I've been giving this a lot of thought over the last few months. My personal insight is that "reasoning" is simply the application of a probabilistic reasoning manifold on an input in order to transform it into constrained output that serves the stability or evolution of a system.
This manifold is constructed via learning a decontextualized pattern space on a given set of inputs. Given the inherent probabilistic nature of sampling, true reasoning is expressed in terms of probabilities, not axioms. It may be possible to discover axioms by locating fixed points or attractors on the manifold, but ultimately you're looking at a probabilistic manifold constructed from your input set.
But I don't think you can untie this "reasoning" from your input data. It's possible you will find "meta-reasoning", or similar structures found in any sufficiently advanced reasoning manifold, but these highly decontextualized structures might be entirely useless without proper recontextualization, necessitating that a reasoning manifold is trained on input whose patterns follow learnable underlying rules, if the manifold is to be useful for processing input of that kind.
Decontextualization is learning, decomposing aspects of an input into context-agnostic relationships. But recontextualization is the other half of that, knowing how to take highly abstract, sometimes inexpressible, context-agnostic relationships and transform them into useful analysis in novel domains.
This doesn't mean a well-trained model can't reason about input it hasn't encountered before, just that the input needs to be in some way causally connected to the same laws which governed the input the manifold was trained on.
I'm sure we could create a fully generalized reasoning manifold which could handle anything, but I don't see how we possibly get that without first considering and encountering all possible inputs. But these inputs still have to have some form of constraint governed by laws that must be learned through sampling, otherwise you'd just be training on effectively random data.
The other commenter who suggested simply generating all possible sentences and training on internal consistency should probably consider Gödel's incompleteness theorems, and that internal consistency isn't enough to accurately model and interpret the universe. One could construct a thought experiment about an isolated brain in a jar with effectively unlimited neuronal connections, but no sensory connection to the outside world. It's possible, with enough connections, that the likelihood of the brain conceiving of true events it hasn't actually encountered does increase meaningfully. But the brain still has nothing to validate against, and can't simply assume that because something is internally logically consistent, that it must exist or have existed.
This manifold is constructed via learning a decontextualized pattern space on a given set of inputs. Given the inherent probabilistic nature of sampling, true reasoning is expressed in terms of probabilities, not axioms. It may be possible to discover axioms by locating fixed points or attractors on the manifold, but ultimately you're looking at a probabilistic manifold constructed from your input set.
But I don't think you can untie this "reasoning" from your input data. It's possible you will find "meta-reasoning", or similar structures found in any sufficiently advanced reasoning manifold, but these highly decontextualized structures might be entirely useless without proper recontextualization, necessitating that a reasoning manifold is trained on input whose patterns follow learnable underlying rules, if the manifold is to be useful for processing input of that kind.
Decontextualization is learning, decomposing aspects of an input into context-agnostic relationships. But recontextualization is the other half of that, knowing how to take highly abstract, sometimes inexpressible, context-agnostic relationships and transform them into useful analysis in novel domains.
This doesn't mean a well-trained model can't reason about input it hasn't encountered before, just that the input needs to be in some way causally connected to the same laws which governed the input the manifold was trained on.
I'm sure we could create a fully generalized reasoning manifold which could handle anything, but I don't see how we possibly get that without first considering and encountering all possible inputs. But these inputs still have to have some form of constraint governed by laws that must be learned through sampling, otherwise you'd just be training on effectively random data.
The other commenter who suggested simply generating all possible sentences and training on internal consistency should probably consider Gödel's incompleteness theorems, and that internal consistency isn't enough to accurately model and interpret the universe. One could construct a thought experiment about an isolated brain in a jar with effectively unlimited neuronal connections, but no sensory connection to the outside world. It's possible, with enough connections, that the likelihood of the brain conceiving of true events it hasn't actually encountered does increase meaningfully. But the brain still has nothing to validate against, and can't simply assume that because something is internally logically consistent, that it must exist or have existed.