## FIRB RESEARCH PROJECT 2012 – 2016

##### Principal Investigator

**Vincenzo Marra**, Dipartimento di Matematica “Federigo Enriques”, Università degli Studi di Milano, Italy

##### Varese Research Unit Coordinator

**Brunella Gerla**, Dipartimento di Scienze Teoriche e Applicate, Università degli Studi dell’Insubria, Varese, Italy

##### The Research Project

It is the eve of the 2006 World Cup Final at Berlin’s Olympic Stadium: Italy is going to play France. Consider the sentence \(X =\) “Italy will score in the match against France”. You do not know for sure whether \(X\) will turn out to be true or false after the match. That is why bookmakers take bets on (the event described by) the sentence \(X\) – because the information conveyed by \(X\) is uncertain. Uncertainty is the realm of the theory of probability. Certainty, by contrast, is the realm of logic. Logic and probability are intimately related. Betting on the compound sentence \(X\vee\neg X =\)“Either Italy will score or Italy will not score in the match against France” is hardly exciting precisely because the logic of the original sentence \(X\) is classical – and so, in particular, \(X\) satisfies the tertium non datur law that \(X\vee\neg X\) is always true. We can reprhase this by saying \(X\) describes a classical, yes/no event: one that either obtains, or it does not. Consider next a slight variant of \(X\), the sentence \(Y =\) “Italy will score early in the match against France”. The information conveyed by \(Y\) now suffers from two types of imperfection: first, you do not know for sure which instants of time in the course of the match should count as ¿early¿; second, even if you knew that, you would still not know whether \(Y\) will turn out to be true or false after the match. In other words, not only is the information conveyed by \(Y\) uncertain, as before; it is also vague. The vague, non-classical proposition \(Y\) denotes a vague, non-classical event. This research project tackles the question, can one develop a substantive, mathematically significant probability theory of non-classical events.

The project is structured into five work packages.

- Foundational issues for the subjective probability theory of non-classical events.
- Duality theory (Stone duality and generalisations).
- Measure theory over dual spaces.
- Modalities over many-valued logics for reasoning about assignments of probabilities.
- Implementation of a software platform to handle bets on non-classical events.