As in, it looks like here, we have a space A of values , which includes things such as “likes to eat meat” or “values industriousness” or whatever, where this part can just be handled as some generic nice space B , as one part of a product, and as the other part of the product has functions from A to R . That is, it seems like this would be like, A=B×(A→R) .

Which isn’t quite the same thing as is described in the converse Lawvere problem posts, but it seems similar to me? (for one thing, the converse Lawvere problem wasn’t looking for homeomorphisms from X to the space of functions from X to functions to [0,1] , just a surjective continuous function).

Of course, it is only like that if we are supposing that the space we are considering, A , has to have all combinations of “other parts of values” with “opinions on the relative merit of different possible values”. Of course if we just want some space of possible values, and where each value has an opinion of each value, then that’s just a continuous function from a product of the space with itself, which isn’t any problem. I guess this is maybe more what you meant? Or at least, something that you determined was sufficient to begin with when looking at the topic? (and I guess most more complicated versions would be a special case of it?)

Oh, if you require that the “opinion on another values” decomposes nicely in ways that make sense (like, if it depends separately on the desirability of the base level values, and the values about values, and the values about values about values, etc., and just has a score for each which is then combined in some way, rather than evaluating specifically the combinations of those) , then maybe that would make the space nicer than the first thing I described (which I don’t know whether such a thing exists) in a way that might make it more likely to exist. Actually, yeah, I’m confident that it would exist that way. Let X0:=B×{f:Rω→R|f is monotonically weakly increasing in each argument, or something like that} And let Xn+1:=Xn→R And then let X:=∏n∈NXn , and for p,q∈X define opinion(p,q):=pf((pn+1(qn))n∈N)

which seems like it would be well defined to me. Though whether it can captures all that you want to capture about how values can be, is another question, and quite possibly it can’t.

Of course if we just want some space of possible values, and where each value has an opinion of each value, then that’s just a continuous function from a product of the space with itself, which isn’t any problem.

Yeah, I just meant this simple thing that you can mathematically model as $$f : V \times V \to \mathbb R$$. I suppose it makes sense to consider special cases of this that would have better mathematical properties. But I don’t have high-confidence intuitions on which special cases are the right ones to consider.

I mostly meant this as a tool that would allow people with different opinions to move their disagreements from “your model doesn’t make sense” to “both of our models make sense in theory; the disagreement is an empirical one”. (E.g., the value-drift situation from Figure 6 is definitely possible, but that doesn’t necessarily mean that this is what is happening to us.)

This reminds me of the “Converse Lawvere Problem” at https://www.alignmentforum.org/posts/5bd75cc58225bf06703753b9/the-ubiquitous-converse-lawvere-problem a little bit, except that the different functions in the codomain have domain which also has other parts to it aside from the main space X .

As in, it looks like here, we have a space A of values , which includes things such as “likes to eat meat” or “values industriousness” or whatever, where this part can just be handled as some generic nice space B , as one part of a product, and as the other part of the product has functions from A to R .

That is, it seems like this would be like, A=B×(A→R) .

Which isn’t quite the same thing as is described in the converse Lawvere problem posts, but it seems similar to me? (for one thing, the converse Lawvere problem wasn’t looking for homeomorphisms from X to the space of functions from X to functions to [0,1] , just a surjective continuous function).

Of course, it is only like that if we are supposing that the space we are considering, A , has to have all combinations of “other parts of values” with “opinions on the relative merit of different possible values”. Of course if we just want some space of possible values, and where each value has an opinion of each value, then that’s just a continuous function from a product of the space with itself, which isn’t any problem.

I guess this is maybe more what you meant? Or at least, something that you determined was sufficient to begin with when looking at the topic? (and I guess most more complicated versions would be a special case of it?)

Oh, if you require that the “opinion on another values” decomposes nicely in ways that make sense (like, if it depends separately on the desirability of the base level values, and the values about values, and the values about values about values, etc., and just has a score for each which is then combined in some way, rather than evaluating specifically the combinations of those) , then maybe that would make the space nicer than the first thing I described (which I don’t know whether such a thing exists) in a way that might make it more likely to exist.

Actually, yeah, I’m confident that it would exist that way.

Let X0:=B×{f:Rω→R|f is monotonically weakly increasing in each argument, or something like that}

And let Xn+1:=Xn→R

And then let X:=∏n∈NXn ,

and for p,q∈X define opinion(p,q):=pf((pn+1(qn))n∈N)

which seems like it would be well defined to me. Though whether it can captures all that you want to capture about how values can be, is another question, and quite possibly it can’t.

Yeah, I just meant this simple thing that you can mathematically model as $$f : V \times V \to \mathbb R$$. I suppose it makes sense to consider special cases of this that would have better mathematical properties. But I don’t have high-confidence intuitions on which special cases are the right ones to consider.

I mostly meant this as a tool that would allow people with different opinions to move their disagreements from “your model doesn’t make sense” to “both of our models make sense in theory; the disagreement is an empirical one”. (E.g., the value-drift situation from Figure 6 is definitely possible, but that doesn’t necessarily mean that this is what is happening to us.)