# 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},
> > 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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