Most attributes of 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 started from a propositional theory, which we dubbed quantitative linear logic (QLL), and we are now moving 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 \(\times \)).
Thomas Flinkow also helped us investigating the role quantitative logic can have in property-driven learning, with very encouraging results.
Reading list
- On Quantifiers for Quantitative Reasoning, by me
- Quantitative Linear Logic, by me, Bob Atkey, Ekaterina Komendantskaya, and Charles Grellois
- Quantitative Linear Logic for Neuro-Symbolic Learning and Verification
- A Taste of Quantitative Logic (Part 1 (slides, video) and Part 2 (slides, video)), by me