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Some theory is tuple of set of axioms (including ones that are statements about data), set of inference rules and set of already deduced theorems (statements) in it. Theory can be discovered by human being by lengthy and partially informal process. But theory can also be learned by neural networks. There are true but unknown theories and there are known theories that approximate the unknown ones to some degrees. There is reasoning with degrees of truth.

My question is - is there some philosophical, epistemic, gnoseological know that measures how much one theory is close to another theory, some kind of distance among theories? Of course, different aspects can be measured (complexity, expressivity, number of statements with the same truth value of both theories) and that is why different measures can exist. I am not seeking the ultimate list of such aspects and measures. I am just seeking the common name, term, keyword that is used in the philosophy to describe the efforts to research the closeness of theories.

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    Judging by the first sentence, you have in mind formal theories. For those the question of proximity is meaningless until one is interpreted in the other (or both in a third), see interpretability. With that in place, see Friend et al., Distances between formal theories for a framework for measuring proximity. This research is recent, even equivalence of theories is a very complex and unsettled subject.
    – Conifold
    Aug 20, 2021 at 7:04
  • Thanks! It is OK, if research is nascent only. I am seeking the entry points.
    – TomR
    Aug 20, 2021 at 7:17
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    It is not about vague vs crisp. Empirical theories include heuristics and procedures that operationally connect them to their subject matter and allow for interpretation, revision, testing, etc. None of that can be captured linguistically because it involves performing actions and interpreting their outcomes. Informal (linguistic) theory lifted from that is only a shadow of the original, and even more so when formalized. Different empirical theories can conceptualize events "incommensurably" and then even the preliminary inter-interpretation process becomes problematic.
    – Conifold
    Aug 20, 2021 at 7:39
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    Consider special relativity and Lorentz's theory of ether. They are formally equivalent, there is a translation that preserves all theorems, but people take them as very different theories. Whatever distinguishes them will not be captured by formal distance, it is lost in translation. Intuitively, they assign different "meanings" to the symbols involved. What "meaning" is is notoriously difficult to pinpoint, but it includes operational aspects that are not captured by symbols and axioms for manipulating them.
    – Conifold
    Aug 20, 2021 at 8:54
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    Artificial intelligence may be able to handle it if it is equipped with operational capabilities but that will not erase the difference between them and linguistic representation. No representation can do the representing itself required for concept specification and interpretation of events. On incommensurability of theories see SEP and references there. In particular, it discusses semantic view of theories that does not conceptualize them as linguistic entities and perhaps can capture more of the empirical originals.
    – Conifold
    Aug 20, 2021 at 9:01

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