Ratio Mathematica
Vol.34, 2018, pp. 35-47

ISSN: 1592-7415
eISSN: 2282-8214

On a Geometric Representation of
Probability Laws and of a Coherent

Prevision-Function According to
Subjectivistic Conception of Probability

Pierpaolo Angelini∗, Angela De Sanctis†

Received: 22-03-2018. Accepted: 30-05-2018. Published: 30-06-2018

doi:10.23755/rm.v34i0.401

c©Pierpaolo Angelini and Angela De Sanctis

Abstract

We distinguish the two extreme aspects of the logic of certainty by identify-
ing their corresponding structures into a linear space. We extend probability
laws P formally admissible in terms of coherence to random quantities. We
give a geometric representation of these laws P and of a coherent prevision-
function P which we previously defined. This work is the foundation of our
next and extensive study concerning the formulation of a geometric, well-
organized and original theory of random quantities.
Keywords: metric; collinearity; vector subspace; convex set; linear depen-
dence
2010 AMS subject classifications: 51A05; 60A02; 60B02.

∗MIUR, Roma, Italia. pierpaolo.angelini@istruzione.it
†DEA, Univerity “G. D’Annunzio” of Chieti-Pescara, Pescara, Italia an-

gela.desanctis@unich.it

35



Pierpaolo Angelini and Angela De Sanctis

1 Introduction

An event E is conceptually a mental separation between subjective sensations:
it is actually a proposition or statement such that, by betting on it, we can establish
in an unmistakable fashion whether it is true or false, that is to say, whether it has
occurred or not and so whether the bet has been won or lost ([9], [15]). For any
individual who does not certainly know the true value of a quantity X, which is
random in a non-redundant usage for him, there are two or more than two possi-
ble values for X. The set of these values is denoted by I(X). In any case, only
one is the true value of each random quantity and the meaning that you have to
give to random is the one of unknown by the individual of whom you consider his
state of uncertainty or ignorance. Thus, random does not mean undetermined but
it means established in an unequivocal fashion, so a supposed bet based upon it
would unmistakably be decided at the appropriate time. When one wonders if infi-
nite events of a set are all true or which is the true event among an infinite number
of events, one can never verify if such statements are true or false. These state-
ments are infinite in number, so they do not coincide with any mental separation
between subjective sensations: they are conceptually meaningless. Hence, we can
understand the reason for which it is not a logical restriction to define a random
quantity as a finite partition of incompatible and exhaustive events, so one and
only one of the possible values for X belonging to the set I(X) = {x1, . . . ,xn} is
necessarily true. A random quantity is dealt with by the logic of certainty as well
as by the logic of probable ([8]). We recognize two different and extreme aspects
concerning the logic of certainty. At first we distinguish a more or less extensive
class of alternatives which appear objectively possible to us in the current state
of our information: when a given individual outlines the domain of uncertainty
he does not use his subjective opinions on what he does not know because the
possible values of X depend only on what he objectively knows or not. After-
wards we definitively observe which is the true alternative to be verified among
the ones logically possible. The probability is an additional notion, so it comes
into play after constituting the range of possibility and before knowing which is
the true alternative to be verified: the logic of probable will fill in this range in a
coherent way by considering a probabilistic mass distributed upon it. An individ-
ual correctly makes a prevision of a random quantity when he leaves the objective
domain of the logically possible in order to distribute his subjective sensations of
probability among all the possible alternatives and in the way which will appear
most appropriate to him ([7], [12], [13]). Given an evaluation of probability pi,
i = 1, . . . ,n, a prevision of X turns out to be P(X) = x1p1 + . . . + xnpn, where
we have 0 ≤ pi ≤ 1, i = 1, . . . ,n, and

∑n
i=1 pi = 1: it is rendered as a func-

tion of the probabilities of the possible values for X and it is admissible in terms
of coherence because it is a barycenter of these values. It is usually called the

36



On a Geometric Representation of Probability Laws and of a Coherent
Prevision-Function

mathematical expectation of X or its mean value ([14]). It is certainly possible
to extend this result by using more advanced mathematical tools such as Stielt-
jes integrals. Nevertheless, such an extension adds nothing from conceptual and
operational point of view and for this reason we will not consider it. Conversely,
the possible values of any possible event are only two: 0 and 1. Therefore, each
event is a specific random quantity. The same symbol P denotes both prevision
of a random quantity and probability of an event ([10]).

2 Space of alternatives as a linear space
When we consider one random quantity X, each possible value of it, for a

given individual at a certain instant, is a real number in the space S of alternatives
coinciding with a line on which an origin, a unit of length and an orientation are
chosen. Every point of this line is assumed to correspond to a real number and
every real number to a point: the real line is a vector space over the field R of real
numbers, that is to say, over itself, of dimension 1. When we consider two random
quantities, X1 and X2, a Cartesian coordinate plane is the space S of alternatives:
possible pairs (x1,x2) are the Cartesian coordinates of a possible point of this
plane. Every point of a Cartesian coordinate plane is assumed to correspond to
an ordered pair of real numbers and vice versa: R2 is a vector space over the
field R of real numbers of dimension 2 and it is called the two-dimensional real
space. When we consider three random quantities, X1, X2 and X3, the three-
dimensional real space R3 corresponds to the set S of alternatives and possible
triples (x1,x2,x3) are the Cartesian coordinates of a possible point of this linear
space. There is a bijection between the points of the vector space R3 over the
field R of real numbers and the ordered triples of real numbers. More generally,
in the case of n random quantities, where n is an integer > 3, one can think of
the Cartesian coordinates of the n-dimensional real space Rn. There is a bijection
between the points of the vector space Rn over the field R and the ordered n-
tuples of real numbers. It is always essential that different pairs of real numbers
are made to correspond to distinct points or different triples of real numbers are
made to correspond to different points or, more generally, distinct n-tuples of real
numbers are made to correspond to dissimilar points ([11]). Those alternatives
which appear possible to us are elements of the set Q and they are embedded in
the space S of alternatives. Such a space is conceptually a set of points whose
subset Q consists of those possible points non-themselves subdivisible for the
purposes of the problem under consideration. Sometimes, the set Q coincides
with S. There is a very meaningful point among the points of Q: it represents
the true alternative, that is to say, the one which will turn out to be verified “a
posteriori”. It is a random point “a priori” and it expresses everything there is to

37



Pierpaolo Angelini and Angela De Sanctis

be said.

3 Two different aspects of the logic of certainty into
a linear space

We study the two aspects of the logic of certainty into a linear space coincid-
ing with the n-dimensional real space Rn where we consider n random quantities
X1, . . . ,Xn. Therefore, we have n orthogonal axes to each other: a same Carte-
sian coordinate system is chosen on every axis. Thus, the real space Rn has a
Euclidean structure and it is evidently our space S of alternatives. Into the logic
of certainty exist certain and impossible and possible as alternatives with respect
to the temporary knowledge of each individual: each random quantity justifies it-
self “a priori” because every finite partition of incompatible and exhaustive events
referring to it shows the possible ways in which a certain reality may be expressed.
A multiplicity of possible values for every random quantity is only a formal con-
struction that precedes the empirical observation by means of which a single value
among the ones of the set Q is realized. The set Q of every random quantity is
a subset of a vector subspace of dimension 1 into the n-dimensional real space
Rn. In general, given X, we have Q = I(X) = {x1, . . . ,xn}. It is absolutely the
same thing if every possible value of each random quantity is viewed as a par-
ticular n-tuple of real numbers or as a single real number. Every possible value
for a random quantity definitively becomes 0 or 1 when we make an empirical
observation referring to it: into the logic of certainty also exist true and false as
final answers ([2], [3]). Logical operations are applicable to idempotent numbers
0 and 1. If A and B are events, the negation of A is Ā = 1 −A and such an event
is true if A is false, while if A is true it is false; the negation of B is similarly
B̄ = 1 −B. The logical product of A and B is A∧B = AB and such an event is
true if A is true and B is true, otherwise it is false; the logical sum of A and B is
(A∨B) =

(
Ā∧ B̄

)
= 1 − (1 −A)(1 −B), from which it follows that such an

event is true if at least one of events is true, where we have A∨B = A + B when
A and B are incompatible events because it is impossible for them both to occur.
Concerning the logical product and the logical sum, we have evidently the same
thing when we consider more than two events. An algebraic structure (L,∧,∨),
where the logical product ∧ and the logical sum ∨ are two binary operations on the
set L whose elements are 0 and 1, is a Boolean algebra because commutative laws,
associative laws, absorption laws, idempotent laws and distributive laws hold for
0 and 1 of L. It admits both an identity element with respect to the logical product
and an identity element with respect to the logical sum, so we have

(x∧ 1) = x, (x∨ 0) = x,

38



On a Geometric Representation of Probability Laws and of a Coherent
Prevision-Function

for all x of L. It admits that every x of L has a unique complement x̄, so we have

(x∧ x̄) = 0, (x∨ x̄) = 1.

We can extend the logical operations into the field of real numbers when we make
the following definitions: x ∧ y = min(x,y), x ∨ y = max(x,y), x̄ = 1 − x.
Therefore, it is not true that the logical operations are applicable only to idempo-
tent numbers 0 and 1 because they are also applicable to all real numbers. On the
other hand, it is not true that the arithmetic operations are applicable only to nat-
ural, rational, real, complex numbers or integers because they are also applicable
to idempotent numbers 0 and 1 identifying events. For instance, the arithmetic
sum of many events coincides with the random number of successes given by
Y = E1 + . . . + En. Therefore, we observe that the set Q of every random
quantity considered into a linear space becomes a Boolean algebra whose two
idempotent numbers are on every axis of Rn. These two numbers are elements
of a subset of a vector subspace of dimension 1 into the n-dimensional real space
Rn over the field R of real numbers. That being so, it is evident that to postulate
that the field over which the probability is defined be a σ-algebra is not natural.
Hence, what we will later say is conceptually and mathematically well-founded.

4 Probability laws P formally admissible in terms
of coherence

The probability P of an event E, in opinion of a given individual, is opera-
tionally a price in terms of gain and a bet is the real or conceptual experiment
to be made oneself in order to obtain its measurement ([4]). If p = P(E) is a
coherent assessment expressed by this individual, then such a bet is fair because
it is acceptable in both senses indifferently. Therefore, he considers as fair an
exchange, for any S positive or negative, between a certain sum pS and the right
to a sum S dependent on the occurrence of E ([5]). From notion of fairness it
follows that the two possible values of the random quantity G′ = (λ−p)S, where
λ is a random quantity whose possible values are 0 and 1, do not have the same
sign. Given S, if these values of G′ are only positive or negative, then we have
an incoherent assessment and the bet on the event under consideration is not fair.
If E is a certain event, then we have p = 1 in a coherent fashion. If E is an
impossible event, then we have p = 0 in a coherent fashion. If E is a possible
event because it is not either certain or impossible, then we have 0 ≤ p ≤ 1 in a
coherent fashion. Even the probability P of the trievent E = E′′|E′ is a price in
terms of gain. It is the price to be paid for a bet that can be won or lost or annulled
if E′ does not occur. Nevertheless, we will not consider the notion of conditional

39



Pierpaolo Angelini and Angela De Sanctis

probability from now on, because it is not essential to this context. Given n events
E1, . . . ,En of the set E of events, a certain individual assigns to them, respec-
tively, the probabilities p1 = P(E1), . . . ,pn = P(En) in a coherent way. Thus,
by betting on E1, . . . ,En, this individual considers as fair an exchange, for any
S1, . . . ,Sn positive or negative, between a certain sum p1S1 + . . . + pnSn and the
right to a sum E1S1 + . . . + EnSn dependent on the occurrence of E1, . . . ,En,
where we have Ei = 1 or Ei = 0, i = 1, . . . ,n, whether Ei occurs or not. Ev-
idently, if p1 = P(E1), . . . ,pn = P(En) are not coherent assessments, then the
possible values of the random quantity G = (λ1 −p1)S1 + . . . + (λn −pn)Sn are
all positive or negative. Probability laws P formally admissible in terms of coher-
ence allow to extend in a logical or coherent way the probabilities of the events
of E which are already evaluated in a subjective way. These laws allow to de-
termine which is the most general set of events whose probabilities are uniquely
determined, in accordance with theorems of probability calculus, because one
knows the probability of each event of E. Moreover, probability laws P allow
to determine which is the most general set of events for which their probabilities
lie between two numbers, which are not 0 and 1, after evaluating the probabil-
ity of each event of E in a subjective way, while for the remaining events can be
said nothing in addition to the banal observation that their probabilities are in-
cluded between 0 and 1. If E is a finite set of incompatible and exhaustive events
E1, . . . ,En, then P is a probability law formally admissible in terms of coherence
with regard to events of E if and only if the theorem of total probability is valid,
so we have P(E1) +. . .+P(En) = 1. Probability laws P formally admissible are
evidently ∞n and a given individual may subjectively choose one of these laws
depending on the circumstances. Given A, its probability P(A) is uniquely deter-
mined when A is a logical sum of two or more than two incompatible events of
E: A is linearly dependent on these events. Otherwise, we can only say that P(A)
is greater than or equal to the sum of the probabilities of the events Ei which
imply A and less than or equal to the sum of the probabilities of the events Ei
which are compatible with A. If E is a finite set of events E1, . . . ,En whatsoever,
then the 2n constituents C1, . . . ,Cs form a finite set of incompatible and exhaus-
tive events for which it is certain that one and only one of them occurs. These
constituents are elementary or atomic events and they are obtained by the logical
product E1 ∧ . . . ∧ En: each time we substitute in an orderly way one event Ei,
i = 1, . . . ,n, or more than one event with its negation Ēi or their negations, we
obtain one constituent of the set of constituents generated by E1, . . . ,En. It is pos-
sible that some constituent is impossible, so the number of possible constituents
is s ≤ 2n. The most general probability law assigns to the possible constituents
C1, . . . ,Cs the probabilities q1, . . . ,qs which sum to 1, while the probability of
an impossible constituent is always 0. Conversely, every probability law which
is valid for the events of E can be extended to the constituents C1, . . . ,Cs, so the

40



On a Geometric Representation of Probability Laws and of a Coherent
Prevision-Function

probabilities p1 = P(E1), . . . ,pn = P(En) are admissible if and only if the non-
negative numbers q1, . . . ,qs satisfy a system of n + 1 linear equations in the s
variables q1, . . . ,qs expressed by



∑(1)
i qi = p1

...∑(n)
i qi = pn∑s
i=1 qi = 1.

The notation
∑(h)

i qi represents the sum concerning those indices i for which Ci
is an event implying Eh. If A is a logical sum of some constituent, then we have
x =

∑(A)
i qi and we can say that the probability of A is uniquely determined

because x =
∑(A)

i qi is linearly dependent on the n + 1 linear equations of the
system under consideration. If A is not a logical sum of constituents, then A′ is
the greatest logical sum of the ones which are contained in A and A′′ is the low-
est logical sum of the ones which contain A, so we have x′ ≤ x ≤ x′′, where
x′ is the lowest admissible probability of A′, while x′′ is the greatest admissible
probability of A′′. If E is an infinite set of events, then P is a probability law
formally admissible with regard to events of E if and only if P is a probability
law formally admissible with regard to any finite subset of E. Therefore, given A,
its probability P(A) is uniquely determined or bounded from above and below or
absolutely undetermined because we have 0 ≤ P(A) ≤ 1. Now we extend proba-
bility laws P formally admissible in terms of coherence to random quantities we
defined in the beginning. The set X can be a finite set of n random quantities
X1, . . . ,Xn or it can be an infinite set of random quantities. In general, given
X, I(X) = {x1, . . . ,xn} is the set of its possible values. Thus, after assigning
to every possible value xi of X its subjective and corresponding probability pi,
with

∑n
i=1 pi = 1, we have infI(X) ≤ P(X) ≤ supI(X) in accordance with

convexity property of P. Given Z = X1 + . . . + Xn which is a linear combination
of n random quantities X1, . . . ,Xn of X , I(Z) = {z1, . . . ,zn} is the set of its
possible values. Therefore, its coherent prevision must satisfy convexity property
of P, so we have infI(Z) ≤ P(Z) ≤ supI(Z), where it turns out to be P(Z)
= P(X1) + . . . + P(Xn) in accordance with linearity property of P. Linearity
property can clearly be of interest to any linear combination of n random quan-
tities. We may also consider less than n random quantities. The possibility of
certain consequences whose unacceptability appears recognizable to everyone is
excluded when convexity property of P and its linearity property are valid. They
are the foundation of the whole theory of probability because they are necessary
and sufficient conditions for coherence: decisions under conditions of uncertainty
lead to a certain loss when linearity and convexity of P are broken ([1]). The

41



Pierpaolo Angelini and Angela De Sanctis

probabilities of every possible value of a given random quantity belonging to a
finite or infinite set of random quantities sum to 1 in a coherent way according to
probability laws P formally admissible in terms of coherence with regard to these
possible values.

5 A coherent prevision-function P
From mathematical point of view, P is a function. We define it by tak-

ing into account its objective coherence. The domain of P is the arbitrary set
X = {X1, . . . ,Xn} consisting of a finite number of random quantities: for each of
them, the set of possible values is I(Xi) = {xi1, . . . ,xin}, with xi1 < .. . < xin,
i = 1, . . . ,n. Moreover, we suppose xi1 6= xj1 and xin 6= xjn, with i 6= j,
i,j = 1, . . . ,n. The codomain of P is the set Y consisting of as many intervals
as random quantities are found into the set X of P, with infI(Xi) ≤ P(Xi) ≤
supI(Xi) for every interval referring to the random quantity Xi, i = 1, . . . ,n.
Therefore, both X and Y are sets whose elements are themselves sets. The coher-
ent function P is called prevision-function and it is a bijective function because
each element of X , Xi ∈X , is paired with exactly one element of Y, for which it
turns out to be infI(Xi) ≤ P(Xi) ≤ supI(Xi), and each element of Y is paired
with exactly one element of X . There are no unpaired elements, with P(Xi) which
is a prevision of Xi on the basis of the state of information of a certain individual
at a given instant. Given the set I(X) = {x1, . . . ,xn}, with x1 < .. . < xn, the
image of X under P is P(X) = x1p1 +. . .+xnpn, with 0 ≤ pi ≤ 1, i = 1, . . . ,n,
and

∑n
i=1 pi = 1: such an image coincides with all weighted arithmetic means

calculated in a coherent fashion when pi varies while xi is constant. All coherent
previsions of X satisfy the inequality infI(X) ≤ P(X) ≤ supI(X). The image
of the entire domain X of P is the image of P and it coincides with the entire
codomain Y. If X is an infinite set of random quantities, we can always con-
sider a restriction of the prevision-function P which is a new function obtained
by choosing a smaller and finite domain. Therefore, the above observations re-
main unchanged because such a new function coincides with P whose domain
is a finite set of random quantities. In the case in which the domain of P is the
arbitrary set E = {E1, . . . ,En} consisting of a finite number of possible events,
its codomain is the set Y consisting of as many intervals as events are found into
the set E of P, with infEi ≤ P(Ei) ≤ supEi, i = 1, . . . ,n, for each of such
intervals. Nevertheless, since we have infEi = 0 and supEi = 1, i = 1, . . . ,n, it
turns out to be 0 ≤ P(Ei) ≤ 1 for every interval of Y. The coherent function P
is called probability-function and it is a bijective function because each element
of E, Ei ∈E, is paired with exactly one element of Y, for which it turns out to be
0 ≤ P(Ei) ≤ 1, and each element of Y is paired with exactly one element of E.

42



On a Geometric Representation of Probability Laws and of a Coherent
Prevision-Function

There are no unpaired elements, with P(Ei) which is an evaluation of probability
of Ei. The image of Ei under P is an interval. If E is an infinite set of events,
we can always consider a restriction of the probability-function P as above. We
admit that P can be evaluated by anybody for every event E or random quan-
tity X. Thus, it is not true that it would make sense to speak of probability only
when all events under consideration are repeatable, as well as it is not true that it
would make sense to speak of prevision only when all random quantities under
consideration belong to a measurable set I. We cannot pretend that P is actually
imagined as determined, by any individual, for all events or random quantities
which could be considered in the abstract. We must recognize if P includes or not
any incoherence. If so the individual, when made aware of such an incoherence,
should eliminate it. Thus, the subjective evaluation is objectively coherent and
can be extended to any larger set of events or random quantities. It is necessary to
interrogate a given individual in order to force him to reveal his evaluation of ele-
ments of the codomain Y of P, P(Xi) or P(Ei), i = 1, . . . ,n: both prevision of
a random quantity and probability of an event always express what an individual
chooses in his given state of ignorance, so it is wrong to imagine a greater degree
of ignorance which would justify the refusal to answer. If a prevision-function
is not understood as an expression of the opinion of a certain individual, we can
interrogate many individuals in order to study their common opinion which is de-
noted by P. Therefore, P will exist in the ambit of those random quantities X for
which all evaluations Pi(X), i = 1, . . . ,n, coincide. Such evaluations will define
P(X) in this way. Evidently, P will not exist elsewhere, for other random quan-
tities X for which the subjective evaluations Pi(X) do not coincide. The above
observations remain valid when a given individual confines himself to evaluations
which conform to more restrictive criteria coinciding with classical definition of
probability and with the statistical one ([6], [16]).

6 Geometric representation of P
Given the set X = {X1, . . . ,Xn} or the set E = {E1, . . . ,En}, the possible

values of each random quantity or random event can geometrically be represented
on n lines for which a Cartesian coordinate system has been chosen. Such lines
belong to the vector space Rn over the field R of real numbers. Rn has a Euclidean
structure characterized by a metric. Hence, the standard basis of Rn is given by
{e1, . . . ,en}, where we have e1 = (1, . . . , 0), . . . , en = (0, . . . , 1), and it consists
of orthogonal vectors to each other having a Euclidean norm equal to 1. The point
of Rn where n lines meet is the origin of Rn given by (0, . . . , 0). We have an one-
to-one correspondence between the points of Rn and the n-tuples of real numbers.
We consider n coordinate subspaces of dimension 1 in the vector space Rn. In fact,

43



Pierpaolo Angelini and Angela De Sanctis

when we project every point of Rn referring to (X1,X2, . . . ,Xn) and expressed
by (x1,x2, . . . ,xn) onto the coordinate axis x1, it becomes (x1, . . . , 0). When we
project the same point onto the coordinate axis x2, it becomes (0,x2, . . . , 0) and
so on. After projecting all the possible points of X1 onto the coordinate axis x1,
. . . , all the possible points of Xn onto the coordinate axis xn, every point onto
the coordinate axis x1, coinciding with a particular n-tuple of real numbers of Rn,
can be viewed as a real number of R, . . . , every point onto the coordinate axis
xn, coinciding with a particular n-tuple of real numbers of Rn, can be viewed as
a real number of R. It is finally clear that n projected points onto the coordinate
axis x1 can be viewed as n real numbers of an one-dimensional vector space, . . . ,
n projected points onto the coordinate axis xn can be viewed as n real numbers
of an one-dimensional vector space. The possible points of Ei projected onto
the coordinate axis xi, i = 1, . . . ,n, are evidently only two. For instance, if n
= 3, we have the points (1, 0, 0) and (0, 0, 0) onto the coordinate axis x1 refer-
ring to E1 which can respectively be viewed as 1 and 0, . . . , the points (0, 0, 1)
and (0, 0, 0) onto the coordinate axis x3 referring to E3 which can respectively
be viewed as 1 and 0. Nevertheless, we have always three real lines, so we do
not get confused. In any case, it is conceptually the same thing if we make use
only of particular n-tuples of real numbers of Rn without seeing them as real
numbers of R. The codomain of P is the set Y consisting of n intervals which
coincide with n line segments belonging to n different real lines. These line seg-
ments could become increasingly larger by virtue of linearity of P extended to
any finite number of random quantities considered on a same line. Indeed, we
observe that all weights or probabilistic masses, which are non-negative and sum
to 1, remain unchanged with respect to starting point characterized by only one
random quantity. Nevertheless, they are paired with real numbers whose absolute
values are evidently greater. Such numbers can be interpreted as the possible val-
ues of one random quantity considered on a same line. It is evident that the set
of all coherent previsions of every random quantity Xi as well as the set of all
coherent probabilities of every random event Ei, i = 1, . . . ,n, is a subset of a
vector subspace of dimension 1. Such a subset is however a convex set while the
set of the possible values for every random quantity considered into Rn is not a
convex set. The same thing goes when we consider the set of the possible values
for every random event represented into Rn. We already saw that it is always pos-
sible to consider a finite number of events or random quantities in order to study
probability laws P formally admissible in terms of coherence. As a first step we
refer to events. Given n events E1, . . . ,En of E, we represent them by means
of n axes of Rn. Nevertheless, instead of concentrating our attention on n axes
of Rn as above, we consider only one of them which we choose in an arbitrary
fashion. Such an axis is an one-dimensional vector subspace of Rn. It is gener-
ated by a vector of the standard basis of Rn. Every point of Rn on a same line

44



On a Geometric Representation of Probability Laws and of a Coherent
Prevision-Function

is obtained multiplying by a real number the vector of the standard basis of Rn
which we have arbitrarily chosen. Therefore, we can always multiply by any real
number a same n-tuple of real numbers in order to obtain points of Rn which are
said to be collinear. Now the real number or coefficient of the linear combination
under consideration, characterized by only one scalar, represents the probability
of an event A into our geometric scheme of representation. Given P(E1), . . . ,
P(En), we know that P(A) can be uniquely determined or bounded from above
and below or absolutely undetermined depending on the circumstances. If it is
uniquely determined, then we have a precise point of Rn on the axis under consid-
eration. If it is bounded from above and below, then we have two points of Rn on
this axis and an admissible probability is found between them. If it is absolutely
undetermined, then we have a larger interval on this axis which is included be-
tween the lowest admissible probability of any event and the greatest admissible
one. In particular, given P(E1) = p1, . . . , P(En) = pn, after choosing the vec-
tor en of the standard basis of Rn, by means of the linear combination given by
[(λ1−p1)S1 + . . . + (λn−pn)Sn]en, with Si 6= 0, i = 1, . . . ,n, we can obtain the
possible values of the random quantity G = (λ1−p1)S1 +. . .+ (λn−pn)Sn refer-
ring to n bets concerning n events as special n-tuples of Rn. The same thing goes
if we choose another vector of the standard basis of Rn. Thus, we even represent
n bets concerning n events into a linear space. By examining n random quan-
tities X1, . . . ,Xn of X , we similarly represent them by means of n axes of Rn.
Nevertheless, by considering another random quantity Z which is again bounded
from above and below, instead of concentrating our attention on n axes of Rn, we
consider only one of them which we choose in an arbitrary way. After individ-
uating two points of Rn on this axis which are respectively the lowest possible
value of the random quantity under consideration and the greatest possible one,
P(Z) can be viewed as a point of Rn coherently included between the two points
of Rn already individuated. Probability laws P formally admissible in terms of
coherence are those laws for which the probabilities of the possible values of the
random quantity under consideration sum to 1.

7 Conclusions
We distinguished the two extreme aspects of the logic of certainty by identi-

fying their corresponding structures into a linear space. We extended probabil-
ity laws P formally admissible in terms of coherence to random quantities. We
proposed a geometric representation of these laws and of a coherent prevision-
function P which we previously defined. We connected the convex set of all
coherent previsions of a random quantity as well as the convex set of all coherent
probabilities of an event with a specific algebraic structure: such a structure is an

45



Pierpaolo Angelini and Angela De Sanctis

one-dimensional vector subspace over the field R of real numbers because events
of any finite set of events can be viewed as special points of a vector space of
dimension n over the field R of real numbers. It is exactly the linear space of ran-
dom quantities having a Euclidean structure characterized by a metric coinciding
with the dot product in a natural way. Overall, we pointed out that linearity is the
most meaningful concept concerning probability calculus whose laws gain a more
extensive rigour by means of the geometric scheme of representation we showed.
On the other hand, it is possible to extend linearity concept in order to formulate a
geometric, well-organized and original theory of random quantities: we will make
this into our next works.

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On a Geometric Representation of Probability Laws and of a Coherent
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