Betting on continuous independent events

Abstract

de Finetti’s Dutch book theorem explains why probability has to be additive on disjunctions of incompatible yes–no (Boolean) events. The theorem holds verbatim also for continuous events, or random variables, as formalized in Łukasiewicz logic. Independence, a subtler notion than incompatibility, has a probability-free definition: two events are logically independent if there is no logical relation between them. Our main result in this paper is that product is the only coherence preserving operation on books on logically independent sets of continuous (as well as yes–no) events. Thus, also the product law for stochastically independent sets of events is implicit in de Finetti’s deeper notion of a coherent betting system.

Appendices

Appendix 1: The maximal spectral space of an MV-algebra

The zerosetZf of a function $f:X\rightarrow {\mathbb {R}}$ is defined by $Z f = f^{-1}(0) = \{x\in X\mid f(x)=0\}.$ More generally, for any set ${\mathfrak {j}} \subseteq {\mathbb {R}}^{X}$ the zeroset$Z {\mathfrak {j}}$ of ${\mathfrak {j}}$ is defined by $Z {\mathfrak {j}} =\bigcap \{\mathcal {Z} f\mid f\in {\mathfrak {j}} \}.$

Definition 4.1

For any MV-algebra A, the set $\varvec{\mu }(A)$ of maximal ideals of A is equipped with the spectral (also known as hull-kernel, or Zariski) topology. A basis of closed sets for this topology is given by all sets of the form $F_a= \{{\mathfrak {m}}\in \varvec{\mu }(A)\mid a\in {\mathfrak {m}}\}$, letting a range over elements of A. We say that $\varvec{\mu }(A)$ is the maximal spectral space of A. As shown in (Mundici 2011, Proposition 4.15), for any MV-algebra A, $\varvec{\mu }(A)$ is a nonempty compact Hausdorff space.

Let $E\not =\emptyset$ be a set and $B\subseteq {{\mathrm{[0,1]}}}^E$ an MV-algebra of ${{\mathrm{[0,1]}}}$-valued functions on E, with the pointwise operations of ${{\mathrm{[0,1]}}}$. We then say that Bseparates points (or, B is separating), if for any two distinct points $x,y\in E$ there is $f\in B$ such that $f(x)\not =f(y)$.

Theorem 4.2

Let A be an MV-algebra.

(i):

For any maximal ideal ${\mathfrak {m}}$ of A there is a unique pair $(\overline{{\mathfrak {m}}}, I_{{\mathfrak {m}}})$ with $I_{{\mathfrak {m}}}$ an MV-subalgebra of the standard MV-algebra ${{\mathrm{[0,1]}}}$ and $\overline{{\mathfrak {m}}}$ an isomorphism of the quotient MV-algebra $A/{\mathfrak {m}}$ onto $I_{{\mathfrak {m}}}$.

(ii):

The map $\ker :\eta \mapsto \ker \eta$ is a one-one correspondence between the set $\hom (A)$ of homomorphisms of A into ${{\mathrm{[0,1]}}}$ and $\varvec{\mu }(A)$. The inverse map sends each ${\mathfrak {m}}\in \varvec{\mu }(A)$ to the homomorphism $\eta _{{\mathfrak {m}}}:A\rightarrow {{\mathrm{[0,1]}}}$ given by $a \mapsto \overline{\mathfrak m}(a/{\mathfrak {m}})$. For each $\theta \in \hom (A)$ and $a\in A$, $\theta (a)=\overline{\ker \theta }\left( {a}/{\ker \theta }\right) .$

(iii):

The map $^*:a \in A \mapsto a^* \in [0,1]^{{\varvec{\mu }}(A)}$ defined by $a^*({\mathfrak {m}}) = \overline{{\mathfrak {m}}}(a/{\mathfrak {m}})$ is a homomorphism of A onto a separating MV-subalgebra $A^{*}$ of continuous ${{\mathrm{[0,1]}}}$-valued functions over ${\varvec{\mu }}(A))$. The map $a\mapsto a^*$ is an isomorphism of A onto $A^*$ iff A is semisimple.

(iv):

Suppose $X\not =\emptyset$ is a compact Hausdorff space and B is a separating subalgebra of C(X), the latter denoting the MV-algebra of all continuous [0, 1]-valued functions on X, with the pointwise operations of the MV-algebra [0, 1]. Then the map $\iota :x\in X\mapsto {\mathfrak {h}}_x = \{f\in B \mid f(x)=0\}$ is a homeomorphism of X onto ${\varvec{\mu }}(B)$. The inverse map $\iota ^{-1}$ sends each ${\mathfrak {m}}\in \varvec{\mu }(B)$ to the only element of ${Z}{\mathfrak {m}}$.

(v):

With the same hypotheses of (iv), for each $f\in B,$$f^*\iota = f.$ Thus, the map $f^*\in B^*\mapsto f^*\iota \in C(X)$ is the inverse of the isomorphism $^{*}:B\cong B^*$ of (iii). In particular, $f(x)=f^*({\mathfrak {h}}_{x})$ for each $x\in X$.

Proof

(Mundici 2011, Theorem 4.16). $\square$

Appendix 2: The semisimple tensor product

Given semisimple MV-algebras A and B one can naturally multiply elements of A by elements of B, by jointly embedding A and B into their semisimple tensor product $A \otimes B$, in a sense that will be made precise in this appendix.

Throughout this appendix we allow the trivial singleton MV-algebra$\{0\}=\{1\}$. The symbols A, B, C will range over MV-algebras with $0 \not =1$. The symbols S and T will range over all MV-algebras, including the trivial one. For any two MV-algebras S and T, $S \times T$ will denote the Cartesian product of their underlying sets.

Definition 5.1

For every element u in an MV-algebra S, let [0, u] denote the set $\{x \in S \mid 0 \le x \le u\}$ equipped with the operations $\lnot _{u} x = u \odot \lnot x$ and $x \oplus _{u} y = u \wedge (x\oplus y)$. Then [0, u] is a (possibly singleton) MV-algebra with zero element 0 and unit element u. We say that [0, u] is the interval MV-algebra of S. The ambient MV-algebra S will always be clear from the context.

Definition 5.2

Let A, B, S be MV-algebras. A bimorphism$\flat$ of the set $A \times B$ into S is a function $\flat :A \times B \rightarrow S$ satisfying the following conditions, for all $a,a_{1}, a_{2} \in A$ and $b,b_{1}, b_{2} \in B$:

• (1) $\flat (1,1) = 1.$

• (2) $\flat (a,0)= 0.$

• (3) $\flat (0,b)=0$.

• (4) $\flat (a,b_{1}\vee b_{2}) =\flat (a,b_{1})\vee \flat (a,b_{2}).$

• (5) $\flat (a,b_{1}\wedge b_{2}) =\flat (a,b_{1})\wedge \flat (a,b_{2}).$

• (6) $\flat (a_{1}\vee a_{2}, b ) =\flat (a_{1},b)\vee \flat (a_{2},b)$.

• (7) $\flat (a_{1}\wedge a_{2}, b ) =\flat (a_{1},b)\wedge \flat (a_{2},b)$.

• (8) $\flat (a,b_{1})\odot \flat (a,b_{2}) = 0\;$ whenever $\;b_{1}\odot b_{2}=0.$

• (9) $\flat (a,b_{1}\oplus b_{2}) =\flat (a,b_{1})\oplus \flat (a,b_{2})$ whenever $\;b_{1}\odot b_{2}=0.$

• (10) $\flat (a_{1},b)\odot \flat (a_{2},b) = 0\;$ whenever $\;a_{1}\odot a_{2}=0.$

• (11) $\flat (a_{1}\oplus a_{2}, b ) = \flat (a_{1},b)\oplus \flat (a_{2},b)$ whenever $\;a_{1}\odot a_{2}=0.$

We denote by    bim(A, B, S)    the set of all bimorphisms $\flat :A \times B \rightarrow S.$

Let A, B, S, T be MV-algebras. If $\eta :S \rightarrow T$ is a homomorphism and $\flat \in \mathrm{bim}(A,B,S)$   then    $\eta \flat \in \mathrm{bim}(A,B,T)$.

A bimorphism $\flat \in \mathrm{bim}(A,B,S)$ is said to be universal if for every MV-algebra T and $\flat ' \in \mathrm{bim}(A,B,T)$ there is a unique homomorphism $\lambda :S \rightarrow T$ such that $\lambda \flat = \flat '$.

The construction of a universal bimorphism of two MV-algebras A and B parallels Definition 5.2:

Construction 5.3

For any MV-algebras A and B, let $\mathsf {FREE}_{A\times B}$ be the free MV-algebra over the free generating set $A \times B$. Let ${\mathfrak {t}}$ be the ideal of $\mathsf {FREE}_{A\times B}$ generated by the following elements, for every $a, a_{1}, a_{2} \in A$ and $b, b_{1}, b_{2} \in B$ (here “${{\mathrm{\mathrm dist}}}$” denotes Chang’s distance function (Cignoli et al. 2000, Definition 1.2.4)):

• (1) ${{\mathrm{\mathrm dist}}}((1,1),1)$,

• (2) ${{\mathrm{\mathrm dist}}}((a,0),0),$

• (3) ${{\mathrm{\mathrm dist}}}((0,b),0)$,

• (4) ${{\mathrm{\mathrm dist}}}((a,b_{1}\vee b_{2}), (a,b_{1})\vee (a,b_{2})),$

• (5) ${{\mathrm{\mathrm dist}}}((a,b_{1}\wedge b_{2}), (a,b_{1})\wedge (a,b_{2})),$

• (6) ${{\mathrm{\mathrm dist}}}((a_{1}\vee a_{2}, b ), (a_{1},b)\vee (a_{2},b))$,

• (7) ${{\mathrm{\mathrm dist}}}((a_{1}\wedge a_{2}, b ), (a_{1},b)\wedge \flat (a_{2},b))$,

• (8) ${{\mathrm{\mathrm dist}}}((a,b_{1})\odot (a,b_{2}), 0)\;$ whenever $\;b_{1}\odot b_{2}=0,$

• (9) ${{\mathrm{\mathrm dist}}}((a,b_{1}\oplus b_{2}), (a,b_{1})\oplus (a,b_{2}))$ whenever $\;b_{1}\odot b_{2}=0,$

• (10) ${{\mathrm{\mathrm dist}}}((a_{1},b)\odot (a_{2},b), 0)\;$ whenever $\;a_{1}\odot a_{2}=0,$

• (11) ${{\mathrm{\mathrm dist}}}((a_{1}\oplus a_{2}, b ), (a_{1},b)\oplus (a_{2},b))$ whenever $\;a_{1}\odot a_{2}=0.$

We now set $A \otimes _\mathrm{MV}B = {\mathsf {FREE}_{A\times B}}/{{\mathfrak {t}}}.$

Without danger of confusion, let the map $\otimes _\mathrm{MV}:(a,b) \mapsto a \otimes _\mathrm{MV}b$ be defined by $a \otimes _\mathrm{MV}b = {(a,b)}/{{\mathfrak {t}}}, \,\,\, \text{ for } \text{ all } \,\,\, (a,b)\in A\times B.$ Then $\otimes _\mathrm{MV}\in \text{ bim }(A,B,A\times B)$. Since $A\times B$ is a generating set of $\mathsf {FREE}_{A\times B}$, the MV-algebra $A \otimes _\mathrm{MV}B$ is generated by the set of elements of the form $a \otimes _\mathrm{MV}b$.

Any element of $A \otimes _\mathrm{MV}B$ has the form $\tau (a_1\otimes _\mathrm{MV}b_1,\ldots ,a_k\otimes _\mathrm{MV}b_k)$ for some $a_i\in A, \,\,b_i\in B$ and MV-term $\tau (X_1,\ldots ,X_k)$.

Proposition 5.4

(Universal property of the $\otimes _\mathrm{MV}$ bimorphism) For any bimorphism $\flat :A \times B \rightarrow S$ there is precisely one homomorphism $\lambda :A \otimes _\mathrm{MV}B \rightarrow S$ such that $\lambda (a \otimes _\mathrm{MV}b) = \flat (a,b)$ for all $(a,b)\in A\times B$. Thus, $A \otimes _\mathrm{MV}B$ is uniquely determined up to isomorphism. $A \otimes _\mathrm{MV}B$ is said to be the MV-algebraic tensor product of A and B.

Proof

(Mundici 2011, Proposition 9.13). $\square$

For any MV-algebra A the radical${{\mathrm{\mathrm Rad}}}(A)$ of A is the intersection of all maximal ideals of A.

Definition 5.5

For any semisimple MV-algebras A and B we write

$$\begin{aligned} A \otimes B = \frac{A \otimes _\mathrm{MV}B}{{{\mathrm{\mathrm Rad}}}(A \otimes _\mathrm{MV}B)} \end{aligned}$$

and say that $A\otimes B$ is the semisimple tensor product of A and B.

As an immediate consequence of the definition we have:

Proposition 5.6

The map

$$\begin{aligned} (a,b)\in A\times B \mapsto a\otimes b = \frac{(a \otimes _\mathrm{MV}b)}{{{\mathrm{\mathrm Rad}}}(A \otimes _\mathrm{MV}B)}\in A\otimes B \end{aligned}$$

is a bimorphism of $A \times B$ into $A \otimes B$ whose range generates $A \otimes B$. Each element $e\in A\otimes B$ has the form $e = \tau (a_1\otimes b_1,\ldots ,a_k\otimes b_k),$ for suitable elements $a_i\in A, \,\, b_j\in B$ and MV-term $\tau (X_1,\ldots ,X_k)$.

Proposition 5.7

(Universal property of the $\otimes$ bimorphism) Let A and B be semisimple MV-algebras. Then $A \otimes B$ is a semisimple MV-algebra. Further, for every bimorphism $\flat$ of $A \times B$ into a semisimple MV-algebra C there is a unique homomorphism $\lambda :A \otimes B \rightarrow C$ such that $\lambda (a \otimes b) = \flat (a,b)$ for all $(a,b)\in A\times B$.

Proof

(Mundici 2011, Proposition 9.16). $\square$

We next provide a concrete representation of semisimple tensor products of MV-algebras. For any topological space W recall the notation C(W) for the MV-algebra of all continuous ${{\mathrm{[0,1]}}}$-valued functions on W:

Theorem 5.8

For X and Y nonempty compact Hausdorff spaces, let the MV-algebras A and B be separating subalgebras of C(X) and C(Y), respectively. In view of Theorem 4.2 let us identify X with ${\varvec{\mu }}(A)$ and Y with ${\varvec{\mu }}(B)$. Let the map $\varvec{\pi }$ transform each pair $(f,g)\in A\times B$ into the function $\varvec{\pi }(f,g) \in C({\varvec{\mu } }(A) \times {\varvec{\mu } }(B))$ given by

$$\begin{aligned} (\varvec{\pi }(f,g))({\mathfrak {l}},{\mathfrak {m}}) = f({\mathfrak {l}}) \cdot g({\mathfrak {m}}) \,\,\, \text{ for } \text{ all } \,\,\, (\mathfrak l,{\mathfrak {m}}) \in {\varvec{\mu } }(A)\times {\varvec{\mu } }(B). \end{aligned}$$

Then the MV-algebra G generated by the range of $\varvec{\pi }$ in $C({\varvec{\mu } }(A) \times {\varvec{\mu } }(B))$ is a separating subalgebra of $C({\varvec{\mu } }(A) \times {\varvec{\mu } }(B))$. Further $\varvec{\pi }(f,g)= \varvec{\jmath }(f \otimes g)$ for a uniquely determined isomorphism $\varvec{\jmath }$ of $A \otimes B$ onto G. For each MV-term $\tau$ in k variables, $\varvec{\jmath }( \tau (a_1\otimes b_1,\ldots ,a_k\otimes b_k)) = \tau (a_1\cdot b_1,\ldots ,a_k\cdot b_k).$

Proof

(Mundici 2011, Theorem 9.17). $\square$

Corollary 5.9

For each $f\in A$ and $g\in B$ let us write $f_\mathrm{cyl\uparrow }=\varvec{\pi }(f,1_B)$ and $g_\mathrm{cyl_{\rightarrow }}=\varvec{\pi }(1_A,g)$. Then the cylindrification maps $f\mapsto f_\mathrm{cyl\uparrow }$ (resp., $g \mapsto g_\mathrm{cyl_{\rightarrow }}$) isomorphically embed A (resp., B) onto a subalgebra $A_\mathrm{cyl\uparrow }$ (resp., onto a subalgebra $B_\mathrm{cyl_{\rightarrow }}$) of $A\otimes B$.

For any $f\in A$ and $g\in B$ the product $f_\mathrm{cyl\uparrow }\cdot g_\mathrm{cyl_{\rightarrow }}$ is said to be a pure tensor and is denoted $f\otimes g$ without fear of ambiguity.