Abbreviating First Order Logic With Identity and Membership
in [Logic]
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From: Transfer Principle on 28 Apr 2010 00:13 On Apr 26, 10:16 pm, zuhair <zaljo...(a)yahoo.com> wrote: > Abbreviating FOL with identity and membership FOL(=,e) When I first saw this zuhair thread and saw how many posts it has accumulated in less than 24 hours, I thought that he was introducing yet another theory. But now I see that he's actually defining a new _notation_. > Criticism: The customary way of writing FOL(=,e) is > ridiculous, for the following reasons > (1) It contains strange symbols like for example > the symbols used for universal quantification > which looks like an upside-down A , and > the symbol used for Existential quantification > which look like a turnaround E. > This gives the impression of a scribble made by > mentally ill patients, rather than a way of writing > a rigorous formal language. I'm of two minds on this issue. On one hand, I know that standard theorists regularly criticize posters who invent their own notation. Some of them go as far as to assign them dubious points on some scale (either the Baez scale, or the Dudley scale). On the other hand, I agree that standard formal notation often looks like a "scribble" -- but zuhair's notation is even _more_ of a "scribble" than the standard theorists'! My opinion is that the formulas should be _longer_, not _shorter_, including the use of English words to make the meaning clearer. Some posters believe that standard formal language is _too_ symbolic, to the point that those who write only symbolic formulas aren't really doing mathematics at all. So here's how I'd write some of the formulas mentioned in zuhair's post here, using English: > Q|P S denote (Q or P) and S English: Both S, and either Q or P. > Q| PS denotes Q or (P and S). English: Either Q, or both P and S. > .x ~yx English: There is a set x such that no y is in x. > Extensionality: zx<>zy > x=y English: If x and y have the same elements, then they are equal. > Foundation: x .yx > .yx .cy cx English: Every set x has an element y disjoint from x. > Pairing: a,b .x yx <> y=a|y=b English: If a and b are sets, then there is another set x whose only elements are a and b. Often times I might use a hybrid between English and symbolic language. For simple formulas such as the Empty Set Axiom, I'd just keep the fully symbolic formula, namely ExAy (~yex). But for something like: > Exist meR For all i Exist k,l,p,q ((0el <-> 0eq) & > ((iekeley)<-> Exist j (jepeqem & for all z (zei<->z=j))))). English: There is an element m of R such that for every i, there exists sets k,l,p,q such that 0el <-> 0eq, and we have iekeley iff there's a set j such that jepeqem and i={j}. If R is intended to be the set of real numbers, then I would have begun with "There is a real number m such that..." (but then it's doubtful that zuhair would write "jepeqem" if m were a real number).
From: zuhair on 28 Apr 2010 09:07 On Apr 27, 11:13 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Apr 26, 10:16 pm, zuhair <zaljo...(a)yahoo.com> wrote: > > > Abbreviating FOL with identity and membership FOL(=,e) > > When I first saw this zuhair thread and saw how many > posts it has accumulated in less than 24 hours, I thought > that he was introducing yet another theory. But now I see > that he's actually defining a new _notation_. > > > Criticism: The customary way of writing FOL(=,e) is > > ridiculous, for the following reasons > > (1) It contains strange symbols like for example > > the symbols used for universal quantification > > which looks like an upside-down A , and > > the symbol used for Existential quantification > > which look like a turnaround E. > > This gives the impression of a scribble made by > > mentally ill patients, rather than a way of writing > > a rigorous formal language. > > I'm of two minds on this issue. On one hand, I know that > standard theorists regularly criticize posters who invent > their own notation. Some of them go as far as to assign > them dubious points on some scale (either the Baez scale, > or the Dudley scale). > > On the other hand, I agree that standard formal notation > often looks like a "scribble" -- but zuhair's notation is > even _more_ of a "scribble" than the standard theorists'! What is your evidence? When I criticized the standard notation I gave four reasons, were do you see a symbol that is like a known letter but in the opposite direction(like the standard quantifiers), were do you see repeated symbols (like the membership symbol and the brackets, the quantified variables etc), were do you see crowded long formulas, were do you see complex symbols, for example the universal quantifier is not a simple symbol, it can be viewed to be composed of at least three componenets, so is the implication symbol, so it the Existential quantifier symbol, actually all the connectives (excpet) negation and Quantifiers are complex symbols, do you see such a thing in my notation. It might look as a scribble at the end, but definitely it less of a scribble than the standard notation. Zuhair > > My opinion is that the formulas should be _longer_, not > _shorter_, including the use of English words to make the > meaning clearer. Some posters believe that standard > formal language is _too_ symbolic, to the point that those > who write only symbolic formulas aren't really doing > mathematics at all. Yea, this can be done, but it is on the other direction of what I am presenting here, so this is another subject. > > So here's how I'd write some of the formulas mentioned in > zuhair's post here, using English: > > > Q|P S denote (Q or P) and S > > English: Both S, and either Q or P. > > > Q| PS denotes Q or (P and S). > > English: Either Q, or both P and S. > > > .x ~yx > > English: There is a set x such that no y is in x. > > > Extensionality: zx<>zy > x=y > > English: If x and y have the same elements, then they > are equal. > > > Foundation: x .yx > .yx .cy cx > > English: Every set x has an element y disjoint from x. > > > Pairing: a,b .x yx <> y=a|y=b > > English: If a and b are sets, then there is another set > x whose only elements are a and b. > > Often times I might use a hybrid between English and > symbolic language. For simple formulas such as the > Empty Set Axiom, I'd just keep the fully symbolic > formula, namely ExAy (~yex). But for something like: > > > Exist meR For all i Exist k,l,p,q ((0el <-> 0eq) & > > ((iekeley)<-> Exist j (jepeqem & for all z (zei<->z=j))))). > > English: There is an element m of R such that for every i, > there exists sets k,l,p,q such that 0el <-> 0eq, and we > have iekeley iff there's a set j such that jepeqem and i={j}. > > If R is intended to be the set of real numbers, then I > would have begun with "There is a real number m such that..." > (but then it's doubtful that zuhair would write "jepeqem" if > m were a real number).
From: MoeBlee on 28 Apr 2010 11:21 On Apr 27, 9:39 pm, Tim Little <t...(a)little-possums.net> wrote: > On 2010-04-27, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > Oops, sorry, to be fair, you did just mention that you're not > > proposing a rigorous syntax but rather informal notational > > conventions. > > Actually in the first post he *did* say that it was proposed as an > alternative for the current notation as a rigorous formal notation. Tim, I don't see where he said that in his first post. He said that it was "an abbreviation" and he claims it has certain virtues including greater clarity. But I don't see where he said that it is rigorous and/ or formal. Yes, my initial impression was that he intended rigor and formality, because I would ordinarily expect that is what one would have in mind when one proposes such alternatives, but he later qualified that he did not, and when I look back at his first post indeed I don't see anyplace he made such a claim of rigor and formality. MoeBlee
From: Chip Eastham on 28 Apr 2010 12:05 On Apr 27, 4:02 pm, zuhair <zaljo...(a)gmail.com> wrote: > Thanks, I just glanced at it, it is a neat use of the dots, perhaps > the best that can > be done using these dots, but honestly the whole methodology is also > crumbled, there is no need for those dots, to me artistic use of > spaces > is much better. > > Zuhair Are you suggesting the use of whitespace is subject to "artistic" choices in your notation? I suspect (strongly) the proposed notation lacks a reliable scoping mechanism. E.g. how would you distinguish: (for all x, P(x)) OR Q(x) for all x, (P(x) OR Q(x)) A problem is that using a combination of prefix and infix notations leads to formal questions about operator priority. Without parentheses the parsing has to be quite tight to avoid ambiguity. regards, chip
From: zuhair on 28 Apr 2010 18:39
On Apr 28, 11:05 am, Chip Eastham <hardm...(a)gmail.com> wrote: > On Apr 27, 4:02 pm, zuhair <zaljo...(a)gmail.com> wrote: > > > Thanks, I just glanced at it, it is a neat use of the dots, perhaps > > the best that can > > be done using these dots, but honestly the whole methodology is also > > crumbled, there is no need for those dots, to me artistic use of > > spaces > > is much better. > > > Zuhair > > Are you suggesting the use of whitespace > is subject to "artistic" choices in your > notation? > > I suspect (strongly) the proposed notation > lacks a reliable scoping mechanism. E.g. > how would you distinguish: > > (for all x, P(x)) OR Q(x) > > for all x, (P(x) OR Q(x)) I think you mean for all x P(x) or for all x Q(x) this can be symbolized as x P(x) | x Q(x) and this is different from x P(x) | Q(x) For example Replacement can be written as: x!yQ > a .b y yb<>.xaQ but this is confusing somewhat, therefore it is better not to quantify the same variable more than once. Zuhair > > A problem is that using a combination > of prefix and infix notations leads to > formal questions about operator priority. > Without parentheses the parsing has to be > quite tight to avoid ambiguity. > > regards, chip |