1. About Probability

Probability theory gives us ways of reasoning when several possibilities remain open.

Logic let us ask which possibilities remained compatible with fixed premises and mappings.

Probability lets us go further when several possibilities remain open and assign different weights to them. Under suitable conditions, the result can feel strongly constrained.

Where does that constraint come from?

The previous investigation of logic began with very small formal systems. In small cases, the relevant possibilities could remain open to inspection. We could see what had been represented, which mappings had been fixed, which possibilities survived, and where a counterexample remained.

Probability offers cases of the same kind.

Dice and cards give us examples in which the relevant possibilities are easy to inspect. A die has only six faces, two throws still give a manageable number of pairs, and after drawing a card we can inspect what remains.

These examples are elementary enough that some of the structure later compressed into probability formulas stays visible.

This essay is not an introduction to probability theory. It will leave almost all of probability theory untouched. Nor does it take sides in the discussion about competing interpretations of probability, or between Bayesian and other statistical methods.

This is also why the essay uses Bayesian updating rather than beginning with a more elaborate statistical method. More elaborate methods usually rely on several layers of probability theory and other mathematical machinery. To inspect their convincing force in the same way, we would first have to reconstruct a considerable number of smaller tools.

Bayes' theorem is far enough along to show how probability can reorganize weights among possibilities, while still being simple enough that much of the relevant structure can be rebuilt from dice, cards, conditional probability, and algebra.

The examples have been chosen for a narrower reason. They let us keep enough of the relevant possibilities and mappings visible to inspect the constraint directly. We will follow a small path from dice to cards and from cards to Bayesian updating, asking where some of the convincing force along that path comes from.

As in the logic essay, we can ask what had to become convincing before the calculation could begin.

And once those things were in place, what stopped being independently open?