# Compound conditionals as random quantities and Boolean algebras

Tommaso Flaminio1, Angelo Gilio2, Lluis Godo1,
Giuseppe Sanfilippo3
1 Artificial Intelligence Research Institute (IIIA), CSIC
08193 Bellaterra, Spain
<EMAIL_ADDRESS>
2 Dept. SBAI, University of Rome “La Sapienza”,
Rome, Italy (retired)
<EMAIL_ADDRESS>
3 Dept. Mathematics and Computer Science, University of Palermo,
Palermo, Italy
<EMAIL_ADDRESS>
###### Abstract

Conditionals play a key role in different areas of logic and probabilistic
reasoning, and they have been studied and formalised from different angles. In
this paper we focus on the de Finetti’s notion of conditional as a three-
valued object, with betting-based semantics, and its related approach as
random quantity as mainly developed by two of the authors. Compound
conditionals have been studied in the literature, but not in full generality.
In this paper we provide a natural procedure to explicitly attach conditional
random quantities to arbitrary compound conditionals that also allows us to
compute their previsions. By studying the properties of these random
quantities, we show that, in fact, the set of compound conditionals can be
endowed with a Boolean algebraic structure. In doing so, we pave the way to
build a bridge between the long standing tradition of three-valued
conditionals and a more recent proposal of looking at conditionals as elements
from suitable Boolean algebras.

##  1 Introduction

Conditional expressions are pivotal in representing knowledge and reasoning
abilities of intelligent agents. Conditional reasoning features in a wide
range of areas spanning non-monotonic reasoning [1, 10, 4, 37, 20, 22, 3],
causal inference [33, 43], and more generally reasoning under uncertainty [32,
7, 41, 45] or conditional preferences [17, 49, 8]. Bruno de Finetti was one of
the first who put forward an analysis of conditionals beyond the realm of
conditional probability theory arguing that they cannot be described within
the bounds of classical logic [13, 14]. He expressed this by referring to them
as trievents: a conditional, denoted as $(a\mid b)$, is a basic object to be
read “$a$ given $b$” that turns out to be true if both $a$ and $b$ are true,
false if $a$ is false and $b$ is true, and void if $b$ is false.

The vast literature on conditionals also includes the study of compound
conditionals, that is to say, those expressions obtained by combining basic
conditionals like $(a\mid b)$ by usual logical operations of “and”, “or”,
“negation” and so forth, see, e.g., [47, 5, 38, 31, 35, 11, 39, 40, 48, 36,
6].

In this line, and based on de Finetti’s original conception, [21, 23, 24, 25,
26, 28] propose and develop an approach to interpret both basic and compound
conditionals as random quantities. This approach has been proved to allow for
a suitable numerical representation of conditionals and some families of
compound conditional expressions. Indeed, as we will recall in Section 2,
starting from trivents $(a\mid b)$ regarded as random quantities taking values
in a three-element set $\\{0,1,x\\}$, where $x$ is a real value in $[0,1]$
representing a conditional probability, one can extend such representation
also to cover more complex conditional formulas.

An alternative, more logically-oriented approach to conditionals has been put
forward in e.g. [16] or [30]’s Conditional Event algebras, and more recently
in [15] in a finitary context. These papers formalise conditional expressions
in an algebraic setting, and therefore provide a symbolic representation of
them. In this approach, as it is common in logico-algebraic representations,
neither basic expressions $(a\mid b)$ nor compound conditional formulas
necessarily have a numerical counterpart, as their interpretation remains at
the symbolic level. However, this does not forbid, as it is also shown in
[15], to consider for instance a fully compatible probabilistic layer on top
of it.

In the present paper we will put forward an analysis that takes inspiration
from both of the above settings and proposes a numerical representation of
conditionals that, in addition, also allows for a logico-symbolic
representation. In particular, we will:

  * •

present a natural and uniform procedure to interpret compound conditionals as
random quantities;

  * •

investigate the numerical and logical properties of such a representation for
compound conditionals via their associated random quantities; and

  * •

prove that, at the logical level, the present approach, leads to the same
results of [15], where the authors showed that compound symbolic conditionals
can be endowed with a Boolean algebra structure.

Let us further remark that the apparent contradiction between the perspective
that looks at three-valued conditionals as numerical random quantities as
done, e.g., in [23, 26], and the Boolean algebraic perspective on conditionals
used in [15] to reason about them, actually dissolves once we precisely set at
which level the numerical and the symbolic representation intervene.

The present paper is organized as follows. In the next Section 2 we will
recall in some more details the original approach of [13] (Subsection 2.1) and
the one that followed of [23] (Subsection 2.2). The interpretation of compound
conditionals in terms of conditional random quantities will be the subject of
Section 3. There we will also provide examples in order for our basic
construction to be clear. In Section 4 we will then prove numerical and also
logical properties of the random quantities that represent compound
conditionals, and in Section 5 we will examine some probabilistic aspects. The
comparison between the algebras arising from the random quantities studied
here and the Boolean algebras of conditionals of [15] will be the topic of
Section 6, where we prove that they are indeed isomorphic. In the last Section
7 we will gather some conclusions and remarks about future work on this
subject.

##  2 Some preliminary comments on conjoined conditionals

In this section, in order to better understand the formalism and results of
this paper, we recall some notions given in the approach by Gilio-Sanfilippo,
and their relation with the notions and results given here.

We will denote here events by capital letters, such as $A,B,H,K,\ldots$.
Moreover, we denote random quantities by capital letters, such as
$X,Y,Z\ldots$. In particular we will denote by $X_{A}$ the indicator function
of the event $A$.

###  2.1 The conditional prevision in the approach of de Finetti

Given a random quantity $Z$ and an event $H\neq\bot$, in the approach of de
Finetti (1935) the conditional prevision $\mathbb{P}(Z|H)$ can be assessed by
applying the following conditional bet:

  * 1)

you are asked to assess the value $z=\mathbb{P}(Z|H)$, by knowing that if $H$
is true then the bet is in effect;

  * 2)

if the bet has effect, then you pay $z$ and you receive the random amount $Z$;
otherwise, if $H$ is false, the bet is null;

  * 3)

the checking of coherence for the assessment $z=\mathbb{P}(Z|H)$ is made by
only considering the cases where the bet has effect.

We observe that for the random quantity $Z{\cdot}X_{H}+yX_{\widebar{H}}$, by
linearity of the prevision, it holds that, for all $y$,

$\begin{array}[]{l}\mathbb{P}(Z{\cdot}X_{H}+yX_{\widebar{H}}\mid
H)=\mathbb{P}(Z{\cdot}X_{H}\mid H)+yP(\widebar{H}\mid H)\\\ =\mathbb{P}(Z\mid
H)=z.\end{array}$

In particular, for $y=z$, one has $\mathbb{P}(ZX_{H}+zX_{\widebar{H}}\mid
H)=\mathbb{P}(Z\mid H)=z$. Moreover, for all $y$,

$\begin{array}[]{l}\mathbb{P}(Z{\cdot}X_{H}+yX_{\widebar{H}})=\mathbb{P}(Z{\cdot}X_{H})+yP(\widebar{H})=\\\
=\mathbb{P}(Z\mid H)P(H)+yP(\widebar{H})=zP(H)+yP(\widebar{H})\,,\end{array}$

and hence, for $z=y$, we have

$\begin{array}[]{l}\mathbb{P}(Z{\cdot}X_{H}+zX_{\widebar{H}})=zP(H)+zP(\widebar{H})=z.\end{array}$

In other words, we have the following equalities:

$\mathbb{P}(Z{\cdot}X_{H}+zX_{\widebar{H}})=\mathbb{P}(Z{\cdot}X_{H}+zX_{\widebar{H}}\mid
H)=\mathbb{P}(Z\mid H)=z.$ (1)

Now, consider the random quantity $Z{\cdot}X_{H}+yX_{\widebar{H}}$, under the
assumptions that:

  * (i)

$y=\mathbb{P}(Z{\cdot}X_{H}+yX_{\widebar{H}})$;

  * (ii)

in order to check the coherence of the assessment $y$, we discard the cases
where you receive back the paid amount $y$ (whatever $y$ be), that is we
discard the case where $H$ is false (bet called off).

Of course, $y=z$ satisfies both $(i)$ and $(ii)$.

  * Question: is it coherent to assess $y\neq z$?

We show below that the answer is NO.
Indeed, if we make a bet on the random quantity

$(Z{\cdot}X_{H}+zX_{\widebar{H}})-(Z{\cdot}X_{H}+yX_{\widebar{H}})=(z-y)X_{\widebar{H}},$

we should agree to pay $z-y$ by receiving
$(z-y)X_{\widebar{H}}\in\\{0,z-y\\}$. As when $H$ is false we receive back the
paid amount $z-y$ (whatever $z-y$ be), this case must be discarded for
checking the coherence of the assessment $z-y$. Then, we should pay $z-y$ by
knowing that (when the bet is not called off) we receive 0. Therefore, by
coherence, $z-y=0$, that is $y=z=\mathbb{P}(Z|H)$.

In other words, in order to assess the conditional prevision
$\mathbb{P}(Z|H)$, it is equivalent to assess the value $z$ such that
$z=\mathbb{P}(Z{\cdot}X_{H}+zX_{\widebar{H}})$; then, we can define the
conditional random quantity $Z|H$ as

$Z|H=ZX_{H}+zX_{\widebar{H}},\text{ where
}z=\mathbb{P}(ZX_{H}+zX_{\widebar{H}}).$ (2)

In this way we can look at $z$ both as the conditional prevision
$\mathbb{P}(Z|H)$ _à la de Finetti_ , and as the prevision of the conditional
random quantity $Z|H=ZX_{H}+zX_{\widebar{H}}$. In particular, given two events
$A$ and $H\neq\emptyset$, if $x=\mathbb{P}(X_{AH}+xX_{\widebar{H}})$, then
$x=P(A|H)$ and hence

$X_{A|H}=X_{AH}+P(A|H)X_{\widebar{H}}.$ (3)

A final remark: from (1) it holds that

$\mathbb{P}[(Z|H)|H]=\mathbb{P}(ZX_{H}+zX_{\widebar{H}}\mid
H)=\mathbb{P}(Z|H)=z;$ (4)

then we formally obtain

$\begin{array}[]{l}(Z|H)|H=\\\
=(ZX_{H}+zX_{\widebar{H}})X_{H}+\mathbb{P}(ZX_{H}+zX_{\widebar{H}}\mid
H)X_{\widebar{H}}=\\\ =ZX_{H}+zX_{\widebar{H}}=Z|H.\end{array}$

###  2.2 The conjunction in the approach by Gilio $\&$ Sanfilippo

We recall that in the approach by Gilio and Sanfilippo the compound
conditionals, like conjunctions and disjunctions, are defined as conditional
random quantities, in the setting of coherence, see e.g. [23, 24, 28]. In this
section, in order to make explicit the numerical aspects, we recall these
notions by using the notations of the current paper. Then, the indicator of an
event $A$, or a conditional event $A|H$ is denoted (not by the same symbol,
but) by $X_{A}$, or $X_{(A|H)}$, respectively. Likewise, the conjunction
$A|H\wedge B|K$ is denoted by $X_{(A|H)\wedge(B|K)}$, and so on.

In the setting of coherence, given a probability assessment
$P(A|H)=x,P(B|K)=y$, the conjunction of $A|H$ and $B|K$ is defined as

$X_{(A|H)\wedge(B|K)}=\left\\{\begin{array}[]{ll}1,&\mbox{if $AHBK$ is
true,}\\\ 0,&\mbox{if $\widebar{A}H{\vee}\widebar{B}K$ is true,}\\\
x,&\mbox{if $\widebar{H}BK$ is true,}\\\ y,&\mbox{if $AH\widebar{K}$ is
true,}\\\ z,&\mbox{if $\widebar{H}\widebar{K}$ is true,}\\\
\end{array}\right.$ (5)

where $z$ is the prevision of $X_{(A|H)\wedge(B|K)}$, which (in the framework
of the betting scheme) is the amount to be paid in order to receive the random
amount $X_{(A|H)\wedge(B|K)}$. We observe that

$X_{(A|H)\wedge(B|K)}=X_{AHBK}+xX_{\widebar{H}BK}+yX_{AH\widebar{K}}+zX_{\widebar{H}\widebar{K}}\,,\;$
(6)

where
$z=\mathbb{P}(X_{AHBK}+xX_{\widebar{H}BK}+yX_{AH\widebar{K}}+zX_{\widebar{H}\widebar{K}})$.
Then, by applying (2), where $H$ is replaced by $H{\vee}K$ and $Z$ is replaced
by $X_{AHBK}+xX_{\widebar{H}BK}+yX_{AH\widebar{K}}$, we have

$\begin{array}[]{l}Z\cdot X_{H\vee
K}=(X_{AHBK}+xX_{\widebar{H}BK}+yX_{AH\widebar{K}})X_{H\vee K}=\\\
=X_{AHBK}+xX_{\widebar{H}BK}+yX_{AH\widebar{K}},\end{array}$

and it follows that

$\begin{array}[]{ll}X_{AHBK}+xX_{\widebar{H}BK}+yX_{AH\widebar{K}}+zX_{\widebar{H}\widebar{K}}=\\\
=(X_{AHBK}+xX_{\widebar{H}BK}+yX_{AH\widebar{K}})|(H{\vee}K).\end{array}$

Hence, from (6) we obtain

$X_{(A|H)\wedge(B|K)}=(X_{AHBK}+xX_{\widebar{H}BK}+yX_{AH\widebar{K}})|(H\vee
K),$

with

$\begin{array}[]{l}z=\mathbb{P}[(X_{AHBK}+xX_{\widebar{H}BK}+yX_{AH\widebar{K}})\mid
H{\vee}K)=\\\ \quad=P(AHBK\mid H{\vee}K)+xP(\widebar{H}BK\mid H{\vee}K)\;+\\\
\qquad+\;yP(AH\widebar{K}\mid H{\vee}K).\end{array}$ (7)

We observe that, if $P(H\vee K)>0$,

$z=\frac{P(AHBK)+xP(\widebar{H}BK)+yP(AH\widebar{K})}{P(H\vee K)},$

which is the well known formula given in [36] and in [38].

Moreover, by observing that

$\begin{array}[]{l}X_{(AHBK\mid H\vee K)}=X_{AHBK}+P(AHBK\mid
H{\vee}K)X_{\widebar{H}\widebar{K}},\\\ xX_{(\widebar{H}BK\mid H\vee
K)}=xX_{\widebar{H}BK}+xP(\widebar{H}BK\mid
H{\vee}K)X_{\widebar{H}\widebar{K}},\\\ yX_{(AH\widebar{K}\mid H\vee
K)}=yX_{AH\widebar{K}}+yP(AH\widebar{K}\mid
H{\vee}K)X_{\widebar{H}\widebar{K}},\\\ \end{array}$

from (7) we obtain

$\begin{array}[]{l}X_{(AHBK\mid H\vee K)}+xX_{(\widebar{H}BK\mid H\vee
K)}+yX_{(AH\widebar{K}\mid H\vee K)}\\\
=X_{AHBK}+xX_{\widebar{H}BK}+yX_{AH\widebar{K}}+zX_{\widebar{H}\widebar{K}}.\end{array}$

Finally,

$\begin{array}[]{l}X_{(A|H)\wedge(B|K)}=\\\ X_{(AHBK\mid H\vee
K)}+xX_{(\widebar{H}BK\mid H\vee K)}+yX_{(AH\widebar{K}\mid H\vee
K)}.\end{array}$

###  2.3 The relation with the approach in this paper

For notational convenience, in the rest of the paper we will use lower case
letters for events involved in (compound) conditionals; hence, for instance,
we will denote by $(a|b)$ a conditional event and by $(a|b)\land(c|d)$ the
conjunction of $(a|b)$ and $(c|d)$. Moreover, in order to distinguish the
logical aspects from the numerical ones, with each compound conditional $t$ we
will associate a suitable random quantity $X_{t}$, whose numerical values are
the conditional previsions of some intermediate objects, called reducts. In
particular, with the conjunction $(a|b)\land(c|d)$ we will associate a random
quantity $X_{(a|b)\land(c|d)}$, whose numerical values are conditional
previsions associated with the reducts determined by the following partition
which corresponds to the one in (5):

$\\{abcd\,,\;\widebar{a}b\lor\widebar{c}d\,,\;\widebar{b}cd\,,\;ab\widebar{d}\,,\;\widebar{c}\widebar{d}\\}\,.$

As we will see more in general later, the reducts associated with the elements
of the partition above are denoted by

$1\,,\;0\,,\;(a|b)\,,\;(c|d)\,,\;(a|b)\wedge(c|d)\,,$

respectively. From them we obtain the respective numerical values $1,0,x,y,z$
of $X_{(a|b)\land(c|d)}$, which are interpreted by the following conditional
previsions:

$\begin{array}[]{c}1=\mathbb{P}(1|\Omega)\,,\;0=\mathbb{P}(0|\Omega)\,,\;x=\mathbb{P}(X_{(a|b)}|b)\,,\\\
y=\mathbb{P}(X_{(c|d)}|d),\;z=\mathbb{P}[X_{(a|b)\wedge(c|d)}|(c\lor
d)]\,.\end{array}$

We observe that, based on (2), it could be verified that

$\begin{array}[]{l}x=\mathbb{P}(X_{(a|b)}|b)=P(a|b)\,,\;y=\mathbb{P}(X_{(c|d)}|d)=P(c|d),\\\
z=\mathbb{P}[X_{(a|b)\wedge(c|d)}|(c\lor
d)]=\mathbb{P}[X_{(a|b)\wedge(c|d)}].\end{array}$

Thus, $X_{(a|b)\wedge(c|d)}$ is nothing else that the conjunction between the
conditional events $(a|b)$ and $(c|d)$ defined in (5). We remark that, by this
approach, each compound conditional $t$ is not explicitly defined, but we
operate on it by means of the associated random quantity $X_{t}$.

##  3 General compound conditionals as conditional random quantities

From now on we will be considering a fix finite Boolean algebra of ordinary
events ${\bf A}=(A,\land,\lor,\neg,\bot,\top)$. For the sake of a lighter
notation, we will also use $ab$ for the conjunction $a\land b$ of events $a$
and $b$, $\bar{a}$ for $\neg a$ of the event $a$, while we will keep denoting
the disjunction of $a$ and $b$ by $a\vee b$.

In this setting, the set of the atoms ${\rm at}({\bf A})$ of the algebra of
events $\bf A$, can be identified with the set $\Omega$ of interpretations for
$\bf A$, i.e. the set of Boolean algebra homomorphisms $w:{\bf
A}\to\\{0,1\\}$. Indeed, we will say that an event $a\in\bf A$ is true (resp.
false) under an interpretation (or possible world) $w\in\Omega$ when $w(a)=1$
(resp. $w(a)=0$), also denoted as $w\models a$ (resp. $w\not\models a$).

As we anticipated in the previous section, conditional events of the form “if
$a$ then $b$’, or “$a$ given $b$”, where $a$ and $b$ are events from $\bf A$
with $b$ different from $\bot$, will be denoted by pairs $(a\mid b)$. We will
also let $A|A^{\prime}=\\{(a\mid b)\mid a\in A,b\in A^{\prime}\\}$, where
$A^{\prime}=A\setminus\\{\bot\\}$, be the set of all conditionals that can be
built from $\bf A$, that will be also called basic conditionals. Compound
conditionals, then, are combinations of basic ones by operations of negation,
conjunction and disjunction, that we will keep denoting them as $\neg,\land$
and $\lor$ without danger of confusion. Denote by $\mathbb{T}(A)$ the term
algebra of type $(\wedge,\vee,\neg,\bot,\top)$ over the set $A|A^{\prime}$, so
that its support $\mathbb{T}(A)$ is the set of arbitrary terms generated from
$A|A^{\prime}$ (taken as variables) over the signature
$(\neg,\land,\lor,\bot,\top)$. For instance, if $a,c,e\in A$ and $b,d,f\in
A^{\prime}$, then $(a\mid b)\land(c\mid d)$ or $(a\mid b)\lor((c\mid
d)\land\neg(e\mid f))$ are examples of compound conditionals from
$\mathbb{T}(A)$. One of our ultimate aims will be to present a Boolean algebra
obtained from ${\mathbb{T}(A)}$ that extends $\bf A$.

In the rest of the section we will further extend the random quantity-based
approach to conditionals and propose an unambiguous procedure to interpret any
compound conditional as a suitable random quantity. Notice that, given a
family of $n$ conditional events $\\{(a_{1}\mid b_{1}),\ldots,(a_{n}\mid
b_{n})\\}$, we can consider the possible values of the random vector
$(X_{(a_{1}|b_{1})},\ldots,X_{(a_{n}|b_{n})})$, the so-called points
$Q_{h}$’s, by means of which we can develop a geometrical approach for
checking coherence and for propagation (see, e.g., [25, 28]).

For every $t\in\mathbb{T}(A)$, let us denote by $Cond(t)=\\{(a_{1}\mid
b_{1}),\ldots,(a_{n}\mid b_{n})\\}$ the set of basic conditionals appearing in
$t$, and by ${\bf b}(t)=b_{1}\lor\ldots\lor b_{n}$ the disjunction of the
antecedents in $Cond(t)$.

######  Definition 3.1.

Let $w\in\Omega$ be a classical interpretation and let $t\in\mathbb{T}(A)$ be
a term. The $w$-reduct of $t$, denoted $t^{w}$, is the term in
$\mathbb{T}(A)$, called the $w$-reduct of $t$, obtained from $t$ by the
following procedure:

  * (1)

replace each $(a_{i}\mid b_{i})\in Cond(t)$ by 1 if $w\models a_{i}b_{i}$, and
by 0 if $w\models\bar{a}_{i}b_{i}$,

  * (2)

apply the following reduction rules to subterms of $t$ until no further
reduction is possible:

\- $\neg 1:=0$, $\neg 0:=1$
\- $r{\land}1=1{\land}r:=r$
\- $r{\land}0=0{\land}r:=0$
\- $r{\lor}1=1{\lor}r:=1$
\- $r{\lor}0=0{\lor}r:=r$

where the $r$ denotes a subterm of $t$.

This symbolic reduction procedure has some interesting properties. First of
all, $w\models\neg{\bf b}(t)$, then no reduction is possible and hence
$t^{w}=t$. Second, the reduction commutes with the operation symbols, in the
following sense:

######  Lemma 3.2.

For every terms $t\in\mathbb{T}(A)$ and for every $w\in\Omega$, the following
hold:

  1. 1.

$(\neg t)^{w}=\neg t^{w}$

  2. 2.

$(t{\land}s)^{w}=t^{w}{\land}s^{w}$

  3. 3.

$(t{\lor}s)^{w}=t^{w}{\lor}s^{w}$

We will denote by $Red(t)=\\{t^{w}\mid w\in\Omega$ the set of $w$-reducts of
$t$, and by $Red^{0}(t)=Red(t)\setminus\\{t\\}$, the set of its proper
$w$-reducts.

######  Example 3.3.

Let $t=(a\mid b){\land}((c\mid d){\lor}\neg(e\mid f))$ and let $w$ such that
$w(a)=1,w(b)=0,w(c)=0,w(d)=0,w(e)=1,w(f)=1$, i.e. $w\models
a\bar{b}\bar{c}\bar{d}ef$. Then

$t^{w}:=(a\mid b){\land}((c\mid d){\lor}\neg 1):=(a\mid b){\land}((c\mid
d){\lor}0)$

$:=(a\mid b){\land}(c\mid d).$

Let $w^{\prime}$ such that $w^{\prime}\models abc\bar{d}ef$. Then

$t^{w^{\prime}}:=1{\land}((c\mid d){\lor}\neg 1):=(c\mid d){\lor}0:=(c\mid
d).$

Further, if $w^{\prime\prime}$ is such that $w^{\prime\prime}\models abcdef$,
then

$t^{w^{\prime\prime}}:=1\land(1\lor\neg 1):=1\land(1\lor 0):=1\land 1:=1.$

In fact, one can check that $Red^{0}(t)=\\{1,0,(a\mid b),$
$(c\mid d),\neg(e\mid f),(a\mid b){\land}(c\mid d),(a\mid b){\land}\neg(e\mid
f),$
$(c\mid d){\lor}\neg(e\mid f)\\}.$

We recall that, given a finite algebra ${\bf A}$ and a conditional probability
$P:A\times B\rightarrow[0,1]$, where $B\subseteq A^{\prime}$, $P$ could not be
coherent [27, 19, 7]. Some sufficient conditions for coherence of $P$ are:
$(i)$ $B=A^{\prime}$; $(ii)$ $B\cup\\{\bot\\}$ is a subalgebra of $A$; $(iii)$
$B$ is an additive [34, 7] or quasi-additive [18, 42, 46] class.

Now, we recall the notion of conditional prevision of a random quantity.

######  Definition 3.4.

Let $X:\Omega\to[0,1]$ be a random quantity, and let $b\in A$ be an event.
Then, given a conditional probability $P:A\times A^{\prime}\to[0,1]$, the
conditional prevision of $X$ given $b$ is defined as:

$\mathbb{P}(X\mid b)=\sum_{w\in\Omega}X(w)\cdot P(w\mid b)=\sum_{w\in
b}X(w)\cdot P(w\mid b).$

The next definition presents a suitable way to associate a random quantity to
every compound conditional $t\in\mathbb{T}(A)$.

######  Definition 3.5.

Let ${\bf A}$ be a finite Boolean algebra and $P:A\times A^{\prime}\to[0,1]$ a
conditional probability on ${\bf A}$. For every term $t$ in $\mathbb{T}(A)$,
we define the random quantity $X_{t}:\Omega\to[0,1]$ as follows: for every
$w\in\Omega$,

$X_{t}(w):=\mathbb{P}(X_{t^{w}}\mid{\bf b}(t^{w})).$

If $t^{w}=1$ or $t^{w}=0$, we take ${\bf b}(t^{w})=\top$, and hence $X_{1}=1$
or $X_{0}=0$ respectively. Thus, $X_{1}(w)=1$ or $X_{0}(w)=0$.

Regarding this definition, some observations are in order here:

  * (i)

As we have observed before, if $w\models\widebar{\bf b}(t)$ then $t^{w}=t$,
and thus $X_{t}(w)=\mathbb{P}(X_{t}\mid{\bf b}(t))$, and hence
$\mathbb{P}(X_{t}\mid\widebar{\bf b}(t))=\mathbb{P}(X_{t}\mid{\bf b}(t))$.

  * (ii)

The above definition of $X_{t}$ strongly depends on the assumed conditional
probability $P$ of $A\times A^{\prime}$. Actually, once we fix the initial
conditional probability $P$ on $A\times A^{\prime}$, all random quantities
$X_{t}$ are fully determined. Indeed, the above Definition 3.5 is in fact a
recursive definition, since

$\mathbb{P}(X_{t^{w}}\mid{\bf
b}(t^{w}))=\sum_{w^{\prime}}X_{t^{w}}(w^{\prime})\cdot P(w^{\prime}\mid{\bf
b}(t^{w})),$

and in turn,

$X_{t^{w}}(w^{\prime})=\mathbb{P}(X_{(t^{w})^{w^{\prime}}}\mid{\bf
b}((t^{w})^{w^{\prime}})),$

and so on, until reaching basic conditionals.

  * (iii)

As a consequence, in general different conditional probabilities $P$ and
$P^{\prime}$ on ${\bf A}$ will define different random quantities $X_{t}$ for
the same term $t$.

  * (iv)

In the case $t$ is of the form $t=(a\mid\top)$, the associated random quantity
$X_{t}$ coincides with the indicator function of $a$, that is, $X_{t}(w)=1$
whenever $w\models a$, and $X_{t}(w)=0$ otherwise. This shows that
$(a\mid\top)$ can indeed be indentified with the plain event $a$. In this
case, for the sake of a lighter notation, we will write $X_{a}$ instead of
$X_{(a\mid\top)}$.

######  Notation 1.

From now on, for simplicity, for any $t\in\mathbb{T}(A)$, we will write
$\mathbb{P}^{c}(X_{t})$ for $\mathbb{P}(X_{t}\mid{\bf b}(t))$.

Displayed in another way, the random quantity $X_{t}$ can be specified as
follows: let $Red^{0}(t)=\\{t^{w}\mid
w\in\Omega\\}=\\{t_{1},t_{2},...,t_{k}\\}$ and let
$\\{E_{1},E_{2},...,E_{k}\\}$ be the corresponding subsets of interpretations
leading to a same element of $Red^{0}(t)$, then

$X_{t}(w)=\mathbb{P}(X_{t^{w}}\mid{\bf b}(t^{w}))=$
$=\left\\{\begin{array}[]{ll}\mathbb{P}^{c}(X_{t_{1}}),&\mbox{if }w\models
E_{1}\\\ \ldots,&\ldots\\\ \mathbb{P}^{c}(X_{t_{k}}),&\mbox{if }w\models
E_{k}\\\\[2.84544pt] \hdashline\\\\[-8.5359pt] \mathbb{P}^{c}(X_{t}),&\mbox{if
}w\models\neg(E_{1}\lor\ldots\lor E_{k})\end{array}\right.$

where the dashed line separates those cases where the interpretation $w$
belongs to ${\bf b}(t)$ from those which do not.

In this setting, it is clear that $X_{t}$ is in fact the following linear
combination of the indicator functions of the events defining an associated
partition:

$X_{t}=\mathbb{P}^{c}(X_{t_{1}})X_{E_{1}}+\ldots+\mathbb{P}^{c}(X_{t_{k}})X_{E_{k}}+\mathbb{P}^{c}(X_{t})X_{E_{k+1}}$

where $E_{k+1}=\widebar{E_{1}}\land\ldots\land\widebar{E_{k}}=\overline{{\bf
b}}(t)$, and hence

$\mathbb{P}^{c}(X_{t})=\mathbb{P}^{c}(X_{t_{1}})\cdot P(E_{1}\mid{\bf
b}(t))+\ldots+\mathbb{P}^{c}(X_{t_{k}})\cdot P(E_{k}\mid{\bf b}(t)).$

We remark that, from (i) above and from (1), it holds that

$\begin{array}[]{ll}\mathbb{P}(X_{t})=\mathbb{P}(X_{t}\cdot X_{{\bf
b}(t)})+\mathbb{P}(X_{t}\cdot X_{\widebar{\bf b}(t)})=\\\
=\mathbb{P}(X_{t}\mid{\bf b}(t))P({\bf
b}(t))+\mathbb{P}(X_{t}\mid\widebar{{\bf b}}(t))P(\widebar{{\bf b}}(t))=\\\
=\mathbb{P}^{c}(X_{t})P({\bf b}(t))+\mathbb{P}^{c}(X_{t})P(\widebar{{\bf
b}}(t))=\mathbb{P}^{c}(X_{t}).\end{array}$ (8)

In other words, formula (8) shows that the prevision of $X_{t}$ coincides with
the conditional prevision $\mathbb{P}^{c}(X_{t})$.

We end this section by exemplifying the above definition of $X_{t}$ for
selected known cases of compound conditionals $t$ that will be helpful in next
sections.

######  Example 3.6.

Let $t=(a\mid b)$. Then, by applying the above definition, we get

$t^{w}=\left\\{\begin{array}[]{ll}1,&\mbox{if }w\models ab\\\ 0,&\mbox{if
}w\models\bar{a}b\\\ (a\mid b),&\mbox{if }w\models\bar{b}\\\
\end{array}\right.\quad{\bf
b}(t^{w})=\left\\{\begin{array}[]{ll}\top,&\mbox{if }w\models ab\\\
\top,&\mbox{if }w\models\bar{a}b\\\ b,&\mbox{if }w\models\bar{b}\\\
\end{array}\right.$

and thus we have:

$X_{(a\mid b)}(w)=\mathbb{P}(X_{t^{w}}\mid{\bf b}(t^{w}))=$
$=\left\\{\begin{array}[]{ll}\mathbb{P}(X_{1}\mid\top)=1,&\mbox{if }w\models
ab\\\ \mathbb{P}(X_{0}\mid\top)=0,&\mbox{if }w\models\bar{a}b\\\
\mathbb{P}(X_{(a\mid b)}\mid b),&\mbox{if }w\models\bar{b}\\\
\end{array}\right..$

Now, since $P(w\mid b)=0$ whenever $w\models\bar{b}$, we have

$\begin{array}[]{l}\mathbb{P}(X_{(a\mid b)}\mid b)=\\\ =1\cdot P(ab\mid
b)+0\cdot P(\bar{a}b\mid b)+\mathbb{P}(X_{(a\mid b)}\mid b)\cdot 0=\\\
=P(ab\mid b)=P(a\mid b).\end{array}$

Therefore we get the following well-known three-valued representation of a
conditional $(a\mid b)$:

$X_{(a\mid b)}(w)=\left\\{\begin{array}[]{ll}1,&\mbox{if }w\models ab\\\
0,&\mbox{if }w\models\bar{a}b\\\ P(a\mid b),&\mbox{if }w\models\bar{b}\\\
\end{array}\right..$

Equivalently, in agreement with (3), $X_{(a\mid b)}$ can be expressed as the
following linear combination of the indicator functions of the events $ab$ and
$\widebar{b}$:

$X_{(a\mid
b)}=1X_{ab}+0X_{\widebar{a}b}+P(a|b)X_{\widebar{b}}=X_{ab}+P(a|b)X_{\widebar{b}}.$

######  Example 3.7.

Now let $t=\neg(a\mid b)$, the negation of $(a\mid b)$. Then, by applying the
above definition, we get:

$t^{w}=\left\\{\begin{array}[]{ll}\neg 1:=0,&\mbox{if }w\models ab\\\ \neg
0:=1,&\mbox{if }w\models\bar{a}b\\\ \neg(a\mid b),&\mbox{if
}w\models\bar{b}\\\ \end{array}\right.,$ ${\bf
b}(t^{w})=\left\\{\begin{array}[]{ll}\top,&\mbox{if }w\models ab\\\
\top,&\mbox{if }w\models\bar{a}b\\\ b,&\mbox{if }w\models\bar{b}\\\
\end{array}\right.,$

and thus we have: $X_{\neg(a\mid b)}(w)=$

$=\mathbb{P}(X_{t^{w}}\mid{\bf
b}(t^{w}))=\left\\{\begin{array}[]{ll}0,&\mbox{if }w\models ab\\\ 1,&\mbox{if
}w\models\bar{a}b\\\ \mathbb{P}(X_{\neg(a\mid b)}\mid b),&\mbox{if
}w\models\bar{b}\\\ \end{array}\right..$

Now, since $P(w\mid b)=0$ whenever $w\models\neg b$, we have

$\begin{array}[]{lll}\mathbb{P}(X_{\neg(a\mid b)}\mid b)=\\\ =0\cdot P(ab\mid
b)+1\cdot P(\bar{a}b\mid b)+\mathbb{P}(X_{\neg(a\mid b)}\mid b)\cdot 0\\\
=P(\bar{a}b\mid b)=P(\bar{a}\mid b)=1-P(a\mid b).\end{array}$

Therefore the final expression for $X_{\neg(a\mid b)}$ is

$X_{\neg(a\mid b)}(w)=\left\\{\begin{array}[]{ll}0,&\mbox{if }w\models ab\\\
1,&\mbox{if }w\models\bar{a}b\\\ 1-P(a\mid b),&\mbox{if }w\models\bar{b}\\\
\end{array}\right..$

That is to say, $X_{\neg(a\mid b)}=1-X_{(a\mid b)}=X_{(\widebar{a}\mid b)}$.

######  Example 3.8.

Let us examine again the case of a conjunction of two conditionals $t=(a\mid
b){\land}(c\mid d)$ from the current perspective. Here we have ${\bf
b}(t)=b\lor d$, and

$X_{t}(w)=\left\\{\begin{array}[]{ll}1,&\mbox{if }w\models abcd\\\ 0,&\mbox{if
}w\models(\bar{a}b)\lor(\bar{c}d)\\\ \mathbb{P}^{c}(a\mid b),&\mbox{if
}w\models\bar{b}cd\\\ \mathbb{P}^{c}(c\mid d),&\mbox{if }w\models
ab\bar{d}\vspace{0.1cm}\\\ \hdashline\\\\[-5.69046pt] \mathbb{P}^{c}((a\mid
b){\land}(c\mid d)),&\mbox{if }w\models\bar{b}\bar{d}\\\ \end{array}\right.$

Now, we know that $\mathbb{P}^{c}(a\mid b)=P(a\mid b)$ and
$\mathbb{P}^{c}(c\mid d)=P(c\mid d)$. Then, using the above definition we get:

$\begin{array}[]{ll}\mathbb{P}^{c}((a\mid b){\land}(c\mid
d))=\mathbb{P}(X_{(a\mid b){\land}(c\mid d)}\mid b\lor d)\\\ \mbox{
}=P(abcd\mid b\lor d)+P(a\mid b)\cdot P(\bar{b}cd\mid b\lor d)+\\\ \mbox{
}\quad+P(c\mid d)\cdot P(ab\bar{d}\mid b\lor d),\end{array}$

that coincides, when $P(b\lor d)>0$, with the formula given in [38, 36].

Let us now consider two particular cases:

  * •

First, consider the case $a\leq b=c\leq d$. Then $t=(a\mid b){\land}(b\mid
d)$, ${\bf b}(t)=b\lor d=d$, and moreover:

\- $abcd=abd=ad=a$

\-
$(\bar{a}b)\lor(\bar{c}d)=(\bar{a}b)\lor(\bar{b}d)=(\bar{a}bd)\lor(\bar{a}\bar{b}d)=\bar{a}d$

\- $\bar{b}cd=\bar{b}b=\bot$

\- $ab\bar{d}=a\bar{d}=\bot$

Hence, the above general definition reduces to:

$X_{(a\mid b){\land}(b\mid d)}(w)=\left\\{\begin{array}[]{ll}1,&\mbox{if
}w\models a\\\ 0,&\mbox{if }w\models\bar{a}d\\\ \mathbb{P}^{c}((a\mid
b)\land(b\mid d)),&\mbox{if }w\models\bar{d}\end{array}\right.$

where $\mathbb{P}^{c}((a\mid b)\land(b\mid d))=1\cdot P(a\mid d)=P(a\mid
d)=\mathbb{P}^{c}((a\mid d))$ and thus $X_{(a\mid b){\land}(b\mid
d)}(w)=X_{(a\mid d)}(w)$ for all $w$. Thus, from the numerical point of view,
the compound conditional $(a\mid b){\land}(b\mid d)$ behaves as the basic
conditional $(a\mid d).$ See also [25, formulas (55) and (56)] where it is
also observed that $P(a|d)=P(a|b)P(b|d)$ (compound probability theorem).

  * •

Consider now the case $b=d$. Then $t=(a\mid b)\land(c\mid b)$ and ${\bf
b}(t)=b$. In this case the above general expression reduces to the following
one:

$X_{t}(w)=\left\\{\begin{array}[]{ll}1,&\mbox{if }w\models abc\\\ 0,&\mbox{if
}w\models(\bar{a}\lor\bar{c})b=\overline{ac}b\\\ \mathbb{P}^{c}((a\mid
b){\land}(c\mid b)),&\mbox{if }w\models\bar{b},\\\ \end{array}\right.$

where $\mathbb{P}^{c}((a\mid b){\land}(c\mid b))=1\cdot P(abc\mid b)=P(ac\mid
b)=\mathbb{P}^{c}((ac\mid b))$, and therefore it is clear that $X_{(a\mid
b)\land(c\mid b)}(w)=X_{(a\land c\mid b)}(w)$ for all $w$. Thus, in this case,
we see that the compound conditional $(a\mid b)\land(c\mid b)$ actually
behaves like the basic conditional $(a\land c\mid b)$.

######  Example 3.9.

Finally, let us consider a more complex compound conditional $t=(a\mid
b){\land}((c\mid d){\lor}\neg(e\mid f))$. Again, by the above definition, we
get its associated random quantity:

$X_{t}(w)=\left\\{\begin{array}[]{ll}1,&\mbox{if }w\models
ab(cd\lor\bar{e}b)\\\ 0,&\mbox{if }w\models\bar{a}b\lor\bar{c}def\\\
\mathbb{P}^{c}((a\mid b)),&\mbox{if }w\models\bar{b}(cd\lor\bar{e}f)\\\
\mathbb{P}^{c}((a\mid b){\land}(c\mid d)),&\mbox{if
}w\models\bar{b}\bar{d}ef\\\ \mathbb{P}^{c}((a\mid b){\land}\neg(e\mid
f)),&\mbox{if }w\models\bar{b}\bar{c}d\bar{f}\\\ \mathbb{P}^{c}((c\mid
d)),&\mbox{if }w\models ab\bar{d}ef\\\ \mathbb{P}^{c}(\neg(e\mid f)),&\mbox{if
}w\models ab\bar{c}d\\\ \mathbb{P}^{c}((c\mid d){\lor}\neg(e\mid f)),&\mbox{if
}w\models ab\bar{d}\bar{f}\\\\[2.84544pt] \hdashline\\\\[-8.5359pt]
y,&\mbox{if }w\models\bar{b}\bar{d}\bar{f}\\\ \end{array}\right.$

where $y=\mathbb{P}^{c}((a\mid b){\land}((c\mid d){\lor}\neg(e\mid f)))$, and
for simplicity, we have written $\mathbb{P}^{c}(s)$ for
$\mathbb{P}^{c}(X_{s})$, i.e. for $\mathbb{P}(X_{s}\mid{\bf b}(s))$.

We end this section with the following remark on a betting interpretation for
compound conditionals $t\in\mathbb{T}(A)$. A bet between a gambler and a
bookmaker on $X_{t}$ is specified as follows:

  * •

The gambler pays to the bookmaker: $\mathbb{P}^{c}(X_{t})$

  * •

In a situation $w\in\Omega$, the gambler receives from the bookmaker:
$X_{t}(w)=\mathbb{P}^{c}(X_{t^{w}})$

Note that if $w\models\neg{\bf b}$ then $t^{w}=t$, and thus
$\mathbb{P}^{c}(X_{t^{w}})=\mathbb{P}^{c}(X_{t})$, i.e. what the gambler
receives is what he has payed, i.e. it represents the situation in which the
bet is called off. We observe that coherence requires that the prevision of
the random gain must be equal to zero. Indeed, given $t$, $X_{t}$ is specified
as:

$X_{t}(w)=\left\\{\begin{array}[]{ll}\mathbb{P}^{c}(X_{E_{1}}),&\mbox{if
}w\models E_{1}\\\ \ldots&\ldots\\\ \mathbb{P}^{c}(X_{E_{k}}),&\mbox{if
}w\models E_{k}\\\\[2.84544pt] \hdashline\\\\[-8.5359pt]
\mathbb{P}^{c}(X_{t}),&\mbox{if }w\models\neg(E_{1}\lor\ldots\lor E_{k})\\\
\end{array}\right..$

In this setting, the balance of the betting for the bookmaker in a given
situation $w$ such that $w\models E_{i}$ is what he receives,
$\mathbb{P}(X_{t})$, minus what he pays to the gambler, $X_{t}$, that is:

$G(w)=\mathbb{P}(X_{t})-X_{t}(w)=\mathbb{P}^{c}(X_{t})-\mathbb{P}^{c}(X_{E_{i}}).$

Moreover, when $w\models\widebar{{\bf b}}(t)$, it holds that
$G(w)=\mathbb{P}^{c}(X_{t})-\mathbb{P}^{c}(X_{t})=0.$ Then, the prevision of
the random quantity $G$ of the balance will be:

$\begin{array}[]{lll}\mathbb{P}(G)&=&\sum_{w\in\Omega}G(w)\cdot P(w\mid{\bf
b}(t))=\\\
&=&\sum^{k}_{i=1}(\mathbb{P}^{c}(X_{t})-\mathbb{P}^{c}(X_{E_{i}}))\cdot
P(E_{i}\mid{\bf b}(t))=\\\
&=&\mathbb{P}^{c}(X_{t})\cdot\sum^{k}_{i=1}P(E_{i}\mid{\bf b}(t))-\\\
&=&\sum^{k}_{i=1}\mathbb{P}^{c}(X_{E_{i}}))\cdot P(E_{i}\mid{\bf b}(t))=\\\
&=&\mathbb{P}^{c}(X_{t})-\mathbb{P}^{c}(X_{t})=0.\end{array}$

##  4 Properties of compound conditionals

In this section we show some key properties of the random quantities
associated to certain forms of compound quantities that will allow us to show
they can be endowed somehow with a Boolean algebraic structure

We start by showing that if two compound conditional terms $t$ and
$t^{\prime}$ are such that ${\bf b}(t)\equiv{\bf b}(t^{\prime})$, to check
whether they have the same associated random quantity, it is enough to check
whether the random quantities take the same value over those worlds satisfying
the disjunction ${\bf b}(t)$. A related result is given in [23, Theorem 4].

######  Lemma 4.1.

Let $t,t^{\prime}\in\mathbb{T}(A)$ such that ${\bf b}(t)\equiv{\bf
b}(t^{\prime})$ and $X_{t}(w)=X_{t^{\prime}}(w)$ for each $w\in\Omega)$ such
that $w\models{\bf b}(t)$. Then $X_{t}=X_{t^{\prime}}$.

###### Proof.

It is enough to show that $X_{t}(w)=X_{t^{\prime}}(w)$ when $w\models\neg{\bf
b}(t)$. Due to the fact that if $w\models\neg{\bf b}(t)$ then
$X_{t}(w)=\mathbb{P}(X_{t}\mid{\bf b}(t))$, this is equivalent to show that
$\mathbb{P}(X_{t}\mid{\bf b}(t))=\mathbb{P}(X_{t^{\prime}}\mid{\bf b}(t))$.
But these previsions only depend on the values of $X_{t}$ and $X_{t^{\prime}}$
on $w$’s such that $w\models{\bf b}(t)$, and by hypothesis they coincide. ∎

Clearly, if ${\bf b}(t)\equiv{\bf b}(t^{\prime})$ and $t^{w}=t^{\prime w}$
then $X_{t}(w)=\mathbb{P}(X_{t^{w}}\mid{\bf b}(t))=\mathbb{P}(X_{t^{\prime
w}}\mid{\bf b}(t))=X_{t^{\prime}}(w)$. Hence we have the following corollary.

######  Corollary 4.2.

If ${\bf b}(t)={\bf b}(t^{\prime})$ and $t^{w}=t^{\prime w}$ for each
$w\in\Omega$ such that $w\models{\bf b}(t)$, then $X_{t}=X_{t^{\prime}}$.

######  Proposition 4.3.

Let $t\in\mathbb{T}(A)$ and let $s$ be a subterm of $t$. Further let
$s^{\prime}\in\mathbb{T}(A)$ such that ${\bf b}(s)={\bf b}(s^{\prime})$ and
$X_{s^{w}}=X_{s^{\prime w}}$ for all $w\models{\bf b}(s)$, and let
$t^{\prime}$ the term obtained by uniformly replacing $s$ by $s^{\prime}$ in
$t$. Then $X_{t}=X_{t^{\prime}}$.

###### Proof.

It follows from the recursive definition of $X_{t}$ and that, for any
$w\in\Omega$, $t^{w}$ can be computed from the $w$-reducts of its subterms
(Lemma 3.2). ∎

Next we show that the random quantities for compound conditionals satisfy
properties very familiar from a Boolean algebraic perspective.

######  Proposition 4.4.

For every $t,s,r\in\mathbb{T}(A)$ the following conditions hold:

  1. 1.

$X_{t}=X_{t{\land}t}$

  2. 2.

$X_{t{\land}s}=X_{s{\land}t}$

  3. 3.

$X_{t{\land}(s{\land}r)}=X_{(t{\land}s){\land}r}$

  4. 4.

$X_{t{\land}\neg t}=0$

  5. 5.

$X_{\neg(t{\land}s)}=X_{\neg t{\lor}\neg s}$

  6. 6.

$X_{t{\land}(s{\lor}r)}=X_{(t{\land}s){\lor}(t{\land}r)}$

  7. 7.

$X_{t{\lor}s}=X_{t}+X_{s}-X_{t{\land}s}$

  8. 8.

$X_{\neg t}=1-X_{t}$ and hence $X_{\neg\neg t}=X_{t}$

  9. 9.

If $a\leq b$, then $X_{(a\mid b)\wedge(a\mid b\vee c)}=X_{a\mid b\vee c}\,$.

###### Proof.

The detailed proof can be found in the Appendix. Items (1)-(8) are proved in a
similar way by induction on the complexity of the terms involved, and where
the base cases are those only involving basic conditionals. Many of these base
cases are proved in previous examples. ∎

Now we show that compound conditionals actually form a Boolean algebra. In
order to do it we first properly arrange the compound conditionals in
equivalence classes each of which contains terms from $\mathbb{T}(A)$ that
provides the same random quantity under any given conditional probability.

More precisely, recalling observation (ii) after Def. 3.5, let us define the
binary relation $\equiv$ on $\mathbb{T}(A)$ as follows: for all
$t,s\in\mathbb{T}(A)$,

$t\equiv s\mbox{ iff }X_{t}=X_{s}$

under any conditional probability $P$ on $A\times A^{\prime}$

It is immediate to check that $\equiv$ is an equivalence relation. Thus, let
$\mathbb{T}(A)$ to be the quotient $\mathbb{T}(A)/\mathord{\equiv}$. If we
denote by $[t]$ the equivalence class of a generic term $t\in\mathbb{T}(A)$
under $\equiv$, define operations on $\mathbb{T}(A)$ as follows: for all
$[t],[s]\in\mathbb{T}(A)$, $[t]{\land}^{*}[s]=[s{\land}t]$,
$[t]{\lor}^{*}[s]=[s{\lor}t]$, $\neg^{*}[t]=[\neg t]$, $0=[(\bot\mid\top)]$,
$1=[(\top\mid\top)]$.

By Proposition 4.3, the above operations are well defined. Moreover, the
following holds:

######  Theorem 4.5.

For every finite Boolean algebra ${\bf A}$, the structure $\mathcal{T}({\bf
A})=(\mathbb{T}(A)/\mathord{\equiv},{\land}^{*},{\lor}^{*},\neg^{*},0,1)$ is a
Boolean algebra.

###### Proof.

By Proposition 4.4, the operations ${\land}^{*},{\lor}^{*},\neg^{*}$ and
constants $0,1$ satisfy all the required equations for $\mathcal{T}({\bf A})$
being a Boolean algebra. ∎

######  Remark 4.6.

Notice that, given $[t]$ and $[s]$ in $\mathcal{T}({\bf A})$, $[t]\leq[s]$ in
the lattice order of $\mathcal{T}({\bf A})$ iff $[t]\wedge^{*}[s]=[t]$ iff
$[t\wedge s]=[t]$ iff $X_{t\wedge s}=X_{t}$.

Next proposition shows that well-known properties of conditionals also hold in
the setting of the present paper.

######  Proposition 4.7.

The following properties hold in $\mathcal{T}({\bf A})$:

  * (i)

$[(a\mid a)]=1$

  * (ii)

$[(a\mid b)\land(c\mid b)]=[(a\land c\mid b)]$

  * (iii)

$[\neg(a\mid b)]=[(\bar{a}\mid b)]$

  * (iv)

$[(a\land b\mid b)]=[(a\mid b)]$

  * (v)

$[(a\mid b)\land(b\mid c)]=[(a\mid c)]$, if $a\leq b\leq c$

  * (vi)

if $a\leq b$, then $[(a\mid b\vee c)]\leq[(a\mid b)]$ for all $c$.

###### Proof.

For each one of the equalities above, of the form $[t]=[s]$, we proved in
previous examples, that $X_{t}=X_{s}$. As for the last claim, apply
Proposition 4.4 (9). Then the claim follows by the definition of
$\mathcal{T}({\bf A})$. ∎

##  5 Compound conditionals and their probability

Since $\mathcal{T}({\bf A})$ is a Boolean algebra, we are allowed to define
probabilities on it. In this section we will show that the previsions
$\mathbb{P}(X_{t})$’s of the random quantities $X_{t}$’s determine in fact a
probability on $\mathcal{T}({\bf A})$.

######  Definition 5.1.

Given a conditional probability $P:A\times A^{\prime}\to[0,1]$, we define the
mapping $P^{*}:\mathbb{T}(A)\to[0,1]$ as follows: for every
$t\in\mathbb{T}(A)$, $P^{*}(t)=_{def}\mathbb{P}(X_{t})$. In other words, based
on (8)

$\begin{array}[]{l}P^{*}(t)=\mathbb{P}(X_{t})=\mathbb{P}^{c}(X_{t})=\mathbb{P}(X_{t}\mid{\bf
b}(t))=\\\ =\sum_{w}X_{t}(w)\cdot P(w\mid{\bf
b}(t))=\sum_{w}\mathbb{P}^{c}(X_{t^{w}})\cdot P(w\mid{\bf b}(t)).\end{array}$

From the above definition, it is cleat that if $t$ and $t^{\prime}$ are terms
such $t\equiv t^{\prime}$, then $P^{*}(t)=P^{*}(t^{\prime})$.

######  Proposition 5.2.

Any given conditional probability $P:A\times A^{\prime}\to[0,1]$ fully
determines the probabilities $P^{*}(t)$ for all compound conditional
$t\in\mathbb{T}(A)$.

###### Proof.

Let $t\in\mathbb{T}(A)$ such that $Cond(t)=\\{(a_{1}\mid
b_{1}),\ldots,(a_{n}\mid b_{n})\\}$. We have seen that $P^{*}((a_{i}\mid
b_{i}))=P(a_{i}\mid b_{i})$. Suppose that for every $s\in Red^{0}(t)$, there
is a family of conditionals $C_{s}\subseteq A\times A^{\prime}$ such that
$P^{*}(s)$ is of the form $P^{*}(s)=f_{s}(\\{P(c_{i}\mid d_{i}):(c_{i}\mid
d_{i})\in C_{s}\\})$ for some function $f_{s}$. Then, by definition, we have

$\begin{array}[]{l}P^{*}(t)=\mathbb{P}(X_{t})=\sum_{w\in
W}\mathbb{P}^{c}(X_{t^{w}})\cdot P(w\mid{\bf b}(t))=\\\ =\sum_{w\in
W}P^{*}(t^{w})\cdot P(w\mid{\bf b}(t))=\\\ =\sum_{w\in W}f_{t^{w}}(\\{P(c\mid
d):(c\mid d)\in C_{t^{w}}\\})\cdot P(w\mid{\bf b}(t)),\end{array}$

the latter expression clearly only depending on $P$. ∎

Moreover, the following is a direct consequence of Proposition 4.4.

######  Corollary 5.3.

For every $t,s,r\in\mathbb{T}(A)$ the following conditions hold:

  1. 1.

$P^{*}(t)=P^{*}(t{\land}t)$

  2. 2.

$P^{*}(t{\land}s)=P^{*}(s{\land}t)$

  3. 3.

$P^{*}(t{\land}(s{\land}r))=P^{*}((t{\land}s){\land}r)$

  4. 4.

$P^{*}(t{\land}\neg t)=0$.

  5. 5.

$P^{*}(\neg(t{\land}s))=P^{*}(\neg t{\lor}\neg s)$

  6. 6.

$P^{*}(t{\land}(s{\lor}r))=P^{*}((t{\land}s){\lor}(t{\land}r))$

  7. 7.

$P^{*}(t)=1$ if $X_{t}=1$

  8. 8.

$P^{*}(t)=0$ if $X_{t}=0$

  9. 9.

$P^{*}(t{\lor}s)=P^{*}(t)+P^{*}(s)-P^{*}(t{\land}s)$

  10. 10.

$P^{*}(\neg t)=1-P^{*}(t)$.

Notice that the last four items of the above corollary together with the above
remarked fact that $t\equiv t^{\prime}$ implies $P^{*}(t)=P^{*}(t^{\prime})$,
show that $P^{*}$ naturally induces a probability on the algebra
$\mathcal{T}({\bf A})$ that we will still denote by the same symbol.

######  Corollary 5.4.

For every conditional probability $P$ on $A\times A^{\prime}$ the map $P^{*}$
is a probability measure on $\mathcal{T}({\bf A})$.

Furthermore, as we have checked in Example 3.6, for every basic conditional
$(a\mid b)$ we have $P^{*}((a\mid b))=P(a\mid b)$, thus complying with the
Equation [11], or Stalnaker’s hypothesis, see, e.g., [9, 44]. The next example
shows how to compute the probability $P^{*}$ of a (more complex) compound
conditional.

######  Example 5.5.

Continuing the above Example 3.9 for $t=(a\mid b){\land}((c\mid
d){\lor}\neg(e\mid f))$, we have ${\bf b}(t)=b\lor d\lor f$ and using the
above Definition 5.1 we get:

$P^{*}(t)$ | $=\mathbb{P}(X_{t})=\mathbb{P}(X_{t}\mid{\bf b}(t))=$
---|---
| $=P(abcd\lor\bar{e}f)\mid b\lor d\lor f)+$
| $+\,P(a\mid b)\cdot P(\bar{b}(cd\lor\bar{e}f)\mid b\lor d\lor f)+$
| $+\,P^{*}((a\mid b){\land}(c\mid d))\cdot P(\bar{b}\bar{d}ef\mid b\lor d\lor
f)+$
| $+\,P^{*}((a\mid b){\land}\neg(e\mid f))\cdot P(\bar{b}\bar{c}d\bar{f}\mid
b\lor d\lor f)+$
| $+\,P(c\mid d)\cdot P(ab\bar{d}ef\mid b\lor d\lor f)+$
| $+\,(1-P(e\mid f))\cdot P(ab\bar{c}d\mid b\lor d\lor f)+$
| $+\,P^{*}((c\mid d){\lor}\neg(e\mid f))\cdot P(ab\bar{d}\bar{f}\mid b\lor
d\lor f).$

##  6 Boolean algebras of random variables and Boolean algebras of
conditionals

The above Theorem 4.5, tells us that the random quantities that represent
conditionals form a Boolean algebra. Actually, in [15], the authors presented
a way to construct Boolean algebras of conditionals that also represent
conditional statements, although following a different line of thoughts.

For the sake of clarity, let us briefly recall how a Boolean algebras of
conditional $\mathcal{C}({\bf A})$ is defined for every Boolean algebra ${\bf
A}$. Consider the free Boolean algebra ${\rm Free}(A|A)$ generated by the set
$A|A$ of all pairs $(a,b)\in A\times A$ such that $b\neq\bot$, in the language
$\land,\lor,\neg,\top^{*}$. Every pair $(a,b)\in A|A$ will be henceforth
denoted by $(a\mid b)$. Then, one considers the following set of basic
requirements the algebra $\mathcal{C}({\bf A})$ should satisfy:

(R1) For all $b\in A^{\prime}$, the conditional $(b\mid b)$ will be the top
element of $\mathcal{C}({\bf A})$ and $(\neg b\mid b)$ will be the bottom;

(R2) Given $b\in A^{\prime}$, the set of conditionals $A\mid b=\\{(a\mid
b):a\in A\\}$ will be the domain of a Boolean subalgebra of $\mathcal{C}({\bf
A})$, and in particular when $b=\top$, this subalgebra will be isomorphic to
${\bf A}$;

(R3) In a conditional $(a\mid b)$ we can replace the consequent $a$ by $a\land
b$, that is, the conditionals $(a\mid b)$ and $(a\land b\mid b)$ represent the
same element of $\mathcal{C}({\bf A})$;

(R4) For all $a\in A$ and all $b,c\in A^{\prime}$, if $a\leq b\leq c$, then
the result of conjunctively combining the conditionals $(a\mid b)$ and $(b\mid
c)$ must yield the conditional $(a\mid c)$.

Notice that (R4) encodes a sort of restricted chaining of conditionals and it
is inspired by the chain rule of conditional probabilities: $P(a\mid b)\cdot
P(b\mid c)=P(a\mid c)$ whenever $a\leq b\leq c$.

One then proceeds by considering the smallest congruence relation
$\equiv_{\mathfrak{C}}$ on ${\rm Free}(A|A)$ satisfying:

  * (C1) $(b\mid b)\equiv_{\mathfrak{C}}\top^{*}$, for all $b\in A^{\prime}$;

  * (C2) $(a_{1}\mid b)\land(a_{2}\mid b)\equiv_{\mathfrak{C}}(a_{1}\wedge a_{2}\mid b)$,
for all $a_{1},a_{2}\in A$, $b\in A^{\prime}$;

  * (C3) ${\neg}(a\mid b)\equiv_{\mathfrak{C}}(\neg a\mid b)$, for all $a\in A$, $b\in A^{\prime}$;

  * (C4) $(a\wedge b\mid b)\equiv_{\mathfrak{C}}(a\mid b)$, for all $a\in A$, $b\in A^{\prime}$;

  * (C5) $(a\mid b)\land(b\mid c)\equiv_{\mathfrak{C}}(a\mid c)$,
for all $a\in A$, $b,c\in A^{\prime}$ such that $a\leq b\leq c$.

Note that (C1)-(C5) faithfully account for the requirements R1-R4 where, in
particular, (C2) and (C3) account for R2. Finally, the algebra
$\mathcal{C}({\bf A})$ is formally defined as follows.

######  Definition 6.1.

For every Boolean algebra ${\bf A}$, the Boolean algebra of conditionals of
${\bf A}$ is the quotient structure $\mathcal{C}({\bf A})={\rm
Free}(A|A)/_{\equiv_{\mathfrak{C}}}.$

By definition, the algebra $\mathcal{C}({\bf A})$ is finite whenever so is
${\bf A}$. In particular, if ${\bf A}$ is atomic with atoms
$\alpha_{1},\ldots,\alpha_{n}$, in [15] it is shown that the atoms of
$\mathcal{C}({\bf A})$ are all in the following form: let
$\langle\beta_{1},\ldots,\beta_{n-1}\rangle$ be a sequence of pairwise
different atoms of ${\bf A}$. Then,
$(\beta_{1}\mid\top)\wedge(\beta_{2}\mid\overline{\beta_{1}})\wedge(\beta_{n-1}\mid\overline{\beta_{1}}\wedge\ldots\wedge\overline{\beta_{n-2}})$
is an atom of $\mathcal{C}({\bf A})$ and indeed all atoms of $\mathcal{C}({\bf
A})$ have that form for some sequence of atoms of ${\bf A}$ of length $n-1$.
For any such sequence $\langle\alpha\rangle$, we will write
$\omega_{\langle\alpha\rangle}$ to denote its associated atom of
$\mathcal{C}({\bf A})$.

For every sequence $\langle\alpha\rangle$, $\omega_{\langle\alpha\rangle}$
clearly belongs to $\mathbb{T}({\bf A})$. Thus it makes sense to look at these
terms inside $\mathcal{T}({\bf A})$.

######  Theorem 6.2.

For every finite Boolean algebra ${\bf A}$ with atoms
$\alpha_{1},\ldots,\alpha_{n}$, the set ${\rm at}(\mathcal{T}({\bf A}))$ of
atoms of $\mathcal{T}({\bf A})$ coincides with the set
$\\{[\omega_{\langle\alpha\rangle}]\in\mathcal{T}({\bf
A})\mid\omega_{\langle\alpha\rangle}$ is an atom of $\mathcal{C}({\bf A})\\}$.

###### Proof.

Direct inspection on the proofs of Proposition 4.3 and Theorem 4.4 from [15]
where the authors proved that ${\rm at}(\mathcal{C}({\bf A}))$ is the set of
atoms of $\mathcal{C}({\bf A})$, shows that the unique properties of
$\mathcal{C}({\bf A})$ needed in their proofs are the basic properties of
Boolean algebras (in particular commutativity and associativity of $\wedge$
and distributivity), plus the following:

  * •

If $a\leq b$, then for all $c$, $[(a\mid b\vee c)]\leq[(a\mid b)]$ (see part
(ii) in the proof of [15, Theorem 4.4]);

  * •

for all $a\neq\bot$, $[(a\mid a)]=1$ and $[(\bot\mid a)]=0$ (see part (b) in
the proof of [15, Proposition 4.3]);

  * •

$[(a\mid b)]=\bigvee_{\alpha\in{\rm at}({\bf A}):\alpha\leq a}[(\alpha\mid
b)]$; (see (a) in the proof of [15, Proposition 4.3]).

All these properties hold in $\mathcal{T}({\bf A})$, whence the same proofs
applies to this case. ∎

######  Corollary 6.3.

For every finite Boolean algebra ${\bf A}$, $\mathcal{T}({\bf A})$ is
isomorphic to $\mathcal{C}({\bf A})$.

###### Proof.

Two Boolean algebras with the same cardinality are isomorphic. ∎

In [15] it is proved that any (unconditional) positive probability $P$ on
${\bf A}$ canonically extends to a positive probability $\mu_{P}$ on
$\mathcal{C}({\bf A})$ such that for every basic conditional $(a\mid b)$,
$\mu_{P}(a\mid b)=P(a\wedge b)/P(b)$.

We can finally apply the latter result and the above Theorem 6.2 to show our
final outcome, for which we still need a previous lemma on the factorization
of the probability $P^{*}$ on the atoms of $\mathcal{T}({\bf A})$, that is in
fact a particular case of [25, Theorem 18]. For the reader’s convenience, we
add its proof in the Appendix.

######  Lemma 6.4.

Let ${\bf A}$ be a finite Boolean algebra with ${\rm at}({\bf
A})=\\{\alpha_{1},\ldots,\alpha_{n}\\}$ and let $P$ be a conditional
probability on $A\times A^{\prime}$. Then,

$P^{*}((\alpha_{1}\mid\top){\land}(\alpha_{2}\mid\neg\alpha_{1}){\land}\ldots{\land}(\alpha_{n-1}\mid\alpha_{n-1}\lor\alpha_{n}))=$

$=P(\alpha_{1})\cdot P(\alpha_{2}\mid\neg\alpha_{1})\cdot\ldots\cdot
P(\alpha_{n-1}\mid\alpha_{n-1}\lor\alpha_{n})$.

We observe that, if $P$ is a positive probability on ${\bf A}$, for each basic
conditional $(a\mid b)$, $P^{*}(a\mid b)=P(a\wedge b)/P(b)=\mu_{P}(a\mid b)$.
Our last result shows that $P^{*}$ and $\mu_{P}$ coincide on every compound
conditional as well.

######  Theorem 6.5.

Given a positive probability $P$ on ${\bf A}$ and any $t\in\mathbb{T}(A)$,
$P^{*}(t)=\mu_{P}(t)$, once we identify the elements of $\mathcal{T}({\bf A})$
with those of $\mathcal{C}({\bf A})$.

###### Proof.

By Lemma 6.4, $P^{*}$ coincides with $\mu_{P}$ on the atoms of
$\mathcal{T}({\bf A})$, and hence on the whole algebra. ∎

##  7 Conclusions

In the present paper we have put forward an investigation on compound
conditionals that, starting from the original setting proposed by de Finetti,
aims at representing them in terms of conditional random quantities.
Technically speaking, we start by a finite Boolean algebra ${\bf A}$ of events
and a (coherent) conditional probability $P$ on $A\times A^{\prime}$, where
$A^{\prime}=A\setminus\\{\bot\\}$. Then, to each term $t$ written in the
language having as variables basic conditionals of the form $(a\mid b)$ (for
$a\in A$ and $b\in A^{\prime}$), we first consider, for each interpretation
$w\in\Omega$ the reduct $t^{w}$ of $t$, and then we associate to $t$ a
conditional random quantity $X_{t}:\Omega\to[0,1]$, that assigns to each
$w\in\Omega$ the value $X_{t}(w)$ given by the conditional prevision
$\mathbb{P}(X_{t^{w}}\mid{\bf b}(t^{w}))$.

By doing this, we have presented a natural and uniform procedure to interpret
compound conditionals as random quantities and we have investigated the
numerical and logical properties of such representation for compound
conditionals via their associated random quantities. Our main contribution
concerns the possibility of defining operations among conditionals by an
iterative procedure. Furthermore, we have proved that these operations allow
us to regard the set of those numerical representations as a Boolean algebra
$\mathcal{T}({\bf A})$. This latter result provides in turn a numerical
counterpart of the construction explored in [15], where the authors showed
that compound symbolic conditionals, satisfying certain identities, can be
endowed with a suitable Boolean algebra structure $\mathcal{C}({\bf A})$. In
particular, we have shown that the two algebraic structures that arise from
these numerical and the symbolic representations of conditionals turn out to
be isomorphic. Moreover, any conditional probability $P$ on $A\times
A^{\prime}$ extends in the same way to both algebras $\mathcal{T}({\bf A})$
and $\mathcal{C}({\bf A})$.

As for future work we plan to generalize Theorem 6.5 to the case in which $P$
is a conditional probability on $A\times A^{\prime}$. Moreover, we aim at
studying the extension of the random quantity-based approach to compound
conditionals developed in this paper to deal with general forms of iterated
conditionals, see e.g. [44, 29]. Finally, we plan to study possible
applications of compound conditionals to non-monotonic reasoning from
conditional bases and to conditional logics more in general, in the line of
possible interrelationships with different areas as discussed in [2]. Another
interesting area of application of compound conditionals to be further
explored is the area of psychology of uncertain reasoning [12].

### Acknowledgments

Flaminio and Godo acknowledge partial support by the MOSAIC project (EU
H2020-MSCA-RISE-2020 Project 101007627) and also by the Spanish project
PID2019-111544GB-C21, funded by MCIN/AEI/10.13039/501100011033. Sanfilippo
acknowledges support by the FFR2020-FFR2021 projects of University of Palermo.

## Appendix

### A1. Axiomatic conditional probability

Given an algebra of events ${\bf A}$ and a non empty subfamily $B$ of
$A^{\prime}$, where $A^{\prime}=A\setminus\\{\bot\\}$, a (finitely-additive)
_conditional probability_ on $A\times B$ is a real-valued function $P$ defined
on $A\times B$ satisfying the following properties, see, e.g., [Renyi55,
Csaszar55, Popper59, Dubins75, 34, Rega85, Cole94, 7]:

  1. 1.

$P(\cdot|b)$ is a finitely additive probability on ${\bf A}$, for every $b\in
B$;

  2. 2.

$P(b|b)$=1, for every $b\in B$;

  3. 3.

$P(ab|c)=P(a|bc)P(b|c)$, for every $a\in A$ and $b,c\in B$.

Notice that, when $c=\top$, from 3 it follows that $P(ab)=P(a|b)P(b)$, where
$P(\cdot)=P(\cdot|\top)$. In particular, $P(a|b)=P(ab)/P(b)$, when $P(b)>0$.

### A2. Proof of Propostion 4.4

Proposition 4.4. For every $t,s,r\in\mathbb{T}(A)$ the following conditions
hold:

  1. 1.

$X_{t}=X_{t{\land}t}$

  2. 2.

$X_{t{\land}s}=X_{s{\land}t}$

  3. 3.

$X_{t{\land}(s{\land}r)}=X_{(t{\land}s){\land}r}$

  4. 4.

$X_{t{\land}\neg t}=0$.

  5. 5.

$X_{\neg(t{\land}s)}=X_{\neg t{\lor}\neg s}$

  6. 6.

$X_{t{\lor}(s{\land}r)}=X_{(t{\land}s){\lor}(t{\lor}r)}$

  7. 7.

$X_{t{\lor}s}=X_{t}+X_{s}-X_{t{\land}s}$

  8. 8.

$X_{\neg t}=1-X_{t}$ and hence $X_{\neg\neg t}=X_{t}$

  9. 9.

If $a\leq b$, then $X_{(a\mid b)\wedge(a\mid b\vee c)}=X_{a\mid b\vee c}$.

###### Proof.

The proofs of the items (1)-(8) are by induction on the complexity of the
terms $t,s,r$ appearing in the expressions. We will show the proofs of (1),
(2), (3), (4), (8) and the basic case of (7) to exemplify the general idea.
Also notice that, by Lemma 4.1, to prove an identity $X_{t}=X_{s}$ where ${\bf
b}(t)={\bf b}(s)$ it is enough to check that $X_{t}(w)=X_{s}(w)$ for all
$w\in{\bf b}(t)$.

(1) $X_{t}=X_{t{\land}t}$. Note that ${\bf b}(t)={\bf b}(t{\land}t)$. We
proceed by induction on the complexity of $t$:

  * •

Let $t=(a\mid b)$, then one can check that $(a\mid b)^{w}=((a\mid
b){\land}(a\mid b))^{w}\in\\{1,0\\}$ for every $w\models b$. Hence $X_{(a\mid
b)}=X_{(a\mid b){\land}(a\mid b)}$

  * •

Assume $X_{s{\land}s}=X_{s}$ for every $s\in Red^{0}(t)$ ands we check that
$X_{t^{w}}=X_{(t{\land}t)^{w}}$ for every $w\in{\rm at}({\bf A})$ such that
$w\models{\bf b}(t)$. But $t^{w}\in Red^{0}(t)$, hence by the inductive
hypothesis, we have $X_{(t{\land}t)^{w}}=X_{t^{w}{\land}t^{w}}=X_{t^{w}}$,
hence $X_{t}=X_{t{\land}t}$.

(2) $X_{t{\land}s}=X_{s{\land}t}$.

  * •

First we check the condition for basic conditionals $t=(a\mid b)$ and
$s=(c\mid d)$ by computing their conjunctions as in Example 3.8.

  * •

Then, assume $X_{t^{\prime}{\land}s^{\prime}}=X_{s^{\prime}{\land}t^{\prime}}$
for every $t^{\prime},s^{\prime}$ such that $t^{\prime}{\land}s^{\prime}\in
Red^{0}(t{\land}s)$. Then, again use Lemma 4.1 and show
$X_{(t{\land}s)^{w}}=X_{(s{\land}t)^{w}}$ for every $w\in{\bf
b}(t{\land}s)={\bf b}(s{\land}t)$, but
$X_{(t{\land}s)^{w}}=X_{t^{w}{\land}s^{w}}=X_{s^{w}{\land}t^{w}}=X_{(s{\land}t)^{w}}$.

(3) $X_{t{\land}(s{\land}r)}=X_{(t{\land}s){\land}r}$. The proof is analogous
to the previous one:

  * •

First check that the condition for basic conditionals $t=(a\mid b)$, $s=(c\mid
d)$ and $r=(e\mid f)$. Indeed, $((a\mid b)\land((c\mid d)\land(e\mid
f)))^{w}=(((a\mid b)\land(c\mid d))\land(e\mid f))^{w}\in\\{(a\mid
b)\land(c\mid d),(a\mid b)\land(e\mid f),(c\mid d)\land(e\mid f),(a\mid
b),(c\mid d),(e\mid f),1,0\\}$ for every $w\models b{\lor}d{\lor}f$. Hence,
$X_{(a\mid b)\land((c\mid d)\land(e\mid f))}=X_{((a\mid b)\land(c\mid
d))\land(e\mid f)}$.

  * •

Then, assume
$X_{t^{\prime}{\land}(s^{\prime}{\land}r^{\prime})}=X_{(t^{\prime}{\land}s^{\prime}){\land}r^{\prime}}$
for every $t^{\prime}\land(s^{\prime}\land r^{\prime})\in
Red^{0}(t{\land}(s{\land}r))$. Then, again use Lemma 4.1 and show
$X_{(t{\land}(s{\land}r))^{w}}=X_{((t{\land}s){\land}t)^{w}}$ for every
$w\in{\bf b}(t{\land}(s{\land}r))={\bf b}((t{\land}s){\land}t)$. But
$X_{(t{\land}(s{\land}r))^{w}}=X_{t^{w}{\land}(s^{w}{\land}r^{w})}=X_{(t^{w}{\land}s^{w}){\land}r^{w}}=X_{(t{\land}s){\land}r)^{w}}$.

(4) $X_{t{\land}\neg t}=0$. Clearly, $X_{(a\mid b){\land}\neg(a\mid b)}(w)=0$
for every $w\models b$. If $t$ is arbitrary, by the inductive hypothesis, we
have $X_{(t{\land}\neg t)^{w}}=X_{t^{w}{\land}\neg t^{w}}=0$, and thus
$X_{t{\land}\neg t}=0$ as well.

(7) $X_{t{\lor}s}=X_{t}+X_{s}-X_{t{\land}s}$. As usual, first we check the
condition for basic conditionals $t=(a\mid b)$ and $s=(c\mid d)$. Recalling
how $X_{(a\mid b){\land}(c\mid d)}(w)$ is defined in Example 3.8, a simple
check gives;

$X_{(a\mid b)}(w)+X_{(c\mid d)}(w)-X_{(a\mid b){\land}(c\mid d)}(w)=$

$\left\\{\begin{array}[]{ll}1,&\mbox{if }w\models ab\lor cd\\\ 0,&\mbox{if
}w\models\bar{a}b\bar{c}d\\\ {P}(a\mid b),&\mbox{if
}w\models\bar{b}\bar{c}d\\\ {P}(c\mid d),&\mbox{if
}w\models\bar{a}b\bar{d}\\\\[2.84544pt] \hdashline\\\\[-8.5359pt] {P}(a\mid
b)+{P}(c\mid d)-\\\ \mathbb{P}^{c}((a\mid b){\land}(c\mid d)),&\mbox{if
}w\models\bar{b}\bar{d}\\\ \end{array}\right.$

while:

$X_{(a\mid b){\lor}(c\mid d)}(w)=\left\\{\begin{array}[]{ll}1,&\mbox{if
}w\models ab\lor cd\\\ 0,&\mbox{if }w\models\bar{a}b\bar{c}d\\\ {P}(a\mid
b),&\mbox{if }w\models\bar{b}\bar{c}d\\\ {P}(c\mid d),&\mbox{if
}w\models\bar{a}b\bar{d}\\\\[2.84544pt] \hdashline\\\\[-8.5359pt]
\mathbb{P}^{c}((a\mid b){\lor}(c\mid d)),&\mbox{if }w\models\bar{b}\bar{d}\\\
\end{array}\right.$

Therefore $X_{(a\mid b)}+X_{(c\mid d)}-X_{(a\mid b){\land}(c\mid d)}=X_{(a\mid
b){\lor}(c\mid d)}$ over $b\lor d$, then it coincides over the whole ${\rm
at}({\bf A})$.

(8) Again by induction on $t$. Notice that the case $t=(a\mid b)$ has been
already proved in Example 3.7. Then, assume $X_{\neg
t^{\prime}}=1-X_{t^{\prime}}$ for every $t^{\prime}$ less complex than $t$.
Note that ${\bf b}(\neg t^{\prime})={\bf b}(t^{\prime})$. Then, if
$w\models{\bf b}(t)$, by definition, $X_{\neg t}(w)=\mathbb{P}(X_{(\neg
t)^{w}}\mid{\bf b}(t^{w}))$, and since we can assume that $t^{w}$ is strictly
less complex than $t$, by the induction hypothesis, $\mathbb{P}(X_{(\neg
t)^{w}}\mid{\bf b}(t^{w}))=\mathbb{P}(1-X_{t^{w}}\mid{\bf
b}(t^{w}))=1-\mathbb{P}(X_{t^{w}}\mid{\bf b}(t^{w}))=1-X_{t}(w)$.

(9) The claim follows by direct computation of the random quantities
$X_{(a\mid b)\wedge(a\mid b\vee c)}$ and of $X_{(a\mid b\vee c)}$, when $a\leq
b$, by using the generic definition of the random quantity $X_{t\wedge s}$ for
any $t$ and $s$ as given in Definition 3.5. ∎

### A3. Proof of Lemma 6.3

Lemma 6.3. Let ${\bf A}$ be a finite Boolean algebra with ${\rm at}({\bf
A})=\\{\alpha_{1},\ldots,\alpha_{n}\\}$ and let $P$ be a conditional
probability on $A\times A^{\prime}$. Then,

$P^{*}((\alpha_{1}\mid\top){\land}(\alpha_{2}\mid\neg\alpha_{1}){\land}\ldots{\land}(\alpha_{n-1}\mid\alpha_{n-1}\lor\alpha_{n}))=$

$P(\alpha_{1})\cdot P(\alpha_{2}\mid\neg\alpha_{1})\cdot\ldots\cdot
P(\alpha_{n-1}\mid\alpha_{n-1}\lor\alpha_{n})$.

###### Proof.

Let
$A_{i}=(\alpha_{i}\mid\alpha_{i}{\lor}\ldots\vee\alpha_{n}){\land}\ldots{\land}(\alpha_{n-1}\mid\alpha_{n-1}{\lor}\alpha_{n})$.
We will prove first that, for every $1\leq i\leq n-1$, the following equality
holds:

$\begin{array}[]{ll}P^{*}(A_{i})=P(\alpha_{i}\mid\alpha_{i}{\lor}...{\lor}\alpha_{n})\cdot
P^{*}(A_{i+1})\end{array}$

Indeed, one can check that the random quantity $X_{A_{i}}$ is defined as
follows:

$X_{A_{i}}(w)=\left\\{\begin{array}[]{ll}P^{*}(A_{i}),&\mbox{if
}w\models\alpha_{1}{\lor}\cdots{\lor}\alpha_{i-1},\\\ P^{*}(A_{i+1}),&\mbox{if
}w\models\alpha_{i},\\\ 0,&\mbox{if
}w\models\alpha_{i+1}\vee\cdots\vee\alpha_{n}.\end{array}\right.$

Hence, since ${\bf b}(A_{i})=\alpha_{i}\lor\ldots\lor\alpha_{n}$, we have

$P^{*}(A_{i})=\mathbb{P}^{c}(X_{A_{i}})=P^{*}(A_{i+1})\cdot
P(\alpha_{i}\mid\alpha_{i}\lor\ldots\lor\alpha_{n}).$

Now, by repeatedly applying this equality, we finally have:

$\begin{array}[]{ll}P^{*}(A_{1})=P(\alpha_{1})\cdot P^{*}(A_{2})=\\\
=P(\alpha_{1})\cdot P(\alpha_{2}\mid\neg\alpha_{1})\cdot
P^{*}(A_{3})=\cdots=\\\ =P(\alpha_{1})\cdot
P(\alpha_{2}\mid\neg\alpha_{1})\cdot\ldots\cdot
P(\alpha_{n-1}\mid\alpha_{n-1}\lor\alpha_{n}).\end{array}$

Then the proposition follows just by observing that
$A_{1}=(\alpha_{1}\mid\top){\land}(\alpha_{2}\mid\neg\alpha_{1}){\land}\ldots{\land}(\alpha_{n-1}\mid\alpha_{n-1}\lor\alpha_{n})$.
∎

## References

  * [1] Adams, E.W.: The logic of conditionals. Reidel, Dordrecht (1975)
  * [2] Aucher, G., Égré, P., Kern-Isberner, G., Poggiolesi, F.: Conditional logics and conditional reasoning: New joint perspectives (Dagstuhl seminar 19032). Dagstuhl Reports 9(1), 47–66 (2019). https://doi.org/10.4230/DagRep.9.1.47, https://doi.org/10.4230/DagRep.9.1.47
  * [3] Beierle, C., Eichhorn, C., Kern-Isberner, G., Kutsch, S.: Properties of skeptical c-inference for conditional knowledge bases and its realization as a constraint satisfaction problem. Ann. Math. Artif. Intell. 83(3-4), 247–275 (2018)
  * [4] Benferhat, S., Dubois, D., Prade, H.: Nonmonotonic reasoning, conditional objects and possibility theory. Artificial Intelligence 92, 259–276 (1997). https://doi.org/10.1016/S0004-3702(97)00012-X
  * [5] Calabrese, P.: An algebraic synthesis of the foundations of logic and probability. Information Sciences 42(3), 187 – 237 (1987). https://doi.org/10.1016/0020-0255(87)90023-5
  * [6] Ciucci, D., Dubois, D.: A map of dependencies among three-valued logics. Information Sciences 250, 162 – 177 (2013). https://doi.org/10.1016/j.ins.2013.06.040
  * [7] Coletti, G., Scozzafava, R.: Probabilistic logic in a coherent setting. Kluwer, Dordrecht (2002)
  * [8] Coletti, G., Petturiti, D., Vantaggi, B.: Dutch book rationality conditions for conditional preferences under ambiguity. Ann. Oper. Res. 279(1-2), 115–150 (2019)
  * [9] Douven, I., Dietz, R.: A puzzle about Stalnaker’s hypothesis. Topoi pp. 31–37 (2011). https://doi.org/10.1007/s11245-010-9082-3
  * [10] Dubois, D., Prade, H.: Conditional objects as nonmonotonic consequence relations: Main results. In: Doyle, J., Sandewall, E., Torasso, P. (eds.) Proceedings of the 4th International Conference on Principles of Knowledge Representation and Reasoning (KR’94). Bonn, Germany, May 24-27, 1994. pp. 170–177. Morgan Kaufmann (1994)
  * [11] Edgington, D.: On conditionals. Mind 104, 235–329 (1995)
  * [12] Elqayam, S., Douven, I., Evans, J.S.B.T., Cruz, N.: Logic and Uncertainty in the Human Mind: A Tribute to David E. Over. Routledge, Oxon (2020). https://doi.org/10.4324/9781315111902
  * [13] de Finetti, B.: La Logique de la Probabilité. In: Actes du Congrès International de Philosophie Scientifique, Paris, 1935, pp. IV 1 – IV 9. Hermann et Cie Éditeurs, Paris (1936)
  * [14] de Finetti, B.: La prévision: ses lois logiques, ses sources subjectives. Annales de l’Institut Henri Poincaré 7(1), 1–68 (1937)
  * [15] Flaminio, T., Godo, L., Hosni, H.: Boolean algebras of conditionals, probability and logic. Artificial Intelligence 286, 103347 (2020). https://doi.org/10.1016/j.artint.2020.103347
  * [16] van Fraassen, B.: Probabilities of conditionals. Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science 1, 261–308 (1976), cited By 89
  * [17] Ghirardato, P.: Revisiting savage in a conditional world. Economic Theory 20(1), 83–92 (2002)
  * [18] Gilio, A.: Classi quasi additive di eventi e coerenza di probabilità condizionate. Rendiconti dell’Istituto di Matematica dell’Università di Trieste XXI(1), 22–38 (1989)
  * [19] Gilio, A.: Algorithms for precise and imprecise conditional probability assessments. Mathematical Models for Handling Partial Knowledge in Artificial Intelligence pp. 231–254 (1995)
  * [20] Gilio, A.: Probabilistic reasoning under coherence in System P. Annals of Mathematics and Artificial Intelligence 34, 5–34 (2002). https://doi.org/10.1023/A:101442261
  * [21] Gilio, A., Sanfilippo, G.: Conjunction, disjunction and iterated conditioning of conditional events. In: Synergies of Soft Computing and Statistics for Intelligent Data Analysis, AISC, vol. 190, pp. 399–407. Springer, Berlin (2013). https://doi.org/10.1007/978-3-642-33042-1_43
  * [22] Gilio, A., Sanfilippo, G.: Quasi conjunction, quasi disjunction, t-norms and t-conorms: Probabilistic aspects. Information Sciences 245, 146–167 (2013). https://doi.org/10.1016/j.ins.2013.03.019
  * [23] Gilio, A., Sanfilippo, G.: Conditional random quantities and compounds of conditionals. Studia Logica 102(4), 709–729 (2014). https://doi.org/10.1007/s11225-013-9511-6
  * [24] Gilio, A., Sanfilippo, G.: Generalized logical operations among conditional events. Applied Intelligence 49(1), 79–102 (Jan 2019). https://doi.org/10.1007/s10489-018-1229-8
  * [25] Gilio, A., Sanfilippo, G.: Algebraic aspects and coherence conditions for conjoined and disjoined conditionals. International Journal of Approximate Reasoning 126, 98 – 123 (2020)
  * [26] Gilio, A., Sanfilippo, G.: On compound and iterated conditionals. Argumenta 6(2), 241–266 (2021). https://doi.org/10.14275/2465-2334/202112.gil, https://www.argumenta.org/article/compound-iterated-conditionals/
  * [27] Gilio, A., Spezzaferri, F.: Knowledge integration for conditional probability assessments. In: Dubois, D., Wellman, M.P., D’Ambrosio, B., Smets, P. (eds.) Uncertainty in Artificial Intelligence, pp. 98–103. Morgan Kaufmann Publishers (1992)
  * [28] Gilio, A., Sanfilippo, G.: Compound conditionals, Fréchet-Hoeffding bounds, and Frank t-norms. International Journal of Approximate Reasoning 136, 168–200 (2021). https://doi.org/https://doi.org/10.1016/j.ijar.2021.06.006
  * [29] Gilio, A., Sanfilippo, G.: Iterated conditionals and characterization of p-entailment. In: Vejnarová, J., Wilson, N. (eds.) Symbolic and Quantitative Approaches to Reasoning with Uncertainty, ECSQARU 2021, LNCS, vol. 12897, pp. 629–643. Springer International Publishing (2021). https://doi.org/https://doi.org/10.1007/978-3-030-86772-0_45
  * [30] Goodman, I.R., Nguyen, H.T.: A theory of conditional information for probabilistic inference in intelligent systems: II. product space approach. Inf. Sci. 76(1-2), 13–42 (1994)
  * [31] Goodman, I.R., Nguyen, H.T., Walker, E.A.: Conditional Inference and Logic for Intelligent Systems: A Theory of Measure-Free Conditioning. North-Holland (1991), www.dtic.mil/dtic/tr/fulltext/u2/a241568.pdf
  * [32] Halpern, J.Y.: Reasoning about Uncertainty. MIT Press, Cambridge, MA, USA (2003)
  * [33] Halpern, J.Y.: Actual Causality. The MIT Press (2016)
  * [34] Holzer, S.: On coherence and conditional prevision. Bollettino dell’Unione Matematica Italiana 4(6), 441–460 (1985)
  * [35] Jeffrey, R.: Matter-of-Fact Conditionals. Proceedings of the Aristotelian Society, Supplementary Volume 65, 161–183 (1991)
  * [36] Kaufmann, S.: Conditionals right and left: Probabilities for the whole family. Journal of Philosophical Logic 38, 1–53 (2009). https://doi.org/10.1007/s10992-008-9088-0
  * [37] Kern-Isberner, G.: Conditionals in Nonmonotonic Reasoning and Belief Revision – Considering Conditionals as Agents. Lecture Notes in Artificial Intelligence, Springer (2001)
  * [38] McGee, V.: Conditional probabilities and compounds of conditionals. Philosophical Review 98(4), 485–541 (1989). https://doi.org/http://dx.doi.org/10.2307/2185116
  * [39] Milne, P.: Bruno de Finetti and the Logic of Conditional Events. British Journal for the Philosophy of Science 48(2), 195–232 (1997), http://www.jstor.org/stable/687745
  * [40] Nguyen, H.T., Walker, E.A.: A history and introduction to the algebra of conditional events and probability logic. IEEE Transactions on Systems, Man, and Cybernetics 24(12), 1671–1675 (1994). https://doi.org/10.1109/21.328924
  * [41] Pfeifer, N., Sanfilippo, G.: Probabilistic squares and hexagons of opposition under coherence. International Journal of Approximate Reasoning 88, 282–294 (2017). https://doi.org/10.1016/j.ijar.2017.05.014
  * [42] Rigo, P.: Un teorema di estensione per probabilità condizionate finitamente additive. Atti della XXXIV Riunione Scientifica S.I.S. pp. 27–34 (1988)
  * [43] van Rooij, R., Schulz, K.: Conditionals, causality and conditional probability. Journal of Logic, Language and Information 28(1), 55–71 (2019). https://doi.org/10.1007/s10849-018-9275-5, https://doi.org/10.1007/s10849-018-9275-5
  * [44] Sanfilippo, G., Gilio, A., Over, D., Pfeifer, N.: Probabilities of conditionals and previsions of iterated conditionals. International Journal of Approximate Reasoning 121, 150 – 173 (2020). https://doi.org/10.1016/j.ijar.2020.03.001
  * [45] Sanfilippo, G., Pfeifer, N., Over, D., Gilio, A.: Probabilistic inferences from conjoined to iterated conditionals. International Journal of Approximate Reasoning 93(Supplement C), 103 – 118 (2018). https://doi.org/10.1016/j.ijar.2017.10.027
  * [46] Sanfilippo, G.: From imprecise probability assessments to conditional probabilities with quasi additive classes of conditioning events. In: Proc. of the Twenty-Eighth Conference on Uncertainty in Artificial Intelligence (UAI-12). pp. 736–745. AUAI Press, Corvallis, Oregon (2012)
  * [47] Schay, G.: An algebra of conditional events. Journal of Mathematical Analysis and Applications 24, 334–344 (1968). https://doi.org/10.1016/0022-247X(68)90035-8
  * [48] Stalnaker, R., Jeffrey, R.: Conditionals as random variables. In: Eells, E., Skyrms, B. (eds.) Probability and Conditionals: Belief Revision and Rational Decision, pp. 31–46. Cambridge University Press, New York, NY, USA (1994), http://dl.acm.org/citation.cfm?id=207895.207899
  * [49] Vantaggi, B.: Incomplete preferences on conditional random quantities: Representability by conditional previsions. Math. Soc. Sci. 60(2), 104–112 (2010). https://doi.org/10.1016/j.mathsocsci.2010.06.002, https://doi.org/10.1016/j.mathsocsci.2010.06.002