The most interesting things we can say about systems aren't qualitative absolutes, but quantitative statements. How far things are from each other, how much error there is in an approximation, what is the probability of a property being satisfied, and so on. Thus adapting the structural methods of logic, category theory, and categorical systems theory to a quantitative setting is crucial for making them practically useful.
Meanwhile, many ideas and phenomena in statistics, machine learning, and functional analysis have a logical flavour. Can we turn this hunch into a tool to understand such settings in a new light?
Together with Bob Atkey and Ekaterina Komendantskaya I am developing quantitative logic in the sense of F. William Lawvere, i.e. as an 'enriched' logic. We are starting from a propositional theory, which we dubbed quantitative linear logic (QLL), and we will soon move on to predicate logic.
We put a strong emphasis on softness, thus relaxing the hard properties of lattice connectives (\(\lor , \land \)) to accomodate more useful quantitative connectives (like \(+\) and \(\times \)).
Reading list
- The Logic of the Reals, by me
- On Quantifiers for Quantitative Reasoning, by me
- A Taste of Quantitative Logic (Part 1 (slides, video) and Part 2 (slides, video)), by me