In this post, I will introduce a new way of thinking about logical uncertainty. The main goal of logical uncertainty is to learn how to assign probabilities to logical sentences which have not yet been proven true or false.
One common approach is to change the question, assume logical omniscience and only try to assign probabilities to the sentences that are independent of your axioms (in hopes that this gives insight to the other problem). Another approach is to limit yourself to a finite set of sentences or deductive rules, and assume logical omniscience on them. Yet another approach is to try to define and understand logical counterfactuals, so you can try to assign probabilities to inconsistent counterfactual worlds.
One thing all three of these approaches have in common is they try to allow (a limited form of) logical omniscience. This makes a lot of sense. We want a system that not only assigns decent probabilities, but which we can formally prove has decent behavior. By giving the system a type of logical omniscience, you make it predictable, which allows you to prove things about it.
However, there is another way to make it possible to prove things about a logical uncertainty system. We can take a program which assigns probabilities to sentences, and let it run forever. We can then ask about whether or not the system eventually gives good probabilities.
At first, it seems like this approach cannot work for logical uncertainty. Any machine which searches through all possible proofs will eventually give a good probability (1 or 0) to any provable or disprovable sentence. To counter this, as we give the machine more and more time to think, we have to ask it harder and harder questions.
We therefore have to analyze the machine's behavior not on individual sentences, but on infinite sequences of sentences. For example, instead of asking whether or not the machine quickly assigns 110 to the probability that the 3↑↑↑↑3rd digit of π is a 5 we look at the sequence:
an:= the probability the machine assigns at timestep 2n to the n↑↑↑↑nth digit of π being 5,
and ask whether or not this sequence converges to 110.
Disclaimer: This sequence is a paper that I am trying to write. I intend to make minor changes to the posts in this sequence over time. I will try not to make changes that change important content, and will provide a warning if I do. There may be large gaps in time between posts. For now, I am trying to get these posts out quickly, so there may be typos.
In this post, I will introduce a new way of thinking about logical uncertainty. The main goal of logical uncertainty is to learn how to assign probabilities to logical sentences which have not yet been proven true or false.
One common approach is to change the question, assume logical omniscience and only try to assign probabilities to the sentences that are independent of your axioms (in hopes that this gives insight to the other problem). Another approach is to limit yourself to a finite set of sentences or deductive rules, and assume logical omniscience on them. Yet another approach is to try to define and understand logical counterfactuals, so you can try to assign probabilities to inconsistent counterfactual worlds.
One thing all three of these approaches have in common is they try to allow (a limited form of) logical omniscience. This makes a lot of sense. We want a system that not only assigns decent probabilities, but which we can formally prove has decent behavior. By giving the system a type of logical omniscience, you make it predictable, which allows you to prove things about it.
However, there is another way to make it possible to prove things about a logical uncertainty system. We can take a program which assigns probabilities to sentences, and let it run forever. We can then ask about whether or not the system eventually gives good probabilities.
At first, it seems like this approach cannot work for logical uncertainty. Any machine which searches through all possible proofs will eventually give a good probability (1 or 0) to any provable or disprovable sentence. To counter this, as we give the machine more and more time to think, we have to ask it harder and harder questions.
We therefore have to analyze the machine's behavior not on individual sentences, but on infinite sequences of sentences. For example, instead of asking whether or not the machine quickly assigns 110 to the probability that the 3↑↑↑↑3rd digit of π is a 5 we look at the sequence:
an:= the probability the machine assigns at timestep 2n to the n↑↑↑↑nth digit of π being 5,
and ask whether or not this sequence converges to 110.
Here is the list of posts in this sequence:
Disclaimer: This sequence is a paper that I am trying to write. I intend to make minor changes to the posts in this sequence over time. I will try not to make changes that change important content, and will provide a warning if I do. There may be large gaps in time between posts. For now, I am trying to get these posts out quickly, so there may be typos.