Re: Fuzzy proofs.

From: Pj (pgroundwater@virgin.net)
Date: Tue May 22 2001 - 13:29:43 MET DST

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    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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