The best way of writing FOL with identity and membership.
in [Math]
|
Prev: Chapt 3, Fiberglass Experiment; Are the quasars nothing more than the last of the Supernova in view? #65; ATOM TOTALITY
Next: Inscribed triangles
From: zuhair on 7 May 2010 22:25 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. I generally understand the rational behind that. But still we can use ">" to denote "greater than" in numeric comparison by using this notation. Numeric comparison is between *terms*, or individual symbols like x>y, certainly this is not the case with the implication which occurs between *formulas*. for example we can even use the symbol | | to represent cardinality, and still we can use >, we can write for example |x| > |y| Now cardinality is a one place *function* symbol, so the above wouldn't be confused for implication, actually there is no ambiguity whatsoever here, since it is understood that |x| and |y| are individual terms, and not formulae, so > between them cannot represent implication! So there is no problem as you thought, you can use this notation to represent numeric comparisons. Zuhair > > > 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
From: zuhair on 8 May 2010 08:44 On May 7, 8:18 am, Marshall <marshall.spi...(a)gmail.com> wrote: > On May 7, 1:50 am, zuhair <zaljo...(a)gmail.com> wrote: > > > > > In my personal opinion the above record is the *best way > > of writing first order logic with identity and membership*, > > known thus far! > > I would disagree. > > You overuse juxtaposition. It means both conjunction and > set membership. How do you distinguish? You also use it, > with a space, to mean universal quantification. Juxtaposition > is the biggest weapon in the syntax designer's arsenal; > it should be fired only with the utmost care for the greatest > need. Juxtaposing formulae must be done with a space to avoid confusion for example. cy xy this is c in y and x in y if you don't leave a space between these formula, the notation would indeed be misleading, since it would look as: cyxy which is c in y and y in x and x in y one can also resort to the use of dot notation of course ,which is part of this system! so we can write it as cy.xy Now regarding the confusion with quantification especially with universal quantification suppose we want to write for all y ( x in y) this would be y xy notice the space between y and x y, without this space the expression would be yxy which is y in x and x in y however the space would serve to remove the confusion with membership. another possible confusion is with conjunction, like in for example for all x for all y ( x in y ) this must be written as x,y xy or can be written as x y xy Now in the first way the coma between x and y serve to remove confusion since without it we may have xy xy which is x in y and x in y. in the second case the space resolve the confusion since x y (i.e. x space y) is not membership since membership is symbolized by intimate juxtaposing i.e. no space left between the terms juxtaposed. Generally speaking that remark of yours is indeed a good one, one must use space to differentiate between different kinds of juxtapositioning symbolizing different concepts. Although I think spacing can do the job fairly well, but for the sake of rigor one may add the coma between the quantificational appearance of the variable and the formula so instead of writing: for all x Q(x) as x Q(x) we may write x,Q(x) Similarly we write _x,Q(x) which is there exist x Q(x) and in case of conjunction when it might appear confusing, then we can use the dot between the formulae to resolve such confusion. Zuhair > > Your dot notation is at least somewhat creative, but it has > some unfortunate characteristics. For one thing, it apparently > requires the writer to distinguish between terms where > there is no requirement to do so. For any operator with > the associative property, there is no meaningful difference > between different orderings. (P|Q)|R = P|(Q|P). Requiring > the distinction to be written down is requiring people to > attend to a difference that makes no difference. Another > problem in that it requires expressions to be rewritten > during substitution. > > Marshall
From: zuhair on 8 May 2010 09:21 On May 8, 7:44 am, zuhair <zaljo...(a)gmail.com> wrote: > On May 7, 8:18 am, Marshall <marshall.spi...(a)gmail.com> wrote: > > > > > > > On May 7, 1:50 am, zuhair <zaljo...(a)gmail.com> wrote: > > > > In my personal opinion the above record is the *best way > > > of writing first order logic with identity and membership*, > > > known thus far! > > > I would disagree. > > > You overuse juxtaposition. It means both conjunction and > > set membership. How do you distinguish? You also use it, > > with a space, to mean universal quantification. Juxtaposition > > is the biggest weapon in the syntax designer's arsenal; > > it should be fired only with the utmost care for the greatest > > need. > > Juxtaposing formulae must be done with a space to avoid confusion > > for example. > > cy xy > > this is > > c in y and x in y > > if you don't leave a space between these formula, the notation > would indeed be misleading, since it would look as: > > cyxy > > which is > > c in y and y in x and x in y > > one can also resort to the use of dot notation of course > ,which is part of this system! > > so we can write it as > > cy.xy > > Now regarding the confusion with quantification especially with > universal quantification > > suppose we want to write > > for all y ( x in y) > > this would be > > y xy > > notice the space between y and x y, > without this space the expression would be > > yxy > > which is > > y in x and x in y > > however the space would serve to remove the confusion with membership. > > another possible confusion is with conjunction, like in for example > > for all x for all y ( x in y ) > > this must be written as > > x,y xy > > or can be written as > > x y xy > > Now in the first way the coma between x and y serve to > remove confusion since without it we may have xy xy > which is x in y and x in y. > > in the second case the space resolve the confusion > since x y (i.e. x space y) is not membership > since membership is symbolized by intimate juxtaposing > i.e. no space left between the terms juxtaposed. > > Generally speaking that remark of yours is indeed a good one, > one must use space to differentiate between different kinds > of juxtapositioning symbolizing different concepts. > > Although I think spacing can do the job fairly well, but for > the sake of rigor one may add the coma between > the quantificational appearance of the variable and the formula > > so instead of writing: for all x Q(x) as > > x Q(x) > > we may write > > x,Q(x) > > Similarly we write > > _x,Q(x) > > which is > > there exist x Q(x) > > and in case of conjunction when it might appear confusing, > then we can use the dot between the formulae to resolve > such confusion. > > Zuhair > > > > > Your dot notation is at least somewhat creative, but it has > > some unfortunate characteristics. For one thing, it apparently > > requires the writer to distinguish between terms where > > there is no requirement to do so. For any operator with > > the associative property, there is no meaningful difference > > between different orderings. (P|Q)|R = P|(Q|P). Requiring > > the distinction to be written down is requiring people to > > attend to a difference that makes no difference. Another > > problem in that it requires expressions to be rewritten > > during substitution. > > > Marshall To make a rule out of that, we may say. Item symbol Membership immediate juxtaposing Conjunction single space juxtaposing or line below juxtaposing (large formulas) Quantification coma before the quantified formula Zuhair
From: zuhair on 8 May 2010 09:30 On May 8, 7:44 am, zuhair <zaljo...(a)gmail.com> wrote: > On May 7, 8:18 am, Marshall <marshall.spi...(a)gmail.com> wrote: > > > > > > > On May 7, 1:50 am, zuhair <zaljo...(a)gmail.com> wrote: > > > > In my personal opinion the above record is the *best way > > > of writing first order logic with identity and membership*, > > > known thus far! > > > I would disagree. > > > You overuse juxtaposition. It means both conjunction and > > set membership. How do you distinguish? You also use it, > > with a space, to mean universal quantification. Juxtaposition > > is the biggest weapon in the syntax designer's arsenal; > > it should be fired only with the utmost care for the greatest > > need. > > Juxtaposing formulae must be done with a space to avoid confusion > > for example. > > cy xy > > this is > > c in y and x in y > > if you don't leave a space between these formula, the notation > would indeed be misleading, since it would look as: > > cyxy > > which is > > c in y and y in x and x in y > > one can also resort to the use of dot notation of course > ,which is part of this system! > > so we can write it as > > cy.xy > > Now regarding the confusion with quantification especially with > universal quantification > > suppose we want to write > > for all y ( x in y) > > this would be > > y xy > > notice the space between y and x y, > without this space the expression would be > > yxy > > which is > > y in x and x in y > > however the space would serve to remove the confusion with membership. > > another possible confusion is with conjunction, like in for example > > for all x for all y ( x in y ) > > this must be written as > > x,y xy > > or can be written as > > x y xy > > Now in the first way the coma between x and y serve to > remove confusion since without it we may have xy xy > which is x in y and x in y. > > in the second case the space resolve the confusion > since x y (i.e. x space y) is not membership > since membership is symbolized by intimate juxtaposing > i.e. no space left between the terms juxtaposed. > > Generally speaking that remark of yours is indeed a good one, > one must use space to differentiate between different kinds > of juxtapositioning symbolizing different concepts. > > Although I think spacing can do the job fairly well, but for > the sake of rigor one may add the coma between > the quantificational appearance of the variable and the formula > > so instead of writing: for all x Q(x) as > > x Q(x) > > we may write > > x,Q(x) > > Similarly we write > > _x,Q(x) > > which is > > there exist x Q(x) > > and in case of conjunction when it might appear confusing, > then we can use the dot between the formulae to resolve > such confusion. > to make a rule out of that, we may say. Item symbol Membership immediate juxtaposing Conjunction single space juxtaposing or line below juxtaposing (large formulas) Regards Zuhair
From: Marshall on 10 May 2010 11:13
On May 7, 2:20 pm, zuhair <zaljo...(a)gmail.com> wrote: > On May 7, 8:18 am, Marshall <marshall.spi...(a)gmail.com> wrote: > > > As regards your latest statement, let met quote it: > > \Another > problem in that it requires expressions to be rewritten > during substitution.\ > > I really don't know what you mean by that, I usually write the whole > formula, then make the punctuation (the dot notation), same thing > is done with the brackets. I regret I don't have time to respond to all of your detailed comments. Let me just explain the bit about substitution. (It's a consideration that's important to me and my applications; it may have no relevance to you at all.) Consider the subset of first order logic in which quantifiers are limited to a single level of universal quantification. This is related to equational logic. In such cases we can omit the quantifiers entirely; every unbound variable is implicitly universally quantified. This allows us to write the associative property like this: x ^ (y ^ z) = (x ^ y) ^ z Quite terse and easy to understand, imho. In such a system, every variable in any true formula can be replaced with any expression, and it will yield a true formula. (Modulo issues of name capture.) The replacement, or substitution, can be simple textual substitution. This is an important property, and one which I don't think you're supporting. But it may not matter to your particular application. Also, and I wish I had thought of this sooner, but here's a paper that you really ought to read: from Boolean Algebra to Unified Algebra Eric C. R. Hehner http://www.cs.toronto.edu/~hehner/BAUA.pdf I was quite taken by this paper. I really didn't think there was much useful to say about syntax, and this paper demonstrates otherwise. In looking it up, I discover that he also has some new papers on similar topic; I can't vouch for them, but I will certainly read them. Who knows, he may have something awesome or he may have gone off the deep end. :-) Oh, another tiny comment: the "rationale" as it were for ^ and v being used for AND and OR comes from lattice theory. Boolean algebra is a kind of lattice algebra, and those symbols make exquisite sense in the context of lattices. Marshall |