To bring probability into the picture, the logic needs to be augmented with enough probabilities of values of variables in the logic that the rest of the probabilities can be derived.
I feel like this treat predicate logic as being “logic with variables”, but “logic with variables” seems more like Aristotelian logic than like predicate logic to me.
Another way to view it: a logic, possibly a predicate logic, is just a compact way of specifying a set of models (in the logician’s sense of the word “models”, i.e. the things a Bayesian would normally call “worlds”). Roughly speaking, to augment that logic into a probabilistic model, we need to also supply enough information to derive the probability of each (set of logician!models/Bayesian!worlds which assigns the same truth-values to all sentences expressible in the logic).
Idk, I guess the more fundamental issue is this treats the goal as simply being assigning probabilities to statements in predicate logic, whereas his point is more about whether one can do compositional reasoning about relationships while dealing with nebulosity, and it’s this latter thing that’s the issue.
What’s a concrete example in which we want to “do compositional reasoning about relationships while dealing with nebulosity”, in a way not handled by assigning probabilities to statements in predicate logic? What’s the use-case here? (I can see a use-case for self-reference; I’m mainly interested in any cases other than that.)
I feel like this treat predicate logic as being “logic with variables”, but “logic with variables” seems more like Aristotelian logic than like predicate logic to me.
Another way to view it: a logic, possibly a predicate logic, is just a compact way of specifying a set of models (in the logician’s sense of the word “models”, i.e. the things a Bayesian would normally call “worlds”). Roughly speaking, to augment that logic into a probabilistic model, we need to also supply enough information to derive the probability of each (set of logician!models/Bayesian!worlds which assigns the same truth-values to all sentences expressible in the logic).
Does that help?
Idk, I guess the more fundamental issue is this treats the goal as simply being assigning probabilities to statements in predicate logic, whereas his point is more about whether one can do compositional reasoning about relationships while dealing with nebulosity, and it’s this latter thing that’s the issue.
What’s a concrete example in which we want to “do compositional reasoning about relationships while dealing with nebulosity”, in a way not handled by assigning probabilities to statements in predicate logic? What’s the use-case here? (I can see a use-case for self-reference; I’m mainly interested in any cases other than that.)