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\title{Fuzzy Interval-Valued Inference System with Para-Consistent 
 and Grey Set Extensions}

\author{Ladislav J. Kohout}


\pagestyle{myheadings}

\markright{APIC'95, El Paso,
Extended Abstracts,
A Supplement to the international journal of {\rm Reliable
Computing}\ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ }

\maketitle

\auffil{The author is with 
Department of Computer Science, 
Florida State University, Tallahassee, Florida 32306, USA.}

Many-valued logic based interval  reasoning plays increasingly 
important role in fuzzy and other many-valued extensions of traditional
two-valued logic.
It is often overlooked that many-valued logic reasoning has richer
inference rule 
base than the classical crisp logics. Indeed some rules that are not
possible in a point-based crisp logic (in which truth values are
numbers) will have come to existence when we accept {\it intervals}
as truth values, i.e., as the basic 
elements of the semantic space, with point values as special cases (that
correspond to degenerate intervals). 
 
Interval logic also naturally arises when to determine 
a fuzzy truth value $a=t(A)$
(from the interval $[0,1]$), we ask a community of $n$ experts to vote,
and take $a=u_1/n$, where $u_1$ is the total number of agents who voted
in favor of $A$ being true. This fuzzy value can be thus viewed as an 
{\it approximation} to the (crisp) $n-$dimensional logical vector that 
represents the actual results of voting.
This interpretation of fuzzy values leads to the following
interpretation of connectives: if we have a logical connective $CON$
(e.g., $AND$, $OR$, etc.), then we can interpret $a\, CON\, b$ as follows:
\begin{itemize}
\item We find statements $A$ and $B$ for which the above procedure leads
to truth values $a$ and $b$, and
\item We interpret the (similarly determined) truth value 
of $A\,CON\,B$ as $a\,CON\,b$.
\end{itemize}
The problem with this natural definition is that 
for one and the same pair of values $(a,b)$, 
different choices of $A$ and $B$ are possible, and for these different
choices, the truth values of $A\,CON\,B$ may be different. As a result,
instead of a {\it single} truth value $a\,CON\,b$, this procedure leads
to an {\it interval} of possible truth values. The lower and upper
bounds of this interval will be denoted by $BOT$ and $TOP$. 
So, for each connective from two-valued logic, 
we now have a pair of connectives from a multi-valued logic 
(e.g., a pair of AND connectives, implication operators etc.) whose
values $CONBOT$ and $CONTOP$ 
form the natural bounds for $a\,CON\,b$ within $[0,1]$ valuation space. 
A formal semantics for such pairs that is derived by means of an 
exact mathematical method has been provided by the  {\bf 
checklist paradigm} \cite{80.2,84.5,85.5,86.8}.

Interplay of formal techniques and conceptual issues has to be examined 
carefully, if one attempts to extend the interval-based inference to include 
essential {\it paraconsistent} features \cite{91.17} (i.e., crudely
speaking, the possibility for a statement and for its negation both have
some degree of belief to be true). One of our aims is to 
analyze the epistemological and ontological status 
of some basic notions of checklist paradigm-based interval logic systems,
so that we can analyze not only the valid inference, but also semantic 
conflicts in fuzzy logics. This requires also a careful foundational
analysis of the 
systems involved. In such a foundational analysis we ought to distinguish 
sharply \cite{88.2}
\begin{itemize}
\item[(1)] mathematical questions, 
\item[(2)] logical questions, 
\item[(3)] ontological, epistemological and metaphysical questions,
\end{itemize}
when looking at the interrelationship of the pairs of connectives that 
form the interval semantic space. 

 In order to bring out the characteristic features of the metatheory of 
paraconsistency in approximate reasoning systems, we need to employ an 
adequate method of conceptual analysis. The purpose of this paper is to 
provide an up-to-date survey of some checklist paradigm techniques and 
extend the results in that direction. 


In addition to the theoretical analysis presented here, some
applications of fuzzy interval 
to knowledge representation will be discussed. The foundational concepts 
are needed as a basis for a sound methodology that could adequately 
trace not only the consistency of our requirements concerning the 
knowledge represented by formal structures, but also systematically 
examine the possible inherent contradictions in alternative designs of 
specific knowledge representation schemes. Once the contradictions and 
inconsistencies are discovered, one has to provide the adequate means 
for removing, or at least isolating and localizing, these. 

 
The interval inference that has been used in Activity Structures based 
Knowledge Processing Systems (such as knowledge based system {\sc Clinaid}) 
has been theoretically justified by the checklist paradigm. The
checklist paradigm 
generates {\it pairs of} distinct connectives of the same logical 
type that determine the end points of intervals, thus providing 
formally and epistemologically justified systems of {\it 
interval-valued} approximate inference. 
The most relevant further references are 
\cite{80.2,84.5,86.8,87.6,93.3}, each addressing 
a different aspect of the problem. 

As said above (and presented in detail elsewhere), an interval  fuzzy
membership 
function can be derived within the checklist  paradigm as an 
approximation of the crisp logical vector that represents a vote of a
community 
of some formal agents. Given a crisp Boolean vector that represent this vote 
(a ``fine structure"), a consensus summarization function for each
vector and a measure 
$m(F)$ operating on a pair of such vectors, a ``coarse structure" of
many valued pair of interval 
connectives can be generated. This coarse structure is captured
algebraically as an interval 
many-valued logic system, consisting of pairs of logical connectives
of the same logical 
type that is used to manipulate the interval valuations of logical sentences. 

 We have shown elsewhere (Bandler and Kohout
\cite{80.2,84.5,85.5,86.8})
that the {\it coarse} structure imposes bounds upon the fine 
structure, without determining it completely. Hence, {\it 
associated with the various logical connectives between 
propositions} {\bf are} {\it their extreme values} that represent the interval 
valuations of logical sentences. Thus we obtain the inequality
restricting the possible 
values of $m(F)$:
$$ CONTOP \geq m(F) \geq CONBOT,$$
where {\it CON} is the name of connective represented by 
$F$. So, there are 16 such inequalities, as there are 16 {\bf 
logical types} of {\it CON}, 10 of which are nontrivially two-argument.

Let us look now at some typical results. Choosing for the logical 
type of the connective {\it CON} the {\it 
implication} and making the assessment of the fuzzy value of the 
truth of a proposition by the formula (proposed, e.g., in \cite{86.8})
$$m_{1}(F)= 1- (u_{10}/n),$$ where $u_{10}$ is the number of experts who
voted ``for'' for the first statement $A$, and ``against'' the second
statement $B$, we obtain:
$$\min(1, 1-a+b) \geq m_{1}(PLY) \geq \max(1-a, b).$$
We can see that for the {\it CONTOP}, the {\it PLYTOP} was chosen, 
and the checklist paradigm produced the Lukasiewicz implication
operator, and the other bound ({\it PLYBOT}) is the Kleene-Dienes implication 
operator. 

 Choosing for {\it CON} the connective type {\it AND}, we get:
$$\min(a,b) \geq m_{1}(AND) \geq \max(0, a+b-1).$$

 Choosing for {\it CON} the connective type {\it OR}, we get:
$$\min(a+b-1, 1) \geq m_{1}(OR) \geq \max(a, b).$$
 
	We have shown \cite{84.5,86.8} that for a given 
contraction/approximation measure $m_{1}(F)$, there are 
16 inequalities linking the $TOP$ and $BOT$ types of connectives, 
as there are 16 logical types of connectives. Out of these 16 possible 
two-argument options, 10 are genuinely dependent on both arguments. The 
latter are of the following logical types: 
\begin{itemize}
\item {\it conjunction (a\,OR\,b)},
\item {\it disjunction (a\,AND\,b)},
\item {\it non-conjunction} ($\neg(a\,OR\,b)$),
\item {\it non-disjunction} ($\neg(a\,AND\,b)$),
\item {\it implications} ($a \rightarrow b$, $a \leftarrow b$), 
\item {\it non-implications} ($\neg(a \leftarrow b)$, $\neg(a \rightarrow b)$), 
\item {\it equivalence} ($a \equiv b$), 
\item {\it non-equivalence} ($a \bigoplus b$), 
also known as {\it exclusive or} ($a\,EOR\, b$).
\end{itemize}

The minimal set 
of necessary conditions that the individual logical types of 
connectives have to satisfy are:

\begin{itemize}
\item[1.] Duality of {\it AND} and {\it OR}.
\item[2.] A non-implication is obtained by negating an implication, and 
vice versa.
\end{itemize}

 These conditions give general logical constraints on the systems 
of connectives. Taking a specific constraint measure, one obtains more 
specific results. For example, $m_{1}$ (defined as above)
will yield 16 pairs $CONBOT \leq CONTOP$ specifying the end points of
the interval 
for all the possible logical types of connective, some of which are 
listed below:
\begin{itemize}
\item
For $\neg (a \rightarrow b)$, the valuations are 
$$\max(b-a) \leq \min(1-a, b).$$ 
\item For $\neg (a \leftarrow b)$, the valuations are
$$\max(a-b) \leq \min(a, 1-b).$$
\item For $a \equiv b$, the valuations are 
$$\max(a+b-1, 1-(a+b)) \leq \min(1-a+b, 1-b+a).$$
\item For $a\, EOR\, b$, the valuations are 
$$\max(a-b, b-a) \leq \min(a+b, 2-(a+b)).$$
\item For $\neg(a\, OR\, b)$, the valuations are
$$\max(0, 1-a-b) \leq \min(1-a,1-b).$$
\item For $\neg(a\, AND\, b)$, the valuations are
$$\max(1-a, 1-b) \leq \min(1, 2-a-b).$$
\end{itemize}
For the exhaustive listing of all 16 connectives see, e.g.,
\cite{84.5,86.8}.

The width of the interval produced by an application of a pair of associated 
connectives  (i.e. $TOP$ and $BOT$ connectives) characterizes the margins
of imprecision 
of an interval logic expression. The interval between the $TOP$
connective and the bottom 
connective is directly linked to the concept of fuzziness $\phi$. 
 If we define the {\it unnormalized fuzziness of x} (cf. Bandler 
and Kohout 1978 \cite{78.5}) as $\phi x = \min(x, 1-x)$, then for $x$ 
in the range 
$[0,1]$, $\phi x$ is in the range $[0, .5]$, with value $0$ if and 
only if x is {\it crisp}, and value $.5$ iff $x$ is $.5$. We have:
\smallskip

\noindent{\bf GAP THEOREM.} (Bandler and Kohout, 1986, \cite{86.8}):
$$a\, ANDTOP\, b - a\, ANDBOT\, b=$$
$$a\, ORTOP b - a\, ORBOT\, b=$$
$$a\, PLYTOP\, b - a\, PLYBOT\, b=\min(\phi a, \phi b).$$
\smallskip

$$a\, IFFTOP\, b - a\, IFFBOT\, b=$$
$$a\, EORTOP\, b - a\, EORBOT\, b =2 \cdot\min(\phi a, \phi b).$$
\smallskip

Hence, the {\it margins of imprecision} can be directly measured 
by the degree of fuzziness $\phi$.

Measures other than $m_{1}$ yield other interesting results as 
demonstrated by Bandler and Kohout in their 1980 paper \cite{80.2}. 
If for {\it F} an implication operator type of connective 
is chosen again, but this time evaluation ``by performance"  
$$m_{2}=u_{11}/(u_{10}+u_{11})$$ is used, this yields the inequality 
$$\min(1, b/a) \geq m_{2}(F) \geq \max(0,(a+b-1)/a),$$
where {\it PLYTOP} is in this instance the well-known G43 
implication of Goguen-Gaines (cf., e.g., \cite{80.2}). 

Still another contracting measure $$m_{3}=u_{11} \bigvee
(u_{00}+u_{01}),$$ where $\bigvee$ stands for $\max$,  
yields \cite{80.2}:
  \[ \max[\min(a, b), 1-a] \geq m_{3}(F) \geq \max(a+b-1, 1-a). \] The
lower ``contrapositivization'' of 
$m_{3}$ yields the measure $$m_{4}=(u_{11} \bigvee (u_{00}+u_{01}))
\bigvee (u_{00} \bigvee (u_{01}+u_{11}))$$ that
gives the following bounds \cite{80.2}:
$$\min[\max (a+b-1, 1-a), \max(b, 1-a-b)] \leq m_{4} \leq$$
$$\min[\max(1-a, b), {\kappa}a, {\kappa}b].$$
The measure $m_{5}=m_{2} \bigvee u_{00}+u_{11}$ yields: 
$$\max[\min(1, b/a), 1-a] \geq m_{5} \geq$$
$$ \max[(a+b-1)/a, 1-a].$$

For the proofs of the results presented in this section and further
explanation see 
\cite{80.2}, Sections 5 and 6.

Logic transformations are useful in investigating the mutual
interrelationships of  interval logic 
connectives.  Let transformations on basic propositional functions $f(x,y)$ 
of 2 arguments be given as follows:
\noindent
$$I(f)=f(x,y), \  D(f)=\neg f(\neg x, \neg y),$$
$$C(f)=f(\neg x,\neg y), \ N(f)=\neg f(x,y).$$

In the set of the above transformations $T_{P}=\{I, D, C, N\}$, the
individual transformations are called {\it 
identity,  dual, contradual, negation} transformation, respectively.
It is known that for the crisp (2-valued) logic these
transformations determine the Piaget group \cite{pink.lsm}. This group of 
transformations is a realization of abstract Klein 4-element group. 

Adding new non-symmetrical transformations to those defined by Piaget 
enriches the algebraic structure of logical transformations 
\cite{79.1r,92.8}.

Adding
$$LC(f)=f(\neg x,y), \ RC(f)=f(x,\neg y),$$
$$LD(f)=\neg f(\neg x, y),\ RD(f)=\neg f(x,\neg y)$$ 
to the above defined four symmetrical
transformations we obtain a new 
8-element group of transformations $$T=\{I, D, C, N, LC, RD, LC, RD \}.$$  

The interval logic system based on $m_{1}$ can be characterized by
such groups of transformations. 
If we are interested in extending the interval logics into the domain
of paraconsistency, we have 
however base our systems on different measures, such as $m_{2}$ and such like.

One area where conflicts may arise is when we separate assertions  and
rejections of  propositions. 
Assertion and rejection of a proposition belong properly to the domain
of meta-operations. In those 
kinds of logic, for which the mutual duality of 
assertion and rejection are their meta-properties the object language
defined ply operators will 
be {\it contrapositive}. The duality of inference rules (such as modus
ponens and tollens) is assured 
by the contrapositive property of the ply operator
\cite{78.5,80.2,80.1} that enters the 
rules, i.e., by the property that for all $a$ and $b$, 
the implications $a\to b$
and $\neg b\to neg a$ have the same truth value:
$a \rightarrow b=\neg b\rightarrow \neg a$.

One such system that is not contrapositive is the one that is 
determined by the measure  
$m_{2}$.
The ${\cal S}_{2 \times 2 \times 2}$ group also brings some order to
the connectives generated by 
this  measure.

\noindent 
\noindent{\bf THEOREM.} {\it The closed set of connectives generated from the
ply-top implication 
operator  $a \rightarrow_{4} b= \min(1, b/a) $ by the transformation
group $T$ is listed below:
$$g_{1}=I(\rightarrow_{4})=\min(1, b/a),$$
$$g_{2}=C(\rightarrow_{4})= \min(1, 1-b/1-a),$$
$$g_{3}=D(\rightarrow_{4})=\max(0, b-a/1-a),$$
$$g_{4}=N(\rightarrow_{4})=\max(0, a-b/a),$$
$$g_{5}=LC(\rightarrow_{4})=\min(1, b/1-a),$$
$$g_{6}=LD(\rightarrow_{4})=\max(0, 1-a-b/1-a),$$ 
$$g_{7}=RC(\rightarrow_{4})=\min(1, 1-b/a),$$
$$g_{8}=RD(\rightarrow_{4})=\max(0, a+b-1/a).$$
 This set of 
connectives together with $T$ is a realization of the abstract group 
${\cal S}_{2 \times 2 \times 2}$.}
\smallskip

The corresponding  implication operator operating on negative
propositions, given by the formula 
$\max(0, (a+b-1)/a)$, also generates a  similar group of transformations.   

\begin{thebibliography}{99}

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\bibitem{80.1}
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\bibitem{87.6}
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\bibitem{88.2}
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\bibitem{79.1r}
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\end{thebibliography}

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