Abbreviating First Order Logic With Identity and Membership
in [Logic]
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From: zuhair on 27 Apr 2010 11:17 On Apr 27, 3:27 am, William Elliot <ma...(a)rdrop.remove.com> wrote: > On Mon, 26 Apr 2010, zuhair wrote: > > Abbreviating FOL with identity and membership FOL(=,e) > > > Criticism: The customary way of writing FOL(=,e) is > > ridiculous, for the following reasons > > Conjunction No symbol, only spacing. > > That was employed by Rosseur in "Logic for Mathematicians". > > > Spacing will act to differentiate between different formulas > > > Examples: for the formulas Q,P,S > > > Q|P S denote (Q or P) and S > > > Q| PS denotes Q or (P and S). > > > so by spacing technique one can differentiate between different > > formulas. > > What about noticing the difference between double and triple spacing? > > For example, parse: A BvC v D E This would be A B|C | D E This would mean: ((A and (B or C)) or D) and E However I think this must be written in a clearer manner like: A B|C | D E of course we can put comas as spacers between them as for example: A,B|C,,|,,D,,,E But this is Ugly. > > If you're using a font with a narrow space, as most fonts, for > ease of reading change to a font with a wide space, if there are > any other than monospaced fonts. > > > So spacing technique would replace the need for brackets. > > I've seen it used in limited, unnested circumstances but in general, the > dot convention is more practical. What is exactly the dot convention? I agree of course there is the technique of using double, triple and quadruple spaces if needed, to me a formula that is so complex that the spacing technique do not manage it, is an indication for breaking down that formula into simpler components, because anyway such a complex formula would be incomprehensible to be written in one block. > > As I read below, I see you're creating a language that'll be as hard, if > not harder, to parse than C++. Yes, it greatly abbreviates FOL so it uses > less space but the cost of packing notation is greater time in use. That > is common computer science, that space and time are inversely conserved. I like this later statement of yours. But I think that with training one would use lesser time as he gets more familiar with using the notation, so at the end we'll have a notation that uses less space and less time as well. Zuhair
From: zuhair on 27 Apr 2010 11:34 On Apr 27, 10:15 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Apr 27, 12:16 am, zuhair <zaljo...(a)yahoo.com> wrote: > > > Criticism: The customary way of writing FOL(=,e) is > > ridiculous, for the following reasons > > One approach is complete with: > > variables > function symbols of arity 0 or greater > predicate symbols of arity 0 or greater > one connective > one quantifier > > Using Polish notation we don't need parentheses (we can use them > informally though for easier reading). And we can define all other 11 > binary connectives if we wish). And we can define other quantifiers. > This is elegant and easy to read. > > > (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. > > Strange until the first page of a book on the subject. > > > This gives the impression of a scribble made by > > mentally ill patients, rather than a way of writing > > a rigorous formal language. > > That's your subjective view. I don't at all agree with it. > > > (2) It contains a lot of repeated symbols, for > > example the brackets, and the symbols for > > membership relation and the symbol of conjunction, > > now these symbols are the most repeated symbols > > and they need not be actually symbolized. > > But it turns out that you yourself resort to an additional symbol (the > space symbol). And the problem with the space symbol is that it is > sometimes not so clear whether a space is intended (especially, for > example in fast writing on a blackboard or on scratch paper). > > > (3) it can contain long sentences that are virtually > > incomprehensible, and better be broken down into > > smaller components. > > It turns out that the notation you propose below is no easier to > parse. Also, long formulas can be presented in "chunk" form using > spaces, indents, and line breaks. But with your system that is not > possible since a space is itself a symbol. There is a great ADVANTAGE > to not having space represent a symbol. > > > (4) The symbols are pretty much complex symbols > > while at the same time they are supposed to be > > denoting simple concepts, for example the symbol > > given to disjunction which is a Large V (clumsy looking actually) > > so is the symbol given for conjunction which is an upside-down V > > ,also the symbol given to implication which is virtually > > composed of three smaller symbols, which is very complex, so > > are the symbols of the quantifiers, they are too complex symbols. > > I see hardly anything very "complex" about such symbols. > > > > > > > Alternative Notation of FOL(=,e) > > > (1) Logical connectives > > > Negation ~ > > > Disjunction | > > > Implication > > > > Biconditional <> > > > Conjunction No symbol, only spacing. > > > so the formula Q and P is written as QP, no need for any symbol > > between them. > > That fails unique readability. In your notation: > > P|Q is a formula. > R|S is a formula. > > So the conjunction of Q|P and R|S is > > P|QR|S > > But that could also be read not just as > > (P|Q)(R|S) > > but also as > > P((QR)|S) > or as > P|(Q(R|S)) > or as > ((P|Q)R)|S > > Fails unique readability. > > > Spacing will act to differentiate between different formulas > > > Examples: for the formulas Q,P,S > > > Q|P S denote (Q or P) and S > > > Q| PS denotes Q or (P and S). > > > so by spacing technique one can differentiate between different > > formulas. > > > So spacing technique would replace the need for brackets. > > Oh, so your syntax rule of juxtaposition is not a rule at all, but > rather you have a more elaborate rule in mind with an additional > character (you're using a space as a symbol instead of parentheses). > > So what does > PQ| RS > stand for? > > Is it > (P and Q) or (R and S) > or is it > P and (Q or (R and S)) ? > > Please give a RECURSIVE (or algorithmically checkable) specification > of your syntax rule and so that your rule provides unique readability > without fail. > > MoeBlee Good point Moe! I agree, but what I was speaking about is to use spaces in a manner that is not confusing, I am rather speaking of an art here rather than a rigorous way really. But still your point is noted! So back to your example P|Q|R|S This notation must be avoided! since it is confusing, any notation that use spacing in such a manner that do not lead to unique readability is to be forbidden in this system, however I didn't yet set rigorous rules for that, I left it open to the spacial imagination of the writer. But as I said any use of this system that cause confusion is not acceptable. So back to your question, the answer would be that P|Q|R|S is NOT acceptable way of writing . If one want to say P or (Q or (R or S) for example then this must be written as P | Q | R|S or if one want to write (P or Q) or (R or S) then this must be written as: P|Q | R|S What I meant to say is that the writer must use the spacing technique in a manner that do not lead to non unique readability. I think spacing can do the job really, and as I said if the formula is too complex for spacing to do the job, then it must be decomposed into simpler components. Zuhair
From: MoeBlee on 27 Apr 2010 11:35 ~PQ read as (~P) and Q or as ~(P and Q) ? MoeBlee
From: MoeBlee on 27 Apr 2010 11:41 On Apr 27, 10:34 am, zuhair <zaljo...(a)gmail.com> wrote: > I think spacing can do the job really But you've not given an unambiguous machine-checkable spacing RULE. Moreover, spacing is very unreliable visually, especially for formulas scribbled on a chalkboard or scratchpad. MoeBlee
From: G. A. Edgar on 27 Apr 2010 11:42
In article <4fbd2971-604c-4b9b-a6a6-7bae207b844c(a)q15g2000yqj.googlegroups.com>, zuhair <zaljohar(a)gmail.com> wrote: > of course we can put comas as spacers between them as for example: > > A,B|C,,|,,D,,,E > > But this is Ugly. Lots of the classic logic texts use dots instead of parentheses. You get used to it after a while. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ |