A complete list of my publications in Journals and Conference Proceedings appears in my updated CV or on DBLP, MathSciNet.

**Boolean algebras of conditionals, probability and logic**

*T. Flaminio, L. Godo, H. Hosni*

This paper presents an investigation on the structure of conditional events and on the probability measures which arise naturally in this context. In particular we introduce a construction which defines a (finite) *Boolean algebra of conditionals* from any (finite) Boolean algebra of events. By doing so we distinguish the properties of conditional events which depend on probability and those which are intrinsic to the logico-algebraic structure of conditionals. Our main result provides a way to regard standard two-place conditional probabilities as one-place probability functions on conditional events. We also consider a logical counterpart of our Boolean algebras of conditionals with links to preferential consequence relations for non-monotonic reasoning. The overall framework of this paper provides a novel perspective on the rich interplay between logic and probability in the representation of conditional knowledge.

* Keywords*: Conditional probability, Conditional events, Boolean algebras, Preferential consequence relations

**Three Characterizations of Strict Coherence on Infinite-****Valued Events**

*T. Flaminio*

This article builds on a recent paper coauthored by the present author, H. Hosni and F. Montagna. It is meant to contribute to the logical foundations of probability theory on many-valued events and, specifically, to a deeper understanding of the notion of strict coherence. In particular, we will make use of geometrical, measure-theoretical and logical methods to provide three characterizations of strict coherence on formulas of infinite-valued Łukasiewicz logic.

* Keywords: *de Finetti’s coherence, strict coherence, faithful states, MV-algebras, Łukasiewicz logic.

DOI: https://doi.org/10.1017/S1755020319000546

**A representation theorem for finite Gödel algebras with operators**

*T. Flaminio, L. Godo, R. O. Rodriguez*

In this paper we introduce and study finite Gödel algebras with operators (GAOs for short) and their dual frames. Taking into account that the category of finite Gödel algebras with homomorphisms is dually equivalent to the category of finite forests with order-preserving open maps, the dual relational frames of GAOs are forest frames: finite forests endowed with two binary (crisp) relations satisfying suitable properties. Our main result is a Jónsson-Tarski like representation theorem for these structures. In particular we show that every finite Gödel algebra with operators determines a unique forest frame whose set of subforests, endowed with suitably defined algebraic and modal operators, is a GAO isomorphic to the original one.

* Keywords: *Finite Gödel algebras; modal operators; finite forests; representation theorem.

**Towards a probability theory for product logic: States, integral representation and reasoning**

*T. Flaminio, L. Godo, S. Ugolini*

The aim of this paper is to extend probability theory from the classical to the product t-norm fuzzy logic setting. More precisely, we axiomatize a generalized notion of finitely additive probability for product logic formulas, called state, and show that every state is the Lebesgue integral with respect to a unique regular Borel probability measure. Furthermore, the relation between states and measures is shown to be one–one. In addition, we study geometrical properties of the convex set of states and show that extremal states, i.e., the extremal points of the state space, are the same as the truth-value assignments of the logic. Finally, we axiomatize a two-tiered modal logic for probabilistic reasoning on product logic events and prove soundness and completeness with respect to probabilistic spaces, where the algebra is a free product algebra and the measure is a state in the above sense.

**Keywords: **Probability theory; Nonclassical events; Free product algebras; States; Riesz representation theorem; Regular Borel measures; Two-tiered modal logics

*International Journal of Approximate Reasoning* 93: 199–218, 2018.

Also check its corrigendum: * Corrigendum: Towards a probability theory for product logic: states, integral representation and reasoning. *International Journal of Approximate Reasoning 103: 267–269, 2018.

Preprint available – https://arxiv.org/abs/1803.03208

**Strict Coherence on Many-Valued Events**

*T. Flaminio, H. Hosni, F. Montagna*

We investigate the property of strict coherence in the setting of many-valued 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.

**Keywords: **Probability logic, strict coherence, MV-algebras, faithful states, many-valued logics.

*The Journal of Symbolic Logic 83(1): 55-69, 2018*.

**Equivalences between subcategories of MTL-algebras via Boolean algebras and prelinear semihoops**

*S. Aguzzoli, T. Flaminio, S. Ugolini*

This article studies the class of strongly perfect MTL-algebras, i.e. MTL-algebras having an involutive co-radical, and the variety they generate, namely SBP0. Once these structures will be introduced, we will first establish categorical equivalences for several of their relevant proper subvarieties by employing a generalized notion of *triplets* whose main components are a Boolean algebra and a prelinear semihoop. When triplets are further expanded by a suitable operation between their semihoop reducts, we define a category of quadruples that are equivalent to the whole category of SBP0-algebras. Finally, we will provide an explicit representation of SBP0-algebras in terms of (weak) Boolean products.

**Keywords**: Strongly perfect MTL-algebras, Boolean Algebras, Prelinear Semihoops, Categorical Equivalence

**Convex MV-algebras: Many-valued logics meet decision theory**

*T. Flaminio, H. Hosni, S. Lapenta*

This paper introduces a logical analysis of convex combinations within the framework of Łukasiewicz real-valued logic. This provides a natural link between the fields of many-valued logics and decision theory under uncertainty, where the notion of convexity plays a central role. We set out to explore such a link by defining convex operators on MV-algebras, which are the equivalent algebraic semantics of Łukasiewicz logic. This gives us a formal language to reason about the expected value of bounded random variables. As an illustration of the applicability of our framework we present a logical version of the Anscombe–Aumann representation result.

**Keywords**: MV-algebras, Convexity, Uncertainty Measures, Anscombe-Aumann

*Studia Logica* 106(5): 913-945, 2018.

**Coherence in the aggregate: a betting method for belief functions on many-valued events. **

*T. Flaminio, L. Godo, H. Hosni*

Betting methods, of which de Finetti’s Dutch Book is by far the most well-known, are uncertainty modelling devices which accomplish a twofold aim. Whilst providing an (operational) interpretation of the relevant measure of uncertainty, they also provide a formal definition of *coherence*. The main purpose of this paper is to put forward a betting method for *belief functions* on MV-algebras of many-valued events which allows us to isolate the corresponding coherence criterion, which we term *coherence in the aggregate*. Our framework generalises the classical Dutch Book method.

**Keywords**: Belief functions, Necessity measures, Subjective probability, Many-valued events, Betting methods, De Finetti

*International Journal of Approximate Reasoning* 58: 71–86, 2015.

**On the logical structure of de Finetti’s notion of event. **

*T. Flaminio, L. Godo, H. Hosni.*

This paper sheds new light on the subtle relation between probability and logic by (i) providing a logical development of Bruno de Finetti’s conception of events and (ii) suggesting that the subjective nature of de Finetti’s interpretation of probability emerges in a clearer form against such a logical background. By making explicit the epistemic structure which underlies what we call *Choice-based probability* we show that whilst all rational degrees of belief must be probabilities, the converse doesn’t hold: some probability values don’t represent decision-relevant quantifications of uncertainty.

* Keywords: *Events; De Finetti’s coherence criterion; Informations frames; Choice-based probability

*Journal of Applied Logic*, 12: 279–301, 2014.

**Logics for belief functions on MV- algebras**

*T. Flaminio, L. Godo, E. Marchioni.*

In this paper we present a generalization of belief functions over fuzzy events. In particular we focus on belief functions defined in the algebraic framework of finite MV-algebras of fuzzy sets. We introduce a fuzzy modal logic to formalize reasoning with belief functions on many-valued events. We prove, among other results, that several different notions of belief functions can be characterized in a quite uniform way, just by slightly modifying the complete axiomatization of one of the modal logics involved in the definition of our formalism.

**Keywords**: Belief functions, Łukasiewicz logic, Modal logics, Fuzzy events

*International Journal of Approximate Reasoning*, 54(4): 491–512, 2013.

*Geometrical aspects of possibility measures on finite domain MV-clans*

*T. Flaminio, L. Godo, E. Marchioni.*

In this paper, we study generalized possibility and necessity measures on MV-algebras of [0, 1]-valued functions (MV-clans) in the framework of idempotent mathematics, where the usual field of reals ℝ is replaced by the max-plus semiring ℝmax. We prove results about extendability of partial assessments to possibility and necessity measures, and characterize the geometrical properties of the space of homogeneous possibility measures. The aim of the present paper is also to support the idea that idempotent mathematics is the natural framework to develop the theory of possibility and necessity measures, in the same way classical mathematics serves as a natural setting for probability theory.

**Keywords**: Possibility Measures, MV-algebras, Idempotent Mathematics, Max-Plus Convexity.

*Soft Computing*, 16(11): 1863–1873, 2012.

**Models for many-valued probabilistic reasoning**

*T. Flaminio, F. Montagna.*

In this article, we compare models for many-valued probabilistic reasoning from the point of view of the sets of satisfiable formulas, positive satisfiable formulas, and tautologies. The results arising from this comparison will be used in the final part of the present article to provide results about the computational complexity for the problem of deciding if a formula belongs to one of the previously discussed sets.

**Keywords:***SMV-algebras, states on MV-algebras, probabilistic Kripke models, PSPACE containment*

*Journal of Logic and Computation*, 21(3): 447–464, 2011.

**The Coherence of Łukasiewicz Assessments is NP- complete**

*S. Bova, T. Flaminio*

The problem of deciding whether a rational assessment of formulas of infinite-valued Łukasiewicz logic is coherent has been shown to be decidable by Mundici and in PSPACE by Flaminio and Montagna. We settle its computational complexity proving an NP-completeness result. We then obtain NP-completeness results for the satisfiability problem of certain many-valued probabilistic logics introduced by Flaminio and Montagna.

**Keywords: **De Finetti’s coherence criterion, Infinite-valued Łukasiewicz logic, NP-completeness, SMV-algebras, Probabilistic Kripke models

*International Journal of Approximate Reasoning*, 51(3): 294–304, 2010.

**MV-algebras with internal states and probabilistic fuzzy logics**

*T. Flaminio, F. Montagna*

In this paper we enlarge the language of MV-algebras by a unary operation σ equationally described so as to preserve the basic properties of a state in its original meaning. The resulting class of algebras will be called *MV-algebras with internal state *(or SMV-algebras for short). After discussing some basic algebraic properties of SMV-algebras, we apply them to the study of the coherence problem for rational assessments on many-valued events. Then we propose an algebraic treatment of the Lebesgue integral and we show that internal states defined on a divisible MVΔ-algebra can be represented by means of this more general notion of integral.

**Keywords: **States, MV-algebras, Coherence, Generalized Lebesgue integral, Integral representation

*International Journal of Approximate Reasoning*, 50(1): 138–152, 2009.

**Strong non-standard completeness for fuzzy logics**

*T. Flaminio*

In this paper we are going to introduce the notion of *strong non-standard completeness *(SNSC) for fuzzy logics. This notion naturally arises from the well known construction by ultraproduct. Roughly speaking, to say that a logic L is strong non-standard complete means that, for any countable theory Γ over L and any formula *φ* such that Γ⊬φ, there exists an evaluation *e* of L-formulas into a L-algebra A such that the universe of A is a non-Archimedean extension [0,1]* of the real unit interval [0,1], *e* is a model for Γ, but *e*(*φ*) < 1. Then we will apply SNSC to prove that various modal fuzzy logics allowing to deal with simple and conditional probability of infinite-valued events are complete with respect to classes of models defined starting from non-standard measures, that is measures taking value in [0,1]*

**Keywords: **Non-standard completeness, Ultraproduct construction, Conditional probability, Fuzzy events

*S**oft Computing*, 12(4): 321–333, 2008.

**NP-Containment for the Coherence Tests of Assessment of Conditional Probability: a Fuzzy-Logical Approach**

*T. Flaminio*

In this paper we investigate the problem of testing the coherence of an assessment of conditional probability following a purely logical setting. In particular we will prove that the coherence of an assessment of conditional probability χ can be characterized by means of the logical consistency of a suitable theory *T*_{χ} defined on the modal-fuzzy logic *FP*_{k}(*R*Ł_{Δ}) built up over the many-valued logic *R*Ł_{Δ}. Such modal-fuzzy logic was previously introduced by Flaminio in order to treat conditional probability by means of a list of simple probabilities following the well known ideas exposed by Halpern and by Coletti and Scozzafava. Roughly speaking, such logic is obtained by adding to the language of *R*Ł_{Δ} a list of *k* modalities for “*probably*” and axioms reflecting the properties of simple probability measures. Moreover we prove that the satisfiability problem for modal formulas of *FP*_{k}(*R*Ł_{Δ}) is NP-complete. Finally, as main result of this paper, we prove *FP*_{k}(*R*Ł_{Δ}) in order to prove that the problem of establishing the coherence of rational assessments of conditional probability is NP-complete.

**Keywords: **Many-valued logics, Conditional probability, Coherence, Computational Complexity

*Archive for Mathematical Logic*, 46(3-4): 301–319, 2007.

**A logic for reasoning about the probability of fuzzy events**

*T. Flaminio, L. Godo*

In this paper we present the logic FP(Łn,Ł) which allows to reason about the probability of fuzzy events formalized by means of the notion of state in a MV-algebra. This logic is defined starting from a basic idea exposed by Hájek. Two kinds of semantics have been introduced, namely the class of *weak* and *strong* probabilistic models. The main result of this paper is a completeness theorem for the logic FP(Łn,Ł) w.r.t. both weak and strong models. We also present two extensions of FP(Łn,Ł): the first one is the logic FP(Łn,RPL), obtained by expanding the FP(Łn,Ł)-language with truth-constants for the rationals in [0,1] while the second extension is the logic FCP(Łn,ŁΠ12) allowing to reason about conditional states.

**Keywords: **Łukasiewicz logic, State and conditional states on MV-algebras, Fuzzy events, Standard completeness

*Fuzzy Sets and Systems*, 158(6): 625–638, 2007.

**T-norm Based Logics with an Independent Involutive Negation**

*T. Flaminio, E. Marchioni*

In this paper we investigate the addition of arbitrary independent involutive negations to t-norm-based logics. We deal with several extensions of MTL and establish general completeness results. Indeed, we will show that, given any t-norm-based logic satisfying some basic properties, its extension by means of an involutive negation preserves algebraic and (finite) strong standard completeness. We will deal with both propositional and predicate logics.

*Many-valued logics, Left-continuous t-norms, Involutive negations, Standard completeness, Predicate calculi*

**Keywords:***Fuzzy Sets and Systems*, 157(24): 3125–3144, 2006.

**A Logical and Algebraic Approach to Conditional Probability**

*T. Flaminio, F. Montagna*

This paper is devoted to a logical and algebraic treatment of conditional probability. The main ideas are the use of non-standard probabilities and of some kind of standard part function in order to deal with the case where the conditioning event has probability zero, and the use of a many-valued modal logic in order to deal probability of an event *φ* as the truth value of the sentence *φ is probable*, along the lines of Hájek’s book. To this purpose, we introduce a probabilistic many-valued logic, called *FP*(*S*ŁΠ), which is sound and complete with respect a class of structures having a non-standard extension [0,1]* of [0,1] as set of truth values. We also prove that the coherence of an assessment of conditional probabilities is equivalent to the coherence of a suitably defined theory over *FP*(*S*ŁΠ) whose proper axioms reflect the assessment itself.

**Keywords: **Many-valued logic, Conditional probability, Coherence

*Archive for Mathematical Logic*, 44(4): 245–262, 2005.