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Fuzzy Logic, Neutrosophic Logic, and Applications

- Florentin Smarandache
- University of New Mexico
- Gallup, NM 87301, USA
- E-mail smarand_at_unm.edu
- M. Khoshnevisan
- Griffith University, Gold Coast
- Queensland 9726, Australia
- E-mail m.khoshnevisan_at_mailbox.edu.au

Contents

- Introduction to Non-Standard Analysis
- Survey of Tripartion Logics and Combination Rules

up to Intuitionistic Fuzzy Logic - Neutrosophic Components
- Neutrosophic Logic - a Generalization of the

Intuitionistic Fuzzy Logic - Differences between Intuitionistic Fuzzy Logic

(IFL) and Neutrosophic Logic (NL) - Operations on Classical Sets
- Neutrosophic Logic Connectors
- Generalizations of Other Logics
- Applications of Fuzzy and Neutrosophic Logics to

Finance

1. Introduction to Non-Standard Analysis

- - in 1960s Abraham Robinson has developed the

non-standard analysis - - formally, x is said to be infinitesimal if and

only if for all positive integers n one has ?x? lt

1/n - - by -a one denotes a monad, i.e. a set of

hyper-real numbers in non-standard analysis - (-a) a-x x?R, x is infinitesimal,
- and similarly b is a monad
- (b) bx x?R, x is infinitesimal
- - combining the two previous mentioned

definitions one gets, what well call, a binad of

-c - (-c)c-x x?R, x is infinitesimal ?

cx x?R, x is infinitesimal - - the left and right borders of a non-standard

interval -a, b are vague, imprecise,

themselves being the non-standard (sub)sets (-a)

and (b) as defined above - - by extension let inf -a, b -a and sup

-a, b b

2. Survey of tripartition logics and combination

rules up to intuitionistic fuzzy logic

- - Hoopers rule of combination of evidence

(1680s), which was a Non-Bayesian approach to

find a probabilistic model - idea of tripartition (truth, falsehood,

indeterminacy) appeared in 1764 when J. H.

Lambert investigated the credibility of one

witness affected by the contrary testimony of

another - - Lukasiewicz considered three values (1, 1/2,

0), Post considered m values in between the

wars - - Koopman in 1940s introduced the notions of

lower and upper probability, followed by Good,

and Dempster (1967) gave a rule of combining two

arguments - - Shafer (1976) extended it to the

Dempster-Shafer Theory of Belief Functions (DST)

by defining the Belief and Plausibility functions

and using the rule of inference of Dempster for

combining two evidences proceeding from two

different sources. Belief function is a

connection between fuzzy reasoning and

probability. DST is a generalization of the

Bayesian Probability (Bayes 1760s, Laplace

1780s) this uses the mathematical probability in

a more general way, and is based on probabilistic

combination of evidence in artificial

intelligence.

2. Survey of tripartition logics and combination

rules up to intuitionistic fuzzy logic

- in Lambert there is a chance p that the witness

will be faithful and accurate, a chance q that he

will be mendacious, and a chance 1-p-q that he

will simply be careless apud Shafer (1986).

Whence three components accurate, mendacious,

careless, which add up to 1. - - The many-valued logic was replaced by Goguen

(1969) and Zadeh (1975) with an Infinite-Valued

Logic (of continuum power, as in the classical

mathematical analysis and classical probability)

called Fuzzy Logic, where the truth-value can be

any number in the closed unit interval 0, 1.

The Fuzzy Set was introduced by Zadeh in 1965. - - Van Fraassen introduced the supervaluation

semantics in his attempt to solve the sorites

paradoxes, followed by Dummett (1975) and Fine

(1975). They all tripartitioned, considering a

vague predicate which, having border cases, is

undefined for these border cases. They all

tripartitioned, considering a vague predicate

which, having border cases, is undefined for

these border cases. Van Fraassen took the vague

predicate heap and extended it positively to

those objects to which the predicate definitively

applies and negatively to those objects to which

it definitively doesnt apply. The remaining

objects border was called penumbra. A sharp

boundary between these two extensions does not

exist for a soritical predicate. Inductive

reasoning is no longer valid too if S is a

sorites predicate, the proposition

?n(San?San1) is false.

2. Survey of tripartition logics and combination

rules up to intuitionistic fuzzy logic

- Thus, the predicate Heap (positive

extension) true, Heap (negative extension)

false, Heap (penumbra) indeterminate. - - Narinyani (1980) used the tripartition to

define what he called the indefinite set - - Atanassov (1982) continued on tripartition and

gave five generalizations of the fuzzy set,

studied their properties and applications to the

neural networks in medicine - a) Intuitionistic Fuzzy Set (IFS)
- Given an universe E, an IFS A over E is a set of

ordered triples ltuniverse_element,

degree_of_membership_to_A(M), degree_of_non-member

ship_to_A(N)gt such that MN ? 1 and M, N ? 0,

1. When M N 1 one obtains the fuzzy set,

and if M N lt 1 there is an indeterminacy I

1-M-N. - b) Intuitionistic L-Fuzzy Set (ILFS)
- Is similar to IFS, but M and N belong to a fixed

lattice L. - c) Interval-valued Intuitionistic Fuzzy Set

(IVIFS) - Is similar to IFS, but M and N are subsets of 0,

1 and sup M sup N ? 1. - d) Intuitionistic Fuzzy Set of Second Type

(IFS2) - Is similar to IFS, but M2 N2 ? 1. M and N are

inside of the upper right quarter of unit circle. - e) Temporal IFS
- Is similar to IFS, but M and N are functions of

the time-moment too.

3. Neutrosophic Logic Components

- Lets consider the non-standard finite numbers 1

1?, - where 1 is its standard part and ? its

non-standard part, - and 0 0-?, where 0 is its standard part and

? its non-standard part. Then, we call -0,

1 a non-standard unit interval. - Let T, I, F be standard or non-standard real

subsets of -0, 1 , - with sup T tsup, inf T tinf,
- sup I isup, inf I iinf,
- sup F fsup, inf F finf,
- and nsup tsupisupfsup,
- ninf tinfiinffinf.
- T, I, F are called neutrosophic components, which

represent the truth value, indeterminacy value,

and falsehood value respectively referring to

neutrosophy, neutrosophic logic, neutrosophic

set, neutrosophic probability and statistics. - The sets T, I, F are not necessarily intervals,

but may be any real sub-unitary subsets

discrete or continuous single-element, finite,

or (countably or uncountably) infinite union or

intersection of various subsets etc. - They may also overlap. The real subsets could

represent the relative errors in determining t,

i, f (in the case when the subsets T, I, F are

reduced to points).

3. Neutrosophic Logic Components

- This representation is closer to the human mind

reasoning. It characterizes/catches the

imprecision of knowledge or linguistic

inexactitude received by various observers

(thats why T, I, F are subsets - not necessarily

single-elements), uncertainty due to incomplete

knowledge or acquisition errors or stochasticity

(thats why the subset I exists), and vagueness

due to lack of clear contours or limits (thats

why T, I, F are subsets and I exists in

particular for the appurtenance to the

neutrosophic sets).

4. Neutrosophic Logic a Generalization of the

Intuitionistic Fuzzy Logic

- Definition
- A logic in which each proposition is estimated to

have the percentage of truth in a subset T, the

percentage of indeterminacy in a subset I, and

the percentage of falsity in a subset F, where T,

I, F are defined above, is called Neutrosophic

Logic. - We use a subset of truth (or indeterminacy, or

falsity), instead of a number only, because in

many cases we are not able to exactly determine

the percentages of truth and of falsity but to

approximate them for example a proposition is

between 0.30-0.40 true and between 0.60-0.70

false, even worst between 0.30-0.40 or 0.45-0.50

true (according to various analyzers), and 0.60

or between 0.66-0.70 false. - The subsets are not necessary intervals, but any

sets (discrete, continuous, open or closed or

half-open/half-closed interval, intersections or

unions of the previous sets, etc.) in accordance

with the given proposition. - A subset may have one element only in special

cases of this logic. - Constants (T, I, F) truth-values, where T, I, F

are standard or non-standard subsets of the

non-standard interval -0, 1 , where ninf

inf T inf I inf F ? -0, and nsup sup T

sup I sup F ? 3. - Atomic formulas a, b, c, .
- Arbitrary formulas A, B, C, .

4. Neutrosophic Logic a Generalization of the

Intuitionistic Fuzzy Logic

- Therefore, we finally generalize the

intuitionistic fuzzy logic to a transcendental

logic, called neutrosophic logic where the

interval 0, 1 is exceeded, i.e. , the

percentages of truth, indeterminacy, and falsity

are approximated by non-standard subsets not by

single numbers, and these subsets may overlap and

exceed the unit interval in the sense of the

non-standard analysis also the superior sums and

inferior sum, nsup sup T sup I sup F ?

-0, 3 , may be as high as 3 or 3, while ninf

inf T inf I inf F ? -0, 3 , may be as low

as 0 or 0. - Lets borrow from the modal logic the notion of

world, which is a semantic device of what the

world might have been like. Then, one says that

the neutrosophic truth-value of a statement A,

NLt(A) 1 if A is true in all possible worlds

(syntagme first used by Leibniz) and all

conjunctures, that one may call absolute truth

(in the modal logic it was named necessary truth,

Dinulescu-C?mpina (2000) names it intangible

absolute truth ), whereas NLt(A) 1 if A is

true in at least one world at some conjuncture,

we call this relative truth because it is

related to a specific world and a specific

conjuncture (in the modal logic it was named

possible truth). - Similarly for absolute and relative falsehood and

absolute and relative indeterminacy. - The neutrosophic inference inference Dezert

(2002), especially for plausible and paradoxist

information, is still a subject of intense

research today.

5. Differences between Neutrosophic Logic and

Intuitionistic Fuzzy Logic

- a) Neutrosophic Logic can distinguish between

absolute truth (truth in all possible worlds,

according to Leibniz) and relative truth (truth

in at least one world), because NL(absolute

truth)1 while NL(relative truth)1. This has

application in philosophy (see the neutrosophy).

Thats why the unitary standard interval 0, 1

used in IFL has been extended to the unitary

non-standard interval -0, 1 in NL. Similar

distinctions for absolute or relative falsehood,

and absolute or relative indeterminacy are

allowed in NL. - b) In NL there is no restriction on T, I, F

other than they are subsets of -0, 1, thus - -0 ? inf T inf I inf F ? sup T sup

I sup F ? 3. This non-restriction allows

paraconsistent, dialetheist, and incomplete

information to be characterized in NL i.e. the

sum of all three components if they are defined

as points, or sum of superior limits of all three

components if they are defined as subsets can be

gt1 (for paraconsistent information coming from

different sources) or lt 1 for incomplete

information, while that information can not be

described in IFL because in IFL the components T

(truth), I (indeterminacy), F (falsehood) are

restricted either to tif1 or to t2 f2 ? 1,

if T, I, F are all reduced to the points t, i, f

respectively, or to sup T sup I sup F 1 if

T, I, F are subsets of 0, 1. - Some researchers normalize the

paraconsistent and incomplete information, but

this procedure is not always suitable. - c) In NL the components T, I, F can be

non-standard subsets included in the unitary

non-standard interval -0, 1, not only standard

subsets included in the unitary standard interval

0, 1 as in IFL. - d) NL, like dialetheism, can describe paradoxes,

NL(paradox) (1, I, 1), while IFL can not

describe a paradox because the sum of components

should be 1 in IFL.

5. Differences between Neutrosophic Logic and

Intuitionistic Fuzzy Logic

- e) The logical operators in IFL (and

similarly the set connectors in IFS) are defined

with respect to T and F components only, and

whats left up to 1 is considered the

Indeterminacy. In NL (and similarly in NS) they

are defined with respect to any of the three

components. - f) Component I, indeterminacy, can be

split into more subcomponents in order to better

catch the vague information we work with, and

such, for example, one can get more accurate

answers to the Question-Answering Systems

initiated by Zadeh (2003). In Belnaps

four-valued logic (1977) indeterminacy is split

into Uncertainty (U) and Contradiction (C).

6. Operations with Classical Sets

- We need to present these set operations in

order to be able to introduce the neutrosophic

connectors. - Let S1 and S2 be two (unidimensional) real

standard or non-standard subsets included in the

non-standard interval -0, 8) then one defines - 6.1 Addition of Sets
- S1?S2 x?xs1s2, where s1?S1 and s2?S2,

- with inf S1?S2 inf S1 inf S2, sup S1?S2 sup

S1 sup S2 - and, as some particular cases, we have
- a?S2 x?xas2, where s2?S2
- with inf a?S2 a inf S2, sup a?S2 a

sup S2. - 6.2 Subtraction of Sets
- S1?S2 x?xs1-s2, where s1?S1 and s2?S2.
- For real positive subsets (most of the cases will

fall in this range) one gets - inf S1?S2 inf S1 - sup S2, sup S1?S2 sup S1 -

inf S2 - and, as some particular cases, we have
- a?S2 x?xa-s2, where s2?S2,
- with inf a?S2 a - sup S2, sup a?S2 a -

inf S2 - also 1?S2 x?x1-s2, where s2?S2,
- with inf 1?S2 1 - sup S2, sup 1?S2 100

- inf S2.

6. Operations with Classical Sets

- 6.3 Multiplication of Sets
- S1?S2 x?xs1?s2, where s1?S1 and s2?S2.
- For real positive subsets (most of the cases will

fall in this range) one gets inf S1?S2 inf S1 ?

inf S2, sup S1?S2 sup S1 ? sup S2 - and, as some particular cases, we have
- a?S2 x?xa?s2, where s2?S2,
- with inf a?S2 a ? inf S2, sup a?S2 a ?

sup S2 - also 1?S2 x?x 1?s2, where s2?S2,
- with inf 1?S2 1? inf S2, sup 1?S2 1 ?

sup S2. - 6.4 Division of a Set by a Number
- Let k ??, then S1?k x?xs1/k, where s1?S1.

7. Neutrosophic Logic Connectors

- One uses the definitions for neutrosophic

probability and neutrosophic set operations. - Similarly, there are many ways to construct such

connectives according to each particular problem

to solve here we present the easiest ones - One notes the neutrosophic logic values of the

propositions A1 and A2 by NL(A1) ( T1, I1, F1 )

and NL(A2) ( T2, I2, F2 ) respectively. - For all neutrosophic logic values below if,

after calculations, one obtains values lt 0 or gt

1, one replaces them by 0 or 1 respectively. - 7.1. Negation
- NL(?A1) ( 1?T1, 1?I1, 1?F1 ).
- 7.2. Conjunction
- NL(A1 ? A2) ( T1?T2, I1?I2, F1?F2 ).
- (And, in a similar way, generalized for n

propositions.) - 7.3 Weak or inclusive disjunction
- NL(A1 ? A2) ( T1?T2?T1?T2, I1?I2?I1?I2,

F1?F2?F1?F2 ). - (And, in a similar way, generalized for n

propositions.)

7. Neutrosophic Logic Connectors

- 7.4 Strong or exclusive disjunction
- NL(A1 ? A2)
- ( T1? (1?T2) ?T2? (1?T1) ?T1?T2?

(1?T1) ? (1?T2), - I1 ? (1?I2) ?I2 ? (1?I1) ?I1

? I2 ? (1?I1) ? (1? I2), - F1? (1?F2) ?F2? (1? F1) ?F1? F2

? (1?F1) ? (1?F2) ). - (And, in a similar way, generalized for n

propositions.) - 7.5 Material conditional (implication)
- NL(A1 ? A2) ( 1?T1?T1?T2, 1?I1?I1?I2,

1?F1?F1?F2 ). - 7.6 Material biconditional (equivalence)
- NL(A1 ? A2) ( (1?T1?T1?T2) ?

(1?T2?T1?T2), - (1? I1? I1? I2) ?

(1?I2? I1 ? I2), - (1?F1?F1? F2) ?

(1?F2?F1? F2) ). - 7.7 Sheffer's connector
- NL(A1 A2) NL(?A1 ? ?A2) ( 1?T1?T2,

1?I1?I2, 1?F1?F2 ). - 7.8 Peirce's connector

8. Generalizations

- When all neutrosophic logic set components are

reduced to one element, then - tsup tinf t, isup iinf i, fsup finf

f, and nsup ninf n tif, therefore

neutrosophic logic generalizes - - the intuitionistic logic, which supports

incomplete theories (for 0 lt n lt 1 and i0, 0 ?

t, i, f ? 1) - - the fuzzy logic (for n 1 and i 0, and 0 ?

t, i, f ? 1) - from "CRC Concise Concise Encyclopedia of

Mathematics", by Eric W. Weisstein, 1998, the

fuzzy logic is "an extension of two-valued logic

such that statements need not to be True or

False, but may have a degree of truth between 0

and 1" - - the intuitionistic fuzzy logic (for n1)
- - the Boolean logic (for n 1 and i 0, with t,

f either 0 or 1) - - the multi-valued logic (for 0 ? t, i, f ? 1)
- definition of ltmany-valued logicgt from "The

Cambridge Dictionary of Phylosophy", general

editor Robert Audi, 1995, p. 461 "propositions

may take many values beyond simple truth and

falsity, values functionally determined by the

values of their components" Lukasiewicz

considered three values (1, 1/2, 0). Post

considered m values, etc. But they varied in

between 0 and 1 only. In the neutrosophic logic

a proposition may take values even greater than 1

(in percentage greater than 100) or less than 0.

8. Generalizations

- - the paraconsistent logic, which support

conflicting information (for n gt 1 and i 0,

with both t, f lt 1) - - the dialetheism, which says that some

contradictions are true (for t f 1 and i 0

some paradoxes can be denoted this way too) - - the faillibilism, which says that uncertainty

belongs to every proposition (for i gt 0) - Compared with all other logics, the neutrosophic

logic and intuitionistic fuzzy logic introduce a

percentage of "indeterminacy" - due to unexpected

parameters hidden in some propositions, or

unknowness, but neutrosophic logic let each

component t, i, f be even boiling over 1

(overflooded), i.e. be 1, or freezing under 0

(underdried), i.e. be 0 in order to be able to

make distinction between relative truth and

absolute truth, and between relative falsity and

absolute falsity in philosophy.

9. Applications of Fuzzy and Neutrosophic Logics

to Finance

- Example 9.1 Reconciliation between theoretical

and market prices of long-term options contracts. - The neutrosophic probability approach makes a

distinction between relative sure event, event

that is true only in certain world(s) NP (rse)

1, and absolute sure event, event that is true

for all possible world(s) NP (ase) 1. Similar

relations can be drawn for relative impossible

event / absolute impossible event and

relative indeterminate event / absolute

indeterminate event. In case where the truth-

and falsity-components are complimentary i.e.

they sum up to unity, and there is no

indeterminacy and one is reduced to classical

probability. Therefore, neutrosophic probability

may be viewed as a generalization of classical

and imprecise probabilities. - When a long-term option priced by the collective

action of the market players is observed to be

deviating from the theoretical price, three

possibilities must be considered - (1) The theoretical price is obtained by an

inadequate pricing model, which means that the

market price may well be the true price, - (2) An unstable rationalization loop has taken

shape that has pushed the market price of the

option out of sync with its true price, or - (3) The nature of the deviation is indeterminate

and could be due to (a) or (b) or a

super-position of both (a) and (b) and/or due to

some random white noise.

9. Applications of Fuzzy and Neutrosophic Logics

to Finance

- However, it is to be noted that in none of these

three possible cases are we referring to the

efficiency or otherwise of the market as a whole.

The market can only be as efficient as the

information it gets to process. Therefore, if the

information about the true price of the option is

misleading (perhaps due to an inadequate pricing

model), the market cannot be expected to process

it into something useful after all, the markets

cant be expected to pull jack-rabbits out of

empty hats! - With T, I, F as the neutrosophic components, let

us now define the following events - H p p is the true option price determined by

the theoretical pricing model and - M p p is the true option price determined by

the prevailing market price

- Then there is a t chance that the event (H ? Mc)

is true, or corollarily, the corresponding

complimentary event (Hc ? M) is untrue, there is

a f chance that the event (Mc ? H) is untrue, or

corollarily, the complimentary event (M ? Hc) is

true and there is a i chance that neither (H ?

Mc) nor (M ? Hc) is true/untrue i.e. the

determinant of the true market price is

indeterminate. This would fit in nicely with

possibility (c) enumerated above that the

nature of the deviation could be due to either

(a) or (b) or a super-position of both (a) and

(b) and/or due to some random white noise. - Illustratively, a set of AR1 models used to

extract the mean reversion parameter driving the

volatility process over time have coefficients of

determination in the range say between 50-70,

then we can say that t varies in the set T (50 -

70).

9. Applications of Fuzzy and Neutrosophic Logics

to Finance

- If the subjective probability assessments of

well-informed market players about the weight of

the current excursions in implied volatility on

short-term options lie in the range say between

40-60, then f varies in the set F (40 - 60).

Then unexplained variation in the temporal

volatility driving process together with the

subjective assessment by the market players will

make the event indeterminate by either 30 or

40. Then the neutrosophic probability of the

true price of the option being determined by the

theoretical pricing model is NP (H ? Mc) (50

70), (40 60), 30, 40. The DSmT (acronym for

Dezert-Smarandache Theory) can be used in cases

like these to fuse the conflicting sources of

information and arrive at a correct and

computable probabilistic assessment of the true

price of the long-term option. - Example 9.2 Extension of the MASS model as a

cost-optimal relative allocation of facilities

technique by the incorporation of neutrosophic

statistics and the DSmT combination rule.

9. Applications of Fuzzy and Neutrosophic Logics

to Finance

- The original CRAFT-type models for cost-optimal

relative allocation of facilities technique as

well as its later extensions are primarily

deterministic in nature. A Modified Assignment

(MASS) model (first proposed by Bhattacharya and

Khoshnevisan in 2003) follows the same iterative,

deterministic logic. However, some amount of

introspection will reveal that the facilities

layout problem is basically one of achieving best

interconnectivity by optimal fusion of spatial

information. In that sense, the problem may be

better modeled in terms of mathematical

information theory whereby the best layout is

obtainable as the one that maximizes relative

entropy of the spatial configuration. Going a

step further, one may hypothesize a neutrosophic

dimension to the problem. Given a DSmT type

combination rule, the layout optimization problem

may be framed as a normalized basic probability

assignment for optimally comparing between

several alternative interconnectivities. The

neutrosophic argument can be justified by

considering the very practical possibility of

conflicting bodies of evidence for the structure

of the load matrix possibly due to conflicting

assessments of two or more design engineers. - If for example we consider two mutually

conflicting bodies of evidence ?1 and ?2,

characterized respectively by their basic

probability assignments ?1 and ?2 and their cores

k (?1) and k (?2) then one has to look for the

optimal combination rule which maximizes the

joint entropy of the two conflicting information

sources. Mathematically, it boils down to the

general optimization problem of finding the value

of min? H (?) subject to the constraints

that the marginal basic probability assignments

?1 (.) and ?2 (.) are obtainable by the summation

over each column and summation over each row

respectively of the relevant information matrix

the sum of all cells of the information matrix is

unity.

9. Applications of Fuzzy and Neutrosophic Logics

to Finance

- Example 9.3 Conditional probability of actually

detecting a financial fraud a neutrosophic

extension to the application of Benfords

first-digit law - In an earlier paper (Kumar and Bhattacharya,

2002), we had proposed a Monte Carlo adaptation

of Benfords first-digit law. There has been some

research already on the application of Benfords

law to financial fraud detection. However, most

of the practical work in this regard has been

concentrated in detecting the first digit

frequencies from the account balances selected on

basis of some known audit sampling method and

then directly comparing the result with the

expected Benford frequencies. We have voiced

slight reservations about this technique in so

far as that the Benford frequencies are

necessarily steady state frequencies and may not

therefore be truly reflected in the sample

frequencies. As samples are always of finite

sizes, it is therefore perhaps not entirely fair

to arrive at any conclusion on the basis of such

a direct comparison, as the sample frequencies

wont be steady state frequencies. - However, if we draw digits randomly using the

inverse transformation technique from within

random number ranges derived from a cumulative

probability distribution function based on the

Benford frequencies then the problem boils down

to running a goodness of fit kind of test to

identify any significant difference between

observed and simulated first-digit frequencies.

This test may be conducted using a known sampling

distribution like for example the Pearsons ?²

distributions. The random number ranges for the

Monte Carlo simulation are to be drawn from a

cumulative probability distribution function

based on the following Benford probabilities

given in Table I below.

9. Applications of Fuzzy and Neutrosophic Logics

to Finance

- The first-digit probabilities can be best

approximated mathematically by the log-based

formula as was derived by Benford P (First

significant digit d) log10 1 (1/d). - 9.4. Computational Algorithm (first proposed by

Kumar and Bhattacharya in 2002) - Define a finite sample size n and draw a sample

from the relevant account balances using a

suitable audit sampling procedure -

- Perform a continuous Monte Carlo run of length ?

? (1/2?)2/3 grouped in epochs of size n using a

customized MS-Excel spreadsheet. Derivation of ?

and other statistical issues have been discussed

in detail in our earlier paper (Kumar and

Bhattacharya, 2002) - Test for significant difference in sample

frequencies between the first digits observed in

the sample and those generated by the Monte Carlo

simulation by using a goodness of fit test

using the ?² distribution. The null and

alternative hypotheses are as follows - H0 The observed first digit frequencies

approximate a Benford distribution - H1 The observed first digit frequencies do not

approximate a Benford distribution

9. Applications of Fuzzy and Neutrosophic Logics

to Finance

- This statistical test will not reveal whether or

not a fraud has actually been committed. All it

does is establishing at a desired level of

confidence that the accounting data may not be

naturally occurring (if H0 can be rejected).

However, given that H1 is accepted and H0 is

rejected, it could possibly imply any of the

following events - I. There is no manipulation - occurrence of a

Type I error i.e. H0 rejected when true. - II. There is manipulation and such manipulation

is definitely fraudulent. - III. There is manipulation and such manipulation

may or may not be fraudulent. - IV. There is manipulation and such manipulation

is definitely not fraudulent. - 9.5. Neutrosophic extension using DSmT

combination rule - Neutrosophic probabilities are a generalization

of classical and fuzzy probabilities and cover

those events that involve some degree of

indeterminacy. It provides a better approach to

quantifying uncertainty than classical or even

fuzzy probability theory. Neutrosophic

probability theory uses a subset-approximation

for truth-value as well as indeterminacy and

falsity values. Also, this approach makes a

distinction between relative true event and

absolute true event the former being true in

only some probability sub-spaces while the latter

being true in all probability sub-spaces.

Similarly, events that are false in only some

probability sub-spaces are classified as

relative false events while events that are

false in all probability sub-spaces are

classified as absolute false events. Again, the

events that may be hard to classify as either

true or false in some probability sub-spaces

are classified as relative indeterminate events

while events that bear this characteristic over

all probability sub-spaces are classified as

absolute indeterminate events.

9. Applications of Fuzzy and Neutrosophic Logics

to Finance

- In classical probability n_sup ? 1 while in

neutrosophic probability n_sup ? 3, where we

have n_sup as the upper bound of the probability

space. In cases where the truth and falsity

components are complimentary, i.e. there is no

indeterminacy, the components sum to unity and

neutrosophic probability is reduced to classical

probability as in the tossing of a fair coin or

the drawing of a card from a well-shuffled deck.

Coming back to our original problem of financial

fraud detection, let E be the event whereby a

Type I error has occurred and F be the event

whereby a fraud is actually detected. Then the

conditional neutrosophic probability NP (F Ec)

is defined over a probability space consisting of

a triple of sets (T, I, U). Here, T, I and U are

probability sub-spaces wherein event F is t

true, i indeterminate and u untrue

respectively, given that no Type I error

occurred. - The sub-space T within which t varies may be

determined by factors such as past records of

fraud in the organization, propensity to commit

fraud by the employees concerned, and

effectiveness of internal control systems. On the

other hand, the sub-space U within which u varies

may be determined by factors like personal track

records of the employees in question, the

position enjoyed and the remuneration drawn by

those employees. For example, if the magnitude of

the embezzled amount is deemed too frivolous with

respect to the position and remuneration of the

employees involved. The sub-space I within which

i varies is most likely to be determined by the

mutual inconsistency that might arise between the

effects of some of the factors determining T and

U. For example, if an employee is for some

reason really irked with the organization, then

he or she may be inclined to commit fraud not so

much to further his or her own interests as to

harm. The DSmT combination rule can be used in

such a circumstance to remove the mutual

inconsistency in the factors deciding T and U.

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