Thanks, but that proof doesn't work for the formulation of Occam's Razor that I was talking about.

For example, if I have a boolean-output function, there are three "simplest possible" (2 bit long) minimum hypotheses as to what it is, before I see the evidence: [return 0], [return 1], and [return randomBit()]. But a "more complex" (longer than 2 bit) hypothesis, like [on call #i to function, return i mod 2] can't be represented as being equivalent to [[one of the previous hypotheses] AND [something else]] so the conjunction rule doesn't apply.

I think the conjunction-rule proof does work for the "minimum entities" formulation, but that one's deeply problematic because, among other things, it assigns a higher prior probability to divine explanations (of complex systems) than physics-based ones.

Thanks, but that proof doesn't work for the formulation of Occam's Razor that I was talking about.

For example, if I have a boolean-output function, there are three "simplest possible" (2 bit long) minimum hypotheses as to what it is, before I see the evidence: [return 0], [return 1], and [return randomBit()]. But a "more complex" (longer than 2 bit) hypothesis, like [on call #i to function, return i mod 2] can't be represented as being equivalent to [[one of the previous hypotheses] AND [something else]] so the conjunction rule doesn't apply.

I think the conjunction-rule proof does work for the "minimum entities" formulation, but that one's deeply problematic because, among other things, it assigns a higher prior probability to divine explanations (of complex systems) than physics-based ones.