The best way of writing FOL with identity and membership.
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From: zuhair on 7 May 2010 17:46 On May 7, 9:50 am, Marshall <marshall.spi...(a)gmail.com> wrote: > On May 7, 1:50 am, zuhair <zaljo...(a)gmail.com> wrote: > > > > > This syntax is both elegant, abbreviating and easily > > readable. > > Maximizing abbreviation has the opposite effect as improving > readability. Agreed, but with training this would minimize, on the other hand I am not claiming that improving readability is due to the abbreviation power of this notation, the improvement comes from removing crowded symbols like brackets, and membership symbol and the universal quantifier symbol, which are the most repeated symbols in most formulae, also the symbols here are simpler than the more complex symbols used in standard method, for example, symbols given to disjunction, conjunction and implication all use two characters, while they are using at most one character here. The quantifiers are odd looking symbols, the universal quantifier is an upside-down A symbol, and I see no creative act in that at all, while the existential quantifier is a rotated E ( a turnaround E) symbol, both look ugly and bizarre, beside they are complex symbols containing many angles and can be viewed as combination of many smaller symbols, while at the same time the are suppose to denote simple concepts. The way how these symbols are written in standard books is totally unacceptable, there is no rational behind such odd characters. (the mirror letters in English, like p and q , are pretty much known symbols and there is no problem with them, n and u are both inverted-rotated symbols, but still they are elegant, but the inverted A and the rotated E are very bizarre, they are simply not edible) > > > The standard notation appears as too long, > > crowded, boring, and clumsy by comparison, that is > > besides it contains inverted symbols and > > rotated symbols without an obvious justification. > > The standard notation is already quite terse, to the > point that compressing it further strikes me as a bad > idea. hmmm......... > > The symbols you describe as rotated aren't. Does the > English alphabet contain rotated symbols? I would say > not, and yet lowercase u and n are the same modulo > rotation. Yes of course, p and q are examples, also as you mentioned n and u are actually inverted-rotated symbols. However upper-case English don't have that phenomena, and as I said above an inverted Capital A and a rotated Capital E are indeed ugly symbols. > > The "shortening" of the implication arrow to a greater-than > symbol is an ASCII-specific optimization. The right arrow > is a single symbol in unicode, and when written by hand. > Also, this makes is difficult/ambiguous to write anything > involving numeric comparison. This may not be a goal > of yours, but it is a common application of standard > notation. > > > In my personal opinion the above record is the *best way > > of writing first order logic with identity and membership*, > > known thus far! > > "Best" is relative to a given set of requirements, and > any design is going to involve making tradeoffs. Agreed. > > Marshall Thanks for your positive contribution here. Zuhair
From: zuhair on 7 May 2010 20:16 On May 7, 9:50 am, Marshall <marshall.spi...(a)gmail.com> wrote: > On May 7, 1:50 am, zuhair <zaljo...(a)gmail.com> wrote: > > > > > This syntax is both elegant, abbreviating and easily > > readable. > > Maximizing abbreviation has the opposite effect as improving > readability. > > > The standard notation appears as too long, > > crowded, boring, and clumsy by comparison, that is > > besides it contains inverted symbols and > > rotated symbols without an obvious justification. > > The standard notation is already quite terse, to the > point that compressing it further strikes me as a bad > idea. > > The symbols you describe as rotated aren't. Does the > English alphabet contain rotated symbols? I would say > not, and yet lowercase u and n are the same modulo > rotation. > > The "shortening" of the implication arrow to a greater-than > symbol is an ASCII-specific optimization. The right arrow > is a single symbol in unicode, and when written by hand. > Also, this makes is difficult/ambiguous to write anything > involving numeric comparison. This may not be a goal > of yours, but it is a common application of standard > notation. > > > In my personal opinion the above record is the *best way > > of writing first order logic with identity and membership*, > > known thus far! > > "Best" is relative to a given set of requirements, and > any design is going to involve making tradeoffs. > > Marshall Just to add to what I wrote previously in reply to your really positive notes, that I appreciate so much. To me personally I see that there must be some kind of rational behind notations. It is true that we can use any notation however bizarre looking to write logic with as far as it is uniquely readable, but I don't like that. Logical symbols must be the simplest possible, they must not be bizarre, they must not contain upside-down symbols or turnaround symbols, there must be some kind of explanation offered behind each symbol. The system that I wrote have an explanation for each symbol in it. ~ to represent negation is an accurate symbol, the symbol have opposing characteristic the first bend is opposite the second one in direction, and this gives opposing impression and it is a good symbolism to represent negation by a symbol that gives opposing impression, that's why I kept this symbol in my system. Conjunction by juxtaposing the two *formulae*, is pretty much intuitive since juxtaposing formulae does indeed give the impression of conjunction. Disjunction using the symbol "|" is also intuitive, since the symbol "|" is a boundary symbol giving the impression of a separator, and it is pretty much intuitive to represent disjunction by a symbol that implies a kind of separation. notice that conjunction and disjunction are symbolized using non directional symbols, you see that | doesn't give any impression of directionality, neither does juxtaposing formulae. Implication, requires a directional symbol, because implication is a directional concept! and the simplest of which is > , there is no need for the stem in -> at all, this is a redundant notation, since > does the job nicely. same goes for the bi-conditional. On the other hand can you tell me the rational behind using a Large V to represent disjunction and an inverted Large V to represent conjunction, I see no rational behind such clumsy symbols at all. Now the quantifiers. I represented existential quantification by the symbol "_", this symbol gives and "expectational" impression, once you see the symbol _ you wander what is after it, and it is good to represent existence by and expectational symbol. Now regarding universal quantification, all what we want to say in universal quantification say of x is "for Any x", this is best represented by just writing x like that, this would give the full impression of any x. no need for any symbol for this quantifier. Can you tell me what is the rational beyond an upside-down A and a turnaround E? Regarding using juxtaposition of terms (variables, constants and functions) to represent the membership relation, then this is evident, since the principal primitive in fol(membership,=) is membership, and it would be the most repeatable symbol, so it is better represented by juxtaposing the terms. Identity relation is an ancient symbol, and it is consistent of two bars identical to each other, so it is OK. The dot notation is much more neat than the bracket system, and it has a good abbreviation power, and it is superior to be the bracket system, and it is a proved notation since Principa Mathematica days. The other rules of abbreviation are only based on simply concepts derived from first order logic itself, and they help to remove the bracket and obviate the need for using excessive dot notation. So in this system, there is rational beyond each symbol, and the net result is a nice looking formulae, clear, not crowded and actually the tell you the core of what is going on in the formulae, you can see the skeleton of reasoning flowing in these formulae, so it strip naked what is going basically their in these formulae ,unlike the traditional notation which serve to get you lost in the crowd. So this notation is indeed in many aspects superior to the standard one. Zuhair
From: zuhair on 7 May 2010 20:33 On May 7, 9:50 am, Marshall <marshall.spi...(a)gmail.com> wrote: > On May 7, 1:50 am, zuhair <zaljo...(a)gmail.com> wrote: > > > > > This syntax is both elegant, abbreviating and easily > > readable. > > Maximizing abbreviation has the opposite effect as improving > readability. > > > The standard notation appears as too long, > > crowded, boring, and clumsy by comparison, that is > > besides it contains inverted symbols and > > rotated symbols without an obvious justification. > > The standard notation is already quite terse, to the > point that compressing it further strikes me as a bad > idea. > > The symbols you describe as rotated aren't. Does the > English alphabet contain rotated symbols? I would say > not, and yet lowercase u and n are the same modulo > rotation. > > The "shortening" of the implication arrow to a greater-than > symbol is an ASCII-specific optimization. The right arrow > is a single symbol in unicode, and when written by hand. > Also, this makes is difficult/ambiguous to write anything > involving numeric comparison. This may not be a goal > of yours, but it is a common application of standard > notation. > > > In my personal opinion the above record is the *best way > > of writing first order logic with identity and membership*, > > known thus far! > > "Best" is relative to a given set of requirements, and > any design is going to involve making tradeoffs. > > Marshall Just to add to what I wrote previously in reply to your really positive notes, that I appreciate so much. To me personally I see that there must be some kind of rational behind notations. It is true that we can use any notation however bizarre looking to write logic with, as far as it is uniquely readable, but I don't like that. Logical symbols must be the simplest possible, they must not be bizarre, they must not contain upside-down symbols or turnaround symbols, there must be some kind of explanation offered behind each symbol. The system that I wrote have an explanation for each symbol in it. ~ to represent negation is an accurate symbol, the symbol have opposing characteristic, the first bend is opposite the second one in direction, and this gives opposing impression, and it is a good symbolism to represent negation by a symbol that gives opposing impression, that's why I kept this symbol in my system. Conjunction by juxtaposing the two *formulae*, is pretty much intuitive since juxtaposing formulae does indeed give the impression of conjunction, and this was actually used before, possibly traced to prinicipa Mathematica. Disjunction using the symbol "|" is also intuitive, since the symbol "|" is a boundary symbol giving the impression of a separator, and it is pretty much intuitive to represent disjunction by a symbol that implies a kind of separation, since disjunction itself implies a kind of separation. notice that conjunction and disjunction are symbolized using non directional symbols, you see that | doesn't give any impression of directionality, neither does juxtaposing formulae. Implication, requires a directional symbol, because implication is a directional concept! and the simplest of which is > , there is no need for the stem in -> at all, this is a redundant notation, since > does the job nicely. same goes for the bi-conditional. On the other hand can you tell me the rational behind using a Large V to represent disjunction and an inverted Large V to represent conjunction, I see no rational behind such clumsy symbols at all. Now the quantifiers. I represented existential quantification by the symbol "_", this symbol gives and "expectational" impression, once you see the symbol _ you wander what is after it, and it is good to represent existence by and expectational symbol. Now regarding universal quantification, all what we want to say in universal quantification say of x is "for Any x", this is best represented by just writing x like that, this would give the full impression of any x. no need for any symbol for this quantifier. If one insist that a symbol must be given to universal quantification, then I would choose the symbol "\", since it gives the impression of a general kind of introduction, and the generality impression of the symbol make it eligible to represent the universal quantifier. However I still insist that only writing x like that is better to represent the meaning of "any x". Can you tell me what is the rational beyond an upside-down A and a turnaround E? Regarding using juxtaposition of terms (variables, constants and functions) to represent the membership relation, this is evident, since the principal primitive in fol(membership,=) is membership, and it would be the most repeatable symbol, so it is better be represented by juxtaposing the terms. Identity relation is an ancient symbol, and it is made up of two bars identical to each other, so it is OK. The dot notation is much more neat than the bracket system, and it has a good abbreviation power, and it is superior to the bracket system, and it is a proved notation since Principa Mathematica days. The other rules of abbreviation are only based on simply concepts derived from first order logic itself, and they help to remove the brackets and obviate the need for using excessive dot notation. So in this system, there is a rational beyond each symbol, and the net result is a nice looking formulae, clear, not crowded and actually tell you the core of what is going on in the formulae, you can see the skeleton of reasoning flowing in these formulae, so it strip naked what is going on basically in these formulae , unlike the traditional notation which serve to get you lost in the crowd. So this notation is indeed in many aspects superior to the standard one. Zuhair
From: zuhair on 7 May 2010 20:43 On May 7, 9:50 am, Marshall <marshall.spi...(a)gmail.com> wrote: > On May 7, 1:50 am, zuhair <zaljo...(a)gmail.com> wrote: > > > > > This syntax is both elegant, abbreviating and easily > > readable. > > Maximizing abbreviation has the opposite effect as improving > readability. > > > The standard notation appears as too long, > > crowded, boring, and clumsy by comparison, that is > > besides it contains inverted symbols and > > rotated symbols without an obvious justification. > > The standard notation is already quite terse, to the > point that compressing it further strikes me as a bad > idea. > > The symbols you describe as rotated aren't. Does the > English alphabet contain rotated symbols? I would say > not, and yet lowercase u and n are the same modulo > rotation. > > The "shortening" of the implication arrow to a greater-than > symbol is an ASCII-specific optimization. The right arrow > is a single symbol in unicode, and when written by hand. > Also, this makes is difficult/ambiguous to write anything > involving numeric comparison. This may not be a goal > of yours, but it is a common application of standard > notation. > > > In my personal opinion the above record is the *best way > > of writing first order logic with identity and membership*, > > known thus far! > > "Best" is relative to a given set of requirements, and > any design is going to involve making tradeoffs. > > Marshall I just wanted to add to what I wrote previously in reply to your really positive notes, that I appreciate so much. To me personally I see that there must be some kind of rational behind notations. It is true that we can use any notation however bizarre looking to write logic with, as far as it is uniquely readable, but I don't like that. Logical symbols must be the simplest possible, they must not be bizarre, they must not contain upside-down symbols or turnaround symbols, there must be some kind of explanation offered behind each symbol. The system that I wrote have an explanation for each symbol in it. ~ to represent negation is an accurate symbol, the symbol have opposing characteristic, the first bend is opposite the second one in direction, and this gives opposing impression, and it is a good symbolism to represent negation by a symbol that gives opposing impression, that's why I kept this symbol in my system. Conjunction by juxtaposing the two *formulae*, is pretty much intuitive since juxtaposing formulae does indeed give the impression of conjunction, and this was actually used before, possibly traced to prinicipa Mathematica. Disjunction using the symbol "|" is also intuitive, since the symbol "|" is a boundary symbol giving the impression of a separator, and it is pretty much intuitive to represent disjunction by a symbol that implies a kind of separation, since disjunction itself implies a kind of separation. notice that conjunction and disjunction are symbolized using non directional symbols, you see that | doesn't give any impression of directionality, neither does juxtaposing formulae. Implication, requires a directional symbol, because implication is a directional concept! and the simplest of which is > , there is no need for the stem in -> at all, this is a redundant notation, since > does the job nicely. same goes for the bi-conditional. On the other hand can you tell me the rational behind using a Large V to represent disjunction and an inverted Large V to represent conjunction, I see no rational behind such clumsy symbols at all. Now the quantifiers. I represented existential quantification by the symbol "_", this symbol gives and "expectational" impression, once you see the symbol _ you wander what is after it, and it is good to represent existence by an expectational symbol. Now regarding universal quantification, all what we want to say in universal quantification say of x is "for Any x", this is best represented by just writing x like that, this would give the full impression of any x. no need for any symbol for this quantifier. If one insist that a symbol must be given to universal quantification, then I would choose the symbol "\", since it gives the impression of a general kind of introduction, and the generality impression of the symbol make it eligible to represent the universal quantifier. However I still insist that only writing x like that is better to represent the meaning of "any x". Can you tell me what is the rational beyond an upside-down A and a turnaround E? Regarding using juxtaposition of terms (variables, constants and functions) to represent the membership relation, this is evident, since the principal primitive in fol(membership,=) is membership, and it would be the most repeatable symbol, so it is better be represented by juxtaposing the terms. Identity relation is an ancient symbol, and it is made up of two bars identical to each other, so it is OK. The dot notation is much more neat than the bracket system, and it has a good abbreviation power, and it is superior to the bracket system, and it is a proved notation since Principa Mathematica days. The other rules of abbreviation are only reflect simple quantification concepts derived from first order logic itself, and they help to remove the brackets and obviate the need for using excessive dot notation. So in this system, there is a rational beyond each symbol, and the net result is a nice looking formulae, clear, not crowded and actually tell you the core of what is going on in the formulae, you can see the skeleton of reasoning flowing in these formulae, so it strip naked what is going on basically in these formulae, unlike the traditional notation which serve to get you lost in the crowd. So this notation is indeed in many aspects superior to the standard one. Zuhair
From: zuhair on 7 May 2010 20:52
On May 7, 9:50 am, Marshall <marshall.spi...(a)gmail.com> wrote: > On May 7, 1:50 am, zuhair <zaljo...(a)gmail.com> wrote: > > > > > This syntax is both elegant, abbreviating and easily > > readable. > > Maximizing abbreviation has the opposite effect as improving > readability. > > > The standard notation appears as too long, > > crowded, boring, and clumsy by comparison, that is > > besides it contains inverted symbols and > > rotated symbols without an obvious justification. > > The standard notation is already quite terse, to the > point that compressing it further strikes me as a bad > idea. > > The symbols you describe as rotated aren't. Does the > English alphabet contain rotated symbols? I would say > not, and yet lowercase u and n are the same modulo > rotation. > > The "shortening" of the implication arrow to a greater-than > symbol is an ASCII-specific optimization. The right arrow > is a single symbol in unicode, and when written by hand. > Also, this makes is difficult/ambiguous to write anything > involving numeric comparison. This may not be a goal > of yours, but it is a common application of standard > notation. > > > In my personal opinion the above record is the *best way > > of writing first order logic with identity and membership*, > > known thus far! > > "Best" is relative to a given set of requirements, and > any design is going to involve making tradeoffs. > > Marshall I just wanted to add to what I wrote previously in reply to your really positive notes, that I appreciate so much. To me personally I see that there must be some kind of rational behind notations. It is true that we can use any notation however bizarre looking to write logic with, as far as it is uniquely readable, but I don't like that. Logical symbols must be the simplest possible, they must not be bizarre, they must not contain upside-down symbols or turnaround symbols, there must be some kind of explanation offered behind each symbol. The system that I wrote have an explanation for each symbol in it. ~ to represent negation is an accurate symbol, the symbol have opposing characteristic, the first bend is opposite the second one in direction, and this gives opposing impression, and it is a good symbolism to represent negation by a symbol that gives opposing impression, that's why I kept this symbol in my system. Conjunction by juxtaposing the two *formulae*, is pretty much intuitive since juxtaposing formulae does indeed give the impression of conjunction, and this was actually used before, possibly traced to prinicipa Mathematica. Disjunction using the symbol "|" is also intuitive, since the symbol "|" is a boundary symbol giving the impression of a separator, and it is pretty much intuitive to represent disjunction by a symbol that implies a kind of separation, since disjunction itself implies a kind of separation. notice that conjunction and disjunction are symbolized using non directional symbols, you see that | doesn't give any impression of directionality, neither does juxtaposing formulae. Implication, requires a directional symbol, because implication is a directional concept! and the simplest of which is > , there is no need for the stem in -> at all, this is a redundant notation, since > does the job nicely. same goes for the bi-conditional. On the other hand can you tell me the rational behind using a Large V to represent disjunction and an inverted Large V to represent conjunction, I see no rational behind such clumsy symbols at all. Now the quantifiers. I represented existential quantification by the symbol "_", this symbol gives an "expectational" impression, once you see the symbol "_ ", you wander what is after it, and it is good to represent existence by an expectational symbol. Now regarding universal quantification, all what we want to say in universal quantification say of x is "for Any x", this is best represented by just writing x like that, this would give the full impression of any x. no need for any symbol for this quantifier. If one insist that a symbol must be given to universal quantification, then I would choose the symbol \ since it gives the impression of a general kind of introduction, and the generality impression of the symbol make it eligible to represent the universal quantifier. However I still insist that only writing x like that is better to represent the meaning of "any x". Can you tell me what is the rational beyond an upside-down A and a turnaround E? Regarding using juxtaposition of terms (variables, constants and functions) to represent the membership relation, this is evident, since the principal primitive in fol(membership,=) is membership, and it would be the most repeatable symbol, so it is better be represented by juxtaposing the terms. Identity relation is an ancient symbol, and it is made up of two bars identical to each other, so it is OK. The dot notation is much more neat than the bracket system, and it has a good abbreviation power, and it is superior to the bracket system, and it is a proved notation since Principa Mathematica days. The other rules of abbreviation only reflect simple quantification concepts derived from first order logic itself, and they help to remove the brackets and obviate the need for using excessive dot notation. So in this system, there is a rational beyond each symbol, and the net result is a nice looking formulae, clear, not crowded and it actually tell you the core of what is going on in the formulae, you can see the skeleton of reasoning flowing in these formulae, so it strip naked what is going on basically in these formulae, unlike the traditional notation which serve to get you lost in the crowd. So this notation is indeed in many aspects superior to the standard one. Zuhair |