Is there a systematic and preferably rigorous way of checking whether a framework of concept interpretations is consistent?
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3Can you elaborate on this a bit? What do you mean exactly by "a framework of concept interpretations"? Like in model theory?– E...Commented Apr 18, 2016 at 8:59
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This really seems to fall in cognitive science or computing, and not to have philosophical content.– user9166Commented Apr 18, 2016 at 17:55
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@jobermark I don't think this falls in cognitive science, because the question is about testing your own models of reality, rather than a description how generally one generates an understanding. It's not the kind of semantics which one talks about in computing.– Matthias Michael EnghCommented Apr 18, 2016 at 19:27
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@Eliran Thanks for asking this and showing interest! :) Yes, it appears I mean something like in model theory. This is already quite interesting to me. Could it be that what I'm really interested in checking whether a topological space is euclidean?– Matthias Michael EnghCommented Apr 18, 2016 at 19:33
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If what you seek is, “checking whether a topological space is Euclidean,” then this sounds like you want to check for some sort of ‘structural equivalence’. This also sounds like a question for the Maths stack.– c.w.chambersCommented Apr 18, 2016 at 20:22
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2 Answers
(You proposed the alternative phrasing "Are there any practical or pragmatic systematic methods to check for contradictions?" and some of this makes more sense in that context.)
The problem is that depending on the kind of system, you have lots of different answers:
Basic Combinatorics says 'Yes, there is one that runs finitely long in most useful, realistic systems' because they tend to be reducible to a set of finite cases by combinatorial tricks.
But Complexity theory says if "practical" is to mean anything, then 'Probably not, the running time of such things (logical constraint satisfaction algorithms) is often NP-complete.'
In the other hand, the experience of Optimization Theory says "Usually, because NP-complete problems often have polynomial time heuristics that guess most real cases correctly."
More abstractly, Computer Science says 'No, not in most cases that model real languages.' because of Turing's theorem on the halting problem.
Even more abstractly, First Order Logic says 'Definitely not if your language tries to model the whole of math" because you have Godel's work referenced in the first answer.
But proofs in Second Order Logic can rescue most of the domains of classical mathematics, and say "yes", for cases even as robust as the Real Numbers and Classical Geometry, as long as they cannot talk explicitly about themselves.
So without knowing what your motivation and intention really is in some detail, we can only guess which kind of answer makes sense. And even then, the mathematicians would be more useful.
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Much better answer than mine. I had just assumed that he was after some natural language, but this encompasses a lot more details.– KKellCommented Apr 19, 2016 at 14:45
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I'm still interpreting this. I don't find this very easy, I will mark it as answered once I can reassure myself I have understood this sufficiently. Sorry about it taking so long Commented May 13, 2016 at 19:58
Not if it is strong enough to express first order arythmetic and states its own consistency. That's Gödel's second incompleteness theorem. I went through the proof once for a class, but here is a quote from wikipedia:
For any formal effectively generated system T including basic arithmetical truths and also certain truths about formal provability, if T includes a statement of its own consistency then T is inconsistent.
I hope that's relevant to your question.
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Oh, my. That's quite something to wrap ones mind around. I need to figure this out before I can give you feedback on the answer. Thanks for answering! :) Commented Apr 19, 2016 at 12:38
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Poking in from Computer Science: bear in mind that most useful formal systems don't state their own consistency. They tend to be about something in particular and have domains of reference that do not include themselves as objects that can be referenced. In particular, they usually let arithmetic intrude from outside, instead of including it as content. Arithmetic itself might well be formally inconsistent, but math, in general, assumes it is not essentially flawed. For most of mathematics, and almost all semantic purposes, it can just be used by, and not contained within the system.– user9166Commented Apr 19, 2016 at 13:20