[FOM] Fwd: Fuzziness

Tue Jun 28 17:49:37 EDT 2016

---------- Forwarded message ----------
From: Libor Behounek <Libor.Behounek at osu.cz>
Subject: Re: [FOM] Fuzziness

=== In reply to Harvey Friedman's FOM post on Fuzziness ===
>
> On Tue Jun 21 2016 at 21:07, Harvey Friedman <hmflogic at gmail.com>
> wrote:
> > Except that the very idea of doing "fuzzy logic" is deeply
> > foundational, and it could be of great interest for the FOM to see a
> > modern account of the basics of fuzzy logic and fuzzy mathematics and
> > fuzziness from the ground up. it would seem likely that f.o.m. people
> > who have not been informed much about it would have something
> > interesting and/or useful to say about it.
>
> I believe that some of the modern developments and problems
> in formal fuzzy logic and logic-based fuzzy mathematics might
> indeed be of a (moderate) interest for the f.o.m. community.
> Let me list just a few examples:
>
> (i) Skolem conjectured that a large part of contemporary
> mathematics could be reconstructed in naive set theory
> over Lukasiewicz logic (Russell's paradox is blocked in
> Lukasiewicz logic by the absence of the derivation rule
> of contraction). Hajek studied the theory in several
> papers; his results are rather discouraging as regards
> Skolem's conjecture, but the theory itself seems worth
> investigating. In particular, the consistency (relative
> to ZF) of naive set theory over Lukasiewicz first-order
> logic has been an open problem for almost 60 years.
>
> To be specific, first-order Lukasiewicz logic is axiomatized
> by the following axiom schemata:
>   A -> (B -> A)
>   (A -> B) -> ((B -> C) -> (A -> C))
>   (~A -> ~B) -> (B -> A)
>   ((A -> B) -> B) -> ((B -> A) -> A)
>   (forall x)P(x) -> P(t)
>   P(t) -> (exists x)P(x)
>   (forall x)(Q -> P) -> (Q -> (forall x)P)
>   (forall x)(P -> Q) -> ((exists x)P -> Q)
> with the derivation rules of modus ponens and
> generalization,
>   A, A -> B / B
>   P / (forall x)P
> where A,B,C,P,Q are first-order formulae, Q does
> not contain free x and t is substitutable for x
> in P. Naive set theory over this logic has the
> only binary predicate \in and the only axiom
> schema of unrestricted comprehension,
>   (exists z)(forall x)(x \in z <-> P)
> for all formulae P in its language.
>
> Unrestricted comprehension is known to be consistent
> over some closely related contraction-free logics,
> such as linear logic, and to be inconsistent with
> extensionality in fuzzy logics. More details on this
> open problem and references to related works and some
> partial results (by Skolem, Chang, and others) can be
> found in section 5 of this paper:
>   http://mujweb.cz/behounek/logic/papers/bh-stafl.pdf
>   (preprint; the published version appeared in F. Montagna,
>   ed.: Petr Hajek on Mathematical Fuzzy Logic, pp. 63-89.)
>
> The lack of the contraction rule in many fuzzy logics
> furthermore enables some constructions which are
> inconsistent classically: e.g., Hajek, Paris and
> Shepherdson showed that it is possible to define
> a fuzzy truth predicate for crisp Peano arithmetic
> (it excludes standard models of PA, though). A survey
> of Hajek's results of this kind can be found in
> section 4 of the paper linked above.
>
> (ii) An intriguing aspect of formal fuzzy mathematics
> is that we gain a certain conceptual simplification of
> mathematical notions in exchange for reasoning in a
> non-classical logic, as sketched in my short conference
> paper on "number-free mathematics", available at
>   http://mujweb.cz/behounek/logic/papers/eusflat09-behounek.pdf
>   (Proceedings of IFSA-EUSFLAT 2009, 449-454)
>
> In particular, since real numbers serve in formal fuzzy
> mathematics both as the real continuum and the set of
> intended truth values, real-valued real functions are
> simultaneously just propositional connectives. This
> gives the structure L of (formal) truth values a central
> position in logic-based fuzzy mathematics, comparable to
> that of natural numbers in intuitionistic mathematics.
>
> A consequence of this conceptual reduction is that in
> formal fuzzy logic, infinitesimal calculus can be based
> on a rather natural (classically inconsistent) fuzzy
> notion of infinitesimal. The limit of a sequence can
> then be defined by a Sigma_2-formula of fuzzy logic
> (unlike the Pi_3- and Pi_1-definitions of classical
> and non-standard analysis).
>
> (iii) Some initial parts of formal fuzzy mathematics
> (mainly the theory of fuzzy relations and some bits
> of fuzzy topology, fuzzy numbers, and set theory)
> have already been developed in formal fuzzy logic
> "from the ground up". Several papers related to this
> enterprise can be found at
>   http://www.mathfuzzlog.org/index.php/FCT_(project)
>
> While we usually use just Russell-style or Church-style
> type theory over fuzzy logic as our foundational theory,
> a ZF-style fuzzy set theory has also been studied a bit
> by Hajek and Hanikova (an overview can be found in
> section 3 of the survey of Hajek's results linked above).
> It might be the case (though it has never been explored)
> that these non-classical set theories can be useful for
> some independence results (as the ZF-style universe of
> fuzzy sets closely resembles Boolean models of classical
> set theory, just over certain non-Boolean algebras).
>
> (iv) As mentioned by Arnon Avron, a recent modern exposition
> of formal fuzzy logic can be found in the Handbook of
> Mathematical Fuzzy Logic, esp. the first two chapters of
> Volume I. Let me just add that these two chapters are freely
> available for download at
>
> http://www.mathfuzzlog.org/index.php/Handbook_of_Mathematical_Fuzzy_Logic
>
> A few more open problems in mathematical fuzzy logic can
> be found in the Handbook. For instance, some fuzzy logics
> are known to be decidable, but their computational complexity
> is as yet unknown; for the fundamental fuzzy logic MTL, the
> problem is open for more than 10 years.
>
> ===
> Libor Behounek
> University of Ostrava, Czech Republic
> libor.behounek at osu.cz
>
>
>
> >>> Harvey Friedman <hmflogic at gmail.com> 06/26/16 12:50 AM >>>
>
>
>
> Thank you for your reply, and it looks like this would make an
>
>
>
> informative FOM posting. The usual procedure is for the moderator to
>
>
>
> post it on your behalf. I have cc'ed the moderator, Martin Davis, to
>
>
>
> this email, so if you are willing to send a version of your email to
>
>
>
> him to post on the FOM, I think that would be quite appropriate for
>
>
>
> FOM.
>
>
>
>
>
>
>
> Harvey Friedman
>
>
>
>
>
>
>
> PS: You might consider explicitly presenting, in your possible FOM
>
>
>
> posting, the system that you mention whose consistency is still open.
>
>
>
> More people might look into it seriously if they get an immediate idea
>
>
>
> as to what it is like.
>
>
>
>
>
>
>
> On Sat, Jun 25, 2016 at 5:56 PM, Libor Behounek <Libor.Behounek at osu.cz>
> wrote:
>
>
>
> > Dear Professor Friedman
>
>
>
> > (CC to Arnon Avron and Vladik Kreinovich,
>
>
>
> > who replied to your FOM post),
>
>
>
> >
>
>
>
> > Thank you for your recent expression of interest in
>
>
>
> > foundational aspects of formal fuzzy logic on FOM.
>
>
>
> > I'm not a member of the mailing list, which is why
>
>
>
> > I bother you with a personal email reply. (I do read
>
>
>
> > FOM's archives ocassionally; this time, however, I
>
>
>
> > had to be alerted of your post by my colleagues).
>
>
>
> >
>
>
>
> > First regarding the second part of your post:
>
>
>
> >
>
>
>
> >> Also, how does it relate to recent breakthroughs in
>
>
>
> >> machine learning, deep leaning, etcetera?
>
>
>
> >
>
>
>
> > I'm afraid that results in fuzzy logic are unrelated
>
>
>
> > to these breakthroughs (as far as I know, they're
>
>
>
> > more related to the theory of neural networks and its
>
>
>
> > applications in pattern recognition; but I'm no expert
>
>
>
> > in this).
>
>
>
> >
>
>
>
> > Nevertheless, some of the modern developments in formal
>
>
>
> > fuzzy logic and logic-based fuzzy mathematics might
>
>
>
> > indeed be of a (moderate) interest for the FOM community;
>
>
>
> > I'll try to give some examples below.
>
>
>
> >
>
>
>
> > (i) Skolem conjectured that a large part of contemporary
>
>
>
> > mathematics could be reconstructed in naive set theory
>
>
>
> > over Lukasiewicz logic (Russell's paradox is blocked in
>
>
>
> > Lukasiewicz logic by the absence of the derivation rule
>
>
>
> > of contraction). The consistency status of unrestricted
>
>
>
> > comprehension in Lukasiewicz logic (or even weaker fuzzy
>
>
>
> > logics) is still unknown; but its (known) consistency
>
>
>
> > (relative to ZF) in closely related logics (e.g., linear
>
>
>
> > logic) gives hope that it might be consistent. Hajek
>
>
>
> > studied the theory in several papers; his results are
>
>
>
> > rather discouraging as regards Skolem's conjecture,
>
>
>
> > but the theory itself seems worth investigating (in
>
>
>
> > particular, its consistency is an open problem for
>
>
>
> > almost 60 years).
>
>
>
> >
>
>
>
> > The lack of the contraction rule in many fuzzy logics
>
>
>
> > furthermore enables some constructions which are
>
>
>
> > inconsistent classically: e.g., Hajek, Paris and
>
>
>
> > Shepherdson showed that it is possible to define
>
>
>
> > a fuzzy truth predicate for crisp Peano arithmetic
>
>
>
> > (it excludes standard models of PA, though).
>
>
>
> >
>
>
>
> > A survey of Hajek's results of this kind can be found
>
>
>
> > in this paper:
>
>
>
> > http://mujweb.cz/behounek/logic/papers/bh-stafl.pdf
>
>
>
> > (preprint; the published version is in F. Montagna, ed.:
>
>
>
> > Petr Hajek on Mathematical Fuzzy Logic, pp. 63-89.)
>
>
>
> >
>
>
>
> > (ii) An intriguing aspect of formal fuzzy mathematics
>
>
>
> > is that we gain a certain conceptual simplification of
>
>
>
> > mathematical notions in exchange for reasoning in a
>
>
>
> > non-classical logic. This has been hinted at in my
>
>
>
> > short conference paper on "number-free mathematics",
>
>
>
> > available at
>
>
>
> > http://mujweb.cz/behounek/logic/papers/eusflat09-behounek.pdf
>
>
>
> > (Proceedings of IFSA-EUSFLAT 2009, 449-454)
>
>
>
> >
>
>
>
> > In particular, since real numbers serve in formal fuzzy
>
>
>
> > mathematics both as the real continuum and the set of
>
>
>
> > intended truth values, real-valued real functions are
>
>
>
> > simultaneously just propositional connectives. This
>
>
>
> > gives the structure L of (formal) truth values a central
>
>
>
> > position in logic-based fuzzy mathematics, comparable to
>
>
>
> > that of natural numbers in intuitionistic mathematics.
>
>
>
> >
>
>
>
> > One consequence of this conceptual reduction is the fact
>
>
>
> > that in formal fuzzy logic, infinitesimal calculus can
>
>
>
> > be based on a rather natural (classically inconsistent)
>
>
>
> > fuzzy notion of infinitesimal. The limit of a sequence
>
>
>
> > can then be defined by a Sigma_2-formula of fuzzy logic
>
>
>
> > (unlike the Pi_3- and Pi_1-definitions of classical and
>
>
>
> > non-standard analysis).
>
>
>
> >
>
>
>
> > (iii) Some initial parts of formal fuzzy mathematics
>
>
>
> > (mainly the theory of fuzzy relations and some bits
>
>
>
> > of fuzzy topology, fuzzy numbers, and set theory)
>
>
>
> > have already been developed in formal fuzzy logic
>
>
>
> > "from the ground up" (to use your words). Some papers
>
>
>
> > related to this enterprise can be found at
>
>
>
> > http://www.mathfuzzlog.org/index.php/FCT_(project)
>
>
>
> >
>
>
>
> > While we usually use just Russell-style or Church-style
>
>
>
> > type theory over fuzzy logic as our foundational theory,
>
>
>
> > a ZF-style fuzzy set theory has also been studied a bit
>
>
>
> > by Hajek and Hanikova (an overview can be found in the
>
>
>
> > aforementioned survey of Hajek's results). While it has
>
>
>
> > never been explored, perhaps these non-classical theories
>
>
>
> > could be used for showing some independence results
>
>
>
> > (as the ZF-style universe of fuzzy sets closely resembles
>
>
>
> > Boolean models of classical set theory, just over non-Boolean
>
>
>
> > algebras).
>
>
>
> >
>
>
>
> > (iv) As mentioned by Arnon Avron, a recent modern exposition
>
>
>
> > of formal fuzzy logic can be found in the Handbook of
>
>
>
> > Mathematical Fuzzy Logic, esp. the first two chapters of
>
>
>
> > Volume I. Let me just add that these two chapters are freely
>
>
>
> > available for download at
>
>
>
> >
> http://www.mathfuzzlog.org/index.php/Handbook_of_Mathematical_Fuzzy_Logic
>
>
>
> >
>
>
>
> > I'm sorry for bothering you with such a long reply to your
>
>
>
> > brief post; hopefully some of it has been of interest for you.
>
>
>
> >
>
>
>
> > With best regards,
>
>
>
> > Libor Behounek
>
>
>
> >
>
>
>
> > Institute for Research and Applications of Fuzzy Modeling
>
>
>
> > University of Ostrava
>
>
>
> > Czech Republic
>
>
>
> >
>
>
>
> >
>
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