JOURNAL OF COMPUTING, VOLUME 2, ISSUE 6, JUNE 2010, ISSN 2151-9617 HTTPS://SITES.GOOGLE.COM/SITE/JOURNALOFCOMPUTING/ WWW.JOURNALOFCOMPUTING.ORG 108 menon of uncertainty is based on the device of probability theory. The vagueness concerns to a way of the descrip‐ tion of the event and does not consider a question about it appearance. The mathematical model of the phenomenon of the vagueness is based on the device of the theory of vague sets. In [3] it is noted, that these phenomena are represented like two sides supplementing each other of the most general phenomenon which authors name *inde*‐ terminacy. From our point of view indeterminacy, consi‐ dered in [3] is very well described by the fuzzy experi‐ ment offered in work by O.Yu.Vorob'ov [4]. A vague experiment is inconceivable without participa‐ tion of set of the individual reasonable subjects making an integral part of vague experiment. Each individual rea‐ sonable subject is the participant of vague experiment and has his own opinion on, whether it is possible to charac‐ terize event in vague experiment by the given syntagma or not. All of their opinions form vague *event* as an out‐ come of vague experiment. Letʹs illustrate formalization of a E‐linguistic variable ʺAgeʺ. Letʹs consider vague experiment in which partici‐ pate M reasonable subjects. Each reasonable subject
  M is characterized by syntagmas (names) from set X. X={x, y}, where x is  ʺthe young manʺ, y is ʺthe young womanʺ for example. Then the intercept [0, 80] of the real axis acts as a set of elementary events Ω. The age of the concrete person corresponds to an elementary event  . Let (Ω, F, P) is probabilistic space. Letʹs define a matrix of the selected random *events* [4] as a set of events XM ={ x, x  X ,  M }, where x  F are measurable random events concerning algebra. Thus, each pair (x,) X  M defines one random event x   . In our example random event
[ , ]  a b x  is a judgment of reason μ, in which it cor‐
relates an age intercept [a,b]   to a syntagma
(name) (tab. 1).

Tab.1. Мatrix of the selected events 

![0_image_1.png](0_image_1.png)

The eventology [4] defines M‐vague event as set of usual Kolmogorov's events, when   M . Thus, sets of elements of rows of a matrix of selected events x {x , M} M     define vague events M x  x ~   and form the set of M‐vague events , } ~{ ~X  x x  X . In our example we consider a set of M‐vague events }~, ~{ ~X  x y , generated by the set X, where
{[ , ], } ~x  a b   M is ʺthe young manʺ and
{[ , ], } ~y  c d   M   M ʺthe young *woman*ʺ.

In the Gau and Buehrer's theory the vague set is defined by membership function on U, which kind cannot be de‐ duced theoretically from more simple concepts, and it is established in each problem, proceeding from external in relation to the theory of consideration. Eventology une‐ quivocally defines eventological membership *function* of va‐ gue event as the indicator of vague event

$$1_{\bar{x}}(\omega)=\frac{1}{\mid M\mid}\sum_{\mu\in M}1_{x_{\mu}}(\omega),\;\;x\in X,$$
$$\left({\mathfrak{I}}{\mathfrak{I}}\right)$$
$$\sigma$$
    (3)
Where   ()  x 1  ‐ indicators of ʺusualʺ events    x which take sets of events x
~ . In our example (3) for correspond‐
ing vague events from the X it will be transformed to a kind (fig. 1):

$$\begin{array}{c}{{\mathbf{1}_{\bar{x}}(\omega)=\frac{1}{\mid M\mid}\sum_{\mu\in M}\mathbf{1}_{[a_{\mu},b_{\mu}]}(\omega)\,,}}\\ {{\mathbf{1}_{\bar{y}}(\omega)=\frac{1}{\mid M\mid}\sum_{\mu\in M}\mathbf{1}_{[c_{\mu},d_{\mu}]}(\omega)\,.}}\end{array}$$

![0_image_0.png](0_image_0.png)

young man" and y
~ - "*the young woman"*, 
constructed as a result of vague experiment in which par‐ ticipated |M | = 71 reasonable subjects.It is necessary to notice, that membership functions of vague sets, con‐ structed by a method of expert estimations, under certain conditions coincide with the   indicator of corresponding vague event. Thus, we formalized atomic syntagmas ʺthe young manʺ and ʺthe young *woman*ʺ by E‐linguistic variable L: *= ʺAge*ʺ by vague events x
~ and y
~ .

Links, uncertainty and the negation defined by the syntac‐ tic procedure GE, are formalized by means of set‐ operations above set of M‐vague events , } ~{ ~X  x x  X , generated by the set X [4] ac‐
cording to Minkovsky. The basic eventological theorem