BELIEF AND FUZZINESS: RESTRUCTURING EPISTEMOLOGY

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BELIEF AND FUZZINESSRE-STRUCTURING EPISTEMOLOGY
I.BURHAN TÜRKSEN
Director, Knowledge / Intelligence Systems
Laboratory Mechanical and Industrial
Engineering University of Toronto Toronto,
Ontario, M5S 3G8 CANADA Tel [phone]
Fax [phone] turksen_at_mie.utoronto.ca http/
/www.mie.utoronto.ca/staff/profiles/turksen.html
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Epistemology and Ontology

• Epistemology
• The study or theory of the nature and grounds of
  knowledge, esp., with reference to its limits and
  validity.
• Ontology
• 1) A branch of metaphysics concerned with the
  nature and relations of being.
• 2) A particular theory about the nature of being
  or the kinds of existents.

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Dimension of Knowledge

• There are three dimensions that are inherent in
  any adequate theory knowledge
• Linguistic,
• Logical, and
• Causal

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Epistemology

• Epistemology lays the ground work for the
  assessment of consistency and believability of a
  set of propositions by evaluating the evidentiary
  basis for each proposition.
• Evidentiary basis could be subjective and/or
  objective.

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General Epistemological Concerns in Fuzzy Theory
(1) What accounts as good, strong, supportive
evidence for belief? Explication of criteria
of evidence or justification. (2) What is the
connection between a belief being well-supported
by good evidence, and the likelihood that it is
true? Ratification, Verification
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General Epistemological Concerns in Fuzzy Theory

• Current Ratification, Verification Criteria
  in most fuzzy system development exercises are
• RMSE Root Mean Square Error
• R2 How successful the fit is in explaining the
  variation in the data
• Accuracy of Prediction
• Power of Prediction

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General Epistemological Concerns in Fuzzy Theory
RMSE , R2 1 -
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General Epistemological Concerns in Fuzzy Theory
Accuracy t () X / P Power () t X / A X
(1) Frequency of the predicted values that are
predicted at the correct
interval (t) Or (2) Total number of predicted
values that are hit correctly.
P Total number of predicted values at
interval t A Total number of actual values at
interval t
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General Epistemological Concerns in Fuzzy Theory
The verification criteria shown above are crisp
theory based. What are the fuzzy equivalents of
these verification criteria?
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ISSUES OF EPISTEMOLOGICAL CONCERNS

• (i) FOUNDATIONALISM
• (ii) COHERENTISM
• (iii) RELIABILISM
• (iv) CRITICAL RATIONALISM

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FOUNDATIONALISM RE-STATED
Foundationalism admits many and various
variations. We re-state classical claims by to
generalizing their scope with fuzziness. (i)
Some basic beliefs are justified to some fuzzy
degree independently of the support of other
beliefs and they are non-empirical in
character. (ii) Some basic beliefs are
justified to some fuzzy degree not by the support
of other beliefs, but by a subjects experience,
i.e., they are empirical.
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FOUNDATIONALISM RE-STATED
(i?) Some basic beliefs are justifie to a some
fuzzy degree, not by the support of other
beliefs, but because of causal or law-like
connection between a subjects belief and the
state of affairs which makes it appear true,
i.e., expert knowledge which are considered crisp
in the classical perspective but which are
intrinsically fuzzy under our sR1oR2s structure
proposed at the Ontological level. (ii?) Some
basic beliefs are justified to some fuzzy degree,
not by the support of other beliefs but by
virtue of its content, its intrinsically
self-justifying character. Again they may be
assumed to be crisp or fuzzy depending on the
agenda we work on.
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FOUNDATIONALISM RE-STATED
(iii?) Some basic beliefs decisively,
conclusively, but approximately are justified
independent of the support of any other
belief. This requires the determination of
critical, affective variables and the belief that
they are independent. Clearly the determination
of these approximately independent critical and
affective variables require crisp or fuzzy
statistical criteria.
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FOUNDATIONALISM Re-Statement of Fuzzy Beliefs
(i??) Some basic beliefs are justified to some
fuzzy degree interacting with other
beliefs. (ii??) Some justified beliefs are
derived and are justified wholly to some fuzzy
degree via direct or indirect support of basic
beliefs that are inherently fuzzy. (iii??) Some
justified beliefs are derived at least in part
via direct or indirect support of basic beliefs
that are inherently fuzzy.
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COHORENTISM
A belief is justified if it belongs to a coherent
fuzzy set of beliefs. (i) Uncompromising
Coherentism A belief is justified iff it belongs
to a coherent fuzzy set of beliefs, no belief
within a coherent fuzzy set has a distinguishing
epistemic status and place. (ii) Moderated
Weighted Coherentism Some beliefs are justified
if they belong to a coherent fuzzy set and they
have a distinguishing initial status and
justification dependent on a weighted mutual
support. (iii) Moderated Fuzzy Coherentism Some
beliefs are justified if they belong to a
coherent fuzzy set and they have a distinguished
initial status and justification by being
embedded to a fuzzy degree within a coherent
fuzzy set.
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Belief Measures
A belief measure is a function Bel P(X) ?
0,1 that satisfies the axioms of fuzzy
measures that are known as (1) Boundary
Condition h(?)0, h(X) 1 (2) Monotonicity For
every A1, A2?P(X) if A1?A2, then h(A1) ?
h(A2). (3) Continuity For ever sequence,
Ai?P(X), i?1,2,..., of subsets of X, if either
A1?A2? ... or A1?A2?..., i.e., the sequence is
monotonic, then
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Belief Measures
More generally, a fuzzy measure is defined
as h B ?0,1 where B? P (X) is a family of
subsets of such that (i) ??B, and X?B, (ii)
If A?B, then c(A)?B, where c(A) is the complement
of A. (ii) B is closed under the operation of
set union, i.e., if A1?B and A2?B, then A1?A2?B.
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Degree of Evidence
Every belief measure and its dual plausibility
measure can be expressed in terms a function m
P(X) ? 0,1 A?P(X) Such that m(?)0, and
, where m(A) is interpreted either as (i) the
degree of evidence supporting the claim that a
specific element of X belongs to a set A but not
to any special subset of A, or (ii) the degree to
which an expert believes that such a claim is
warranted and it is called the basic assignment.
A?P(X)
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Degree of Evidence
The basic assignment has the following
properties (i) It is not required that m(X)
1 (ii) It is not required that m(A) ? m(B) when
A?B. (iii) No relationship between m(A) and
m(c(A)) is required. Given a basic assignment m,
a belief measure and plausibility measure are
uniquely determined by formulas Bel(A)
, and Pl(A) for all A?
P(X).
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Precisiated Natural Language, PNL Language of
Fuzziness (LF)

• Fuzzy Sets, FS
• Computing with Words, CWW
• Computing with Perceptions, CWP
• (Zadeh, 1965 2003)

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Precisiated Natural Language, PNL Language of
Fuzziness (LF)

• All forms of Language have both a communications
  and informatics dimension that facilitates human
  thoughts and decision making.
• Beyond meta-languages,
• Speech, writing, mathematics, science and
  computing form the five links of an evolutionary
  chain of languages.
• PNL Precisiated Natural Language is the sixth
  link in this evolutionary chain.

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Hierarchy of Levels of Theoretical Inquiry

• APPLICATION LEVEL
•  vii. How do people, decision-makers, feel,
  think, behave, and interact? How can we provide
  them with better decision-making tools?
• How can we provide them with a better language?
•  DOMAIN-SPECIFIC EPISTEMOLOGICAL LEVEL
•  vi. How do we validate knowledge appropriately
  in this domain specific field? What
  methodological approaches are appropriate to it?
• What ought to be Domain-Specific language?
•  v. What can we know or hope to learn within this
  domain-specific field or discipline?
• What specific expression of Domain-Specific
  language could and should we use to specify the
  limits or boundaries?

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Hierarchy of Levels of Theoretical Inquiry

• GENERAL EPISTEMOLOGICAL LEVEL
•  iv. How do we validate our knowledge? How do we
  know it is true? What criteria do we use to
  assess its truth-value?
• What linguistic expressions cause the assessment
  of truth?
• iii. What is our access to truth or knowledge in
  general? Where is truth to be found? How or from
  what is it constituted?
• What linguistic encoding allows us to access
  truth or knowledge?
• ONTOLOGICAL LEVEL
• ii. What is our position or relation to that
  Reality (if we do assume that it exists on level
  i below)?
• What linguistic expressions capture our position
  to reality?
• i. Is there any reality independent or partially
  independent of us? Does any absolute truth exist?
  Does fuzziness exists?
• What language explicates reality? Is it crisp or
  fuzzy representation of linguistic variables and
  their connectives?

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Hierarchy of Levels of Theoretical Inquiry

• A Human is a denizen of two worlds
• A biological organism, and
• A universe of symbols (ii)? A universe of
  languages
• Humans create, develop and apply a universe of
  languages.
• (i) However, while on the one hand, they dominate
  such languages,
• (ii) In turn, they are dominated by such universe
  of languages.
• Re-phrased from L.Von Bertalanffy (1901-1972)
• A Systems View of Man.

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Classical Theory

• Table. Positions Taken by Classical Set and Logic
  Theorists on the Hierarchy of Levels of
  Theoretical Inquiry.
• Application vii. Emphasis on mechanistic
  Super Additive systems
• Level theory of interactions, relations,
  equations, etc.
• Domain Specific vi. Validity and methodology
  dictated by meta-physical Epistemological
  theories. e.g., principle of
  determinism, symmetry, Level invariance
  and randomness.
• v. Objective facts or truth accessible, but
  limited only by
• subjective
  distortions (introduction of uncertainty)
• General iv. Correspondence theory of Validity
  only Objective
• Epistemological iii. Objectivist, empiricists,
  certain
• Level
• Ontological ii. sRo Cartesian dualism
• Level i. Realism, crisp meaning
  representation of linguistic variables and
  connectives are defined with two valued
  sets.

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Classical Theory

• On the Ontological Level, in classical theory it
  is assumed, briefly, every element belongs to a
  concept class, say A, either with full membership
  or none, i.e.,
• mA X ? 0,1, mA(x) a ? 0,1, x?X, where
  mA(x) is the membership assignment of an element
  x?X to a concept class A in a proposition.
• Linguistic Variables are assumed to have a
  precise meaning representation.

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Classical Theory
In classical theory, a question raises on the
Conjunctive and Disjunctive Normal (Canonical)
Form representations have equivalence, hence
symmetry and invariance in the combination of
concepts. CNF (.) DNF(.)
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Laws of Equivalence and Conservation in Classical
Set and Logic
Equivalence (OR SYMMETRY) DNF(A OR c(A)) CNF(A
OR c(A)) DNF(A AND c(A)) CNF(A AND
c(A)) CONSERVATION ?DNF(A OR c(A)) CNF(A
OR c(A)) ?DNF(A AND c(A)) CNF(A AND
c(A)) 1
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Classical Theory
DNF(A AND B) CNF(A AND B) DNF(A OR B) CNF(A
OR B) T(a,b) 1 S(n(a), n(b)) Bel (A) Pl
(c(A)) 1 Pl (A) Bel (c(A)) 1 Bel (A) Bel
(c(A)) ? 1 Bel (A OR B) Bel (A ? B) ?
Bel (A) Bel (B) - Bel (A?B) Pr(A) Pr(c(A))
1
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Fuzzy Theory

• Where as in fuzzy theory, it is assumed that
• mA X ? 0,1, mA(x) a ? 0,1,
• Linguistic Variables and linguistic Connectives
  are assumed to be imprecise and hence meaning
  representation is precisiated with fuzzy sets and
  fuzzy connectives.
• (I dont mean t-normsconorms to be fuzzy
  connectives!)

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Fuzzy Theory

• Table. Positions Taken by Fuzzy Set and Logic
  Theorists on the Hierarchy of
• Levels of Theoretical Inquiry.
• Application vii. Emphasis on humanistic
  Decision and Control Systems that contain
• Level highly complex non-linear
  interactions, relations, equations, etc.
• Domain Specific vi. Validity and methodology
  dictated by Meta theories of Modal Logics.
• Etimological e.g., principle of
  non-determinism and overlapping patterns.
• Level v. Subjective and objective facts
  accessible by perceptions and meaning
• representation of linguistic terms of
  linguistic variables, linguistic
  quantifiers and linguistic connectives.
• General iv. Correspondence theory of Validity
  both objective and subjective.
• Epistemological iii. Subjective-objective,
  experimental and empiricist, e.g., expert and
  
• Level fuzzy data mining based.
• Ontological ii. sRoRs schema gives credence both
  the subject and the object
• interaction.
• Level i. Realism fuzzy and uncertain

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Classical Theory

• In classical theory, the descriptive assignment
  of Linguistic Variables, D0,1, are verified or
  asserted to be absolutely True, T, or False, F,
  i.e., mV mA ? T,F, where VT,F, is the
  veristic assignment which is the atomic building
  block of two-valued logic.

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Fuzzy Theory

• In fuzzy theory, the descriptive assignment of
  Linguistic Variables, D0,1, is verified either
  crisply or fuzzily, i.e., we have mV mA ? T,F
  or
• mV mA ? T,F. (Türksen, 1999-2002)
• In addition, in fuzzy theory, linguistic
  connectives AND. OR, IMP, etc., are also
  imprecise. Thus AND does not correspond to
  t-norm and OR does not correspond to a
  t-co-norm in a one-to-one isomorphism. (Türksen,
  1986-2003)

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Fuzzy Theory
Thus the equivalence (symmetry) and hence
invariance properties are broken down FDCF (.)
? FCCF(.) For special cases Archimedean t-norms
and co-norms that are strict and nilpotent, we
have FDCF (.) ? FCCF(.)
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Fuzzy Theory
However, they are re-established in a new
way For example, the (symmetry), equivalence,
and Laws of Conservation are re-established. Fo
r example, there are now two new Laws of
Conservation 1) ?FDCF(A OR c(A)) ?FCCF(A
AND c(A)) 1 2) ?FDCF(A AND c(A)) ?FCCF(A
OR c(A)) 1
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TS Norm in Fuzzy Theory
1) 2)
(Well known)
(Generated from, the breakdown of
FDCF(.)FCCF(.)
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Belief and Plausibility over Fuzzy
Sets Re-Establish Laws of Conservation
(1) Pl FDCF(A AND B) Bel FCCF(c(A) OR c(B)
1 Pl (A?B) Bel (c(A) ? c(B)) 1 (2)
Pl FCCF(A AND B) Bel FDCF(c(A) OR c(B)) 1
Pl (A?B) ? (c(A) ? B) ? (A ? c(B))
Bel(c(A) ? c(B)) ? (A ? c(B)) ? (c(A) ? B)
1
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Probability over Fuzzy Sets Re-Establish Laws of
Conservation
Pr (A AND B) Pr(c(A) OR c(B) 1 (1) Pr
FDCF(A AND B Pr FCCF(c(A) OR c(B) 1
Pr (A?B) Pr(c(A) ? c(B)) 1 (2) PrFCCF(A
AND B) Pr FDCF(c(A) OR c(B)) 1 Pr
(A?B) ? (c(A) ? B) ? (A ? c(B)) Pr
(c(A) ? c(B)) ? (A ? c(B)) ? (c(A) ? B) 1
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Rules in Nuclear Medicine

• Possible Rules for Target Localization and
  Treatment of Cancer cells
• If PET uptake index is very bright and the Volume
  is large, Then dosage should be high
• If PET uptake index is bright and the Volume is
  small, Then dosage should be low

. . .
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Belief on Type 1 Combination of Fuzzy Sets

• Suppose we are concerned with the assessment of
  Belief on the Type 1 combination of two Type 1
  fuzzy sets, say,
• x?X isr A1, OR x?X isr A2.
• Example Identification of Target Lung Carcinoma
• Suppose in a PET uptake index analysis, there
  appears to be three candidates, say,
  SS1,S2,S3, which are suspected to be cancer
  cell targets. Furthermore, PET image analysis of
  gray scales with fuzzy c-means, FCM, reveals that
  PET uptake index could be classified over the
  three suspected targets as fellows
• very bright (.3, S1)?(.6,S2) ? (1,S3)
• bright (.4, S1)?(.7,S2) ? (1,S3)

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Belief on Type 1 Combination of Fuzzy Sets

• Therefore, we get A very bright OR
  bright
• when we construct a Type 1 fuzzy set combination
  we get, assuming (?, ?, -) De Morgan Triple
• A very bright ? bright (.4,S1)?(.7,S2) ?
  (1,S3)
• (Remark at times, we write
• A (.4,S1) (.7,S2) (1,S3)
• where is interpreted as a set aggregation)

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Belief on Type 1 Combination of Fuzzy Sets

• Table 1. Basic Assignments provided by two
  independent sources of evidence, i.e., oncologist
  on the focal elements.

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Belief on Type 1 Combination of Fuzzy Sets

• In order to determine the belief over Type 1
  fuzzy set A, we first express it in terms of its
  ?-cuts and ?-level sets as
• A(0.4)S1 ? S2 ? S3 ? (0.7)S2 ? S3 ? (1)S3
• Recall, with ?-cuts we have
• Bel(A1 OR A2) Bel(A1? A2)
• Bel(A)
• (note that ??, i.e., it is set aggregation and ?
  is a scalar)

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• Table 2. Combination of Degrees of Evidence from
  Two Independent Sources, i.e., Oncologists.

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Belief on Type 1 Combination of Fuzzy Sets

• Bel1,2(A) (0.4), Bel1,2 S1 ? S2 ? S3
• ?(0.7), Bel1,2 S2 ? S3
• ?(1), Bel1,2 S3
• From Table 2, we have
• Bel1,2 S1 ? S2 ? S31,
• Bel1,2 S2 ? S30.22, and
• Bel1,2 S30.09.

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Belief on Type 1 Combination of Fuzzy Sets

• On the other hand, by the application of the
  belief axiom, we are able to show that for (?, ?,
  -)
• Bel(A) Bel0.4 S1?S2 ?S3?(0.7)S2
  ?S3?(1)S3
• ? (0.4), Bel S1?S2 ?S3
• (0.7), Bel S2?S3
• (1), BelS3
• - (0.4?0.7), BelS2?S3 (0.4?1), BelS3
• - (0.7?1), BelS3 (0.4?0.7?1), BelS3
• (0.4), BelS1?S2?S3 (0.3), BelS2?S3
• (0.3), BelS3.

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Algebraic Combination Belief on Type 1
Combination of Fuzzy Sets

• For the case of Avery bright ? bright and
  (?,?,-) gives us
• A((.3.4-.12), S1) ?((.7.6-.42), S2) ?
  ((11-1), S3)
• Thus we get
• A (.58, S1) ?(.88, S2) ? (1, S3)
• Therefore, with the ?-cuts, we get
• A (.58)S1, S2, S3 ? (.88)S1, S2? (1)S3.

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Belief on Interval-Valued Type 2 Fuzzy Sets
In our interval-valued Type 2 theory, which
recently is shown to correspond to a restricted
and modified multi-valued mapping of Dempster,
which we call T-formalism(Türksen, 2001, 2002),
the membership values of the meta-linguistic
combination of A1 OR A2A are mapped into
(1) the upper bound set approximation FCCF(A)
A1?A2 AU (2) the lower bound set
approximation FDCF(A) (A1?A2)?(c(A1)?A2)?(A1?c(A
2)) AL
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Upper Lower Beliefs
BelLA(x)a BelFDCF(A(x)a)
BelUA(x)a BelFCCF(A(x)a)
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Upper Lower Beliefs
Lower bound expression of interval-valued Type 2
representation of the combination If we assume
(?, ?, -) De Morgan Triple, for AL (very
bright OR bright)L (very
bright?bright)?(c(very bright)?bright)
?(very bright
?c(bright)) We get AL (.3)?(.4)?(.3),
S1?(.6)?(.3)?(.4), S2?(1)?(0)?(0), S3
(.4, S1) ? (.6, S2) ? (1, S3)
(0.4)S1,S2,S3 ? (0.6)S2,S3 ? (1)S3
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Upper Lower Beliefs
Bel1,2(AL) (.4), Bel1,2S1, S2, S3 Recall
Again ? (.6), Bel1,2S2, S3 Bel1,2S1, S2,
S3 1 ? (1), Bel1,2S3 Bel1,2S2, S3
0.22 Bel1,2S3 0.09 Bel1,2(AU) (0.4),
Bel1,2S1, S2, S3 ? (0.7), Bel1,2S2,
S3 ? (1), Bel1,2S3 Therefore Bel1,2(A)
(0.4), Bel1,2S1, S2, S3 ? (0.6 - 0.7),
Bel1,2S2, S3 ? (1), Bel1,2S3
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Upper and Lower Probabilities over Interval
Valued Type 2 Fuzzy Sets
Therefore there is a membership uncertainty
interval of 0.6, 0.7 where our belief is 0.22
over the set S2, S3 one of which is discerned
to be the target lung carcinoma. We present an
example for the computation of upper and lower
probabilities over Type 2 fuzzy sets in analogy
to Dempster's upper and lower probabilities.
Consider a three element set SS1, S2, S3, say,
three persons. Suppose we are interested how
cancerous" they would be if we know how very
bright" they are and how bright" they are if
"cancerous"very bright" "AND" bright". Let
very bright" (.3, S1) (.6, S2) (1,
S3) and bright (.4, S1) (.7, S2) (1, S3).
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Upper and Lower Probabilities over Interval
Valued Type 2 Fuzzy Sets
From Interval-valued Type 2 Fuzzy sets, we have
FDCFcancerous(Si?S) (very bright) ?
(bright) FCCFcancerous(Si?S) (very
bright?bright)?(c(very bright) ?bright)?(very
bright?c(bright))
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Upper and Lower Probabilities over Interval
Valued Type 2 Fuzzy Sets
For (?, ?, -), we get FDCFcancerous(Si?S)
(.3, S1)?(.6, S2)?(1, S3) ? FCCFcancerous(Si?S)
(.4, S1)?(.6, S2)?(1, S3)
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Upper and Lower Probabilities over Interval
Valued Type 2 Fuzzy Sets
Thus, the ?-cut representations would
be FDCFcancerous(Si?S)(.3)S1, S2,
S3(.6)S2, S3(1)S3 FCCFcancerous(Si?S)(.4)
S1, S2, S3(.6)S2, S3(1)S3
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Upper and Lower Probabilities over Interval
Valued Type 2 Fuzzy Sets
Table. Upper and lower probabilities when we have
(.3) S1, S2, S3we also have the same upper and
lower probabilities when we have (.4) S1, S2,
S3.
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Upper and Lower Probabilities over Interval
Valued Type 2 Fuzzy Sets
Table. Upper and lower probabilities when we have
(.6) S2, S3, i.e., at ?(.6). Note For two and
single element sets their upper and lower sets.
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Upper and Lower Probabilities over Interval
Valued Type 2 Fuzzy Sets
S' ?,S1 S'' S2, S3
Table. Upper and lower probabilities at ?1,
(1)S3.
TÜRKSEN, December 2003 U of T
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