This is the first post in my Effective-Altruism-funded project aiming to deconfuse goal-directedness. Comments are welcomed. All opinions expressed are my own, and do not reflect the attitudes of any member of the body sponsoring me.

In my preliminary post, I described my basic intuitions about goal-directedness, and focussed on explainability. Concisely, my initial, informal working definition of goal-directedness is that an agent's behaviour is goal-directed to the extent that it is better explained by the hypothesis that the agent is working towards a goal than by other types of explanation.

In this post I'm going to pick away at the most visible idea in this formulation: the concept of an explanation (or at least the aspect of it which is amenable to formalization with minimal effort), and especially the criteria by which an explanation is judged. More precisely, this post is a winding path through some mathematical ideas that could be applied to quantitatively judge explanations; this collection of ideas shall be examined more closely in subsequent posts once some desirata are established. By the end I'll have a naïve first picture of how goal-directedness might be measured in terms of goal-based explanations, while having picked out some tools for measuring explanations which goal-directedness can be contrasted with.

What constitutes a good explanation?

In science (and rational discourse more broadly), explanations are judged empirically: they are used to generate predictions, and these are compared with the results of existing or subsequent observations. The transition from explanations to predictions will be covered in the next section. Here we'll break down the criteria for comparison.

The most obvious way an explanation can fail is if it predicts phenomena which are not observed, or conversely if it fails to predict phenomena which are observed. We can present this criterion as:

Accuracy. The more accurately the observations match the predictions, the better the explanation is considered to be.

That is not the end of the story. For instance, more value is attributed to explanations which produce a greater variety of predictions for future observations. An exhaustive account of existing observations with no extra consequences barely qualifies as an explanation, whereas an explanation that appears to reveal the fundamental nature of the object(s) of scienti...