De Finetti's conception of events is one of the most distinctive aspects of his theory of probability, yet it appears to be somewhat elusive. The purpose of this note is to set up a formal framework in which a rigorous characterisation of this notion, and its cognate modelling assumptions, gives rise to a detailed formalisation of the betting problem which underlies the celebrated Dutch Book Argument. As our main result shows, this re nement captures an intuitive condition which de Finetti imposed on the betting problem, namely that it is irrational to bet on an event which may be true, but whose truth will never be ascertained by the players.
Let E1; : : : ; En be of events of interest. De Finetti's betting problem is the choice that an idealised agent called bookmaker must make when publishing a book, i.e. when making the assignment B = f(Ei; i) : i = 1; : : : ; ng such that each Ei is given value i 2 [0; 1]: Once a book has been published, a gambler can place bets on event Ei by paying i i to the bookmaker. In return for this payment, the gambler will receive i, if Ei occurs and nothing otherwise. De Finetti constructs the betting problem in such a way as to force the bookmaker to publish fair betting odds for book B. To this end, two modelling assumptions are built into the problem, namely that (i) the bookmaker is forced to accept any number of bets on B and (ii) when betting on Ei, gamblers can choose the sign, as well as the magnitude of (monetary) stakes i. Conditions (i-ii) force the bookmaker to publish books with zero-expectation, for doing otherwise may o er the gambler the possibility of making a sure pro t, possibily by choosing negative stakes thereby unilaterally imposing a payo swap to the bookmaker. As the game is zero-sum, this is equivalent to forcing the bookmaker into sure loss. In this context, de Finetti proves that the axioms of probability are necessary and su cient to secure the bookmaker against this possibility.
The crux of the Dutch book argument is the identi cation of the agent's degrees of belief with the price they are willing to pay for an uncertain reward which depends on the future truth value of some presently unknown propositions { the events on which the agents are betting. This clearly suggests that the semantics of events, which bears directly on the de nition of probability, is implicitly endowed with an epistemic structure. The purpose of this paper is to give this structure an explicit formal characterisation and to show how the resulting framework helps us making de Finetti's elusive notion of event much clearer. In particular we shall be able to give a formal setting to the following remark: [T]he characteristic feature of what I refer to as an \event" is that the circumstances under which the event will turn out to be \veri ed" or \disproved" have been xed in advance. [1] 1.1
Let L = fq1; : : : ; qng be a classical propositional language. The set of sentences SL = f'; ; ; : : :g is inductively built up from L through the propositional connectives ^; _; !, and :, as usual. We use ? to denote falsum. Valuations are mappings ! form L into f0; 1g that naturally extend to SL by the truthfunctionality of the propositional connectives (with the usual stipulation that !(?) = 0, for all valuations !). We denote by (L) the class of all valuations over L.
A partial valuation on L is a map : X L ! f0; 1g which, for ! 2 and for every q 2 L is de ned by (q) = !(q) if q 2 X; unde ned otherwise.
The class of all partial valuations over L is denoted by P (L). For : X ! f0; 1g and : Y ! f0; 1g in P (L) we say that extends (and we write ) if X Y , and for every x 2 X, (x) = (x).
Finally, for every formula ' we set ['] = f 2 SL : `' $ g, where as usual, ` denotes the classical provability relation. We conform to the custom of referring to the equivalence classes ['] as to the proposition '.
Note that de Finetti's notion of event is captured by propositions, rather than
sentences. In fact, in the present logical framework, the \circumstances" under
which the propositions turn out to be true or false are nothing but the valuations
in (L). As a consequence, de Finetti's notion of book is formally de ned by
(probability) assignments on a ( nite) set of propositions. We will reserve the
expression propositional books to refer to these particular assignments.
De nition 1. Let ' 2 SL, and let 2 P (L). (
(
Let W be a nite set of nodes interpreted, as usual, as possible worlds. Let e : W ! P (L) such that for every w 2 W , e(w) = w : Xw L ! f0; 1g is a partial valuation. Note that to avoid cumbersome notation we will write w instead of e(w) to denote the partial valuation associated to w, and similarly, we denote by Xw the subset of L for which w is de ned. Finally, let R W W be an accessibility relation. We call a triplet (W; e; R) a partially evaluated frame (pef for short).
Let w 2 W and let ['] be a proposition. We say that that w decides ['], if w ['].
De nition 2 (Events and Facts). Let (W; e; R) be a pef, w 2 W and ' 2 SL. Then we say that a proposition ['] is: { A w-event i w 6 ['], and for every 2 P (L) such that ['], there exists w0 such that R(w; w0), and w0 = . { A w-fact i w [']. w and For every w 2 W we denote by E (w) and F (w) the classes of w-events, and w-facts respectively.
An important consequence of De nition 2 is that in pef events (and facts) are relativised to a speci c state of the world.
A pef K = (W; e; R) is said to be monotonic if satis es: (M) for every w; w0 2 W , if R(w; w0), then w w0 .
Recall that in de Finetti's intuitive characterisation, at the time (i.e. world in W ) at which the contract is signed, bookmaker and gambler agree on which conditions will realise the events in the book. This clearly presupposes some form of monotonic persistence of the underlying structure, which Property (M) guarantees. In particular, in every monotonic pef, w-facts are w0-facts in each w0 which is accessible from w. In addition their truth value once determined, is xed throughout the frame.
De nition 3. Let (W; e; R) be any monotonic pef, and let w 2 W . A w-book is a propositional book B = f['i] = i : i = 1; : : : ; ng where the propositions ['i] are w-events.
Finally, we say that a pef (W; e; R) is complete if the following property is satis ed: (C) for every ' 2 SL and for every w['] 2 W such that =
2 P such that ['], i.e. ['] is a w[']-fact.
w['] De nition 4 (Inaccessible propositions). Let (W; e; R) be a pef and w 2 W . A proposition ['] is said to be w-inaccessible if w 6 ['] and for every w0 such that w0 ['], :R(w; w0).
Inaccessible propositions relativise the betting problem to the speci c information available to the gambler and the bookmaker. This is clearly in consonance with de Finetti's rather strict subjectivism according to which all that matters for the determination of the nal payo is that the agents agree on which events are realised. ['], there exists a
Our rst result singles out the conditions under which a coherent w-book B can be extended, either by w-facts or by w-inaccessible propositions to a coherent propositional book B0.
Theorem 1. Let B = f['i] = i : i = 1; : : : ; ng be a coherent w-book, let
[ 1]; : : : ; [ r] be propositions that are not w-events, and let B0 = B [ f[ j ] = j :
j = 1; : : : ; rg be a propositional book extending B. Then the following hold:
(
Proof. We only prove the direction from left-to-right, the converse being is
immediate in both cases.
(
Theorem 1 captures the key property identi ed by de Finetti in his informal characterisation of events, namely that no monetary betting is rational unless the conditions under which the relevant events will be decided are known to the bookmaker and the gambler. 3
A gambler and a bookmaker interpreted on a complete and monotonic pef are
guaranteed that: (
Example 1 ([2]). Consider an electron , and a world w. We are interested the position and the energy of at w. Let ['] and [ ] be the propositions expressing those measurements, respectively. Moreover let us assume that both ['] and [ ] are w-events. Indeed if at w we are uncertain about the position and the energy of , we can certainly perform experiments to determine them. But, what about ['] ^ [ ]? Position and energy are represented by non-commuting operators in quantum theory, and we can assign an electron a de nite position and a de nite energy, but not both. This fact can be modelled in complete and monotonic pef K, by forcing ['] ^ [ ] = [' ^ ] to be w-inaccessible.
De nition 5. A pef (W; e; R) is fully accessible if R satis es: (A) for all w; w0 2 W , if w
w0 , then R(w; w0).
It is customary [3] to characterise the coherence of a propositional book B in terms of its extension to a ( nitely additive) probability measure on the Boolean algebra spanned by the events in B. As our second result shows, completeness and full accessibility are su cient conditions in order for the algebra generated by the w-events in a w-book B to contain only w-events, and therefore, situations like that of Example 1 cannot be modelled in this context.
Theorem 2. Let K = (W; e; R) be a complete pef. If K is fully accessible, then for every w 2 W , E (w) is closed under the classical connectives. Proof. Let w be any world such that E (w) 6= ; and let ['1] and ['2] be wevents. Without loss of generality we can assume that '1 and '2 are written in conjunctive normal form. We want to prove that ['1] ^ ['2] = ['1 ^ '2] is a w-event. Clearly w 6 ['1 ^ '2], and hence we only need to show that for every partial valuation 0 2 P such that 0 w, and 0 ['1 ^ '2], there exists a world w such that R(w; w ), and 0 = w .
For every 0 2 P such that 0 ['1 ^ '2], (C) ensures the existence of a w (that we would have denoted w['1^'2] using the terminonlogy of (C)) such that ['1 ^ '2], and w w = 0: Since (W; e; R) satis es (A), w w ensures R(w; w ) and hence our claim is settled.
Flaminio acknowledges partial support from the Spanish projects TASSAT (TIN201020967-C04-01), ARINF (TIN2009-14704-C03-03), as well as from the Juan de la Cierva Program of the Spanish MICINN.