Krzysztof Piasecki
An axiomatic definition of fuzzy present value
from the economic point-view
Department of Operations Research, Poznań University of Economics,
al. Niepodległości 10, Poznań, 61-875 Poland, k.piasecki@ue.poznan.pl ,
tel.:+48721771572
Abstract: The concept of present value (PV) may be defined as the cash flow utility which
additionally satisfies that: 1 the utility of current payment is equal to its nominal value; 2
the utility is odd function of payment nominal value. Obtained in this way axiomatic PV is
called generalized PV definition because of it is more general than with the Peccati’s
definition of PV. Due behavioural reasons the PV may be imprecisely valued. The fuzzy set
theory is applied as effective tool of this imprecision description. Proposed by Calzi’s
axiomatic definition of fuzzy PV is strictly connected with the Peccati’s definition which is
weakly coherent with the economics. It is shown that from economical point view the general
PV definition is better than the Peccati’s one. Thus we take into account the generalized
definition of PV as the starting point for construction of a new axiomatic fuzzy PV definition.
All axioms of generalized PV definition are extended to the case when the imprecise present
value is given as fuzzy number. Attached at the end mathematical Appendix contains explicit
definitions of all concepts of mathematics for fuzzy systems which are applied in this paper.
Keywords: imprecision, present value, fuzzy number,
1. Research problem
The current equivalent value of payments available in a fixed point in time is called the
present value (PV for short) of this payment. In financial arithmetic any PV is used as
discounting function for dynamic assessment of the money value. The starting point for the
financial arithmetic development was the interest theory. Further development of the financial
arithmetic theoretical foundations has resulted in the formulation axioms of the financial
arithmetic theory (Peccati, 1972). Using this theoretical approach Peccati has defined PV as
an additive function of payment value. This theory has been extensively developed in recent
years (Janssen et al, 2009).
The Peccati’s definition of PV is generalized in (Piasecki, 2012)
where PV is defined as a utility of the multicriterion comparison determined by the temporal
preference (Mises, 1962) and the wealth preference. If PV is defined as utility, then it may be
non-additive function of payment value.
The considerations set out in this article had their genesis in acceptance of view that without the interest theory - PV of future cash flow may be imprecise. The natural
consequence of this approach is the assessment PV using fuzzy numbers.
Ward (1985) defines a fuzzy PV as a discounted fuzzy cash flow. The fuzzy cash flow
used here is interpreted as an imprecise forecast of future crisp cash flow. Ward’s definition is
generalized to the case of a fuzzy duration by Greenhut et al. (1995).
Sheen (2005)
generalizes Ward’s definition to the case of a fuzzy interest rate. Buckley (1987, 1992),
Gutierrez (1989), Kuchta (2000) and Lesage (2001) discuss some problems connected with
the application of fuzzy arithmetic to calculating fuzzy PV.
According to the uncertainty thesis expressed by Mises (1961) and Kaplan et al. (1967),
each future cash flow is uncertain. In the crisp case, this uncertainty is usually modeled in
such a way that cash flow is described as a random variable. Therefore, Huang (2007a)
generalizes Ward’s definition for the case when future cash flow is given as a fuzzy random
variable in the sense given by Kwakernaak (1978, 1979). A more general definition of a fuzzy
PV is proposed by Tsao (2005), who assumes that the future cash flow is a fuzzy probabilistic
set in the sense given by Hiroto (1981). All these authors present the PV, as the discount of
imprecisely estimated future cash flow value.
A different approach is presented in (Piasecki, 2011), where imprecisely estimated PV
was based on the current market price of a financial asset. The lack of precise in PV
estimation was there justified by behavioural premises. Then imprecise assessment of PV is
given as fuzzy trapezoidal fuzzy number in the sense given by Campos and Gonzales (1989)
An axiomatic definition of fuzzy PV was given by Calzi (1990). The Calzi’s definition
is closely related to the Peccati’s definition of PV. Hence, the Calzi’s definition is valid only
for cases when PV is additive function of payment.
The main aim of this article is to propose such an axiomatic definition of PV which will
be valid also for the case when the PV may be non-additive function of payment. The starting
point for discussion will be the Piasecki’s generalization of Peccati’s definitions. An
important premise for giving final form to the axiomatic definition will be experience gained
by the author during his research.
This article is addressed to two groups of readers. The results obtained may be of
interest to quantitative finance theorists and to practitioners constructing support systems for
investment decisions.
2. Axiomatic definitions of present value
The subject of financial arithmetic is dynamic estimation of the value of money. The
fundamental assumption of financial arithmetic is the certainty that the nominal value of
money in circulation increases with time. In general, this assumption is justified by the
analysis of the quantity of money equation proposed by Irving Fisher (Begg et al, 2005). This
analysis is based on the additional assumption that the total amount of money is constant. This
is a typical normative assumption. Therefore, the value of money considered in financial
arithmetic is called the normative value of money. The growth process of the normative value
of money is called the process of capital appreciation. On the other hand, economic-financial
practice generally results in the increase in the amount of money being faster than the increase
in production volume. Thus we observe a decrease in the real value of money. This means
that the normative value of money cannot be identified with the real value of money. This
raises a question about the essence of the concept of normative value. One consequence of
this question is another question about the essence of the basic functions of financial
arithmetic.
The answer to this question was sought through the development of an axiomatic theory
for financial arithmetic. Let the set of moments Θ ⊆ [0, +∞[ be given. In this particular case,
it may be interpreted as the set of possible capitalization moments or the non-negative halfline depicting time. In financial market analysis, each payment is represented by a financial
instrument described as financial flow (𝑡, 𝐶) ∈ Φ = Θ × ℝ, where the symbol 𝑡 ∈ Θ denotes
the moment of the flow and the symbol 𝐶 ∈ ℝ denotes the nominal value of this flow. Each of
these financial flows can be either an executed receivable or matured liability. The nominal
value of any receivable is non-negative. The debtor's liabilities are always the creditor’s
receivables. In this situation, the value of each liability is equal to the negative of the value of
corresponding receivable. The set Φ is called the set of all payments. In addition, by the
symbol Φ+ = Θ × ℝ0+ we denote the set of all receivables.
For any set Φ of all payments Peccati (1972) has defined1 future value as a function
𝐹𝑉: Φ ⟶ ℝ satisfying the following properties:
∀𝐶∈ℝ : 𝐹𝑉(0, 𝐶) = 𝐶,
(1)
∀(𝑡,𝐶)∈Φ+ ∀Δ𝑡>0 : 𝐹𝑉(𝑡 + Δ𝑡, 𝐶) > 𝐹𝑉(𝑡, 𝐶),
∀(𝑡,𝐶1 ),(𝑡,𝐶2)∈Φ :
𝐹𝑉(𝑡, 𝐶1 ) + 𝐹𝑉(𝑡, 𝐶2 ) = 𝐹𝑉(𝑡, 𝐶1 + 𝐶2 ).
(2)
(3)
Peccati defined PV as the function 𝑃𝑉: Φ ⟶ ℝ, uniquely determined by the identity
𝐹𝑉(𝑡, 𝑃𝑉(𝑡, 𝐶)) = 𝐶.
(4)
Defined above PV fulfils the conditions
∀𝐶∈ℝ : 𝑃𝑉(0, 𝐶) = 𝐶,
(5)
∀(𝑡,𝐶)∈Φ+ ∀Δ𝑡>0 : 𝑃𝑉(𝑡 + Δ𝑡, 𝐶) < 𝑃𝑉(𝑡, 𝐶),
∀(𝑡,𝐶1 ),(𝑡,𝐶2)∈Φ :
𝑃𝑉(𝑡, 𝐶1 ) + 𝑃𝑉(𝑡, 𝐶2 ) = 𝑃𝑉(𝑡, 𝐶1 + 𝐶2 ).
(6)
(7)
Peccati proves that if the fixed PV satisfies conditions (5), (6) and (7) then uniquely
determined by the identity (4) future value satisfies conditions (1), (2) and (3). This means
that the Peccati’s axiomatic theory may be equivalently developed on the basis of PV defined
as any function 𝑃𝑉: Φ ⟶ ℝ satisfying the conditions (5), (6) and (7). Thus a coherent theory
of financial arithmetic was created. Among other things we have here the following theorem:
“Any function 𝑃𝑉: Φ ⟶ ℝ satisfies the conditions (5), (6) and (7) iff it is given by the
identity
𝑃𝑉(𝑡, 𝐶) = 𝐶 ∙ 𝑣(𝑡),
(8)
where the discount factor 𝑣: Θ → ]0; 1] is a decreasing function of time satisfying
𝑣(0) = 1.
(Peccati, 1972) “
(9)
For a fixed value of payment the PV notion is reduced to the discounted utility notion
considered by many researchers. Multithreaded results of studies on the discounted utility
were competently discussed in (Doyle, 2013). Among other things, there are pointed out that
all discussed discounting models can be represented as the product of the discounted payment
1
The conditions (2) and (6) are equivalent to analogous original conditions of temporal monotonicity used by
Peccati. Applied here the descriptions of temporal monotonicity are more convenient for our further
considerations.
and the discount factor dependent on the payment time only. This means that the problem of
PV depending on the interaction of payment value and payment time is omitted in the
Peccati’s theory.
The effect of capital synergy means that an increase in the capital value causes an
increase in the relative speed of capital appreciation. Using the Peccati’s theory we over the
synergy capital effect (Piasecki, 2012).
Investment diversification is understood as the allocation of resources among different
investments. The diversification principle has been popularized by the portfolio theory
introduced by (Markowitz, 1952).
This principle states that diversification should be
preferred. Applying the Peccati theory we ignore the diversification principle (Piasecki,
2012).
All the above facts point to weak coherence the Peccati’s theory with the theory and
practice of economics. Among other things it implies that the interest theory is weakly
coherent with economy.
The Peccati’s axiomatic approach has been extensively studied by many researchers.
Current knowledge on the consequences of this axiomatic approach to the concept of PV is
presented in (Janssen et al, 2009).
The process of capital appreciation is described above by the axiom (2). On the other
hand, the Peccati’s theory does not explain the phenomenon of capital appreciation. This
explanation was obtained by showing that any payment PV is identical with the utility of the
financial flow representing this payment (Piasecki, 2012). Then the PV is defined as any
function 𝑃𝑉: Φ ⟶ ℝ satisfying the conditions (5), (6) and
∀(𝑡,𝐶)∈Φ ∀Δ𝐶>0 :
𝑃𝑉(𝑡, 𝐶) < 𝑃𝑉(𝑡, 𝐶 + Δ𝐶) .
∀(𝑡,𝐶)∈Φ : 𝑃𝑉(𝑡, −𝐶) = −𝑃𝑉(𝑡, 𝐶) ,
(10)
(11)
Fulfillment of condition (11) may be obtained via the assumption that the utility of any
liability is non-positive2. Presented above definition is a generalization of the Peccati’s
definition of PV. Therefore above definition is called generalized one.
2
The notion of negative utility was discussed in the papers (Becker et al, 1960), (Cooper et al, 2001) and
(Rabin, 1993)
In (Piasecki, 2015) we can find examples of such PV which are not additive function of
payment value. The variability of these PV is fully justified by economic reasons.
The
significance of generalization the Peccati’s definition to the PV generalized definition is
shown in this way.
The synergy capital effect and the diversification principle may be considered in the
framework of financial arithmetic based on the generalized definition of PV (Piasecki, 2012).
Any payment PV is identical with the utility of the financial flow representing this
payment. Therefore the Gossen’s First Law3 may be considered as an additional PV feature.
For the case of any function 𝑃𝑉: Φ ⟶ ℝ tis law is described by the inequality
∀(𝑡,𝐶1 ),(𝑡,𝐶2)∈Φ+ ∀α∈]0;1[ ∶ 𝛼 ∙ 𝑃𝑉(𝑡, 𝐶1 ) + (1 − 𝛼)𝑃𝑉(𝑡, 𝐶2 ) < 𝑃𝑉(𝑡, 𝛼𝐶1 + (1 − 𝛼)𝐶2 ) (12)
The example of PV satisfying the Gossen’s first Law is given in (Piasecki, 2014). On the
other side each PV satisfying the conditions (1), (2) and (3) does not fulfil the condition (12).
As we see the Peccati’s theory rejects the Gossen’s First Law, too.
The relationship presented above between the utility of wealth and present value shows
the logical consistency of formal economics and financial models. Finding such relationships
is especially important now when we have become participants in the global financial crisis
triggered by financial management in isolation from the fundamental base created by the
economy.
All above facts show the usefulness of this new generalized theory of financial
arithmetic. The generalized definition of PV is more coherent with economics than the
Peccati’s definition. Therefore we take into account the generalized definition of PV as the
starting point for our further considerations.
3. Fuzzy present value
The notion of PV is subjective in its nature, because it is identical to the utility of
financial flow. Subjective evaluations always depend on behavioral factors. In essence, the
behavioural environment states are defined imprecisely. For this reason it is necessary to
3
The Gossen’s First Law says that the marginal utility of wealth is diminishing (Begg et al, 2005).
disclose the lack of precision in estimating the PV. Effective tool of the imprecision
description is the fuzzy set theory4.
The first axiomatic approach to fuzzy PV was given by Calzi (1990) who takes into
account the condition (8) as start-point for his considerations. He assumed that future flow
value, flow moment and interest rate are imprecise. All of these values are described by
means of fuzzy appraisal5. At the first Calzi proved that in this situation the discounting factor
is also fuzzy appraisal. Calzi finally defines fuzzy PV by applying the Zadeh’s extension
principle for the condition (8). Obtained in this way fuzzy PV is fuzzy appraisal.
Due the condition (8), the Calzi’s approach to fuzzy PV is strictly connected with the
Peccati’s definition which is weakly coherent with the economics. Moreover, the Peccati’s
axioms (5), (6) and (7) are passed over in the Calzi’s definition. These facts show that the
reconsideration of the fuzzy PV definition is necessary.
Any imprecise PV should be an approximation of a value determined by the normative
PV function. Therefore we consider the financial space (Φ, 𝑃𝑉), where 𝑃𝑉: Φ ⟶ ℝ is any
fixed function fulfilling the axioms (5), (6), (10) and (11) of generalized PV definition. This
function assigns to the financial stream (𝑡, 𝐶) ∈ Φ its exact normative PV. Then imprecise PV
may be expressed, as fuzzy number ℛ̃ (𝑃𝑉(𝑡, 𝐶)) ∈ ℱ(ℝ). In this way, all imprecise
evaluations of PV financial flows constitute a family Ξ = {𝜇(∙ |𝑡, 𝐶): (𝑡, 𝐶) ∈ Φ} ⊂ [0; 1]ℝ of
fuzzy number memberships function. From the point-view of multi-valued logics, for fixed
𝑥 ∈ ℝ the value 𝜇(𝑥|𝑡, 𝐶) may be interpreted as truth value of the sentence
𝑥 = 𝑃𝑉(𝑡, 𝐶) .
(13)
Therefore the value 𝜇(𝑥|𝑡, 𝐶) is called the degree of fulfilment the above equality.
According to the definition of fuzzy numbers (Dubois, Prade, 1979), each membership
function 𝜇(∙ |𝑡, 𝐶): ℝ ⟶ [0; 1] satisfies the conditions
𝜇(𝑃𝑉(𝑡, 𝐶)|𝑡, 𝐶) = 1 ,
∀𝑥,𝑦,𝑧𝜖ℝ : 𝑥 ≤ 𝑦 ≤ 𝑧 ⇒ 𝜇(𝑦|𝑡, 𝐶) ≥ 𝑚𝑖𝑛{𝜇(𝑥|𝑡, 𝐶), 𝜇(𝑧|𝑡, 𝐶)}.
4
Attached at the end Appendix contains explicit definitions of all these concepts of mathematics for fuzzy
systems which are used in this Section.
5
fuzzy interval in original Calzi’s work
(14)
(15)
When we are aiming to replace the exact normative assessment 𝑃𝑉(𝑡, 𝐶) by its
approximation, then for the family of membership functions Ξ we impose the conditions
which are the generalization axioms (5), (6), (10) and (11) to the fuzzy case.
The condition (5) may be written in an equivalent way as follows
1
∀𝐶∈ℝ : 𝜇(𝑥|0, 𝐶) = {
0
𝑥=𝐶
.
𝑥≠𝐶
(16)
Using the Zadeh’s extension principle, we generalize the condition (11) to the condition
∀(𝑡,𝐶)∈Φ : 𝜇(𝑥|𝑡, −𝐶) = 𝜇(−𝑥|𝑡, 𝐶).
(17)
Unlike thing happens with the conditions (6) i (10) determining the PV, as the discounted
utility of cash flow. Any of these conditions can be replaced respectively by the inequalities
∀(𝑡,𝐶)∈Φ+ ∀Δ𝑡>0 : ℛ̃ (𝑃𝑉(𝑡 + Δ𝑡, 𝐶)) ≺ ℛ̃ (𝑃𝑉(𝑡, 𝐶)),
∀(𝑡,𝐶)∈Φ ∀Δ𝐶>0 :
ℛ̃ ( 𝑃𝑉(𝑡, 𝐶)) ≺ ℛ̃ (𝑃𝑉(𝑡, 𝐶 + Δ𝐶)) .
(18)
(19)
Each of the above-described relationship is a fuzzy relation. In this situation, the truth value
of the sentences (18) and (19) may be assessed only in terms of multi-valued logics. It means
the necessity of taking this into account when the conditions (6) and (10) are extended to the
fuzzy case. To meet this postulate we apply the membership function 𝜈≺ : [ℱ(ℝ)]2 → [0,1] of
relations used in the inequalities (18) i (19).
For any fixed pair (𝑡, 𝐶) ∈ Φ+ let us consider the function 𝑓𝑡 : ℝ+ ⟶ [0; 1] given by
the identity
𝑓𝑡 (Δ𝑡) = 𝜈≺ (ℛ̃ (𝑃𝑉(𝑡 + Δ𝑡, 𝐶)), ℛ̃ (𝑃𝑉(𝑡, 𝐶)))
(20)
From the point-view of multi-valued logics, for fixed (𝑡, 𝐶) ∈ Φ+ the value 𝑓𝑡 (Δ𝑡) may be
interpreted as truth value of the sentence (18). Therefore the value 𝑓𝑡 (Δ𝑡) is called the degree
of fulfilment this inequality. We can expect that with the increase in change Δ𝑡 the degree of
fulfilment the inequality (18) is not decreasing. According to (33), this condition is equivalent
to the condition:
[A] For any fixed pair (𝑡, 𝐶) ∈ 𝛷+ the function 𝑔𝑡 : ℝ+ ⟶ [0; 1] given by the identity
𝑔𝑡 (Δ𝑡) = sup{min{𝜇(𝑥|𝑡 + Δ𝑡, 𝐶), 𝜇(𝑥|𝑡, 𝐶)}: 𝑥 ∈ ℝ}
(21)
is nonincreasing function.█
In the crisp case of non-fuzzy PV the condition [A] is reduced to the condition (6). All of the
above insights cause that the condition [A] may act out as extension of the condition (6) to the
fuzzy case.
For any fixed pair (𝑡, 𝐶) ∈ Φ let us consider the function 𝑓𝐶 : ℝ+ ⟶ [0; 1] given by the
identity
𝑓𝐶 (Δ𝐶) = 𝜈≺ (ℛ̃ (𝑃𝑉(𝑡, 𝐶)), ℛ̃ (𝑃𝑉(𝑡, 𝐶 + Δ𝐶)))
(22)
From the point-view of multi-valued logics, for fixed (𝑡, 𝐶) ∈ Φ the value 𝑓𝐶 (Δ𝐶) may be
interpreted as truth value of the sentence (19). Therefore the value 𝑓𝐶 (Δ𝐶) is called the degree
of fulfilment this inequality. We can expect that with the increase in change Δ𝐶 the degree of
fulfilment the inequality (19) is not decreasing. According to (33), this condition is equivalent
to the condition:
[B] For any fixed pair (𝑡, 𝐶) ∈ Φ the function 𝑔𝐶 : ℝ+ ⟶ [0; 1] given by the identity
𝑔𝐶 (Δ𝐶) = sup{min{𝜇(𝑥|𝑡, 𝐶), 𝜇(𝑥|𝑡, 𝐶 + Δ𝐶)}: 𝑥 ∈ ℝ}
(23)
is nonincreasing function.█
In the crisp case of non-fuzzy PV the condition [B] is reduced to the condition (10). All of the
above insights cause that the condition [B] may act out as extension of the condition (10) to
the fuzzy case.
̃ : Φ ⟶ ℱ(ℝ) given by the identity
In this way we define estimated PV as function 𝑃𝑉
̃ (𝑡, 𝐶) = ℛ̃ (𝑃𝑉(𝑡, 𝐶)),
𝑃𝑉
(24)
where ℛ̃ (𝑃𝑉(𝑡, 𝐶)) is such fuzzy number that its membership function 𝜇(∙ |𝑡, 𝐶): ℝ ⟶ [0; 1]
fulfills the conditions (14), (15), (16), (17) [A] and [B].
4. Concluding remarks
According to the uncertainty thesis (Mises, 1961; Kaplan et al., 1967), each
anticipated future value is uncertain. Thus future value is usually described as a random
variable. In this situation, if the present value is fuzzy then any return rate is given as fuzzy
probabilistic set. Despite this modernization, in the proposed model it is possible to apply the
collected rich empirical knowledge about probability distribution of return rate. In this way
we can simultaneously take into account the subjective and empirical circumstances to make
investment decisions. When the rate of return is given as a fuzzy probabilistic set then it is at
the composition of non-knightian uncertainty risk and imprecision risk. The concept such
defined return rate may be a starting point for the development of a general theory of financial
markets at imprecision risk. Elements of such theory are formulated in (Piasecki 2011 b,
2014a) for the case of imprecise PV defined as any fuzzy number. Application the conditions
(16), (17) [A] and [B] of the estimated PV definition will make the thesis of this theory more
specific. Then we can obtain new original conclusions. All this will allow us to propose and
explore new investment strategies.
Estimated PV model may be used not only to determine return rate as fuzzy
probabilistic set. This model may be applied wherever there fuzzy evaluation of PV is used.
Examples of such applications can be found in the works (Boussabaine, Elhag, 1999), (Chiu,
Park, 1994), (Fang Yong et al, 2008), (Huang, 2007 a, b) and (Haifeng et al, 2012).
Summing up, the use of estimated PV significantly increases the possibility of analysis
of financial markets. This is a highly advantageous feature of the proposed model, since it
brings the possibility of real applications. Thus proposed in this article axiomatic definition of
estimated PV is a significant contribution to the development of a financial markets formal
theory.
4. Appendix
The symbol ℱ(ℝ) indicates the family of all fuzzy subsets in the real numbers space ℝ.
Each fuzzy set 𝐴̃ ∈ ℱ(𝕏) jest described by means of its membership function 𝜇𝐴 : ℝ → [0; 1].
Any subset 𝐴̃ ∈ ℱ(ℝ) represents the imprecise estimation. In addition, if its
membership function 𝜇𝐴 ∈ [0,1]ℝ satisfies the condition
∀𝑥,𝑦,𝑧𝜖ℝ : 𝑥 ≤ 𝑦 ≤ 𝑧 ⇒ 𝜇𝐴 (𝑦) ≥ min{𝜇𝐴 (𝑥), 𝜇𝐴 (𝑧)},
(25)
then this imprecise estimation is called fuzzy appraisal.
Any real number 𝑘𝜖ℝ may be approximately estimated using fuzzy number ℛ̃ (𝑘) (Dubois,
Prade, 1979) which is a particular case of fuzzy appraisal represented by its membership
function 𝜇(∙ |𝑘): ℝ → [0; 1] fulfilling the condition
𝜇(𝑘|𝑘) = 1 ,
(26)
Imprecise estimates can be used to ordering objects represented by these estimates. At the
outset, we define the order relations on the set of all imprecise estimates ℱ(ℝ). These order
relations are defined only by means of the Zadeh’s extension principle. In a situation when
compared estimates are fuzzy sets on the real line, this ordering should be designated as fuzzy
preorder (Orlovsky, 1978).
Let us take into account the estimations 𝐴̃, 𝐵̃ 𝜖ℱ(ℝ) represented respectively by their
membership functions 𝜇𝐴 , 𝜇𝐵 𝜖[0,1]ℝ . Fuzzy preorder ≼ is defined by the equivalence
̃”.
𝐴̃ ≼ 𝐵̃ ⟺ „The estimation 𝐴̃ is less or equal than the estimation B
(27)
The preorder ≼ is determined by its membership function 𝜈≼ : [ℱ(ℝ)]2 → [0,1] given by the
identity
𝜈≼ (𝐴̃, 𝐵̃ ) = sup{min{𝜇𝐴 (𝑥), 𝜇𝐵 (𝑦)}: 𝑥 ≤ 𝑦}.
(28)
Fuzzy strict order ≺ is defined by the equivalence
̃”.
𝐴̃ ≺ 𝐵̃ ⟺ „The estimation 𝐴̃ is less than the estimation B
(29)
The strict ≺ is determined by its membership function 𝜈≺ : [ℱ(ℝ)]2 → [0,1] given by the
identity
𝜈≺ (𝐴̃, 𝐵̃ ) = min{𝜈≼ (𝐴̃, 𝐵̃ ), 1 − 𝜈≼ (𝐵̃ , 𝐴̃)}.
(30)
For Let us consider the pair of real numbers 𝑘, 𝑙 ∈ ℝ such that 𝑘 < 𝑙 . We take into account a
pair of estimations given as fuzzy numbers ℛ̃ (𝑘), ℛ̃ (𝑙) determined respectively by their
membership functions (∙ |𝑘), 𝜇(∙ |𝑙) ∈ [0; 1]ℝ . From the condition (26) we obtain
1 ≥ sup{min{𝜇(𝑥|𝑘), 𝜇(𝑦|𝑙)}: 𝑥 ≤ 𝑦} ≥ min{𝜇(𝑘|𝑘), 𝜇(𝑙|𝑙)} = 1
,
(31)
On the other side, from the condition (25) we have
sup{min{𝜇(𝑥|𝑙), 𝜇(𝑦|𝑘)}: 𝑥 ≤ 𝑦} = sup{min{𝜇(𝑥|𝑙), 𝜇(𝑥|𝑘)}: 𝑥 ∈ ℝ} .
(32)
Due these facts we can say that
𝜈≺ (ℛ̃ (𝑘), ℛ̃ (𝑙)) = 1 − sup{min{𝜇(𝑥|𝑙), 𝜇(𝑥|𝑘)}: 𝑥 ∈ ℝ} .
The last identity will be applied for determining the axiomatic definition of fuzzy PV.
(33)
5. Acknowledgments
The project was financed by funds of National Science Center - Poland granted on the basis
the decision number DEC-2012/05/B/HS4/03543
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