What could be an appropriate name for this logical operation?
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The logical operation, for which I think it would be good to have a name, could be referenced as "negation of implication" in classical propositional logic, where this operation can be defined like this:
$x setminus y ; {buildrelrm defover=} ; lnot (x to y)$.
But in classical propositional logic this operation can be also defined by another formula
$x setminus y ; {buildrelrm defover=} ; x land lnot y$,
and (possibly) in other manners, which shows that naming an operation by making reference to the manner in which the operation is expressed through other operations is not a good idea.
Obviously, a name for a logical operation can be most useful when it expresses a logical meaning, rather than when it reflects a manner of reading a formula. I used above the operator of set difference "$setminus$" for this logical operation specifically to suggest that I am interested in the meaning of the logical operation, which is usually denoted as "$setminus$" in the signature of of generalized Boolean algebras (GBA) - i.e. Boolean algebras with an optional top element.
The logic of GBAs seems to be a kind of "logic without negation" because one cannot define the negation in this logic - this logic has an operation of "relative negation", and this is the operation intended in the title of this message.
It would be interesting to read any article about a "logic without negation" - please send a reference if you heard about any. Also, any attempts to express in words the meaning of this operation would be appreciated.
Notice, the symbol of this operation alone can be posited in the signature of a special kind of algebras called semi-boolean algebras, which are of two kinds: subtraction algebras and implication algebras. It was a great surprise to me to find that subtraction and implication are some kind of "symmetric" or "dual" operations (https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/j-c-abbott-semiboolean-algebra-matematicki-vesnik-vol-4-1967-pp-177198/66499A95F7D1F1CEA896C3CD366BBC45 ). But such "symmetry" or "duality" emphasizes, that a name for the operation intended in my question would be most useful.
abstract-algebra logic
asked Nov 20 at 9:02
Ioachim Drugus
355111
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up vote
2
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The logical operation, for which I think it would be good to have a name, could be referenced as "negation of implication" in classical propositional logic, where this operation can be defined like this:
$x setminus y ; {buildrelrm defover=} ; lnot (x to y)$.
But in classical propositional logic this operation can be also defined by another formula
$x setminus y ; {buildrelrm defover=} ; x land lnot y$,
and (possibly) in other manners, which shows that naming an operation by making reference to the manner in which the operation is expressed through other operations is not a good idea.
Obviously, a name for a logical operation can be most useful when it expresses a logical meaning, rather than when it reflects a manner of reading a formula. I used above the operator of set difference "$setminus$" for this logical operation specifically to suggest that I am interested in the meaning of the logical operation, which is usually denoted as "$setminus$" in the signature of of generalized Boolean algebras (GBA) - i.e. Boolean algebras with an optional top element.
The logic of GBAs seems to be a kind of "logic without negation" because one cannot define the negation in this logic - this logic has an operation of "relative negation", and this is the operation intended in the title of this message.
It would be interesting to read any article about a "logic without negation" - please send a reference if you heard about any. Also, any attempts to express in words the meaning of this operation would be appreciated.
Notice, the symbol of this operation alone can be posited in the signature of a special kind of algebras called semi-boolean algebras, which are of two kinds: subtraction algebras and implication algebras. It was a great surprise to me to find that subtraction and implication are some kind of "symmetric" or "dual" operations (https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/j-c-abbott-semiboolean-algebra-matematicki-vesnik-vol-4-1967-pp-177198/66499A95F7D1F1CEA896C3CD366BBC45 ). But such "symmetry" or "duality" emphasizes, that a name for the operation intended in my question would be most useful.
abstract-algebra logic
asked Nov 20 at 9:02
Ioachim Drugus
355111
2
How about just $xsetminus y$ meaning "$x$ but not $y$" or "$x$ without $y$"? That way you even keep close to the meaning of the set difference ($setminus$) operator :)
– vrugtehagel
Nov 20 at 10:01
Just like $p NAND q = Not p AND q' = neg (p land q)$, maybe you can call it the 'NIF'.... the problem is that if $p NIF q = ' Not p IF q'$, then that would translate as $neg (q rightarrow p)$ ... so maybe $NFI$ .. or $NONLYIF$? ... ugh... all ugly ....
– Bram28
Nov 20 at 14:42
Why not just call it "relative complement"? I don't really see what you asking us to add to the question you asked years ago on MO about this.
– Rob Arthan
Nov 20 at 22:08
Probably, in my question, I should have put emphasis on "reading of the expression $x setminus y$ ", rather than on "operation's name", since it is the reading which elucidates the "intuitive meaning" of this operation. "$x$ but not $y$" sounds good, but the use of the word "not" is slightly bothersome since it is also used for the negation. "$x$ without $y$" sounds good for quantities, but less good for the assertions (used in assertional logic).
– Ioachim Drugus
Nov 21 at 14:45
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
The logical operation, for which I think it would be good to have a name, could be referenced as "negation of implication" in classical propositional logic, where this operation can be defined like this:
$x setminus y ; {buildrelrm defover=} ; lnot (x to y)$.
But in classical propositional logic this operation can be also defined by another formula
$x setminus y ; {buildrelrm defover=} ; x land lnot y$,
and (possibly) in other manners, which shows that naming an operation by making reference to the manner in which the operation is expressed through other operations is not a good idea.
Obviously, a name for a logical operation can be most useful when it expresses a logical meaning, rather than when it reflects a manner of reading a formula. I used above the operator of set difference "$setminus$" for this logical operation specifically to suggest that I am interested in the meaning of the logical operation, which is usually denoted as "$setminus$" in the signature of of generalized Boolean algebras (GBA) - i.e. Boolean algebras with an optional top element.
The logic of GBAs seems to be a kind of "logic without negation" because one cannot define the negation in this logic - this logic has an operation of "relative negation", and this is the operation intended in the title of this message.
It would be interesting to read any article about a "logic without negation" - please send a reference if you heard about any. Also, any attempts to express in words the meaning of this operation would be appreciated.
Notice, the symbol of this operation alone can be posited in the signature of a special kind of algebras called semi-boolean algebras, which are of two kinds: subtraction algebras and implication algebras. It was a great surprise to me to find that subtraction and implication are some kind of "symmetric" or "dual" operations (https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/j-c-abbott-semiboolean-algebra-matematicki-vesnik-vol-4-1967-pp-177198/66499A95F7D1F1CEA896C3CD366BBC45 ). But such "symmetry" or "duality" emphasizes, that a name for the operation intended in my question would be most useful.
abstract-algebra logic
asked Nov 20 at 9:02
Ioachim Drugus
355111
The logical operation, for which I think it would be good to have a name, could be referenced as "negation of implication" in classical propositional logic, where this operation can be defined like this:
$x setminus y ; {buildrelrm defover=} ; lnot (x to y)$.
But in classical propositional logic this operation can be also defined by another formula
$x setminus y ; {buildrelrm defover=} ; x land lnot y$,
and (possibly) in other manners, which shows that naming an operation by making reference to the manner in which the operation is expressed through other operations is not a good idea.
Obviously, a name for a logical operation can be most useful when it expresses a logical meaning, rather than when it reflects a manner of reading a formula. I used above the operator of set difference "$setminus$" for this logical operation specifically to suggest that I am interested in the meaning of the logical operation, which is usually denoted as "$setminus$" in the signature of of generalized Boolean algebras (GBA) - i.e. Boolean algebras with an optional top element.
The logic of GBAs seems to be a kind of "logic without negation" because one cannot define the negation in this logic - this logic has an operation of "relative negation", and this is the operation intended in the title of this message.
It would be interesting to read any article about a "logic without negation" - please send a reference if you heard about any. Also, any attempts to express in words the meaning of this operation would be appreciated.
Notice, the symbol of this operation alone can be posited in the signature of a special kind of algebras called semi-boolean algebras, which are of two kinds: subtraction algebras and implication algebras. It was a great surprise to me to find that subtraction and implication are some kind of "symmetric" or "dual" operations (https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/j-c-abbott-semiboolean-algebra-matematicki-vesnik-vol-4-1967-pp-177198/66499A95F7D1F1CEA896C3CD366BBC45 ). But such "symmetry" or "duality" emphasizes, that a name for the operation intended in my question would be most useful.
abstract-algebra logic
abstract-algebra logic
asked Nov 20 at 9:02
Ioachim Drugus
355111
asked Nov 20 at 9:02
Ioachim Drugus
355111
asked Nov 20 at 9:02
Ioachim Drugus
355111
asked Nov 20 at 9:02
Ioachim Drugus
355111
asked Nov 20 at 9:02
Ioachim Drugus
355111
355111
2
How about just $xsetminus y$ meaning "$x$ but not $y$" or "$x$ without $y$"? That way you even keep close to the meaning of the set difference ($setminus$) operator :)
– vrugtehagel
Nov 20 at 10:01
Just like $p NAND q = Not p AND q' = neg (p land q)$, maybe you can call it the 'NIF'.... the problem is that if $p NIF q = ' Not p IF q'$, then that would translate as $neg (q rightarrow p)$ ... so maybe $NFI$ .. or $NONLYIF$? ... ugh... all ugly ....
– Bram28
Nov 20 at 14:42
Why not just call it "relative complement"? I don't really see what you asking us to add to the question you asked years ago on MO about this.
– Rob Arthan
Nov 20 at 22:08
Probably, in my question, I should have put emphasis on "reading of the expression $x setminus y$ ", rather than on "operation's name", since it is the reading which elucidates the "intuitive meaning" of this operation. "$x$ but not $y$" sounds good, but the use of the word "not" is slightly bothersome since it is also used for the negation. "$x$ without $y$" sounds good for quantities, but less good for the assertions (used in assertional logic).
– Ioachim Drugus
Nov 21 at 14:45
add a comment |
2
How about just $xsetminus y$ meaning "$x$ but not $y$" or "$x$ without $y$"? That way you even keep close to the meaning of the set difference ($setminus$) operator :)
– vrugtehagel
Nov 20 at 10:01
Just like $p NAND q = Not p AND q' = neg (p land q)$, maybe you can call it the 'NIF'.... the problem is that if $p NIF q = ' Not p IF q'$, then that would translate as $neg (q rightarrow p)$ ... so maybe $NFI$ .. or $NONLYIF$? ... ugh... all ugly ....
– Bram28
Nov 20 at 14:42
Why not just call it "relative complement"? I don't really see what you asking us to add to the question you asked years ago on MO about this.
– Rob Arthan
Nov 20 at 22:08
Probably, in my question, I should have put emphasis on "reading of the expression $x setminus y$ ", rather than on "operation's name", since it is the reading which elucidates the "intuitive meaning" of this operation. "$x$ but not $y$" sounds good, but the use of the word "not" is slightly bothersome since it is also used for the negation. "$x$ without $y$" sounds good for quantities, but less good for the assertions (used in assertional logic).
– Ioachim Drugus
Nov 21 at 14:45
2
2
How about just $xsetminus y$ meaning "$x$ but not $y$" or "$x$ without $y$"? That way you even keep close to the meaning of the set difference ($setminus$) operator :)
– vrugtehagel
Nov 20 at 10:01
How about just $xsetminus y$ meaning "$x$ but not $y$" or "$x$ without $y$"? That way you even keep close to the meaning of the set difference ($setminus$) operator :)
– vrugtehagel
Nov 20 at 10:01
Just like $p NAND q = Not p AND q' = neg (p land q)$, maybe you can call it the 'NIF'.... the problem is that if $p NIF q = ' Not p IF q'$, then that would translate as $neg (q rightarrow p)$ ... so maybe $NFI$ .. or $NONLYIF$? ... ugh... all ugly ....
– Bram28
Nov 20 at 14:42
Just like $p NAND q = Not p AND q' = neg (p land q)$, maybe you can call it the 'NIF'.... the problem is that if $p NIF q = ' Not p IF q'$, then that would translate as $neg (q rightarrow p)$ ... so maybe $NFI$ .. or $NONLYIF$? ... ugh... all ugly ....
– Bram28
Nov 20 at 14:42
Why not just call it "relative complement"? I don't really see what you asking us to add to the question you asked years ago on MO about this.
– Rob Arthan
Nov 20 at 22:08
Why not just call it "relative complement"? I don't really see what you asking us to add to the question you asked years ago on MO about this.
– Rob Arthan
Nov 20 at 22:08
Probably, in my question, I should have put emphasis on "reading of the expression $x setminus y$ ", rather than on "operation's name", since it is the reading which elucidates the "intuitive meaning" of this operation. "$x$ but not $y$" sounds good, but the use of the word "not" is slightly bothersome since it is also used for the negation. "$x$ without $y$" sounds good for quantities, but less good for the assertions (used in assertional logic).
– Ioachim Drugus
Nov 21 at 14:45
Probably, in my question, I should have put emphasis on "reading of the expression $x setminus y$ ", rather than on "operation's name", since it is the reading which elucidates the "intuitive meaning" of this operation. "$x$ but not $y$" sounds good, but the use of the word "not" is slightly bothersome since it is also used for the negation. "$x$ without $y$" sounds good for quantities, but less good for the assertions (used in assertional logic).
– Ioachim Drugus
Nov 21 at 14:45
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2
How about just $xsetminus y$ meaning "$x$ but not $y$" or "$x$ without $y$"? That way you even keep close to the meaning of the set difference ($setminus$) operator :)
– vrugtehagel
Nov 20 at 10:01
Just like $p NAND q = Not p AND q' = neg (p land q)$, maybe you can call it the 'NIF'.... the problem is that if $p NIF q = ' Not p IF q'$, then that would translate as $neg (q rightarrow p)$ ... so maybe $NFI$ .. or $NONLYIF$? ... ugh... all ugly ....
– Bram28
Nov 20 at 14:42
Why not just call it "relative complement"? I don't really see what you asking us to add to the question you asked years ago on MO about this.
– Rob Arthan
Nov 20 at 22:08
Probably, in my question, I should have put emphasis on "reading of the expression $x setminus y$ ", rather than on "operation's name", since it is the reading which elucidates the "intuitive meaning" of this operation. "$x$ but not $y$" sounds good, but the use of the word "not" is slightly bothersome since it is also used for the negation. "$x$ without $y$" sounds good for quantities, but less good for the assertions (used in assertional logic).
– Ioachim Drugus
Nov 21 at 14:45