So far as this is true that all formalizations of belief are full subcategories of davidad!beliefs… why has this only been posted today???
Also, what about stuff like … various logics? AGM theory? Do you not count them as “beliefs”? They are less probability-like but Dempster-Shafer functions are very probability-like and I don’t see them here. I recall that Halpern (and a coauthor?)’s Generalized Expected Utility generalized almost every decision rule (?) but (some decision rule using?) Dempster-Shafer was the main interesting exception. Is something similar going on here?
Also, it seems to me that you’re equating beliefs with functions judging consistency of probability distributions with them. So if you believe that X and Y are independent then this is expressed by a belief function that judges accordingly. But I would expect that in some cases (not in the probabilistic independence case) this equates differently expressed propositional attitudes (different senses/intensions) because of having the same references/extensions, for the purpose of determining a belief function. Do you consider this an issue at all? My guess is that keeping the intentional difference in mind is relevant for handling ontological crisis-shaped stuff.
To your second point, very much yes. I am also working on a much bigger framework for world-modeling, still along the lines of Safeguarded AI TA1.1, which takes this notion of beliefs as a central ingredient in its semantics. The syntax is most of the work. By syntax/semantics I mean the same thing as sense/referent and intension/extension.
The purpose of the semantics is to ground judgments about observational equivalence about belief states, but when we are exploring the hypothesis space about infinite-dimensional as finite creatures, we need rich languages for expressing our beliefs with finite sequences of bits.
Richardson’s PDGs are in my opinion the best syntax published to date.
Dempster-Shafer functions express beliefs via their credal sets. AGM theory and epistemic logics carry only true/false beliefs about propositions, which are quite degenerate but can still be embedded as full subcategories of credal sets ().
Not all credal sets satisfy the inclusion/exclusion rule that characterizes DS belief functions. Also, they arguably combine differently (Dempster’s rule vs conditioning pointwise).
Definitely the right way of turning a DS function into a belief is not the way you capture credal sets, since they are notoriously not idempotent (i.e., for most basic DS mass functions ).
So far as this is true that all formalizations of belief are full subcategories of davidad!beliefs… why has this only been posted today???
Also, what about stuff like … various logics? AGM theory? Do you not count them as “beliefs”? They are less probability-like but Dempster-Shafer functions are very probability-like and I don’t see them here. I recall that Halpern (and a coauthor?)’s Generalized Expected Utility generalized almost every decision rule (?) but (some decision rule using?) Dempster-Shafer was the main interesting exception. Is something similar going on here?
Also, it seems to me that you’re equating beliefs with functions judging consistency of probability distributions with them. So if you believe that X and Y are independent then this is expressed by a belief function that judges accordingly. But I would expect that in some cases (not in the probabilistic independence case) this equates differently expressed propositional attitudes (different senses/intensions) because of having the same references/extensions, for the purpose of determining a belief function. Do you consider this an issue at all? My guess is that keeping the intentional difference in mind is relevant for handling ontological crisis-shaped stuff.
In any case, hooray for expanding the domain fo discourse to reveal the structure already there but hidden!
To your second point, very much yes. I am also working on a much bigger framework for world-modeling, still along the lines of Safeguarded AI TA1.1, which takes this notion of beliefs as a central ingredient in its semantics. The syntax is most of the work. By syntax/semantics I mean the same thing as sense/referent and intension/extension.
The purpose of the semantics is to ground judgments about observational equivalence about belief states, but when we are exploring the hypothesis space about infinite-dimensional
Richardson’s PDGs are in my opinion the best syntax published to date.
fwiw Dempster-Shafer belief functions are a very nice special class of PDGs :)
Dempster-Shafer functions express beliefs via their credal sets. AGM theory and epistemic logics carry only true/false beliefs about propositions, which are quite degenerate but can still be embedded as full subcategories of credal sets (
Not all credal sets satisfy the inclusion/exclusion rule that characterizes DS belief functions. Also, they arguably combine differently (Dempster’s rule vs conditioning pointwise).
Definitely the right way of turning a DS function into a belief