Una caratterizzazione della coerenza forte per eventi non-classici

Immaginiamo la seguente situazione: dati gli eventi $a_1,\ldots, a_n$, un allibratore $A$ fissa un {\em book} $\beta: a_i\mapsto \alpha_i\in [0,1]$, mentre uno scommettitore $S$ sceglie $\lambda_1,\ldots, \lambda_n\in \mathbb{R}$ e paga ad $A$ la quota di $\sum_{i=1}^n \lambda_i\alpha_i$ al fine di ricevere, nel mondo possibile $v$, $\sum_{i=1}^n \lambda_i v(a_i)$. Un book $\beta$ \`e detto {\em coerente}, se non esiste, per $S$, una strategia di scommesse che porti $A$ ad una perdita sicura, ovvero tale per cui il bilancio del gioco calcolato da $\sum_{i=1}^n\lambda_i(\alpha_i-v(a_i))$ non sia positivo. % Un famoso teorema dovuto a Bruno de Finetti [1], stabilisce che un book \`e coerente, se e soltanto se si estende ad una misura di probabilit\`a finitamente additiva definita sull'algebra dagli eventi. %$a_1,\ldots, a_n$. Questo concetto di coerenza %introdotto in precedenza pu\`o essere rafforzato nel seguente modo: un book $\beta$ \`e detto {\em fortemente coerente} se per ogni scelta delle quote $\lambda_1,\ldots, \lambda_n$ da parte dello scommettitore, se esiste un mondo possibile $v$ in cui il bilancio dell'allibratore \`e strettamente negativo, allora esiste anche un altro mondo $w$, in cui il bilancio \`e strettamente positivo [5]. In questo contributo, daremo una caratterizzazione dei book fortemente coerenti su eventi non-classici. In particolare ci concentreremo su quegli eventi che possono essere rappresentati come funzioni da un insieme finito $X$ a valori nell'intervallo unitario reale $[0,1]$. Il contesto algebrico su cui ci concentreremo \`e quindi quello delle MV-algebre [2] e il nostro risultato principale afferma che un book \`e fortemente coerente se e solo se esiste uno {\em stato fedele} [3] di una MV-algebra di funzioni $[0,1]^X$ che lo estende. La nostra caratterizzazione estende e generalizza quella di Shimony [5] per eventi classici, e quella di Mundici [4] per eventi MV-algebrici

Betting on Quantum Objects

Dutch book arguments have been applied to beliefs about the outcomes of measurements of quantum systems, but not to beliefs about quantum objects prior to measurement. In this paper, we prove a quantum version of the probabilists' Dutch book theorem that applies to both sorts of beliefs: roughly, if ideal beliefs are given by vector states, all and only Born-rule probabilities avoid Dutch books. This theorem and associated results have implications for operational and realist interpretations of the logic of a Hilbert lattice. In the latter case, we show that the defenders of the eigenstate-value orthodoxy face a trilemma. Those who favor vague properties avoid the trilemma, admitting all and only those beliefs about quantum objects that avoid Dutch books.Comment: 26 pages, 3 figures, 1 table; improved operational semantics, results unchange

Lukasiewicz logic and Riesz spaces

We initiate a deep study of {\em Riesz MV-algebras} which are MV-algebras endowed with a scalar multiplication with scalars from $[0,1]$. Extending Mundici's equivalence between MV-algebras and $\ell$-groups, we prove that Riesz MV-algebras are categorically equivalent with unit intervals in Riesz spaces with strong unit. Moreover, the subclass of norm-complete Riesz MV-algebras is equivalent with the class of commutative unital C$^*$-algebras. The propositional calculus ${\mathbb R}{\cal L}$ that has Riesz MV-algebras as models is a conservative extension of \L ukasiewicz $\infty$-valued propositional calculus and it is complete with respect to evaluations in the standard model $[0,1]$. We prove a normal form theorem for this logic, extending McNaughton theorem for \L ukasiewicz logic. We define the notions of quasi-linear combination and quasi-linear span for formulas in ${\mathbb R}{\cal L}$ and we relate them with the analogue of de Finetti's coherence criterion for ${\mathbb R}{\cal L}$.Comment: To appear in Soft Computin

State morphism MV-algebras

We present a complete characterization of subdirectly irreducible MV-algebras with internal states (SMV-algebras). This allows us to classify subdirectly irreducible state morphism MV-algebras (SMMV-algebras) and describe single generators of the variety of SMMV-algebras, and show that we have a continuum of varieties of SMMV-algebras

The coherence of Łukasiewicz assessments is NP-complete

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Layers of zero probability and stable coherence over Łukasiewicz events

The notion of stable coherence has been recently introduced to characterize coherent assignments to conditional many-valued events by means of hyperreal-valued states. In a nutshell, an assignment, or book, β on a finite set of conditional events is stably coherent if there exists a coherent variant β of β such that β maps all antecedents of conditional events to a strictly positive hyperreal number, and such that β and β differ by an infinitesimal. In this paper, we provide a characterization of stable coherence in terms of layers of zero probability for books on Łukasiewicz logic events. © 2016, Springer-Verlag Berlin Heidelberg.The authors would like to thank there referee for the valuable comments that considerably improved the presentation of this paper. Flaminio has been funded by the Italian project FIRB 2010 (RBFR10DGUA_002). Godo has been also funded by the MINECO/FEDER Project TIN2015-71799-C2-1-P.Peer Reviewe

Sure-wins under coherence: a geometrical perspective

In this contribution we will present a generalization of de Finetti's betting game in which a gambler is allowed to buy and sell unknown events' betting odds from more than one bookmaker. In such a framework, the sole coherence of the books the gambler can play with is not sucient, as in the original de Finetti's frame, to bar the gambler from a sure-win opportunity. The notion of joint coherence which we will introduce in this paper characterizes those coherent books on which sure- win is impossible. Our main results provide geometric characterizations of the space of all books which are jointly coherent with a xed one. As a consequence we will also show that joint coherence is decidable

Paraconsistent probabilities: consistency, contradictions and bayes' theorem

2010/51038-0sem informaçãoThis paper represents the first steps towards constructing a paraconsistent theory of probability based on the Logics of Formal Inconsistency (LFIs). We show that LFIs encode very naturally an extension of the notion of probability able to express sophisticated probabilistic reasoning under contradictions employing appropriate notions of conditional probability and paraconsistent updating, via a version of Bayes' theorem for conditionalization. We argue that the dissimilarity between the notions of inconsistency and contradiction, one of the pillars of LFIs, plays a central role in our extended notion of probability. Some critical historical and conceptual points about probability theory are also reviewed.This paper represents the first steps towards constructing a paraconsistent theory of probability based on the logics of formal inconsistency (LFIs). We show that LFIs encode very naturally an extension of the notion of probability able to express sophisticated probabilistic reasoning under contradictions employing appropriate notions of conditional probability and paraconsistent updating, via a version of Bayes' theorem for conditionalization. We argue that the dissimilarity between the notions of inconsistency and contradiction, one of the pillars of LFIs, plays a central role in our extended notion of probability. Some critical historical and conceptual points about probability theory are also reviewed.189FAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOCNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTIFICO E TECNOLOGICOFAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOCNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTIFICO E TECNOLOGICO2010/51038-0sem informaçã