First, the notation ξ∈□c(Φ,Θ) makes no sense. The prior is over hypotheses, each of which is an element of □(Γ×Φ). Θ is the notation used to denote a single hypothesis.
Second, having a prior just over Γ doesn’t work since both the loss function and the counterfactuals depend on 2Γ×Γ.
Third, the reason we don’t just start with a prior over 2Γ×Γ, is because it’s important which prior we have. Arguably, the correct prior is the image of a simplicity prior over physicalist hypotheses by the bridge transform. But, come to think about it, it might be about the same as having a simplicity prior over 2Γ×Γ, where each hypothesis is constrained to be invariant under the bridge transform (thanks to Proposition 2.8). So, maybe we can reformulate the framework to get rid of Φ (but not of the bridge transform). Then again, finding the “ultimate prior” for general intelligence is a big open problem, and maybe in the end we will need to specify it with the help of Φ.
Fourth, I wouldn’t say that Φ is supposed to solve the ontology identification problem. The way IBP solves the ontology identification problem is by asserting that 2Γ×Γ is the correct ontology. And then there are tricks how to translate between other ontologies and this ontology (which is what section 3 is about).
First, the notation ξ∈□c(Φ,Θ) makes no sense. The prior is over hypotheses, each of which is an element of □(Γ×Φ). Θ is the notation used to denote a single hypothesis.
Second, having a prior just over Γ doesn’t work since both the loss function and the counterfactuals depend on 2Γ×Γ.
Third, the reason we don’t just start with a prior over 2Γ×Γ, is because it’s important which prior we have. Arguably, the correct prior is the image of a simplicity prior over physicalist hypotheses by the bridge transform. But, come to think about it, it might be about the same as having a simplicity prior over 2Γ×Γ, where each hypothesis is constrained to be invariant under the bridge transform (thanks to Proposition 2.8). So, maybe we can reformulate the framework to get rid of Φ (but not of the bridge transform). Then again, finding the “ultimate prior” for general intelligence is a big open problem, and maybe in the end we will need to specify it with the help of Φ.
Fourth, I wouldn’t say that Φ is supposed to solve the ontology identification problem. The way IBP solves the ontology identification problem is by asserting that 2Γ×Γ is the correct ontology. And then there are tricks how to translate between other ontologies and this ontology (which is what section 3 is about).