P is intended to be like objective reality, exactly analogously with the predicate “True.” So we can adjoin P as a symbol and the reflection principle as an axiom schema, and thereby obtain a more expressive language. Depending on architecture, this also may increase the agent’s ability to formulate or reason about hypotheses.
Statements without P’s in them, are indeed either true or false with probability 1. I agree it is a bit odd for statements with P in them to have probabilities, but I don’t see a strong argument it shouldn’t happen. In particular, it seems irrelevant to anything meaningful we would like to do with a truth predicate. In subsequent versions of this result, the probabilities have been removed and the core topological considerations exposed directly.
The relationship between a truth predicate and the kind of reasoning you discuss (a MIM that believes its own computations are trustworthy) is that truth is useful or perhaps necessary for defining the kind of correspondence that you want the MIM to accept, about a general relationship between the algorithm it is running and what is “true”. So having a notion of “truth” seems like the first step.
P is intended to be like objective reality, exactly analogously with the predicate “True.” So we can adjoin P as a symbol and the reflection principle as an axiom schema, and thereby obtain a more expressive language. Depending on architecture, this also may increase the agent’s ability to formulate or reason about hypotheses.
Statements without P’s in them, are indeed either true or false with probability 1. I agree it is a bit odd for statements with P in them to have probabilities, but I don’t see a strong argument it shouldn’t happen. In particular, it seems irrelevant to anything meaningful we would like to do with a truth predicate. In subsequent versions of this result, the probabilities have been removed and the core topological considerations exposed directly.
The relationship between a truth predicate and the kind of reasoning you discuss (a MIM that believes its own computations are trustworthy) is that truth is useful or perhaps necessary for defining the kind of correspondence that you want the MIM to accept, about a general relationship between the algorithm it is running and what is “true”. So having a notion of “truth” seems like the first step.