looking for a predicate hierarchy
in [OOP]
|
From: Dmitry A. Kazakov on 28 Dec 2006 08:27 On Wed, 27 Dec 2006 20:10:48 +0100 (CET), V.J. Kumar wrote: > "Dmitry A. Kazakov" <mailbox(a)dmitry-kazakov.de> wrote in > news:av8llm95a8p5$.jjkecdqw8xub$.dlg(a)40tude.net: > >> On Tue, 26 Dec 2006 19:38:29 +0100 (CET), V.J. Kumar wrote: >> >> Right, but we have to loose that anyway. To me four-valued logic is >> just a necessary step to a fuzzy one. There you will never have a >> chance to save it. > > Well, not quite. It depends what you mean by fuzzy logic. There are > quite a few of those. The most known varieties like Lukasiewicz's, > Godel's, product logic, all have the fuzzy implication connective, > deduction system which is purely syntactical, are sound and complete (see > Petr Hajek's online articles for an introduction, e.g. Basic Fuzzy Logic > and BL-algebra", or his book "Metamathematics of Fuzzy Logic"). These are systems which don't stand the question you pose later. The only consistent fuzzy logic can be based on the possibility as the set measure. That inevitably leads to intuitionistic fuzzy logic with Belnap's four values as the bounds. That is my view on the things. There could be alternatives based on other measures, like probability (Pr), for instance, but they usually cannot produce a logic in the sense that V and /\ cannot be defined as functions of the arguments. Pr (A U B) /= Pr (A) + Pr (B), only when A and B don't intersect. > It's a controversial issue but I find FL account of vagueness simplistic > and not convincing, for example how does one determine the value of > membership function for borderline cases; combining fuzzy truth values > seems naive and so does FL's truth functionality as whole; at some point > one has to make a decision whether or not to do something in which case > logic collapses to binary, et). Defuzification is outside FL. That's no different from "de-randomization" in probability theory. (You have x probability of shock reading comp.object. Would you read it here and now?) Fundamentally, if we could deduce from fuzzy anything but fuzzy, then that would kill/explain/trivialize fuzzy. Applicability of fuzzy values (as well as their sources) is not an issue and the very question is wrong (outside FL). > You may want to read Parikh's "Test for > Fuzzy Logic" and Hajek's opposing view in "Ten Questions on Fuzzy Logic" > where Hajek himself states that FL gives relative, not asolute conclusions > (comparative truth). >> When it rains to 0.9 degree and does not to 0.6, >> then what? > > How did you arrive at the crisp value of 0.9 or 0.6 in your fuzzy system ? > ;) A good question. The answer is that they are need not to be. Here I mean intuitionistic fuzzy logic based on a four-valued one. A pure fuzzy logic based on [0,1] has no satisfactory answer to your question. But with four values as the bounds the answer is that 0.9 and 0.6 are estimations. The inference does not produce crisp values. It produces crisp estimations of. I.e. pos(A)>= 0.9, nec(A)<=0.6 (equivalently pos(~A)>=0.4). This eliminates the apparent contradiction. >>> The disjunctive syllogism will hold for the formula (A=>A). Just >>> substitute (A->A) for the letter and follow the reasoning. >> >> But >> >> ((A or B) and not A) => B >> >> is not universally true (for example in A=T, B=0). And we cannot use >> -> and => interchangeable. One should stick to one of them. > > It's a typo: I should have written just substitute (A=>A) for ... OK, for =>, when A=T and B=0 the above becomes: ((T V 0) /\ not T) => 0 = (T /\ T) => 0 = T => 0 = _|_ i.e. it does not hold. >>> As long as the logic can handle contradiction and does not explode, >>> I do not care how the implication is defined. >>> >>> For a possible definition, see Avron's articles, e.g. "Value in four >>> values" and others. He defines 'a implies b' to be 1 for 0 and _|_ >>> and b otherwise. >> >> Is this what you mean? >> >> a> | T 0 1 _|_ >> ---+----------------- >> T | 0 0 _|__|_ >> 0 | 1 1 1 1 >> 1 | 0 0 _|__|_ >> _|_| 0 0 _|__|_ >> >> So 1=>1 were _|_. That's too strange. > > No, a implies b equals 1 if a in {0, _|_} and b otherwise: > > | b T 0 1 _|_ > a | > ---+----------------- > T | T 0 1 _|_ > 0 | 1 1 1 1 > 1 | T 0 1 _|_ > _|_| 1 1 1 1 > > T implies T evaluates to T which prevents explosion with any arbitrary > formula. I see. [...] > So the idea with defining the implication is to prevent explosion which is > ensured by T a> T evaluating to T, and the rest of the table is cooked so > that MP would work. But it does not! (A /\ (A a> B)) a> B evaluates T in A=T, B=T and in A=1, B=T. Further ((A V B) a> C) a> ((A a>C) V (B a> C)) also does not (in A,B,C=T). Maybe /\ was also prepared? Oller for example uses consensus instead of V and gullibility instead of /\. -- Regards, Dmitry A. Kazakov http://www.dmitry-kazakov.de
From: V.J. Kumar on 28 Dec 2006 12:50 "Dmitry A. Kazakov" <mailbox(a)dmitry-kazakov.de> wrote in news:o4bdwl2r39pm.33vcggeeyv1a$.dlg(a)40tude.net: > On Wed, 27 Dec 2006 20:10:48 +0100 (CET), V.J. Kumar wrote: > >> "Dmitry A. Kazakov" <mailbox(a)dmitry-kazakov.de> wrote in >> news:av8llm95a8p5$.jjkecdqw8xub$.dlg(a)40tude.net: >> >>> On Tue, 26 Dec 2006 19:38:29 +0100 (CET), V.J. Kumar wrote: >>> >>> Right, but we have to loose that anyway. To me four-valued logic is >>> just a necessary step to a fuzzy one. There you will never have a >>> chance to save it. >> >> Well, not quite. It depends what you mean by fuzzy logic. There >> are quite a few of those. The most known varieties like >> Lukasiewicz's, Godel's, product logic, all have the fuzzy >> implication connective, deduction system which is purely >> syntactical, are sound and complete (see Petr Hajek's online >> articles for an introduction, e.g. Basic Fuzzy Logic and >> BL-algebra", or his book "Metamathematics of Fuzzy Logic"). > > These are systems which don't stand the question you pose later. The > only consistent fuzzy logic can be based on the possibility as the set > measure. That inevitably leads to intuitionistic fuzzy logic with > Belnap's four values as the bounds. It's unclear what you mean by "intuitionistic fuzzy logic". Is it Takeuti's logic or Atanassov's ? If it's the latter, "intuitionistic" is a misnomer because it has been shown that, although based on different insights, IFL is mathematically equivalent to interval-valued fuzzy logic. IVL laso has a host of philosophical issues like how does one substantiate *two* fuzzy membership functions ? > That is my view on the things. > There could be alternatives based on other measures, like probability > (Pr), for instance, but they usually cannot produce a logic in the > sense that V and /\ cannot be defined as functions of the arguments. > Pr (A U B) /= Pr (A) + Pr (B), only when A and B don't intersect. Right, it's well known that probablity theory 'connectives' are not truth-functional. > >> It's a controversial issue but I find FL account of vagueness >> simplistic and not convincing, for example how does one determine the >> value of membership function for borderline cases; combining fuzzy >> truth values seems naive and so does FL's truth functionality as >> whole; at some point one has to make a decision whether or not to do >> something in which case logic collapses to binary, et). > > Defuzification is outside FL. That's no different from > "de-randomization" in probability theory. (You have x probability of > shock reading comp.object. Would you read it here and now?) > Fundamentally, if we could deduce from fuzzy anything but fuzzy, then > that would kill/explain/trivialize fuzzy. Applicability of fuzzy > values (as well as their sources) is not an issue and the very > question is wrong (outside FL). > >> You may want to read Parikh's "Test for >> Fuzzy Logic" and Hajek's opposing view in "Ten Questions on Fuzzy >> Logic" where Hajek himself states that FL gives relative, not >> asolute conclusions (comparative truth). > >>> When it rains to 0.9 degree and does not to 0.6, >>> then what? >> >> How did you arrive at the crisp value of 0.9 or 0.6 in your fuzzy >> system ? ;) > > A good question. The answer is that they are need not to be. Here I > mean intuitionistic fuzzy logic based on a four-valued one. A pure > fuzzy logic based on [0,1] has no satisfactory answer to your > question. But with four values as the bounds the answer is that 0.9 > and 0.6 are estimations. So how does one arrive at the estimations ? Now, you have to substantiate *two* fuzzy interval boundaries intead of one fuzzy number. It's hardly better. >> No, a implies b equals 1 if a in {0, _|_} and b otherwise: >> >> | b T 0 1 _|_ >> a | >> ---+----------------- >> T | T 0 1 _|_ >> 0 | 1 1 1 1 >> 1 | T 0 1 _|_ >> _|_| 1 1 1 1 >> >> T implies T evaluates to T which prevents explosion with any >> arbitrary formula. > > I see. > > [...] >> So the idea with defining the implication is to prevent explosion >> which is ensured by T a> T evaluating to T, and the rest of the >> table is cooked so that MP would work. > > But it does not! > > (A /\ (A a> B)) a> B > > evaluates T in A=T, B=T and in A=1, B=T. > But that's OK because T being a designated truth value means that the formula holds ("has a model") ! Or did I misunderstand you ? > Further > > ((A V B) a> C) a> ((A a>C) V (B a> C)) > > also does not (in A,B,C=T). See above. That's the whole idea behind the desire to be able to handle contradictory information: to have more than one 'designated' truth value. the simple trick also works in the case of three-valued logic, where depending on interpretation of the third symbol (whether it belongs to the designated set or not), a logic can be considered paraconsistent or just multi-valued. > > Maybe /\ was also prepared? Oller for example uses consensus instead > of V and gullibility instead of /\. No, it was not. >
From: Dmitry A. Kazakov on 29 Dec 2006 09:06 On Thu, 28 Dec 2006 18:50:21 +0100 (CET), V.J. Kumar wrote: > "Dmitry A. Kazakov" <mailbox(a)dmitry-kazakov.de> wrote in > news:o4bdwl2r39pm.33vcggeeyv1a$.dlg(a)40tude.net: > >> On Wed, 27 Dec 2006 20:10:48 +0100 (CET), V.J. Kumar wrote: >> >>> "Dmitry A. Kazakov" <mailbox(a)dmitry-kazakov.de> wrote in >>> news:av8llm95a8p5$.jjkecdqw8xub$.dlg(a)40tude.net: >>> >>>> On Tue, 26 Dec 2006 19:38:29 +0100 (CET), V.J. Kumar wrote: >>>> >>>> Right, but we have to loose that anyway. To me four-valued logic is >>>> just a necessary step to a fuzzy one. There you will never have a >>>> chance to save it. >>> >>> Well, not quite. It depends what you mean by fuzzy logic. There >>> are quite a few of those. The most known varieties like >>> Lukasiewicz's, Godel's, product logic, all have the fuzzy >>> implication connective, deduction system which is purely >>> syntactical, are sound and complete (see Petr Hajek's online >>> articles for an introduction, e.g. Basic Fuzzy Logic and >>> BL-algebra", or his book "Metamathematics of Fuzzy Logic"). >> >> These are systems which don't stand the question you pose later. The >> only consistent fuzzy logic can be based on the possibility as the set >> measure. That inevitably leads to intuitionistic fuzzy logic with >> Belnap's four values as the bounds. > > It's unclear what you mean by "intuitionistic fuzzy logic". Is it > Takeuti's logic or Atanassov's ? If it's the latter, Yes > "intuitionistic" > is a misnomer because it has been shown that, although based on > different insights, IFL is mathematically equivalent to interval-valued > fuzzy logic. Surely they are. However, IFL and IVL were introduced as continuations of three state logic based on 0, 1, _|_. When I consider IFL, I would also allow contradiction, i.e. intervals with the lower bound greater than the upper one. That would be a continuation of a four-valued logic. Contradiction is needed for many reasons. > IVL laso has a host of philosophical issues like how does > one substantiate *two* fuzzy membership functions ? That's easy. You have a [fuzzy] subset A of some universe U and a set of "focal" elements X={xi}, normally crisp, xi/\xj=�, \/xi=U. Each focal element is a crisp subset of the universe. Let you have some set-measure s. The upper set of A is a fuzzy subset of X such that s(A/\xi). The lower set is s(~A/\xi). Now a hobby philosopher would say, the universe can be sensed in terms of only X. Let us forget about the nature of xi, which cannot be studied, and accept IFS as "facts." >>> How did you arrive at the crisp value of 0.9 or 0.6 in your fuzzy >>> system ? ;) >> >> A good question. The answer is that they are need not to be. Here I >> mean intuitionistic fuzzy logic based on a four-valued one. A pure >> fuzzy logic based on [0,1] has no satisfactory answer to your >> question. But with four values as the bounds the answer is that 0.9 >> and 0.6 are estimations. > > So how does one arrive at the estimations? Through inference rules from "fuzzy facts." > Now, you have to > substantiate *two* fuzzy interval boundaries intead of one fuzzy number. > It's hardly better. It is better because it handles uncertainty and contradiction. One number, or anything else with an order cannot do that. The reason is same as why we go four-valued. >>> No, a implies b equals 1 if a in {0, _|_} and b otherwise: >>> >>> | b T 0 1 _|_ >>> a | >>> ---+----------------- >>> T | T 0 1 _|_ >>> 0 | 1 1 1 1 >>> 1 | T 0 1 _|_ >>> _|_| 1 1 1 1 >>> >>> T implies T evaluates to T which prevents explosion with any >>> arbitrary formula. >> >> I see. >> >> [...] >>> So the idea with defining the implication is to prevent explosion >>> which is ensured by T a> T evaluating to T, and the rest of the >>> table is cooked so that MP would work. >> >> But it does not! >> >> (A /\ (A a> B)) a> B >> >> evaluates T in A=T, B=T and in A=1, B=T. > > But that's OK because T being a designated truth value means that the > formula holds ("has a model") ! T is "neither," it is "closer" to 0. It seems that it actually was: a>| T 0 1 _|_ ---+----------------- T | 1 1 1 1 0 | 1 1 1 1 1 | T 0 1 _|_ _|_| T 0 1 _|_ > Or did I misunderstand you ? Usually inference should be made only under certain truth. I.e. when x a> y does not evaluate 1, then x|=y is wrong. Arguably one could consider the case when it evaluates _|_ (both true and false) as both x|=y and not(x|=y), and then continue inference along both paths. However, because not(x|=y) ["x does not imply y"] would be a quite weak statement in any four-valued logics, it might deliver nothing useful. In any case we cannot just consider 1 and _|_ same in inference, if we haven't proved that all possible consequences were indeed same. With a> they don't look same! The two lower rows of the truth table aren't identical. Then I have a vague suspicious that this would trivialize logic. Having said that, I actually very like the idea to use all possible [four and more in IFL] outcomes of evaluation of x=>y. That means that "imply" would have no any special meaning, just some relation between x and y. I think it would be a natural consequence of four-valued approach. -- Regards, Dmitry A. Kazakov http://www.dmitry-kazakov.de
From: V.J. Kumar on 29 Dec 2006 14:40 "Dmitry A. Kazakov" <mailbox(a)dmitry-kazakov.de> wrote in news:9n4itzdwg52n.8juaqnsmr8vg.dlg(a)40tude.net: > On Thu, 28 Dec 2006 18:50:21 +0100 (CET), V.J. Kumar wrote: > >> IVL also has a host of philosophical issues like how does >> one substantiate *two* fuzzy membership functions ? > > That's easy. You have a [fuzzy] subset A of some universe U and a set > of "focal" elements X={xi}, normally crisp, xi/\xj=�, \/xi=U. Each > focal element is a crisp subset of the universe. Let you have some > set-measure s. The upper set of A is a fuzzy subset of X such that > s(A/\xi). The lower set is s(~A/\xi). Now a hobby philosopher would > say, the universe can be sensed in terms of only X. Let us forget > about the nature of xi, which cannot be studied, and accept IFS as > "facts." Well, it's too a strong "let us". The usual philosophical objection to the standard FL membership function was its crispness. Say you have a statement like Red(x) which is evaluated by an expert to e.g. 0.9. Then it's natural to think that the statement itself is vague with a degree of vagueness e.g. 0.7, then the statement about the statement about the statement about the vagueness is vague itself, and so on. So you get an infinite chain of vague statements and cannot in principle reason about the fuzzy truth. This idea usually goes under the name of "higher-order vagueness". The FL folks answer to that is, ok, what we had until now was "type-1" membership function, F1:X->[0,1] where X is some set, henceforth we'll use "type-2" membership, F2:X->[0,1]^[0,1]. It turns out that using type- 2 FL is computationally infeasible, so the FL people use greatly simplified interval-valued fuzzy logic, which is sometimes misnamed as 'intuitionistic' FL, instead of 'real' type-2 FL. So now we have two simplifications, and it's unclear whether or not such simplification is legitimate. > >>>> How did you arrive at the crisp value of 0.9 or 0.6 in your fuzzy >>>> system ? ;) >>> >>> A good question. The answer is that they are need not to be. Here I >>> mean intuitionistic fuzzy logic based on a four-valued one. A pure >>> fuzzy logic based on [0,1] has no satisfactory answer to your >>> question. But with four values as the bounds the answer is that 0.9 >>> and 0.6 are estimations. >> >> So how does one arrive at the estimations? > > Through inference rules from "fuzzy facts." Here's an inference for you from "fuzzy facts" courtesy of Edgington ;) Let x, y, z be three balls that an 'expert' determined to be red to some degree and small to some other degree: v(Red(x)) = 1 v(Small(x)) = 0.5 v(Red(y)) = 0.5 v(Small(y)) = 0.5 v(Red(z)) = 0.5 v(Small(z)) = 0 Now using Zadeh blessed definition for 'and' as min(x, y), we'll get the conclusion that all the balls are equally red and small ! 'Red and Small' equals 0.5 in all the cases which clearly contradicts the intuition that x being red and small has to have a higher degree of truth than y, and z has to have the lowest. > >> Now, you have to >> substantiate *two* fuzzy interval boundaries intead of one fuzzy >> number. It's hardly better. > > It is better because it handles uncertainty and contradiction. One > number, or anything else with an order cannot do that. The reason is > same as why we go four-valued. > >>>> No, a implies b equals 1 if a in {0, _|_} and b otherwise: >>>> >>>> | b T 0 1 _|_ >>>> a | >>>> ---+----------------- >>>> T | T 0 1 _|_ >>>> 0 | 1 1 1 1 >>>> 1 | T 0 1 _|_ >>>> _|_| 1 1 1 1 >>>> >>>> T implies T evaluates to T which prevents explosion with any >>>> arbitrary formula. >>> >>> I see. >>> >>> [...] >>>> So the idea with defining the implication is to prevent explosion >>>> which is ensured by T a> T evaluating to T, and the rest of the >>>> table is cooked so that MP would work. >>> >>> But it does not! >>> >>> (A /\ (A a> B)) a> B >>> >>> evaluates T in A=T, B=T and in A=1, B=T. >> >> But that's OK because T being a designated truth value means that the >> formula holds ("has a model") ! > > T is "neither," it is "closer" to 0. It seems that it actually was: > > a>| T 0 1 _|_ > ---+----------------- > T | 1 1 1 1 > 0 | 1 1 1 1 > 1 | T 0 1 _|_ > _|_| T 0 1 _|_ > No it was not, it was exacly as I specified. > > Usually inference should be made only under certain truth. I.e. when x > a> y does not evaluate 1, then x|=y is wrong. If you think so, then you've just destroyed the whole area of paraconsistent logic, perhaps deservedly, but that's another question ;) To treat 'T' as a designated truth value is exactly what various paraconsistent logics do to avoid explosion.
From: Dmitry A. Kazakov on 30 Dec 2006 06:09
On Fri, 29 Dec 2006 20:40:20 +0100 (CET), V.J. Kumar wrote: > "Dmitry A. Kazakov" <mailbox(a)dmitry-kazakov.de> wrote in > news:9n4itzdwg52n.8juaqnsmr8vg.dlg(a)40tude.net: > >> On Thu, 28 Dec 2006 18:50:21 +0100 (CET), V.J. Kumar wrote: >> >>> IVL also has a host of philosophical issues like how does >>> one substantiate *two* fuzzy membership functions ? >> >> That's easy. You have a [fuzzy] subset A of some universe U and a set >> of "focal" elements X={xi}, normally crisp, xi/\xj=�, \/xi=U. Each >> focal element is a crisp subset of the universe. Let you have some >> set-measure s. The upper set of A is a fuzzy subset of X such that >> s(A/\xi). The lower set is s(~A/\xi). Now a hobby philosopher would >> say, the universe can be sensed in terms of only X. Let us forget >> about the nature of xi, which cannot be studied, and accept IFS as >> "facts." > > Well, it's too a strong "let us". The usual philosophical objection to > the standard FL membership function was its crispness. Say you have a > statement like Red(x) which is evaluated by an expert to e.g. 0.9. Then > it's natural to think that the statement itself is vague with a degree of > vagueness e.g. 0.7, then the statement about the statement about the > statement about the vagueness is vague itself, and so on. So you get an > infinite chain of vague statements and cannot in principle reason about > the fuzzy truth. This idea usually goes under the name of "higher-order > vagueness". > > The FL folks answer to that is, ok, what we had until now was "type-1" > membership function, F1:X->[0,1] where X is some set, henceforth we'll > use "type-2" membership, F2:X->[0,1]^[0,1]. It turns out that using type- > 2 FL is computationally infeasible, so the FL people use greatly > simplified interval-valued fuzzy logic, which is sometimes misnamed as > 'intuitionistic' FL, instead of 'real' type-2 FL. Well, integers are "simplified" reals. That alone does not devaluate integers. IFS can be extended to type-2, but I don't think it would be worth of efforts. Especially when it gets to specifying a distribution along the Y-axis. That quickly becomes as empiric as statistic distributions. In fact it would be interesting to mix IFS with probability distributions to get random fuzzy sets. Anyway both solve the issue of "crispness." You have an interval (or other compact set) of possible values and whether the bounds were crisp plays no role. >>> So how does one arrive at the estimations? >> >> Through inference rules from "fuzzy facts." > > Here's an inference for you from "fuzzy facts" courtesy of Edgington ;) > > Let x, y, z be three balls that an 'expert' determined to be red to some > degree and small to some other degree: > > v(Red(x)) = 1 v(Small(x)) = 0.5 > v(Red(y)) = 0.5 v(Small(y)) = 0.5 > v(Red(z)) = 0.5 v(Small(z)) = 0 > > Now using Zadeh blessed definition for 'and' as min(x, y), we'll get the > conclusion that all the balls are equally red and small ! 'Red and > Small' equals 0.5 in all the cases which clearly contradicts the > intuition that x being red and small has to have a higher degree of truth > than y, and z has to have the lowest. 1. For z it is 0. 2. Why x should be more of Red/\Small than y? This is all about the set measure, which determines how the conditional (Red/\Small | x) were related to (Red | x) and (Small | x). Zadeh system is obtained when the membership function of a set A in the element x, A(x) were defined as pos(A|{x}). >>>>> No, a implies b equals 1 if a in {0, _|_} and b otherwise: >>>>> >>>>> | b T 0 1 _|_ >>>>> a | >>>>> ---+----------------- >>>>> T | T 0 1 _|_ >>>>> 0 | 1 1 1 1 >>>>> 1 | T 0 1 _|_ >>>>> _|_| 1 1 1 1 >>>>> >>>>> T implies T evaluates to T which prevents explosion with any >>>>> arbitrary formula. >>>> >>>> I see. >>>> >>>> [...] >>>>> So the idea with defining the implication is to prevent explosion >>>>> which is ensured by T a> T evaluating to T, and the rest of the >>>>> table is cooked so that MP would work. >>>> >>>> But it does not! >>>> >>>> (A /\ (A a> B)) a> B >>>> >>>> evaluates T in A=T, B=T and in A=1, B=T. >>> >>> But that's OK because T being a designated truth value means that the >>> formula holds ("has a model") ! >> >> T is "neither," it is "closer" to 0. It seems that it actually was: >> >> a>| T 0 1 _|_ >> ---+----------------- >> T | 1 1 1 1 >> 0 | 1 1 1 1 >> 1 | T 0 1 _|_ >> _|_| T 0 1 _|_ >> > No it was not, it was exacly as I specified. > >> Usually inference should be made only under certain truth. I.e. when x >> a> y does not evaluate 1, then x|=y is wrong. > > If you think so, then you've just destroyed the whole area of > paraconsistent logic, perhaps deservedly, but that's another question > ;) To treat 'T' as a designated truth value is exactly what various > paraconsistent logics do to avoid explosion. Hmm, without going into philosophical issues about merits of contradictory inference (not to be mixed with inference from contradiction), but purely technically, less inference paths you take, smaller is the set of consequences. So inference under certainty cannot explode more than one under certainty + contradiction. -- Regards, Dmitry A. Kazakov http://www.dmitry-kazakov.de |