6:37
YES: the "negation operation" swaps the truth value of a statement: from TRUE to FALSE and vice evrsa.
– Mauro ALLEGRANZA
Jul 29 at 15:03
2
2
The negation of "2 is a rational number" is simply "2 is not a rational number".
– é«˜ç”°èˆª
Jul 28 at 5:27
The negation of "2 is a rational number" is simply "2 is not a rational number".
– é«˜ç”°èˆª
Jul 28 at 5:27
That's obvious, but the question is about other statements that the obvious one, or even some statements unrelated to the number 2.
– tia
Jul 28 at 6:37
That's obvious, but the question is about other statements that the obvious one, or even some statements unrelated to the number 2.
– tia
Jul 28 at 6:37
YES: the "negation operation" swaps the truth value of a statement: from TRUE to FALSE and vice evrsa.
– Mauro ALLEGRANZA
Jul 29 at 15:03
YES: the "negation operation" swaps the truth value of a statement: from TRUE to FALSE and vice evrsa.
– Mauro ALLEGRANZA
Jul 29 at 15:03
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
1
down vote
If you are talking about the mathematical logic, the answer of your question is true. As this logic is two-valued (binary) and it means the negation of a false statement must be true and vice versa. A statement that all the time is false, is denoted by $bot$ and the all the time true statement is called tautology and denoted by $top$, in the mathematical logic.
In addition, you can find three-valued logic and more like temporal logic. However, I don't think you are seeking about them.
answered Jul 28 at 5:45
OmG
1,740617
add a comment |Â
up vote
1
down vote
The answer to your question depends on which logic framework you think corresponds the most the everyday life. There are two such ones which answer differently to your question :
Classical logic : the framework in which most people think naturally. As OmG said, you can think of it as 2-valued, that is, that statements can be interpreted as either true or false. In this logic, $ neg neg P $ is equivalent to $ P $ ($neg$ is negation). So the answer to your question would be yes.
Intuitionistic logic : here we have something that could be considered less intuitive, as we have all the classical axioms except the axiom of the excluded middle which is that for any proposition $ P $, $ P lor neg P $ ($lor$ is logical or). The reasoning behind it is that it would be impossible to have a way construct a proof of either $ P $ or $neg P$ for any possible $ P $, and in that way it is the correct framework for constructive logic. For example, reductio ad absurdum is not possible in this, because you have $ P Rightarrow neg neg P $ but not $ neg neg P Rightarrow P $. This logic is also not 2-valued, and this is of interest in model theory, with possible models being Heyting algebras, in which you could find "statements" that are never the negation of another one. As an example, take the Heyting algebra of open sets of $ mathbbR $. Then $ mathbbR^*$ is an open set that is never the negation of another one (that is, it is never equal to the interior of $ A^c $ for $ A $ an open set).
answered Jul 28 at 7:33
FreeSalad
374110
This is high-school problem so I think classical logic is what we are talking about. By the way, the logical system without law of the excluded middle is very interesting to me, but I still haven't been through it. All I can imagine is that there might be some proposition P that neither be proven true or false with axiomatic system that we choose.
– tia
Jul 28 at 9:56
1
@tia that a statement is provable or that a statement is true aren't actually the same thing, the difference being one of syntax versus semantics (proof theory versus model theory). For example, what you're describing is perfectly possible in classical logic as well: the axiom of choice is independent of ZF, which means we can't prove $ C $ or $ neg C $ in ZF. However, in any model of ZF, AC is either true or false.
– FreeSalad
Jul 29 at 17:57
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
If you are talking about the mathematical logic, the answer of your question is true. As this logic is two-valued (binary) and it means the negation of a false statement must be true and vice versa. A statement that all the time is false, is denoted by $bot$ and the all the time true statement is called tautology and denoted by $top$, in the mathematical logic.
In addition, you can find three-valued logic and more like temporal logic. However, I don't think you are seeking about them.
answered Jul 28 at 5:45
OmG
1,740617
add a comment |Â
up vote
1
down vote
If you are talking about the mathematical logic, the answer of your question is true. As this logic is two-valued (binary) and it means the negation of a false statement must be true and vice versa. A statement that all the time is false, is denoted by $bot$ and the all the time true statement is called tautology and denoted by $top$, in the mathematical logic.
In addition, you can find three-valued logic and more like temporal logic. However, I don't think you are seeking about them.
answered Jul 28 at 5:45
OmG
1,740617
add a comment |Â
up vote
1
down vote
up vote
1
down vote
If you are talking about the mathematical logic, the answer of your question is true. As this logic is two-valued (binary) and it means the negation of a false statement must be true and vice versa. A statement that all the time is false, is denoted by $bot$ and the all the time true statement is called tautology and denoted by $top$, in the mathematical logic.
In addition, you can find three-valued logic and more like temporal logic. However, I don't think you are seeking about them.
answered Jul 28 at 5:45
OmG
1,740617
If you are talking about the mathematical logic, the answer of your question is true. As this logic is two-valued (binary) and it means the negation of a false statement must be true and vice versa. A statement that all the time is false, is denoted by $bot$ and the all the time true statement is called tautology and denoted by $top$, in the mathematical logic.
In addition, you can find three-valued logic and more like temporal logic. However, I don't think you are seeking about them.
answered Jul 28 at 5:45
OmG
1,740617
answered Jul 28 at 5:45
OmG
1,740617
answered Jul 28 at 5:45
OmG
1,740617
answered Jul 28 at 5:45
OmG
1,740617
1,740617
add a comment |Â
add a comment |Â
up vote
1
down vote
The answer to your question depends on which logic framework you think corresponds the most the everyday life. There are two such ones which answer differently to your question :
Classical logic : the framework in which most people think naturally. As OmG said, you can think of it as 2-valued, that is, that statements can be interpreted as either true or false. In this logic, $ neg neg P $ is equivalent to $ P $ ($neg$ is negation). So the answer to your question would be yes.
Intuitionistic logic : here we have something that could be considered less intuitive, as we have all the classical axioms except the axiom of the excluded middle which is that for any proposition $ P $, $ P lor neg P $ ($lor$ is logical or). The reasoning behind it is that it would be impossible to have a way construct a proof of either $ P $ or $neg P$ for any possible $ P $, and in that way it is the correct framework for constructive logic. For example, reductio ad absurdum is not possible in this, because you have $ P Rightarrow neg neg P $ but not $ neg neg P Rightarrow P $. This logic is also not 2-valued, and this is of interest in model theory, with possible models being Heyting algebras, in which you could find "statements" that are never the negation of another one. As an example, take the Heyting algebra of open sets of $ mathbbR $. Then $ mathbbR^*$ is an open set that is never the negation of another one (that is, it is never equal to the interior of $ A^c $ for $ A $ an open set).
answered Jul 28 at 7:33
FreeSalad
374110
This is high-school problem so I think classical logic is what we are talking about. By the way, the logical system without law of the excluded middle is very interesting to me, but I still haven't been through it. All I can imagine is that there might be some proposition P that neither be proven true or false with axiomatic system that we choose.
– tia
Jul 28 at 9:56
1
@tia that a statement is provable or that a statement is true aren't actually the same thing, the difference being one of syntax versus semantics (proof theory versus model theory). For example, what you're describing is perfectly possible in classical logic as well: the axiom of choice is independent of ZF, which means we can't prove $ C $ or $ neg C $ in ZF. However, in any model of ZF, AC is either true or false.
– FreeSalad
Jul 29 at 17:57
add a comment |Â
up vote
1
down vote
The answer to your question depends on which logic framework you think corresponds the most the everyday life. There are two such ones which answer differently to your question :
Classical logic : the framework in which most people think naturally. As OmG said, you can think of it as 2-valued, that is, that statements can be interpreted as either true or false. In this logic, $ neg neg P $ is equivalent to $ P $ ($neg$ is negation). So the answer to your question would be yes.
Intuitionistic logic : here we have something that could be considered less intuitive, as we have all the classical axioms except the axiom of the excluded middle which is that for any proposition $ P $, $ P lor neg P $ ($lor$ is logical or). The reasoning behind it is that it would be impossible to have a way construct a proof of either $ P $ or $neg P$ for any possible $ P $, and in that way it is the correct framework for constructive logic. For example, reductio ad absurdum is not possible in this, because you have $ P Rightarrow neg neg P $ but not $ neg neg P Rightarrow P $. This logic is also not 2-valued, and this is of interest in model theory, with possible models being Heyting algebras, in which you could find "statements" that are never the negation of another one. As an example, take the Heyting algebra of open sets of $ mathbbR $. Then $ mathbbR^*$ is an open set that is never the negation of another one (that is, it is never equal to the interior of $ A^c $ for $ A $ an open set).
answered Jul 28 at 7:33
FreeSalad
374110
This is high-school problem so I think classical logic is what we are talking about. By the way, the logical system without law of the excluded middle is very interesting to me, but I still haven't been through it. All I can imagine is that there might be some proposition P that neither be proven true or false with axiomatic system that we choose.
– tia
Jul 28 at 9:56
1
@tia that a statement is provable or that a statement is true aren't actually the same thing, the difference being one of syntax versus semantics (proof theory versus model theory). For example, what you're describing is perfectly possible in classical logic as well: the axiom of choice is independent of ZF, which means we can't prove $ C $ or $ neg C $ in ZF. However, in any model of ZF, AC is either true or false.
– FreeSalad
Jul 29 at 17:57
add a comment |Â
up vote
1
down vote
up vote
1
down vote
The answer to your question depends on which logic framework you think corresponds the most the everyday life. There are two such ones which answer differently to your question :
Classical logic : the framework in which most people think naturally. As OmG said, you can think of it as 2-valued, that is, that statements can be interpreted as either true or false. In this logic, $ neg neg P $ is equivalent to $ P $ ($neg$ is negation). So the answer to your question would be yes.
Intuitionistic logic : here we have something that could be considered less intuitive, as we have all the classical axioms except the axiom of the excluded middle which is that for any proposition $ P $, $ P lor neg P $ ($lor$ is logical or). The reasoning behind it is that it would be impossible to have a way construct a proof of either $ P $ or $neg P$ for any possible $ P $, and in that way it is the correct framework for constructive logic. For example, reductio ad absurdum is not possible in this, because you have $ P Rightarrow neg neg P $ but not $ neg neg P Rightarrow P $. This logic is also not 2-valued, and this is of interest in model theory, with possible models being Heyting algebras, in which you could find "statements" that are never the negation of another one. As an example, take the Heyting algebra of open sets of $ mathbbR $. Then $ mathbbR^*$ is an open set that is never the negation of another one (that is, it is never equal to the interior of $ A^c $ for $ A $ an open set).
answered Jul 28 at 7:33
FreeSalad
374110
The answer to your question depends on which logic framework you think corresponds the most the everyday life. There are two such ones which answer differently to your question :
Classical logic : the framework in which most people think naturally. As OmG said, you can think of it as 2-valued, that is, that statements can be interpreted as either true or false. In this logic, $ neg neg P $ is equivalent to $ P $ ($neg$ is negation). So the answer to your question would be yes.
Intuitionistic logic : here we have something that could be considered less intuitive, as we have all the classical axioms except the axiom of the excluded middle which is that for any proposition $ P $, $ P lor neg P $ ($lor$ is logical or). The reasoning behind it is that it would be impossible to have a way construct a proof of either $ P $ or $neg P$ for any possible $ P $, and in that way it is the correct framework for constructive logic. For example, reductio ad absurdum is not possible in this, because you have $ P Rightarrow neg neg P $ but not $ neg neg P Rightarrow P $. This logic is also not 2-valued, and this is of interest in model theory, with possible models being Heyting algebras, in which you could find "statements" that are never the negation of another one. As an example, take the Heyting algebra of open sets of $ mathbbR $. Then $ mathbbR^*$ is an open set that is never the negation of another one (that is, it is never equal to the interior of $ A^c $ for $ A $ an open set).
answered Jul 28 at 7:33
FreeSalad
374110
answered Jul 28 at 7:33
FreeSalad
374110
answered Jul 28 at 7:33
FreeSalad
374110
answered Jul 28 at 7:33
FreeSalad
374110
374110
This is high-school problem so I think classical logic is what we are talking about. By the way, the logical system without law of the excluded middle is very interesting to me, but I still haven't been through it. All I can imagine is that there might be some proposition P that neither be proven true or false with axiomatic system that we choose.
– tia
Jul 28 at 9:56
1
@tia that a statement is provable or that a statement is true aren't actually the same thing, the difference being one of syntax versus semantics (proof theory versus model theory). For example, what you're describing is perfectly possible in classical logic as well: the axiom of choice is independent of ZF, which means we can't prove $ C $ or $ neg C $ in ZF. However, in any model of ZF, AC is either true or false.
– FreeSalad
Jul 29 at 17:57
add a comment |Â
This is high-school problem so I think classical logic is what we are talking about. By the way, the logical system without law of the excluded middle is very interesting to me, but I still haven't been through it. All I can imagine is that there might be some proposition P that neither be proven true or false with axiomatic system that we choose.
– tia
Jul 28 at 9:56
1
@tia that a statement is provable or that a statement is true aren't actually the same thing, the difference being one of syntax versus semantics (proof theory versus model theory). For example, what you're describing is perfectly possible in classical logic as well: the axiom of choice is independent of ZF, which means we can't prove $ C $ or $ neg C $ in ZF. However, in any model of ZF, AC is either true or false.
– FreeSalad
Jul 29 at 17:57
This is high-school problem so I think classical logic is what we are talking about. By the way, the logical system without law of the excluded middle is very interesting to me, but I still haven't been through it. All I can imagine is that there might be some proposition P that neither be proven true or false with axiomatic system that we choose.
– tia
Jul 28 at 9:56
This is high-school problem so I think classical logic is what we are talking about. By the way, the logical system without law of the excluded middle is very interesting to me, but I still haven't been through it. All I can imagine is that there might be some proposition P that neither be proven true or false with axiomatic system that we choose.
– tia
Jul 28 at 9:56
1
1
@tia that a statement is provable or that a statement is true aren't actually the same thing, the difference being one of syntax versus semantics (proof theory versus model theory). For example, what you're describing is perfectly possible in classical logic as well: the axiom of choice is independent of ZF, which means we can't prove $ C $ or $ neg C $ in ZF. However, in any model of ZF, AC is either true or false.
– FreeSalad
Jul 29 at 17:57
@tia that a statement is provable or that a statement is true aren't actually the same thing, the difference being one of syntax versus semantics (proof theory versus model theory). For example, what you're describing is perfectly possible in classical logic as well: the axiom of choice is independent of ZF, which means we can't prove $ C $ or $ neg C $ in ZF. However, in any model of ZF, AC is either true or false.
– FreeSalad
Jul 29 at 17:57
add a comment |Â
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2864992%2fnegation-of-a-true-proposition%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
2
The negation of "2 is a rational number" is simply "2 is not a rational number".
– é«˜ç”°èˆª
Jul 28 at 5:27
That's obvious, but the question is about other statements that the obvious one, or even some statements unrelated to the number 2.
– tia
Jul 28 at 6:37
YES: the "negation operation" swaps the truth value of a statement: from TRUE to FALSE and vice evrsa.
– Mauro ALLEGRANZA
Jul 29 at 15:03