Strict Coherence on Many-Valued Events

Abstract

Abstract. We investigate the property of strict coherence in the setting of manyvalued logics. Our main results read as follows: (i) a map from an MV-algebra to [0,1] is strictly coherent if and only if it satisfies Carnap's regularity condition and, (ii) a [0, 1]-valued book on a finite set of many-valued events is strictly coherent if and only if it extends to a faithful state of an MV-algebra that contains them. Remarkably this latter result allows us to relax the rather demanding conditions for the Shimony-Kemeny characterisation of strict coherence put forward in the mid 1950s in this Journal. §1. Introduction and motivation. This paper contributes to the logical foundations of probability by investigating strict coherence on many-valued events. The notion of strict coherence was introduced in this Journal by Abner Shimony and John Kemeny as a logically inspired refinement of the notion of coherence used by Bruno de Finetti to ground his subjective interpretation of probability. Informally, coherence demands that a rational agent avoids the logical possibility of "sure loss" in suitably specified betting situations. Its strict counterpart, in addition, demands that each prospect of losing should be balanced by a prospect of gaining.Interest in the condition of strict coherence was prompted by Carnap's analysis of what he termed "regular" probability functions in [4] (see also [37, Chapter 10 ]). Informally those functions arise by strengthening the usual normalisation axiom of probability in the right-to-left direction. That is to say that 1 (respectively, 0) is assigned only to tautologies (respectively, contradictions). The rationale for Carnap-regular functions is that, however unlikely, possible events may happen.In [38] Shimony proved that imposing strict coherence to a map from finite boolean algebras to [0, 1] was sufficient to single out Carnap-regular functions. Shortly after, the converse was established by Kemeny in [21], a result proved independently in [24]. The Shimony-Kemeny characterisation is obtained under far more restrictive conditions than those yielding de Finetti's theorem, which 1991 Mathematics Subject Classification. Primary 03B48, Secondary 60A05, 06D35, 06F20, 52B11.

Three Characterizations of Strict Coherence on Infinite-Valued Events

“…In this article we will investigate strictly coherent books on Łukasiewicz events, i.e., elements of a finitely generated free MV-algebra. Our results sensibly extend the results of [11]. In particular, we will provide three characterizations of strict coherence by adopting geometrical, measure-theoretical and logical methods.…”

“…This rather odd feature of coherence was questioned in the mid 1950's first by Shimony [32] and then by Kemeny [16]. These authors studied a refinement of de Finetti's coherence that nowadays goes under the name of strict coherence (see [11]). Intuitively, a choice of prices is strictly coherent if every possibility of loss, for the bookmaker, is paired by a possibility of gain.…”

“…Those functions arise from the axiomatization of finitely additive probabilities by strengthening the usual normalization axiom in the right-to-left direction: 1 (respectively, 0) is assigned only to tautologies (respectively, contradictions). In other words, a probability function P is Carnap, if it is normalized, finitely additive and it satisfies P(e) = 0 for every noncontradictory event e. In [11] the authors characterized strictly coherent books on boolean events in terms of their extendability to Carnap probabilities. 1 Several authors proposed generalizations of de Finetti's coherence criterion and his Dutch-Book theorem to events not pertaining to boolean logic.…”

“…In the MV-algebraic realm valuations are [0, 1]-valued and hence they correspond to homomorphisms into the standard MV-algebra defined on the unit interval [0, 1]. De Finetti's coherence criterion immediately translates to the MV-setting with no extra 1 Carnap probabilities are the same as Carnap-regular probabilities of [11]. In the present article we adopt this simplified notation in order to avoid any misleading interpretation of the adjective "regular" which has indeed different meanings if referred to probability functions or to Borel measures which will be discussed in §4.…”

Probabilities of conditionals and previsions of iterated conditionals

“…Among the various interpretations of probability (see, e.g., [32]), we adopt the subjective analysis, which is due to de Finetti ( [22,23]) and Ramsey ([84]). The probabilistic theory of de Finetti, based on the well-known coherence principle, has been studied and extended by many authors (see, e.g., [3,4,7,8,9,10,15,16,19,20,34,37,38,41,46,47,59,71,72]). We recall that the coherence principle of de Finetti plays a key role in probabilistic reasoning.…”