# Numerically Definite Logic

I will present variations of propositional and predicate logic in which the judgements take the form $A$ holds with proportionately fewer than $q$ exceptions’ and $A$ holds in at least $q$ of the cases’ where models consist of a collection of possible worlds $w_i$ for $i\in I$, $q\in\IQ$ is between $0$ and $1$, and these judgements refer to a normalised measure on the index set $I$. Rules for this logic will be given for which Soundness and Completeness Theorems hold.

Note that the logic itself is not probabilistic: conclusions of the rules are correct and say what can be definitely deduced about the family of possible worlds. This seems to be a useful starting point for describing the scientific experimental method. The rules themselves are rather simple (and no doubt have been observed before, but I don’t have any reference) but have some curious properties. For example, they pay close attention to “not discarding information” (in the way that the traditional AND -elimination rule discards information). The logic does not allow a finite set of judgements to be combined as a single judgement, and this has possible implications for the question of synthetic/analytic nature of mathematics.

As a framework for the experimental method, the logic sheds some light on the nature of induction. In particular I will explain how classical logic is in some sense a “limiting case” of this logic, and how proofs in classical logic can be converted to proofs in a situation where not all worlds satisfy the premises, and why short proofs in classical logic are better than long proofs in this respect.

This talk is part of the Theoretical computer science seminar series.