# Market Truth

In [Infinitesimally False](https://www.lesswrong.com/posts/j3Cv94eEZTDFiLREW/infinitesimally-false), Adrià and I argued that Łukasiewicz logic with hyperreal truth-values is a viable logic for a self-referential theory of truth. In Market Logic ([part 1](https://www.lesswrong.com/posts/g8RBkTf9uGTq9ypub/market-logic-i), [part 2](https://www.lesswrong.com/s/HmANELvkhAZ9eDxFS/p/2dsTdcZL6a3EAxdbS)) I provide an independent motivation for Łukasiewicz logic; namely, that it naturally emerges if we approach the problem of [computational uncertainty](https://www.lesswrong.com/w/logical-uncertainty) with an approach similar to that of [Garrabrant Induction](https://www.lesswrong.com/w/logical-induction), but without forcing classical logic on it. Those posts are prerequisites to this one.

Here, I consider how we might want to combine those ideas.

Evidence
========

In [Market Logic I](https://www.lesswrong.com/posts/g8RBkTf9uGTq9ypub/market-logic-i), I sketched a mathematical model of markets, but I left some options open as to the details. In particular, I said that we *could* model evidence coming in from outside the system as price-fixing, *or* we could instead imagine evidence comes into the market through the action of traders (who may have their own knowledge of the world).

In [Judgements: Merging Prediction and Evidence](https://www.lesswrong.com/posts/3hs6MniiEssfL8rPz/judgements-merging-prediction-and-evidence), I made the same remarks in a more elaborate manner, but without the detailed market design to put math behind it.

We can use hyperreal currency to bridge the gap between the two perspectives. Outside evidence streams which fix prices can be modeled perfectly via traders with infinite wealth relative to the other traders. (In theory, this could also allow modeling of hierarchies of evidence, via larger infinities.) This eliminates the unfortunate implication of the framework in Judgements that evidence-sources should eventually run out of money. It doesn't particularly make sense that, EG, a camera hooked up to a robot should have a finite budget which eventually runs out.

Math
====

In particular, let's give infinite currency to a trader who enforces the theorems of Peano Arithmetic (PA) on a subset of goods corresponding to that language. This includes enforcing classical logic on that subset of goods. We can imagine this trader as a theorem-prover who enforces whatever has been proven so far.

We're doing this so that we can develop a theory of truth, since theories of truth are commonly formalized as extensions of PA (since it is a well-established way of getting the diagonal lemma).

Naive Truth
===========

So, we've got a subset of goods for pure PA, and then a larger subset of goods will include a truth-predicate $T()$ which obeys the T-schema, intermingled with the language of PA. This has consistent fixed-points at finite time, since there are only finitely many sentences in the market at once. The hyperfinite $N $ from [Infinitesimally False](https://www.lesswrong.com/posts/j3Cv94eEZTDFiLREW/rescuing-truth-in-mathematics-from-the-liar-s-paradox-using) becomes a standard finite number, the market day $n$.

This seems to work (more consideration needed, of course) but I'm not sure I endorse such a theory. As I discussed in some of the previous posts, I want a theory of how we *come to think of concepts as real.* This *might* mean I don't want the T-schema to apply to everything equally?

Pragmatic Truth
===============

One natural theory to investigate here is what I'll call *pragmatic truth* (I believe it has some natural connection with the pragmatist theory of truth in philosophy, but I don't know enough to speak authoritatively on that).

My idea is to set the value of $T(``x")$ as the *limit of rational deliberation*, IE whatever price the market sets for $x$ in the limit.

This has some potentially appealing properties, EG, if we can prove that evidence can't change once it comes in, then whatever is evident is also true (it will be the same in the limit).

Because the limit behavior of the market is undecidable, however, I believe the T-schema *will not* hold for all cases. We could examine the degree to which specific instances of the T-schema are believed.

Unfortunately, I have no reason to think this will line up with what I'm trying to model. I don't really think truth is just the limit of rational deliberation; the limit of rational deliberation could still be wrong about some things.

Furthermore, the theory still marks all market goods as having truth values (due to having some limit); I would like to investigate a picture where some things have truth values and other things don't (due to being nonsense). (And perhaps there are intermediate cases, which '''have a truth value to some degree'''.)

Informality
===========

Perhaps truth can be connected to my sketchy ideas in [Informality](https://www.lesswrong.com/posts/q9fynSKxbafu9hSfY/informality). The concept of interpretability discussed there is like having some sort of semantic value -- true, false, penguin, aardvark, etc. If a string is interpretable, it can be mapped to a transparent context, within which we've got a context-independent description of the string's semantic value. We can characterize true and false (and the intermediate truth values) via substitution behavior, so perhaps this can give us a notion of "interpretable as having a truth value"?? (But perhaps all strings will turn out to be interpretable in this weak sense.)