Re: Fuzzy proofs.

From: Ulrich Bodenhofer (ulrich.bodenhofer@scch.at)
Date: Thu May 31 2001 - 12:50:14 MET DST

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    It does not make a difference whether you consider tautologies with respect
    to single truth values or membership functions over some non-empty universe
    X,
    since "standard" fuzzy logic assumes truth functionality. In particular,
    this means
    that you may check whether a formula is a tautology (or that an equality
    holds)
    simply by checking the truth values assumed by the membership functions for
    each x in X independently.

    Best regards,
    Ulrich

    "Pj" <pgroundwater@virgin.net> wrote in message
    news:ejcO6.8552$m93.1157038@news6-win.server.ntlworld.com...
    > Thanks for your help. I understand what you are saying but I am looking
    for
    > proofs of said tautologies in terms of membership functions. For example
    > using the set operators:
    >
    > min(UB(x) - UA(x))
    >
    > I need to see examples in this form so I can apply them to whatever
    > tautology may arise in an exam.
    >
    > Thanks again.
    >
    >
    > Ulrich Bodenhofer <ulrich.bodenhofer@scch.at> wrote in message
    > news:3b08c352@alijku02.edvz.uni-linz.ac.at...
    > > Hm, in any case you have to be aware which kind of fuzzy logic you are
    > > assuming.
    > > There is NOT a single unique kind of fuzzy logic. There are infinitely
    > many
    > > ways
    > > to define the three connectives /\, \/, and =>. Note that fuzzy logics
    are
    > > not even
    > > limited to the unit interval or a linearly ordered domain of truth
    values.
    > >
    > > A/\T=A: This equivalence holds in all settings that are considered as
    > > meaningful,
    > > in particular, in the frameworks of triangular norms on
    the
    > > unit interval,
    > > GL-monoids (a general algebraic structure, the "standard
    > > case" of which
    > > are left-continuous t-norms), and BL-algebras (a general
    > > algebraic structure,
    > > the "standard case" of which are continuous t-norms).
    > >
    > > A\/(B\/C) = (A\/B) \/ C: The law of associativity for a kind of
    > generalized
    > > disjunction
    > > is fulfilled in all practically
    > > relevant logical systems.
    > > Note that it holds for general
    > > triangular conorms and for
    > > the lattice join which is used
    in
    > > GL-monoids and BL-algebras
    > > to model a kind of weak
    > disjunction.
    > >
    > > (A/\(A=>B)) => B: This law strongly depends on the very relationship
    > between
    > > the
    > > conjunction and the implication. In the
    > > t-norm-based setting, this law
    > > can only be guaranteed if => is the
    > residual
    > > implication of /\.
    > > In the settings of GL-monoids and
    > > BL-algebras, this correspondence
    > > is assumed by default (forcing the
    > residual
    > > implication in the unit-interval-
    > > based special case).
    > >
    > > I would like to recommend the following literature:
    > >
    > > @book{Gottwald:01,
    > > author = {S. Gottwald},
    > > title = {A Treatise on Many-Valued Logics},
    > > publisher = {Research Studies Press},
    > > address = {Baldock},
    > > year = {2001},
    > > series = {Studies in Logic and Computation}
    > > }
    > >
    > > @book{Hajek:98,
    > > author = {P. H\'ajek},
    > > title = {Metamathematics of Fuzzy Logic},
    > > publisher = {Kluwer Academic Publishers},
    > > volume = {4},
    > > series = {Trends in Logic},
    > > address = {Dordrecht},
    > > year = {1998}
    > > }
    > >
    > > @book{NovakPerfilievaMockor:99,
    > > author = {V. Nov\'ak and I. Perfilieva and J. Mo\v{c}ko\v{r}},
    > > title = {Mathematical Principles of Fuzzy Logic},
    > > publisher = {Kluwer Academic Publishers},
    > > address = {Boston},
    > > year = {1999}
    > > }
    > >
    > > Best regards,
    > > Ulrich
    > >
    > >
    > > "Groundy" <groundy@ukgateway.net> wrote in message
    > > news:NpRN6.6431$yA4.1129509@news2-win.server.ntlworld.com...
    > > > To help with my artificial intelligence exam revision I am looking for
    > > fuzzy
    > > > proofs of the following laws,
    > > >
    > > > A/\T=A
    > > > A\/(B\/C) = (A\/B) \/ C
    > > > MODUS PONENS
    > > >
    > > > Any help would be greatly appreciated
    > > > Paul.
    > > >
    > > >
    > >
    > >
    >
    >

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