Abbreviating First Order Logic With Identity and Membership
in [Logic]
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From: zuhair on 27 Apr 2010 13:20 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 hmmm.... Yea, there is no problem as far as I can see with the traditional notation of function symbols, predicate symbols of any arity, so I shall keep them as they are. Possibly one might consider removing the spacers between the n- variables So for example instead of writing F(x0,...,xn) one may write F(x0...xn) Also for example the n-tuple (x1,...,xn) can be written as (x1...xn) so the ordered pair is writtin as (a b). Same as for predicates. > > 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
From: zuhair on 27 Apr 2010 13:21 On Apr 27, 10:35 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > ~PQ > > read as > > (~P) and Q > > or as > > ~(P and Q) > > ? > > MoeBlee See my further comments, that's why ~PQ must be avoided!
From: zuhair on 27 Apr 2010 13:22 On Apr 27, 10:56 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Apr 27, 1:10 am, zuhair <zaljo...(a)gmail.com> wrote: > > > Regarding the set notation { : } , I think it must be maintained as it > > is, but the notation { | } must be avoided, since | would be confused > > for disjunction. > > Not really. > > The form for { | } can be confined to: > > {term | formula}. > > So that can't be read as > > {formula | formula}. > > MoeBlee Granted!
From: zuhair on 27 Apr 2010 13:25 On Apr 27, 3:55 am, Phil Carmody <thefatphil_demun...(a)yahoo.co.uk> wrote: > William Elliot <ma...(a)rdrop.remove.com> writes: > > 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). > > Far easier to just use RPN: > QP|S^ > QPS^| > > >> so by spacing technique one can differentiate between different > >> formulas. > > Anyone who relies on whitespace to carry essential information > is living in a state of ignorance about how lossy communication is. > > Phil You can if the formula is not long enough, and not so complex. as I say any formula which is long or complex such as to defy this methodology then to avoid any possible confusion one must break it down to component formula. Anyway these complex long formulas would be incomprehensible to the human reader if they are presented in one block. Zuhair > -- > I find the easiest thing to do is to k/f myself and just troll away > -- David Melville on r.a.s.f1
From: zuhair on 27 Apr 2010 13:26
On Apr 27, 10:57 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > "G. A. Edgar" <ed...(a)math.ohio-state.edu.invalid> writes: > > > Lots of the classic logic texts use dots instead of parentheses. > > And colons, and... > > > You get used to it after a while. > > Turing wrote a paper on it. Do you have the source pointing to this paper, especially on-line. > > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, darüber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |