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   \begin{document}
   \title{Interpreting GUHA Data Mining Logic in Paraconsistent Fuzzy Logic Framework}
   \author{Esko Turunen}
   \institute{Tampere University of Technology, Finland \\
   \email{esko.turunen@tut.fi}}
   \maketitle


   {\bf Abstract:} A natural interpretation of GUHA style data mining logic in paraconsistent fuzzy logic framework is introduced.
    Significance of this interpretation is discussed.

   \vspace {\baselineskip}

   {\bf Key words:} Data mining, fuzzy logic, paraconsistent logic


   \section{Introduction}
    Classical Boolean logic is the logic of mathematics. In pure mathematical world things are binary: either a number is a prime number or is not,
    either a theorem is proved or not, {\it tertium non datur} is valid. Outside mathematics in the real world, in data analysis and decision
    making, however, applying Boolean logic causes anomalies: the law of the excluded middle is problematic, the use of classical quantifiers
    $\forall$ (for all) and $\exists$ (there exists) is clumsy and truth and falsehood need not to be each others complements. To overcome these problems several
    non--classical logics were born. In various many--valued logics such as mathematical fuzzy logic \cite{Turunen} the law of the excluded middle
    does not hold in general,  in GUHA data mining logic \cite{Hajek1} there are several non--classical quantifiers e.g.  'in most cases',
    'above average' etc, and in paraconsistent logic \cite{Belnap}, besides true or false, a statement can be unknown or contradictory, too.
    In this paper we explore the mutual relation of these non--classical logics. We show, in particular, how GUHA logic is related to
    paraconsistent fuzzy logic.


   \section{The GUHA method in data mining}
    GUHA - General Unary Hypotheses Automaton - introduced in \cite{Hajek1} and still developing,
    is a method of automatic generation of hypotheses based on empirical data, thus a method
    of data mining. GUHA is a kind of automated exploratory data analysis: it generates
    systematically hypotheses supported by the data.

    The GUHA method is based on well--defined first order monadic logic containing generalized
    quantifiers on finite models. A GUHA procedure generates statements on association
    between complex Boolean attributes. These attributes are constructed from the predicates
    corresponding to the columns of the data matrix.

    GUHA is primary suitable for exploratory analysis of large data. The processed data forms
    a rectangle matrix, where rows correspond to objects belonging to the sample and each
    column corresponds to one investigated variable. A typical data matrix processed by
    GUHA has hundreds or thousands of rows and tens of columns. Exploratory analysis
     means that there is no single specific hypothesis that should be tested by our data; rather,
     the aim is to get orientation in the domain of investigation, analyze the behavior of
     chosen variables, interactions among them etc. Such inquiry is not blind but directed by
     some general direction of research. GUHA is not suitable for testing a single hypothesis:
      routine packages are good for this.

     GUHA systematically creates all hypotheses interesting from the point of view of a given
     general problem and on the base of given data. This is the main principle: “all interesting
      hypotheses”. Clearly, this contains a dilemma: "all'' means most possible, "only
     interesting'' means "not too many''. To cope with this dilemma, one may use different
    GUHA procedures and, having selected one, by fixing in various ways its numerous  parameters.

    GUHA procedures not only hypotheses relating one variable with another one, but
    expressing relations among single variables, pairs, triples, quadruples of variables etc.
    GUHA offers hypotheses. Exploratory character implies that the hypotheses produced by
    the computer (numerous in number: typically tens or hundreds of hypotheses) are just
   supported by the data, not verified. You are assumed to use this offer as inspiration, and
   possibly select some few hypotheses for further testing.

    For a complete description of   the  GUHA method for data mining, see \cite{Hajek1},  \cite{HajekHavranek}.
   A software implementation of GUHA called LispMiner is available freely from http://lispminer.vse.cz/

   Since mathematical fuzzy logic (cf. \cite{Turunen}) and the GUHA method are both extension of classical Boolean logic
   and are related to vagueness and partial truth,  it is not a surprising news that there are several approaches
   to connect mathematical fuzzy logic to the   GUHA method. Here we mention Hole\v na
   who has introduced a fuzzy version of the GUHA method in \cite{Holena1,Holena2}.  Nov\'ak et al. show in
   \cite{Novak3} that, by evaluating real--valued data by linguistic expressions and then using the GUHA method,
   one obtains data mining outcomes that are easily understandable as they are close to human way of thinking.
   -- Such a target we, too, set when writing this paper. We aim to show that GUHA method has a natural interpretation
   in paraconsistent mathematical fuzzy logic which is our \cite{Turunen6} recent extension of Belnap's
  paraconsistent logic \cite{Belnap}.

   \section{Paraconsistent fuzzy logic}
    Quoting from Stanford Encyclopedia of Philosophy \cite{SEP}: {\it The contemporary logical orthodoxy has it that, from
    contradictory premises, anything can be inferred. To be more precise, let $\models$ be a relation of logical consequence,
    defined either semantically or proof--theoretically. Call $\models$ {\rm explosive} if it validates $\{ \alpha,\neg\alpha \} \models \beta$
     for every $\alpha$ and $\beta$ ({\rm ex contradictione quodlibet}). The contemporary orthodoxy, i.e., classical logic, is explosive,
    but also some non-classical logics such as intuitionist logic and most other standard logics are explosive.
    The major motivation behind paraconsistent logic is to challenge this orthodoxy. A logical consequence relation, $\models$, is said to
    be {\rm paraconsistent} if it is not explosive. Thus, if $\models$ is paraconsistent, then even if we are in certain circumstances
    where the available information is inconsistent, the inference relation does not explode into triviality. Thus, paraconsistent logic
    accommodates inconsistency in a sensible manner that treats inconsistent information as informative.}


    Four possible values associated with a statement (logic formula) $\Phi$ in Belnap's first order paraconsistent logic \cite{Belnap}
    are {\tt true}, {\tt false}, {\tt contradictory} and {\tt unknown}: if there is evidence for $\Phi$ and no
    evidence against $\Phi$, then $\Phi$ obtains the value {\tt true} and  if there is no evidence for $\Phi$
    and evidence against $\Phi$, then $\Phi$ obtains the value {\tt false}. A value {\tt contradictory}
    corresponds to a situation where there is simultaneously evidence for $\Phi$ and against $\Phi$ and, finally,
    $\Phi$ is labeled by value {\tt unknown} if there is no evidence for $\Phi$ nor evidence against $\Phi$.
    More formally, the values are associated with ordered couples $\langle 1, 0 \rangle$,  $\langle 0, 1 \rangle$,
    $\langle 1, 1 \rangle$ and  $\langle 0, 0 \rangle$, respectively.

   In \cite{Tsoukias} Tsoukias  introduced an extension of Belnap's logic (named DDT) most importantly because
   the corresponding algebra of Belnap's original logic is not a Boolean algebra, while the extension is. Indeed, in that paper it was
   introduced and defined the missing connectives in order to obtain a Boolean algebra. Moreover, it was explained
   why we get such a structure. Among others it was shown that negation, which was reintroduced
   in \cite{Tsoukias} in order to recover some well known tautologies in reasoning,  is not a complementation.

    In \cite{PernyTsoukias} and \cite{OzturkTsoukias}, a continuous valued extension of  DDT
    logic was studied. The authors imposed reasonable conditions this continuous valued extension should obey and,
    after a careful analysis, they came to the conclusion that  the graded values are to be computed via
   \begin{eqnarray}
     t(\Phi) = \mbox{min}\{\alpha, 1- \beta  \},          \label{t}\\
     k(\Phi) = \mbox{max}\{\alpha + \beta - 1, 0  \}, \label{k}\\
     u(\Phi) = \mbox{max}\{1- \alpha - \beta, 0 \},   \label{u}\\
     f(\Phi) = \mbox{min}\{1- \alpha, \beta \}.           \label{f}
   \end{eqnarray}
    where an ordered couple $\langle \alpha, \beta \rangle$, called {\it evidence couple}, is given. The intuitive meaning of  $\alpha$ and $\beta$
    is the degree of evidence of a statement $\Phi$ and against $\Phi$, respectively. Moreover, the set of $2 \times 2$ {\it evidence matrices} of a form
    \begin{center}
    $\left[ \begin{array}{lll}
            f(\Phi)      & k(\Phi)     \\
            u(\Phi)     & t(\Phi)
            \end{array}  \right]$
   \end{center}
    is denoted by $\cal M$. The values $ f(\Phi), k(\Phi), u(\Phi)$ and $t(\Phi)$ are values on the real unit interval $[0,1]$  such that
    $ f(\Phi)+k(\Phi)+u(\Phi)+t(\Phi) =1$.  Their   intuitive meaning is {\it falsehood, inconsistency, unknown} and {\it truth}, respectively, of the statement $\Phi$.
    One of the most important features of paraconsistent logic is that truth and falsehood are not each others complements.

    In \cite{PernyTsoukias} it is shown how such a fuzzy version of Belnap's logic can be applied in
    preference modeling. However, several open problems related to this fuzzy extension of paraconsistent logic were posed in \cite{OzturkTsoukias},
    most notably a need of a complete truth calculus and  a more thorough investigation of  the set  ${\cal M}$ and its algebraic structure.

      We continued Tsoukias' et al. work in \cite{Turunen6}.
     To our understanding, the algebraic structure of  ${\cal M}$ should {\it not} be a Boolean algebra.
     This opinion is based on our basic observation that the
      algebraic operations in (\ref{t}) -- (\ref{f}) are expressible by the Lukasiewicz $t$--norm and
      the corresponding residuum, i.e. in the Lukasiewicz structure, which is an example of an injective MV--algebra
      and is not, in general, a Boolean algebra.
      It is known \cite{Turunen} that Lukasiewicz--Pavelka style fuzzy sentential logic is a complete logic
      in a sense that if the truth value set {\bf L} forms an injective MV--algebra, then the set of $a$--tautologies
     and the set of $a$--provable formulae coincide for all $a \in {\bf L}$. We therefore considered the problem that, given a truth value
     set which is an injective MV--algebra, is it possible to transfer an injective MV--structure to the
     set $\cal M$, too. The answer turned out to be affirmative, consequently,  the corresponding paraconsistent
     sentential logic is essentially Pavelka style fuzzy logic. This means that having any set of injective MV-algebra valued evidence couples
     $\langle \alpha, \beta \rangle$, the structure of the evidence matrices
     \begin{eqnarray}
     \left[ \begin{array}{lll}
            \alpha^{\ast}\wedge \beta      & \alpha \odot \beta     \\
            \alpha^{\ast}\odot \beta^{\ast}     &  \alpha\wedge \beta^{\ast}
            \end{array}  \right] \label{matriisi}
    \end{eqnarray}
     forms an injective MV--algebra, too.
     Here the operations $\odot, \wedge$ and  $^{\ast}$ are the algebraic operations {\it product, meet}  and {\it complement},
     respectively, of the original injective MV--algebra. If, in particular, the original MV--algebra is the Lukasiewicz structure on
     the real unit interval, then $a \odot b = \mbox{max}\{0, a+b-1 \}$,  $a \wedge b = \mbox{min}\{a,b \}$, $a^{\ast} = 1-a$
     for all $a,b \in [0,1]$. Moreover, in MV--algebras there is an additional operation $\oplus$, in the Lukasiewicz structure
     it is defined by $a \oplus b = \mbox{min}\{1, a+b \}$, $a,b \in [0,1]$.

     Our result that continuous valued paraconsistent logic can be seen as a special case of Lukasiewicz--Pavelka style fuzzy logic
     has a consequence that a rich logical semantics and syntax is available. For example, all Lukasiewicz tautologies as well as
     Intuitionistic tautologies  can be expressed in the framework of this logic. This follows by the fact that we have
     two sorts of logical connectives conjunction, disjunction, implication and negation interpreted
     either by the monoidal operations $\bigodot, \bigoplus, \longrightarrow, ^{\ast}$ or by the
     lattice operations $\wedge, \vee, \Rightarrow, ^{\star}$, respectively (however, neither $^{\star}$ nor $^{\ast}$
     is a lattice complementation).  Besides, there are many other logical connectives available.

   \section{GUHA in paraconsistent logic framework}
    Assume we have a data file composed  of $k$ columns and $m$ rows, for example the following  fancied allergy matrix:

  \begin{center}
   \begin{tabular}{|c|c|c|c|c|c|} \hline
   Child    & Tomato    & Apple & Orange    & Cheese    & Milk \\  \hline
   Anna & 1 & 1 & 0 & 1 & 1  \\
   Aina & 1 & 1 & 1 & 0 & 0  \\
   Naima    & 1 & 1 & 1 & 1 & 1  \\
   Rauha    & 0 & 1 & 1 & 0 & 1  \\
   Kai  & 0 & 1 & 0 & 1 & 1  \\
   Kille    & 1 & 1 & 0 & 0 & 1  \\
   Lempi    & 0 & 1 & 1 & 1 & 1 \\
   Ville    & 1 & 0 & 0 & 0 & 0 \\
   Ulle & 1 & 1 & 0 & 1 & 1 \\
   Dulle    & 1 & 0 & 1 & 0 & 0 \\
   Dof  & 1 & 0 & 1 & 0 & 1 \\
   Kinge    & 0 & 1 & 1 & 0 & 1 \\
   Laade    & 0 & 1 & 0 & 1 & 1 \\
   Koff & 1 & 1 & 0 & 1 & 1 \\
   Olavi    & 0 & 1 & 1 & 1 & 1 \\  \hline
   \end{tabular}
  \end{center}
   The meaning of '0's and '1's is obvious: Anna for example is allergic to tomato, apple, cheese and milk and is not allergic to orange.
   Now consider two Boolean attributes $\phi$ and $\psi$ (in the above allergic matrix $\phi$ could mean 'child is allergic to tomato and
   apple' and $\psi$ could mean 'child is allergic to milk').  A {\it four--fold contingency table}
   $\langle a,b,c,d \rangle$ related to these attributes is composed from numbers of objects in the data
   satisfying four different binary combinations of these attributes:
   \begin{center}
   \begin{tabular}{r|c|c}
              & $\psi$ & $\neg\psi$ \\
   \hline
   $\phi$     &  $a$     & $b$ \\
   \hline
   $\neg\phi$ & $c$      & $d$
   \end{tabular}
   \end{center}
   where\\
   $\bullet$ $a$ is the number of objects satisfying both $\phi$ and $\psi$,\\
   $\bullet$ $b$ is the number of objects satisfying $\phi$ but not $\psi$,\\
   $\bullet$ $c$ is the number of objects not satisfying $\phi$ but satisfying $\psi$,\\
   $\bullet$ $d$ is the number of objects not satisfying $\phi$ nor $\psi$,\\
   $\bullet$ $m=a+b+c+d$.

   \vspace{0.1in}

   Various relations between $\phi$ and $\psi$ can be measured in the data by different {\it four--fold table quantifiers},
   denoted by $\phi \sim \psi$, $\sim (a,b,c,d)$, $\sim (a,b,c)$ or $\sim (a,b)$ depending on context,
   which here are understood as functions with values in the real unit interval $[0,1]$.
   Among the most well-known are the following two quantifiers:
    \paragraph{1. Basic implicational quantifier}
    A statement connecting two attributes $\phi$ and $\psi$ by {\it basic implicational quantifier} is
    defined to be true in a given data if
   \begin{eqnarray*}
    a \geq n \mbox{ and } \frac{a}{a+b} \geq p,
   \end{eqnarray*}
    where $n \in \cal N$ and $p \in [0,1]$ are
    parameters given by user.
   Notice that basic implicational quantifier is corresponding to association rules obtained by apriori algorithm intoduced by Agrawal et al. in 1993
   \cite{Agrawal}.  A fuzzy logic interpretation of this quantifier is the following

   \vspace{0.1in}
   {\it Given a data, the determining subset $A$ is formed of cases that satisfy $\phi$; there must be enough of them.
    The data supports a relation '$\phi$ implies $\psi$' if there are few cases in $A$ not satisfying $\psi$}.


    \vspace{0.1in}
    We recognize that a proposition {\it Cases in $A$ not satisfying $\psi$} has a truth value $\frac{b}{a+b}$
    which should be low, therefore, its complement -- in terms of Lukasiewicz logic -- should be large enough, i.e.


   \begin{eqnarray*}
   \left(\frac{b}{a+b}\right)^{\ast} = 1-\frac{b}{a+b} =\frac{a+b-b}{a+b} = \frac{a}{a+b} \geq p.
   \end{eqnarray*}


    \paragraph{2. Basic double implicational quantifier}
    A statement connecting two attributes $\phi$ and $\psi$ by {\it basic double implicational quantifier} is
    defined to be true in a given data if
   \begin{eqnarray*}
   a \geq n \mbox{ and } \frac{a}{a+b+c} \geq p,
   \end{eqnarray*}
    where $n \in \cal N$ and $p \in [0,1]$ are
    parameters given by user.

    A fuzzy logic interpretation of this quantifier is now the following

    \vspace{0.1in}
    {\it Given a data, the determining subset $A$ is formed of cases that satisfy $\phi$ or $\psi$;
     there must be enough cases satisfying both of them.
    The data supports a relation '$\phi$ implies $\psi$ and $\psi$ implies $\phi$' if there
    are few cases in $A$ not satisfying $\psi$ or few cases in $A$ not satisfying $\phi$}.

     \vspace{0.1in}
     It is easy to see that a proposition {\it Cases in $A$ not satisfying $\psi$} has a truth value $\frac{b}{a+b+c}$ and
     a proposition {\it Cases in $A$ not satisfying $\phi$} has a truth value $\frac{c}{a+b+c}$, therefore
     a proposition {\it Cases in $A$ not satisfying $\psi$ or not satisfying $\phi$} is related to a
     Lukasiewicz logic truth value $\frac{b}{a+b+c} \oplus \frac{c}{a+b+c}$ which should be low enough,
     therefore, its complement should be high enough, i.e.
     \begin{eqnarray*}
     \left(\frac{b}{a+b+c} \oplus \frac{c}{a+b+c}\right)^{\ast}
                     & = &\left( \frac{b}{a+b+c}\right)^{\ast} \odot \left( \frac{c}{a+b+c}\right)^{\ast} \\
                     & = &\left(1-\frac{b}{a+b+c}\right) \odot \left(1-\frac{c}{a+b+c}\right)  \\
                     & = &\mbox{max} \left\{ \left(1-\frac{b}{a+b+c}\right) + \left(1-\frac{c}{a+b+c}\right) - 1, 0\right\} \\
                     & = & \frac{a+b+c-b-c}{a+b+c}\\
                     & = & \frac{a}{a+b+c} \geq p.
    \end{eqnarray*}
    These two examples introduced in \cite{Ylirinne} show that GUHA data mining logic is related to Lukasiewicz--Pavelka logic.
    However, there are several other quantifiers in LispMiner software implementation that have rather a statistic than logic character.


   Our novel observation is that a value $\alpha = \frac{a}{m}$ can be seen as the degree that  $\phi$ and $\psi$ occur simultaneously,
   a value $\beta = \frac{b+c}{m}$ can be seen as the degree that  $\phi$ and $\psi$ do not occur simultaneously
   and a value $\frac{d}{m}$ the degree that $\phi$ and $\psi$ do not occur at all -- a kind of indifferent situation.
   Recalling $m = a+b+c+d$ and using the Lukasiewicz operations (equations (\ref{t}) -- (\ref{f}))  it is easy to see that
   \begin{center}
   \begin{tabular}{rcl}
   $\alpha^{\ast}\wedge \beta$ & = & $\mbox{min}\{1- \frac{a}{m}, \frac{b+c}{m} \} = \frac{b+c}{m} = \beta$,\\
   $\alpha \odot \beta  $ & = &  $\mbox{max}\{0, \frac{a}{m}+ \frac{b+c}{m}-1 \} = 0$,\\
   $\alpha^{\ast}\odot \beta^{\ast} $ & = &  $\mbox{max}\{0, 1-\frac{a}{m}+1- \frac{b+c}{m}-1 \} =  \frac{d}{m}$, \\
   $\alpha\wedge \beta^{\ast} $ & = & $\mbox{min}\{ \frac{a}{m}, 1- \frac{b+c}{m} \} = \frac{a}{m}= \alpha$.\\
   \end{tabular}
   \end{center}
   Therefore $\langle \alpha, \beta \rangle= \langle \frac{a}{m}, \frac{b+c}{m} \rangle$ can be seen as an evidence couple for a statement $\Phi$:
   '$\phi$ and $\psi$ occur simultaneously'. The correspondent evidence matrix is then
   \begin{center}
    $\left[ \begin{array}{lll}
            f(\Phi)      & k(\Phi)     \\
            u(\Phi)     & t(\Phi)
            \end{array}  \right] =
     \left[ \begin{array}{lll}
            \beta      & 0     \\
            \frac{d}{m}         & \alpha
            \end{array}  \right]  $.
   \end{center}
     In practical data mining tasks run by LispMiner it often happens that 'indifferent cases rule over interesting cases', that is to say, value $d$
     in a four--fold contingency table is much bigger that values $a, b, c$. However, even in such cases it is useful to look for statements $\Phi$
     such that the truth value of $\Phi$ is, say at least $k$ times bigger than the falsehood of $\Phi$, i.e. $\alpha \geq k \beta$, which is equivalent to
     $a \geq k(b+c)$. On the other hand such a statement $\Phi$ is stamped by label {\tt true} if
      \begin{center}
       $\frac{a}{a+b+c} \geq p$ iff $a \geq p(a+b+c)$ iff $a(1-p) \geq p(b+c)$ iff $a \geq \frac{p}{1-p}(b+c)$.
    \end{center}
     This means that $k= \frac{p}{1-p}$, $p \neq 1$, or equivalently $p = \frac{k}{k+1}$.   We have proved
     \begin{theorem}
     Given a data, all statements $\Phi$ such that the truth value of $\Phi$ is at least $k$ times
     bigger than the falsehood of $\Phi$ in the sense of paraconsistent logic,  can be found by the basic double implicational quantifier and
     setting $p = \frac{k}{k+1}$.
     \end{theorem}


    \subsubsection*{Examples}
     Consider the above data about children's allergies. \\
     (a) Let $\phi$ stand for 'child is allergic to tomato and
     apple' and $\psi$ stand for 'child is allergic to milk'. Compute the corresponding contingency table, the evidence
    couple and the evidence matrix for a statement $\Phi$: '$\phi$ and $\psi$ occur simultaneously'.

    \vspace{0.1in}
    Solution. First write the corresponding table where the connective '$\&$' is interpreted as a Boolean conjunction.
  \begin{center}
   \begin{tabular}{|c|c|c|} \hline
   Child    & Tomato    $\&$ Apple  & Milk \\  \hline
   Anna & 1 & 1  \\
   Aina & 1 & 0  \\
   Naima    & 1 &  1  \\
   Rauha    & 0 & 1  \\
   Kai  & 0 & 1  \\
   Kille    & 1 &  1  \\
   Lempi    & 0 & 1 \\
   Ville    & 0 & 0 \\
   Ulle & 1 & 1 \\
   Dulle    & 0 & 0 \\
   Dof  & 0 & 1 \\
   Kinge    & 0 &1 \\
   Laade    & 0 & 1 \\
   Koff & 1 & 1 \\
   Olavi    & 0 & 1 \\  \hline
   \end{tabular}
  \end{center}
   This leads to
   \begin{center}
   \begin{tabular}{r|c|c}
              & $\psi$ & $\neg\psi$ \\
   \hline
   $\phi$     &  $5$     & $1$ \\
   \hline
   $\neg\phi$ & $7$      & $2$
   \end{tabular}
   \end{center}
   Thus, the evidence couple is $\langle \frac{5}{15}, \frac{7+1}{15} \rangle$ and the correspondent evidence matrix is
   \begin{center}
    $\left[ \begin{array}{lll}
            f(\Phi)      & k(\Phi)     \\
            u(\Phi)     & t(\Phi)
            \end{array}  \right] =
     \left[ \begin{array}{lll}
            \frac{8}{15}      & 0     \\
            \frac{2}{15}         & \frac{5}{15}
            \end{array}  \right]$
   \end{center}
    Since $f(\Phi)$, the degree of falsehood of $\Phi$, is larger that $t(\Phi)$, the degree of truth of $\Phi$, we
      conclude that the given data does not support the statement that childen who are allergic to tomato and
     apple are simultaneously allergic to milk, too.

     \vspace{0.1in}
        (b) Let $\phi$ stand for 'child is allergic to cheese' and $\psi$ stand for 'child is allergic to milk'.
    Compute the corresponding contingency table, the evidence  couple and the evidence matrix for the statement $\Phi$: '$\phi$ and $\psi$ occur simultaneously'.

     \vspace{0.1in}
     Solution. From the original data matrix we get the following contingency table
   \begin{center}
   \begin{tabular}{r|c|c}
                              & Milk     & $\neg$Milk \\
   \hline
   Cheese               &  $8$     & $0$ \\
   \hline
   $\neg$Cheese     & $4$      & $3$
   \end{tabular}
  \end{center}
   Thus, the evidence couple is $\langle \frac{8}{15}, \frac{4+0}{15} \rangle$, and the correspondent evidence matrix is
   \begin{center}
    $\left[ \begin{array}{lll}
            f(\Phi)      & k(\Phi)     \\
            u(\Phi)     & t(\Phi)
            \end{array}  \right] =
     \left[ \begin{array}{lll}
            \frac{4}{15}      & 0     \\
            \frac{3}{15}         & \frac{8}{15}
            \end{array}  \right]$
   \end{center}
    We conclude that, based on the given data, the paraconsistent truth value of the statement $\Phi$:
   'cheese allergy and milk allergy occur simultaneously' is two times
    bigger than the paraconsistent falsehood of $\Phi$ and,  thus, the data supports $\Phi$.


   \section {Conclusion and future work}
    Various non--classical logics are useful in data analysis and algorithmic decision making.
   We have deepened the connection between fuzzy logic, paraconsistent logic and GUHA data mining logic.
    Basic implicational quantifier and  basic double implicational quantifier, for example, have a natural interpretation in
   Lukasiewicz--Pavelka fuzzy logic. In this study we have shown that paraconsistent logic, too, has an interpretation
   and connection to GUHA data mining logic; indeed, first we proved that paraconsistent degrees of  truth and falsehood,
   which are not mutually each other complements, of  simultaneous occurrence of two GUHA attributes can be calculated by
   basic double implicational quantifier, and then  we gave an example showing how to use this connection to express
   more sophisticated relationships between properties that could be possible by classical logic or even standard fuzzy logic.
   Lukasiewicz--Pavelka fuzzy logic as well as paraconsistent logic and
   GUHA logic are sound and theoretically well established, widely studied and acknowledged non--classical logics. Therefore
   all connections between these approaches clear the road for future investigation into better understanding of
   data analysis, knowledge extraction and decision theory.

   \begin{thebibliography}{9}
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    Mining Association Rules between Sets of Items in Large Databases. SIGMOD. June 1993, 22(2), 207 -- 16.
    \bibitem {Belnap} Belnap, N.D.: A useful four--valued logic, in Epstein, G. and Dumme, J. (eds.)
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