Garrabrant Induction provides a somewhat plausible sketch of reasoning under computational uncertainty, the gist of which is "build a prediction market". An approximation of classical probability theory emerges. However, this is only because we assume classical logic. The version of Garrabrant Induction in the paper does this by allowing bets to be placed on all boolean combinations of sentences visible to the market. An earlier draft accomplished the same thing via special arbitrage rules, EG, if you own $\frac{1}{5}$ of $S$ , and you own $\frac{1}{5}$ of $\neg S$ , then you can trade these to the market-maker for $\frac{1}{5}$ of a dollar. (So for example, if the current price of a share of $S$ is $50 ¢$ , and the current price of a share of $\neg S$ is $45 ¢$ , then you can arbitrage by buying $\frac{1}{5}$ of a share of both (cost $10 ¢$ + $9 ¢$ = $19 ¢$ ) and cashing these in to the market maker for $20 ¢$ .) This forces the market to converge towards $price (S) + price (\neg S) = $ 1$ for all $S$ , mimicing classical probability. Similar arbitrage rules can be formulated for the other logical connectives.

This is nice for what it does, but it means Garrabrant Induction provides no normative evidence for approximating classical probability theory or classical logic. These are baked-in assumptions, not conditions which emerge naturally from beautiful math.

Sam Eisenstat has (privately) formulated a version of Garrabrant Induction based on intuitionistic logic instead. The result does not require the price of a sentence and its negation to sum to 1 (and also violates some other things we expect of probabilities). It has the nice property that conditional on excluded middle $S \lor \neg S$ , we recover classical probabilities. (This is a natural result to want, since intuitionistic logic plus excluded middle = classical logic.)

This leaves open the question: what logic naturally emerges if we let the math do the talking, rather than imposing one logic or another?

The following modifies a proposal by Sam Eisenstat, and also takes inspiration from yesterday's post.

To lay a foundation for this investigation, I'll sketch something resembling Garrabrant induction, but avoid baking in a specific logic.

I won't go all the way to defining a proper computable thing. Indeed, I won't even define everything completely. I am doing this in compressed time due to Inkhaven, so I may...

Implications of Logical Induction