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Suppose the natural world and supernatural world differ from each other by premises that both share, i.e. they are dependent on the same axioms.

Then the supernatural should also be a subsection of the universe with similar "morphology".

But that is contrary to the common, and religious, understanding of the supernatural.

Thus, in order to be conformal with that, we shall demand the supernatural to contain logically independent axioms from nature.

The problem is then that independent statements have a mismatch in quantified variable type or predicates, as they can consistently violate the modus tollens that holds equivalently for implication, thus they have no complete causal relation.

And if that is the case, all axioms for the supernatural that are independent of those within nature contribute to no causal result in nature, thus either the supernatural already contains natural laws and extens them with new independent laws, thus collapsing natural and supernatural into one, or it is totally independent and contributes no information to nature at all.

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    Reality is not formed from axioms. Commented Jan 29 at 12:43
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    In a nutshell, "supernatural" means something that breaks the laws of nature e.g. an heavy body that float on water instead of sinking. In order to judge that this phenomena is a "real" supernatural effect we have to check that no hidden cause is acting on it in a natural way that can explain the purported breaking of the natural law causing its unexpected behaviour. Commented Jan 29 at 13:48
  • Why would independent "laws of the supernatural" collapse it into the natural if they are qualitatively distinct? And why should the supernatural follow any laws at all that are expressible as axioms? The usual take on it is that it works by rare lawless interventions into the natural (miracles and such). So it can "contribute information to nature" by following distinct laws that are hard for us to capture due to rarity of their manifestations, or by following no laws at all.
    – Conifold
    Commented Jan 29 at 21:15

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What does modus tollens have to do with causality? I don't fully grasp the details of your question.

But, yes, you're correct. This was for example the argument that Willard Van Orman Quine made against mind-body dualism: as far as we can say anything about the mind, it can be made a part of the natural reality, i.e. any theory under which it is a distinct entity is empirically equivalent to some other theory where it is a part of the natural world (it is identical to, e.g. the brain). As far as we cannot say anything about it, it is redundant and doesn't contribute to our understanding of the world.

One response would be to posit some supernatural way of accesing this supersenbile realm. We have, afterall, mathematical knowledge so it's not unthinkable that the Platonic realm which mathematical entities can be said to inhabit is inhabited by some further supernatural creatures. This, however, falls under problem of the criterion like any "intuitive knowledge".

It's, however, completly unnecessary to make it about axiomatic systems.

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  1. You say "dependent on the same axioms", but as I understand, theories in physics, biology, chemistry etc all have different axioms. You cannot describe nature with a coherent and unified set of axioms yet; so why expect to describe super-natural too? (with the same set of axioms).

  2. You say "all axioms for the supernatural ... contribute to no causal result in nature", but a causal result is not based on axioms; axioms are human constructs that (are meant to/used to) justify/explain causal results.

Thus, the "axiomatic (in-)dependance" you say is not a fact, nor you have proved it in any way; it's too an axiom you have inferred.

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