Speaker: Alexandru Baltag
Date and Time: Thursday, May 14th 2020, 16:30-18:00, Amsterdam time.
Title: From known correlations to the logic of continuous dependence
An empirical variable is one whose exact value might not be knowable, and instead only inexact approximations can be observed. Examples are in natural sciences, economics etc (where the inexact observations are some form of measurements), but also in the semantics of questions in natural language (where the inexact observations are partial answers). This leads to a topological conception of empirical variables, as maps from the state space into a topological space. Here, the exact value of the variable is represented by the output of the map, while the open neighborhoods of this value represent the knowable approximations of the exact answer.
A central tenet in empirical sciences is establishing functional correlations between variables, with a view towards (1) establishing causality, but also (2) predicting the (approximate) value of a hard-to-measure variable Y when given (approximate) value(s) of easier-to-measure variable X. In interrogative terms, this is related to inquisitive implication: every partial answer to question Y is entailed by some partial answer to X. In this talk, I argue that knowability of a dependency amounts to the continuity of the given functional correlation. I give a learning-theoretic justification of this claim, connecting with Kevin Kelly’s notion of gradual learnability, then I give some concrete examples. Next, I present a complete and decidable axiomatization of the logic of continuous dependence, and briefly skech the ideas behind the proofs.
Further, I discuss the distinction between knowing the dependence between X and Y, and knowing how to determine Y (with any desired accuracy) from X: the later is a stronger notion of knowability, that requires the ability to find the accuracy that is needed for X-measurements (to determine Y with the given accuracy). I formalize this distinction in terms of continuity versus uniform continuity of the underlying dependence map, and go on to propose an axiomatization of strongly known dependence, in the framework of uniform spaces (-Andrè Weil’s qualitative generalization of metric spaces).
Time-permitting, I may go back to the problem of learning true causal relations from observed functional correlations. I will end with a number of open questions, some technical and some conceptual.
This is ongoing joint work with Johan van Benthem, embodied in a follow-up draft to our joint work on the Logic of Functional Dependence, presented at a previous LIRa seminar. (But my presentation will be self-contained.)