looking for a predicate hierarchy
in [OOP]
|
From: V.J. Kumar on 22 Dec 2006 12:59 "Dmitry A. Kazakov" <mailbox(a)dmitry-kazakov.de> wrote in news:jctw61m4fitf.or6bzp94wggz.dlg(a)40tude.net: > On Fri, 22 Dec 2006 12:40:46 +0100 (CET), V.J. Kumar wrote: > >> "Dmitry A. Kazakov" <mailbox(a)dmitry-kazakov.de> wrote in >> news:1lepbpsij9lm3$.13c2j961bhgkr.dlg(a)40tude.net: >> >>> On Wed, 20 Dec 2006 20:44:22 +0100 (CET), V.J. Kumar wrote: >>> >>>> "Dmitry A. Kazakov" <mailbox(a)dmitry-kazakov.de> wrote in >>>> news:1iraa1mnvtcji.oh5bsnrjjcdw$.dlg(a)40tude.net: >>>> >>>>> On Tue, 19 Dec 2006 21:20:53 +0100 (CET), V.J. Kumar wrote: >>>>> >>>>>> "Dmitry A. Kazakov" <mailbox(a)dmitry-kazakov.de> wrote in >>>>>> news:12cnousl5msxh.1anmyqm356hwb$.dlg(a)40tude.net: >>>>>> >>>>>>> (in logic "uncertain" is usually denoted as _|_, flipped T) >>>>>> >>>>>> In what logic ? >>>>> >>>>> Ah, there are so many. Even for a tri-state logic one could take >>>>> "contradictory" T instead of "uncertain" _|_ as the third element. >>>>> >>>>>> Without implication, your three-valued logic is not fully >>>>>> specified. >>>>> >>>>> Right. That depends on the definition of implication. (not x) V y is >>>>> well defined in tri-state logic because not _|_ = _|_. But it would >>>>> be a bad implication to take. >>>> >>>> It the Kleene logic implication. Whether it's "good" or "bad " >>>> surely depends on the application of such logic. >>> >>> For Booleans not and ~ are equivalent. >>> >>>>> A better one is ~xVy, where ~_|_=T. That is >>>>> not closed in tri-state logic. It is in four-state Belnap logic: >>>>> >>>>> x y x=>y >>>>> ------------------ >>>>> 0 0 1 >>>>> 0 1 1 >>>>> 0 _|_ 1 >>>>> 1 0 0 >>>>> 1 1 1 >>>>> 1 _|_ _|_ >>>>> _|_ 0 T >>>>> _|_ 1 1 >>>>> _|__|_ 1 >>>>> >>>> >>>> What is the truth table for ~ ? >>> >>> x ~x >>> --------- >>> 0 1 >>> 1 0 >>> _|_ T >>> T _|_ >> >> If this is the case, your logic is trivializable becaus it has formulas >> that do not have a model, and in the 4-valued logic all the formulas >> are supposed to have models. Consider for example ^(A=>A) for any >> valuation where ^ is the ordinary negation (0->1, 1->0, _|_->_|_, T-> T). > > not(A=>A) is false for any A. So what? Formulas with your implication potentially cannot handle contradiction, that's what. The whole point of having 'T' as a designated truth value is to allow models for expressions like (F /\ ^F). Now, with your implication it's no longer possible in the general case. In other words, it's easy to see that <FOUR, \/, /\, ^> has a model for every formula, whilst your <FOUR, \/, /\, ~, =>) clearly does not. It is not surprizing: the nasty property of admitting empty models is inherited from the '~' connective that you liked so much ! >
From: V.J. Kumar on 22 Dec 2006 13:08 "Dmitry A. Kazakov" <mailbox(a)dmitry-kazakov.de> wrote in news:jctw61m4fitf.or6bzp94wggz.dlg(a)40tude.net: > > On Fri, 22 Dec 2006 12:40:46 +0100 (CET), V.J. Kumar wrote: >> If this is the case, your logic is trivializable because it has >> formulas that do not have a model, and in the 4-valued logic all the >> formulas are supposed to have models. Consider for example ^(A=>A) >> for any valuation where ^ is the ordinary negation (0->1, 1->0, >> _|_->_|_, T->T). > > not(A=>A) is false for any A. So what? > WIth your implication connective, the modified logic cannot potentially handle contradiction with formulas using the connective, that's what. Originally, every formula in the logic had a model, with the implication it's no longer the case. It's not surprising because the defect is inherited from the '~' connective ! The whole point if using 'b' as a designated truth value was to block the explosion principle/avoid trivialization.
From: Dmitry A. Kazakov on 23 Dec 2006 05:32 [ Sorry guys, for it became a shameless off-topic in comp.object ] On Fri, 22 Dec 2006 18:59:17 +0100 (CET), V.J. Kumar wrote: > "Dmitry A. Kazakov" <mailbox(a)dmitry-kazakov.de> wrote in > news:jctw61m4fitf.or6bzp94wggz.dlg(a)40tude.net: > >> On Fri, 22 Dec 2006 12:40:46 +0100 (CET), V.J. Kumar wrote: >>> If this is the case, your logic is trivializable becaus it has formulas >>> that do not have a model, and in the 4-valued logic all the formulas >>> are supposed to have models. Consider for example ^(A=>A) for any >>> valuation where ^ is the ordinary negation (0->1, 1->0, _|_->_|_, T-> T). >> >> not(A=>A) is false for any A. So what? > > Formulas with your implication potentially cannot handle contradiction, > that's what. The whole point of having 'T' as a designated truth value > is to allow models for expressions like (F /\ ^F). You forgot that everything is in the inference rules. Yes, a contradiction cannot be constructed from 0 and 1 using /\ (AND), or any conventional logic operators. This was a *desired* property, that 1 /\ 0 = 0, 1 V 0 = 1, after all. That alone does not make it trivial, because the contradiction and uncertainty can still be produced. For example by operations like consensus(+) and gullibility(*): 1 + 0 = _|_ 1 * 0 = T for further information see: http://www.dmitry-kazakov.de/ada/fuzzy.htm#fuzzy_proposition -- Regards, Dmitry A. Kazakov http://www.dmitry-kazakov.de
From: V.J. Kumar on 23 Dec 2006 13:38 "Dmitry A. Kazakov" <mailbox(a)dmitry-kazakov.de> wrote in news:o68o7slu8pj7$.1cpgy64c622lt$.dlg(a)40tude.net: >> Formulas with your implication potentially cannot handle >> contradiction, that's what. The whole point of having 'T' as a >> designated truth value is to allow models for expressions like (F /\ >> ^F). > > You forgot that everything is in the inference rules. Not necessarily. The consequence relation { |- ) can be defined either semantically (e.g. with truth tables for the connectives) or syntactically (axioms and inference rules). For a logic to be able to handle contradictions like (F /\ ~ F) for example, the consequence relation F,~F|- P should fail for some P (the non-explosion principle) otherwise the logic becomes trivial by the same semantic argument as for the classical logic. With your '~' connective it's just the case (trivialization) whilst it is not with the '^' connective defined as 0->1, 1->0, _|_->_|_, T->T. Now, are you claiming that you can produce a system of axioms and inference rules that would show that the 4-valued logic with your "~" is not trivial ? >Yes, a > contradiction cannot be constructed from 0 and 1 using /\ (AND), or > any conventional logic operators. This was a *desired* property, that > 1 /\ 0 = 0, 1 V 0 = 1, after all. That alone does not make it trivial, > because the contradiction and uncertainty can still be produced. For > example by operations like consensus(+) and gullibility(*): > > 1 + 0 = _|_ 1 * 0 = T ??? What has it got to do with the price of fish ? What make the 4-valued logic trivial is your "~" because as was said before, (F /\ ~F) has an empty model (there is no valuation v(F) such that v(F /\ ~F) would be in the designated truth set {t, T}) so any arbitrary P vacuously is a semantic consequence of {F,~F}. > > for further information see: > > http://www.dmitry-kazakov.de/ada/fuzzy.htm#fuzzy_proposition > We'll tackle the fuzzy stuff after we've done with the simpler four-valued case first ;)
From: Dmitry A. Kazakov on 24 Dec 2006 05:33
On Sat, 23 Dec 2006 19:38:48 +0100 (CET), V.J. Kumar wrote: > "Dmitry A. Kazakov" <mailbox(a)dmitry-kazakov.de> wrote in > news:o68o7slu8pj7$.1cpgy64c622lt$.dlg(a)40tude.net: > >>> Formulas with your implication potentially cannot handle >>> contradiction, that's what. The whole point of having 'T' as a >>> designated truth value is to allow models for expressions like (F /\ >>> ^F). >> >> You forgot that everything is in the inference rules. > > Not necessarily. Necessarily. The properties of a logical system are determined by the inference rules. Your argumentation seems to be based on an assumption that (A /\ not A) => B were somehow equivalent to { A, not A } |= B but that's wrong. Firstly the former by no means implies the latter. Secondly the former is not trivially true: A (A /\ not A) => B ------------------------------ 0 1 1 1 _|_ 1 if B=1 or B=_|_, otherwise T T 1 if B=1 or B=T, otherwise _|_ > The consequence relation { |- ) can be defined either > semantically (e.g. with truth tables for the connectives) or syntactically > (axioms and inference rules). > > For a logic to be able to handle contradictions like (F /\ ~ F) for > example, This is not considered as a contradiction. Contradiction out of certain evidences {0,1} is not constructed with either V or /\. >>Yes, a >> contradiction cannot be constructed from 0 and 1 using /\ (AND), or >> any conventional logic operators. This was a *desired* property, that >> 1 /\ 0 = 0, 1 V 0 = 1, after all. That alone does not make it trivial, >> because the contradiction and uncertainty can still be produced. For >> example by operations like consensus(+) and gullibility(*): >> >> 1 + 0 = _|_ 1 * 0 = T > > ??? What has it got to do with the price of fish ? What make the 4-valued > logic trivial is your "~" because as was said before, Note, that ~ is not used in place of negation. This is again a jumpy assumption. For negation "not" is still used. The idea behind this x=>y was to define it in terms a set inclusion over subsets {0,1}^2. The background is to make it compatible with a possibility measure, for further extension into intuitionistic fuzzy logic and interpretation of x=>y as a conditional y|x. Merry Christmas, -- Regards, Dmitry A. Kazakov http://www.dmitry-kazakov.de |