Abbreviating First Order Logic With Identity and Membership
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From: zuhair on 27 Apr 2010 01:16 Abbreviating FOL with identity and membership FOL(=,e) 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. (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. (3) it can contain long sentences that are virtually incomprehensible, and better be broken down into smaller components. (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. 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. 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. (2) Quantifiers Existential quantifier . Unique Existential quantifier ! Universal quantifier no symbol only write the quantified variable. Example: x .y is read as for all x there exist y The quantifiers need not be actually written if writing them makes no difference to the meaning of the formula Example: writing the Extensionality axiom (see below) The order of the quantifiers if x is written to the left of y for example then the order of quantification is the same For example ..x ~yx this is the Empty set axiom, it is read as There exist x for all y not y in x. We don't need to write ..x y ~yx because the y in the middle add nothing to the meaning of the sentence, so it is redundant. However this method is tricky, for example ~yx alone would be read as for all y for all x not y in x. also ..y ~yx is read as, there exist y for all x not y in x. While x .y ~yx is read as: For all x there exist y not y in x. Quantification strings To write for example Exist x0, Exist x1,...,Exist xn or usually written as Exist x0,...,xn We write ..x0...xn However this must be differentiated from for example ..x0,,,xn which actually means there exist x0 for all x1, for all x2,..., for all xn. However when we wright for all x0,...,xn we must wright it as x0,,,xn . (3) Primitives: Membership No symbol only juxtapositioning the two symbols. Example: xy mean x is a member of y Identity = Now lets go to examples of writing FOL(=,e) using this technique: Lets write the axioms of ZF _______________________________________________ Extensionality: zx<>zy > x=y Foundation: x .yx > .yx .cy cx Empty: .x ~yx Pairing: a,b .x yx <> y=a|y=b Union: a .x yx <> .z yza Power: a .x yx <> zy>za Separation: a .x yx <> yaQ Replacement: x!yQ(x,y) > a .b zb<>.xaQ(x,z) _______________________________________________ Using this technique greatly abbreviate FOL(=,e) See the following formula: Exist meR For all i Exist k,l,p,q ((0el <-> 0eq) & ((iekeley)<-> Exist j (jepeqem & for all z (zei<->z=j))))). Look how this is shortened to: ..mR i .k.l.p.q 0l<>0q ikly<>.jpqm zi<>z=j As one can see, this method affords a clearer way of writing matters, no need for clumsy brackets or repetitions of the membership symbol and the conjunction symbol no clumsy notation at all. Breaking down complex formulas: Complex formulas that require more than one line to write must be actually decomposed to smaller formulas enough to reduced the whole formula in one line. Example: for all y ( y e t <-> for all w( Exist k ( for all u ( u e w <-> (u is a Wiener ordered pair & for all i Exist sr(iesereu -> i subset k) & for all j Exist pq(jepeqeu&0eq -> j=x)))) -> y e w)) ". Now this is a long formula, so it is better to break it to Q(u) <> u is a wiener ordered pair P(u,k) <> i .s.r isru > i subset k S(u,x) <> j .p.q jpqu 0eq > j=x Now the above formula can be written as: yt <> w .k uw<>Q(u)P(u,k)S(u,x) > yw Representing relations from Domains to Co domains the symbol "to" is shortened to -> instead of -->. so R: A -> B means R is a relation from the domain A to the co domain B. I think that the above technique serve as a good abbreviation of writing FOL with identity and membership. Zuhair
From: zuhair on 27 Apr 2010 02:10 On Apr 27, 12:16 am, zuhair <zaljo...(a)yahoo.com> wrote: > Abbreviating FOL with identity and membership FOL(=,e) > > 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. > > (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. > > (3) it can contain long sentences that are virtually > incomprehensible, and better be broken down into > smaller components. > > (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. > > 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. > > 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. > > (2) Quantifiers > > Existential quantifier . > > Unique Existential quantifier ! > > Universal quantifier no symbol only write the quantified variable. > > Example: > > x .y is read as for all x there exist y > > The quantifiers need not be actually written if writing them > makes no difference to the meaning of the formula > > Example: writing the Extensionality axiom (see below) > > The order of the quantifiers > > if x is written to the left of y for example > then the order of quantification is the same > > For example > > .x ~yx > > this is the Empty set axiom, it is read as > > There exist x for all y not y in x. > > We don't need to write > > .x y ~yx > > because the y in the middle add nothing to the meaning > of the sentence, so it is redundant. > > However this method is tricky, for example > > ~yx alone would be read as for all y for all x not y in x. > > also > > .y ~yx > > is read as, there exist y for all x not y in x. > > While > > x .y ~yx is read as: > > For all x there exist y not y in x. > > Quantification strings > > To write for example > > Exist x0, Exist x1,...,Exist xn > > or usually written as > > Exist x0,...,xn > > We write > > .x0...xn > > However this must be differentiated from for example > > .x0,,,xn > > which actually means > > there exist x0 for all x1, for all x2,..., for all xn. > > However when we wright for all x0,...,xn > > we must wright it as > > x0,,,xn . > > (3) Primitives: > > Membership No symbol > > only juxtapositioning the two symbols. > > Example: xy mean x is a member of y > > Identity = > > Now lets go to examples of writing FOL(=,e) using this > technique: > > Lets write the axioms of ZF > > _______________________________________________ > Extensionality: zx<>zy > x=y > > Foundation: x .yx > .yx .cy cx > > Empty: .x ~yx > > Pairing: a,b .x yx <> y=a|y=b > > Union: a .x yx <> .z yza > > Power: a .x yx <> zy>za > > Separation: a .x yx <> yaQ > > Replacement: x!yQ(x,y) > a .b zb<>.xaQ(x,z) > _______________________________________________ > > Using this technique greatly abbreviate FOL(=,e) > > See the following formula: > > Exist meR For all i Exist k,l,p,q ((0el <-> 0eq) & > ((iekeley)<-> Exist j (jepeqem & for all z (zei<->z=j))))). > > Look how this is shortened to: > > .mR i .k.l.p.q 0l<>0q ikly<>.jpqm zi<>z=j > > As one can see, this method affords a clearer way > of writing matters, no need for clumsy brackets > or repetitions of the membership symbol and > the conjunction symbol > no clumsy notation at all. > > Breaking down complex formulas: > > Complex formulas that require more than one line to write > must be actually decomposed to smaller formulas enough > to reduced the whole formula in one line. > > Example: > > for all y ( y e t <-> for all w( > Exist k ( for all u ( u e w <-> (u is a Wiener ordered pair & > for all i Exist sr(iesereu -> i subset k) & > for all j Exist pq(jepeqeu&0eq -> j=x)))) > > -> y e w)) ". > > Now this is a long formula, so it is better to break it to > > Q(u) <> u is a wiener ordered pair > > P(u,k) <> i .s.r isru > i subset k > > S(u,x) <> j .p.q jpqu 0eq > j=x > > Now the above formula can be written as: > > yt <> w .k uw<>Q(u)P(u,k)S(u,x) > yw > > Representing relations from Domains to Co domains > > the symbol "to" is shortened to -> instead of -->. > > so R: A -> B > > means > > R is a relation from the domain A to the co domain B. > > I think that the above technique serve as a good abbreviation > of writing FOL with identity and membership. > > Zuhair 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. Zuhair
From: William Elliot on 27 Apr 2010 04:27 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 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. 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. > (2) Quantifiers > > Existential quantifier . > > Unique Existential quantifier ! > > Universal quantifier no symbol only write the quantified variable. > > Example: > > x .y is read as for all x there exist y > > The quantifiers need not be actually written if writing them > makes no difference to the meaning of the formula > > Example: writing the Extensionality axiom (see below) > > > The order of the quantifiers > > if x is written to the left of y for example > then the order of quantification is the same > > For example > > .x ~yx > > this is the Empty set axiom, it is read as > > There exist x for all y not y in x. > > We don't need to write > > .x y ~yx > > because the y in the middle add nothing to the meaning > of the sentence, so it is redundant. > > However this method is tricky, for example > > ~yx alone would be read as for all y for all x not y in x. > > also > > .y ~yx > > is read as, there exist y for all x not y in x. > > While > > > x .y ~yx is read as: > > For all x there exist y not y in x. > > > Quantification strings > > To write for example > > Exist x0, Exist x1,...,Exist xn > > or usually written as > > Exist x0,...,xn > > We write > > .x0...xn > > However this must be differentiated from for example > > .x0,,,xn > > which actually means > > there exist x0 for all x1, for all x2,..., for all xn. > > However when we wright for all x0,...,xn > > we must wright it as > > x0,,,xn . > > > (3) Primitives: > > Membership No symbol > > only juxtapositioning the two symbols. > > Example: xy mean x is a member of y > > > Identity = > > > Now lets go to examples of writing FOL(=,e) using this > technique: > > > Let�s write the axioms of ZF > > _______________________________________________ > Extensionality: zx<>zy > x=y > > Foundation: x .yx > .yx .cy cx > > Empty: .x ~yx > > Pairing: a,b .x yx <> y=a|y=b > > Union: a .x yx <> .z yza > > Power: a .x yx <> zy>za > > Separation: a .x yx <> yaQ > > Replacement: x!yQ(x,y) > a .b zb<>.xaQ(x,z) > _______________________________________________ > > > Using this technique greatly abbreviate FOL(=,e) > > See the following formula: > > > Exist meR For all i Exist k,l,p,q ((0el <-> 0eq) & > ((iekeley)<-> Exist j (jepeqem & for all z (zei<->z=j))))). > > > Look how this is shortened to: > > .mR i .k.l.p.q 0l<>0q ikly<>.jpqm zi<>z=j > > > As one can see, this method affords a clearer way > of writing matters, no need for clumsy brackets > or repetitions of the membership symbol and > the conjunction symbol > no clumsy notation at all. > > > > Breaking down complex formulas: > > Complex formulas that require more than one line to write > must be actually decomposed to smaller formulas enough > to reduced the whole formula in one line. > > > Example: > > for all y ( y e t <-> for all w( > Exist k ( for all u ( u e w <-> (u is a Wiener ordered pair & > for all i Exist sr(iesereu -> i subset k) & > for all j Exist pq(jepeqeu&0eq -> j=x)))) > > -> y e w)) ". > > > Now this is a long formula, so it is better to break it to > > Q(u) <> u is a wiener ordered pair > > P(u,k) <> i .s.r isru > i subset k > > S(u,x) <> j .p.q jpqu 0eq > j=x > > Now the above formula can be written as: > > yt <> w .k uw<>Q(u)P(u,k)S(u,x) > yw > > > Representing relations from Domains to Co domains > > the symbol "to" is shortened to -> instead of -->. > > so R: A -> B > > means > > R is a relation from the domain A to the co domain B. > > > I think that the above technique serve as a good abbreviation > of writing FOL with identity and membership. > > > Zuhair > > > > > > > On Mon, 26 Apr 2010, zuhair wrote: > On Apr 27, 12:16�am, zuhair <zaljo...(a)yahoo.com> wrote: >> Abbreviating FOL with identity and membership FOL(=,e) >> >> 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. >> >> (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. >> >> (3) it can contain long sentences that are virtually >> incomprehensible, and better be broken down into >> smaller components. >> >> (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. >> >> 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. >> >> 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. >> >> (2) Quantifiers >> >> Existential quantifier � �. >> >> Unique Existential quantifier � ! >> >> Universal quantifier �no symbol only write the quantified variable. >> >> Example: >> >> x .y �is read as �for all x there exist y >> >> The quantifiers need not be actually written if writing them >> makes no difference to the meaning of the formula >> >> Example: writing the Extensionality axiom (see below) >> >> The order of the quantifiers >> >> if x is written to the left of y for example >> then the order of quantification is the same >> >> For example >> >> .x ~yx >> >> this is the Empty set axiom, it is read as >> >> There exist x for all y not y in x. >> >> We don't need to write >> >> .x y ~yx >> >> because the y in the middle add nothing to the meaning >> of the sentence, so it is redundant. >> >> However this method is tricky, for example >> >> ~yx alone would be read as for all y for all x not y in x. >> >> also >> >> .y ~yx >> >> is read as, there exist y for all x not y in x. >> >> While >> >> x .y ~yx is read as: >> >> For all x there exist y not y in x. >> >> Quantification strings >> >> To write for example >> >> Exist x0, Exist x1,...,Exist xn >> >> or usually written as >> >> Exist x0,...,xn >> >> We write >> >> .x0...xn >> >> However this must be differentiated from for example >> >> .x0,,,xn >> >> which actually means >> >> there exist x0 for all x1, for all x2,..., for all xn. >> >> However when we wright for all x0,...,xn >> >> we must wright it as >> >> x0,,,xn . >> >> (3) Primitives: >> >> Membership �No symbol >> >> only juxtapositioning the two symbols. >> >> Example: �xy �mean x is a member of y >> >> Identity �= >> >> Now lets go to examples of writing FOL(=,e) using this >> technique: >> >> Let�s write the axioms of ZF >> >> _______________________________________________ >> Extensionality: zx<>zy > x=y >> >> Foundation: x .yx > .yx .cy cx >> >> Empty: .x ~yx >> >> Pairing: �a,b .x yx <> y=a|y=b >> >> Union: �a .x yx <> .z yza >> >> Power: �a .x yx <> zy>za >> >> Separation: a .x yx <> yaQ >> >> Replacement: x!yQ(x,y) > �a .b zb<>.xaQ(x,z) >> _______________________________________________ >> >> Using this technique greatly abbreviate FOL(=,e) >> >> See the following formula: >> >> Exist meR For all i Exist k,l,p,q ((0el <-> 0eq) & >> ((iekeley)<-> Exist j (jepeqem & for all z (zei<->z=j))))). >> >> Look how this is shortened to: >> >> .mR i .k.l.p.q �0l<>0q �ikly<>.jpqm zi<>z=j >> >> As one can see, this method affords a clearer way >> of writing matters, no need for clumsy brackets >> or repetitions of the membership symbol and >> the conjunction symbol >> no clumsy notation at all. >> >> Breaking down complex formulas: >> >> Complex formulas that require more than one line to write >> must be actually decomposed to smaller formulas enough >> to reduced the whole formula in one line. >> >> Example: >> >> for all y ( y e t <-> for all w( >> Exist k ( for all u ( u e w <-> (u is a Wiener ordered pair & >> for all i Exist sr(iesereu -> i subset k) & >> for all j Exist pq(jepeqeu&0eq -> j=x)))) >> >> � -> y e w)) ". >> >> Now this is a long formula, so it is better to break it to >> >> Q(u) <> u is a wiener ordered pair >> >> P(u,k) <> i .s.r isru > i subset k >> >> S(u,x) <> j .p.q jpqu 0eq �> j=x >> >> Now the above formula can be written as: >> >> �yt <> �w .k uw<>Q(u)P(u,k)S(u,x) > yw >> >> Representing relations from Domains to Co domains >> >> the symbol "to" is shortened to -> instead of -->. >> >> so R: A -> B >> >> means >> >> R is a relation from the domain A to the co domain B. >> >> I think that the above technique serve as a good abbreviation >> of writing FOL with identity and membership. >> >> Zuhair > > 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. > > Zuhair >
From: Phil Carmody on 27 Apr 2010 04:55 William Elliot <marsh(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 -- I find the easiest thing to do is to k/f myself and just troll away -- David Melville on r.a.s.f1
From: MoeBlee on 27 Apr 2010 11:15
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 |