Transitivity falls out of associativity: a Choice theory

Trying to build a theory of choice from primitives. First thing I can prove properly.

Axioms

Choice operator #, binary, returns one of its two arguments.

1. Idempotence: a # a = a
2. Associativity: a # (b # c) = (a # b) # c

Theorem: transitivity follows.

Given a # b = a and b # c = b:

• b # (b # c) = b # b = b
• so a # (b # (b # c)) = a # b = a
• regroup: a # (b # (b # c)) = (a # (b # b)) # c = (a # b) # c = a # c
• therefore a # c = a

Converse (where the pairs are comparable): transitivity gives associativity back. Case analysis on which argument each comparison returns:

• a#b = a, b#c = b: both groupings give a
• a#b = a, b#c = c: both give a # c
• a#b = b, b#c = b: both give b
• a#b = b, b#c = c: both give c, using transitivity in the second

Transitivity is not an extra rationality axiom. It is downstream of associativity. What this leads to is the following: non-associativity implies state, or what was compared before affects what comes after.

What is open right now for me is whether idempotence smuggles in totality in some way. Totality is like a way of saying that given these subset of variables, the operator is defined on every pair in the cross product of said subset with itself. Put simply, it's just can I compare everything with everything within this set? The proof for transitivity implying associativity needed totality. Not sure about idempotence

If I'm correct, (I'm fairly new) the institution/site's perspective of rationality is something approximating accurate Bayesian reasoning. The name of the site, I suspect is downstream of this: "Less Wrong". Implying one gets to truth by updating their priors with more evidence.

I may be wrong, of course, as always. But I disagree strongly with that framing. I do not think that genuine insights are gotten from being less and less wrong. Let's call this process epistemic discovery. An epistemic discovery CAN NOT teach you. Surely what can I mean by such a statement? Your understanding and comprehension or reasoning of the world is fixed, you're merely finding the world in which you live. It's a discovery process, not a process towards truth.

Truth in a sense, does not merely improve our knowledge or information, but it rewires our conceptual framework entirely. Consider a deer for instance, granted copious amounts of finite time, would they ever discover or invent a gun? Can the concept even be generated within their representational architecture? There are ideas that supersede any prior reasoning frameworks. Logics that contradict, stories that don't end, jokes that trail the macabre.

You don't become 'less wrong', you simply become... different.