# 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},
> 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},
> 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},
> 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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