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RichardKennaway comments on Thought experiments on simplicity in logical probability - Less Wrong Discussion
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More complicated hypotheses have to be on average at least exponentially less likely, because there are exponentially many of them. There are probability measures that decline with length faster than exponential, but none that are slower. One could even say that the Solomonoff probabilities decline as slowly as possible.
The analogy between Solomonoff induction and a simplicity prior on logical sentences is not perfect.
EDIT: What I mean is: and so I'm not sold that the reasons Solomonoff induction uses an exponential distribution also applies to logical sentences.
For example, in thought experiment 1, it is totally allowed that there are only 3 options. If this was solomonoff induction, there are always an infinite number of programs that fit the data. But in thought experiment 1, the only options realio trulio are A, B and C. There are no bad consequences if you give them probabilities proportional to a power law - or at least not the same bad consequences.