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Vladimir_Nesov comments on Notion of Preference in Ambient Control - Less Wrong
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Comments (45)
I think that I'm missing your point. Of course, the interpretation doesn't affect what the agent or its theory can prove. Is that all you're saying?
The reason that I'm led to think in terms of semantics is that your post appeals to properties of the agent that aren't necessarily encoded in the agent's theory. At least, the current post doesn't explicitly say that these properties are encoded in the theory. (Maybe you made it clear how this works in one of your previous post. I haven't read all of these closely.)
The properties I'm thinking of are (1) the agent's computational constraints and (2) the fact that the agent actually does the action represented by the action-constant that yields the highest computed utility, rather than merely deducing that that constant has the highest computed utility.
For example, you claim that [A=1] must be derivable in the theory if the agent actually does A. The form of your argument, as I understand it, is to note that [A=1] is true in the standard interpretation, and to show that [A=1] is the sort of formula which, if true under one interpretation, must be true in all, so that [A=1] must be a theorem by completeness. I'm still working out why [A=1] is the required kind of formula, but the form of your argument does seem to appeal to a particular interpretation before generalizing to the rest.
If the agent actually does 1 (I assume you meant to say). I don't see what you are trying to say again. I agree with the last paragraph (you could recast the argument that way), but don't understand the third paragraph.
Whoops. Right.
Okay. Let me try to make my point by building on the last paragraph, then. According to my understanding, you start out knowing that v(A) = v(1) for a particular interpretation v. Then you infer that v'(A) = v'(1) for an arbitrary interpretation v'. Part of my reason for using the v(.) symbol is to help myself keep the stages of this argument distinct.
If v is an interpretation, it maps (all) terms to elements of corresponding universe, while possible actions are only some formulas, so associated mapping K would map some formulas to the set of actions (which don't have to do anything with any universe). So, we could say that K(1)=1', but K(A) is undefined. K is not an interpretation.
Maybe we're not using the terminology in exactly the same way.
For me, an interpretation of a theory is an ordered pair (D, v), where D is a set (the domain of discourse), and v is a map (the valuation map) satisfying certain conditions. In particular, D is the codomain of v restricted to the constant symbols, so v actually contains everything needed to recover the interpretation. For this reason, I sometimes abuse notation and call v itself the interpretation.
The valuation map v
maps constant symbols to elements of D,
maps n-ary function symbols to maps from D^n to D,
maps n-ary predicate symbols to subsets of D^n,
maps sentences of the theory into {T, F}, in a way that satisfies some recursive rules coming from the rules of inference.
Now, in the post, you write
(Emphasis added.) I've been working with the bolded option, which I understand to be saying that A and 1 are constant symbols. Hence, given an interpretation (D, v), v(A) and v(1) are elements of D, so we can ask whether they are the same elements.
I agree with everything you wrote here...
What was your "associated mapping K"? I took it to be what I'm calling the valuation map v. That's the only map that I associate to an interpretation.
K has a very small domain. Say, K("2+2")=K("5")="pull the second lever", K("4") undefined, K("A") undefined. Your v doesn't appear to be similarly restricted.