Formal systems are known deductive systems, representing some aspects of the environment, nature, thinking or, more frequently, abstract representations of the former subjects.
But, assuming the existence of some syntatically correct representation of the real world, could be infered from them some set of axioms, complete and consistent? Is there any approach to this task? Or at least: any clue?
Peter Breuer Thank you for your comment, it clarified, pointed out the precise idea that I had in mind when I asked the question; This general theoretical question arose from a practical problem: the completion of (knowledge) graphs and the need to create an ontology, based on a set of axioms, to admit or reject new links (arcs) and objects (nodes). In fact, we are talking about a monotonic extension of the given original set.
In fact, the original set of statements is implicitly assumed to be consistent (nothing to say otherwise). But here's a small detail: speaking of "consistency" could be interpreted as assuming the existence of some kind of formal system. Probably the problem should be posed as emerging from a situation in which, there being some relation (binary, for simplicity), written (of course) in some formalized language (hence w.f.f.), representing a set of non-contradictory statements (here, not contradictory means the absence of a statement and its -semantically understood- negation).
And yes, the set of all statements is a (trivial) system of axioms.
As for the assumption of the presence of some kind of logic, it is not assumed. Here, significantly, it is relevant to take into account what is meant by logic; As far as I know, despite their historical and genetic relationships (in the sense of genesis), they are not synonymous, logic (syntactic approach) is (mainly) a kind of formal system.
Well, here I end my ramblings. Thanks for the comment.
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