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What does Occam's razor look like written rigorously?

Motivated by the desire to develop this idea further: https://philosophy.stackexchange.com/questions/41728/

I'm curious to see how a formal version of Occam's razor might look with minimal axiomatic construction, but ideally not just as a comparative tool within current systems, as the fundamental axiom of a system of logic i.e. as an alternative to current systems of logic or with minimal dependence on them.

My effort is along these lines:

For any pair of sentences $s_1,s_2$, define $>$ to intuitively mean "more believable" i.e. $s_1>s_2\implies$ $s_1$ is more believable than $s_2$. For example, $s_1=\text{I exist as a real physical being}$ and $s_2=\text{The flying spaghetti monster exists as a real physical being}$ then $s_1>s_2$.

Clearly every set of equivalent sentences $s_1\sim s_2$ will represent some truth or confidence threshold $t$ and we might define this collection of thresholds $T=\{t_0,t_1,t_2,\ldots\}$ such that $s_1\sim s_2\iff s_1,s_2\in t_m$ for some $m$.

For example, and this may be a personal judgement but a person may choose some $t_1$ to be the threshold they require in order to accept something as true while another person chooses $t_2$. Thus truth is defined only as being equally true, or more or less true, than something else, and is also a personal choice.

Then we might define the minimal collection of assumptions required to achieve some truth threshold $t_j$ for some sentence $s_i$ as: $A(s_i,t_j)$

Then Occam's razor is the identity $s_1>s_2\iff A(s_1,t_j)\subset A(s_2,t_j)$

In this way the choice of truth by Occam's razor may be different depending upon the truth threshold being employed. Clearly there will be different possible orderings $>$ so maybe it is appropriate to augment these with the truth threshold being employed so we have $>_{t_j}$.

Does that look reasonable foundation for such a system, should this be different, and does similar already exist?

Can such a framework can accommodate current modes of logic by incorporating their axioms into some pair $A,t$?

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  • $\begingroup$ $1)$ We should allow $s_1=s_2$ $2)$ The numerical value of the $s's$ can only be defined subjectively $3)$ I am not sure whether it makes sense to define such an order , but surely it is an interesting idea. $\endgroup$
    – Peter
    Jul 28, 2017 at 10:59
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    $\begingroup$ @Peter I used $\sim$ for equivalence to indicate they're different sentences but equivalent in truth since I thought it better to reserve $=$ to imply they're the same sentence. Re the subjective definition I had the same thought as you since some sentence pairs will have alternative orderings e.g. when each requires some assumption the other does not. Hence I added the $>_{t_j}$ part which allows for subjectivity in relation to believability/truth. $\endgroup$ Jul 28, 2017 at 11:08
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    $\begingroup$ I am not sure to what extend this might help you, but there is a rigorous formalization of Occam's razor that uses probability theory: see en.wikipedia.org/wiki/…. It relies on Turing machines to form a prior probability, and more complicated turin machines have a lower prior probability. $\endgroup$
    – supinf
    Jul 28, 2017 at 11:11
  • $\begingroup$ @RobertFrost We should make sure that the implication $$s_1>s_2\ and \ s_2>s_3\implies s_1>s_3$$ always makes sense semantically. I do not know concrete sentences violating it , but it will be difficult to ensure that there actually are none. $\endgroup$
    – Peter
    Jul 28, 2017 at 11:14
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    $\begingroup$ You might want to investigate fuzzy logic for your formulation. $\endgroup$ Jan 18, 2018 at 17:49

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