# PNCE

## 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.

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