## Logic, Uncertainty and Rationality: A Research Profile

My PhD thesis , titled “”Rationality as conformity”, investigates the foundations of the Objective bayesian model of **Maximum entropy probability logic** which was developed between the late 1980s and the end of the 1990s by my supervisor, J.B Paris in collaboration with A. Vencovska. Rationality-as-conformity captures the idea that an essential aspect of rationality is interpersonal, and that in certain specific situations, rationality can be identified with one’s tendency to make the same choices as other like-minded agents who are facing essentially similar problems. The main results of my thesis appear in (Hosni and Paris 2005). Building directly on those results, I have used the Rationality-as-conformity framework to link the **selection of multiple Nash equilibria** in coordination games to Davidson’s problem of radical interpretation in (Hosni 2009). This framework has been recently applied in the foundations of peer-to-peer governance protocols and cryptocurrency modelling . In (Devetag, Hosni and Sillari 2013) the logical investigation on efficient equilibrium selection is combined with an experimental analysis of the conditions which facilitate coordination. The experimental results confirm the formal predictions put forward in my doctoral work. Back to a logical setting, the coordination facilitated in weak-link games provides a novel interpretation of **possibilistic randomisation ** which is investigated in (Hosni and Marchioni 2019).

While carrying out my PhD work, I developed a strong side interest in the **connection between quantitative (probabilistic) and qualitative (non-monotonic and modal) representations of rational reasoning under uncertainty.** This led to several research papers, including (Casini and Hosni 2009). In recent years Tommaso Flaminio, Lluis Godo and I have been investigating this qualitative/quantitative connection in the context of **conditional probabilities. **More specifically we asked in (Flaminio, Godo and Hosni 2015b) how the properties of conditional probability can be divided into the properties of the logic of conditionals and those of the conditional measure. This led to the definition of the **Logic of Boolean Algebras (BAC) **introduced in (Flaminio, Godo and Hosni 2017) which provide a suitable logic to interpret conditional probabilities a single probability functions on conditional logics.

During 2006-7 I translated into English the transcription of Bruno de Finetti’s 1979 INDAM course “”Probabilità”, which was edited by A. Mura and published by Springer in 2008 under the title “*Philosophical Lectures on Probability*. This project gave me the opportunity to delve into the **foundations of probability**, its history and its relevance in epistemology. The question of putting some of its extensions on firm logical footing paved the way to (Fedel, Hosni and Montagna 2011). This paper extends, within a logico-algebraic framework, de Finetti’s subjective foundation of probability to** many-valued events** and **imprecise probabilities**. The ideas and results of this paper have been crucial in putting forward the research project “Rethinking Uncertainty, which was funded by the European Commission during 2013-15. In this project I obtained results relating epistemology, uncertainty modelling and logic. (Hosni 2014) outlines a general framework in which **higher-order uncertainty** can be accommodated within subjective probability by building on the methodology of non-classical logics. This led to a novel characterisation of coherence for non-additive uncertainty measures generalising probability (Flaminio, Godo and Hosni 2015a) . In the same spirit, (Hosni and Montagna 2014) provides an extension of de Finetti’s coherent to **non-standard, imprecise probabilities on many-valued events**. In (Flaminio, Godo and Hosni 2014) **a** **rigorous characterisation of the logical framework assumed implicitly by de Finett**i is provided, showing how his results can be refined to exclude certain probability distributions.

As an independent line of research of the “Rethinking Uncertainty project I developed a strong interest in the **logical foundations of mathematical modelling in economics**. A logical analysis of the hypotheses leading to the “Fundamental theorem of welfare economics” is carried out in the monograph (Giaquinta and Hosni 2015a) and a more general point of view concerning the role played by logic in Rational Choice Theory is put forward in the paper (Giaquinta and Hosni 2015b). This led to a more systematic investigation of the link between **logic and decision theory**. In (Flaminio, Hosni and Lapenta 2017) a logico-algebraic characterisation of convexity is proved to yield naturally the Anscombe-Aumann representation theorem.

I am extremely keen on reaching multidisciplinary research audiences. Since September 2011 I’ve been columnist for The Reasoner a gazette edited at Kent’s Centre for Reasoning. My 1000-word monthly column reports on “What’s hot in Uncertain Reasoning”. In addition, since 2009 I am the editorial manager of Edizione Nazionale Mathematica Italiana. The project aims at providing the largest open-access repository of mathematical texts and bibliographies in Italy and is being co-funded by the Italian Ministry of Cultural Heritage and the Scuola Normale Superiore di Pisa.

### Recent Projects

- Rethinking Uncertainty (2013-2015)
- The logical foundations of mathematical modelling in economics (2014-)

## The logical foundations of mathematical modelling in economics (2014-)

This is joint work with Mariano Giaquinta. Our goal is to re-read the classic results of mathematical economics, especially the mathematical theory of general equilibrium, with an eye to isolating their conceptual underpinnings.

Publications

- M. Giaquinta, M. and H. Hosni. (2015).
*Teoria della scelta sociale e teorema fondamentale dell’economia del benessere*. Edizioni della Normale. - 2015. Mathematics in the social sciences: reflections on the theory of social choice and welfare. Lettera Matematica International ,

## Re-thinking uncertainty: A choice-based approach (2013-2015)

The overarching aim of this research project is to improve our understanding of uncertainty, its quantification and its communication. It sets out to do so by addressing a set of key epistemological questions that arise in the construction of formal models of rational decision- making. Its expected results will throw new light one of the most fundamental problems in the multidisciplinary field of uncertain reasoning: How to identify an efficient trade-off between foundational robustness and expressive power in the quantification of uncertainty.

**Background**

Our need to quantify uncertainty arises primarily in relation to choice problems, i.e. when we must select one from a set of alternatives each yielding a (partially) unknown outcome. Yet, some choice problems lead to an easier quantification of the related uncertainty than others. Suppose a normal-looking die is about to be rolled. In the absence of other relevant information it is entirely plausible to believe that a particular side, say “3”, will show with probability 1/6. The underlying reasoning goes along the following lines: The problem at hand clearly admits of six mutually exclusive outcomes, one of which will certainly occur, and none of which appears to be more likely than the others – hence it would seem irrational of us to give a particular side a probability other than 1/6. Compare this with a situation in which we must decide whether to buy stocks of a certain bank who has invested heavily in Greek bonds. We are told that in the event of Greece exiting the Eurozone, Greek bonds will be completely worthless. Yet the stocks offer very good prospects of profit in a six-month time horizon. In order to make a rational decision we must estimate the likelihood (i.e. choose a representation for our uncertainty) that Greece will exit the Eurozone within the next six months – an event which is often referred to as “GREXIT” in the financial sector. The reasoning we confidently used to choose our probability for the event “3 by rolling a normal-looking die” certainly doesn’t seem to be applicable to GREXIT. In particular there seems no unique way of telling a priori what set should be partitioned in order to define a standard probability distribution.

*Rethinking Uncertainty,*and was part of the LSE’s Managing Severe Uncertainty project.

**Output**

- Flaminio, T., Godo, L., & Hosni, H. (2015) “On the algebraic structure of conditional events”, In S. Destercke and T. Denoeux (eds) Symbolic and Quantitative Approaches to Reasoning with Uncertainty 2015,
*Lecture Notes in Artificial Intelligence*vol. 9161, doi:10.1007/978-3-319-20807-7. - Flaminio, T., Godo, L., & Hosni, H. (2015). “Coherence in the aggregate: A betting method for belief functions on many-valued events”. International Journal of Approximate Reasoning, 58, 71–86. doi:http://dx.doi.org/10.1016/j.ijar.2015.01.001
- Flaminio, T., Godo, L. and Hosni H., (2014) “On the logical structure of de Finetti’s notion of event,”
*Journal of Applied Logic,*12(3), 279–301. doi:http://dx.doi.org/10.1016/j.jal.2014.03.001 - Hosni, H and Montagna, F, (2014) “Stable Non-standard Imprecise Probabilities” In A. Laurent et al. (Ed.), IPMU 2104, Communications in Computer and Information Sciences, 444 (pp. 436–445). Springer. doi:10.1007/978-3-319-08852-5_45
- Hosni, H. (2014). “Towards a Bayesian theory of second-order uncertainty: Lessons from non-standard logics”. In S. O. Hansson (Ed.), David Makinson on Classical Methods for Non-Classical Problems (pp. 195–221). Outstanding Contributions to Logic Volume 3, Springer. doi:10.1007/978-94-007-7759-0_11