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jedishrfu

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http://en.m.wikipedia.org/wiki/Fuzzy_logic

The other thought that comes to mind is Goedels theorem where systems of logic can't prove all their propositions that there will always be something that is unprovable.

http://en.m.wikipedia.org/wiki/Gödel's_incompleteness_theorems

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HallsofIvy

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Indeed, rather than "true-false based symbolic logic", much of the recent work in logic has been the other way- "multi-valued logics" and "fuzzy logic" where statement are NOT just "true or false" but may have varying degrees of "trueness".

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Any set of axioms powerful enough to do arithmetic. Proposition logic is an exception, for example.given any set of axioms, there exist a statement that can neither be proved nor disproved from those axioms.

The numerically-based math reduced to symbolic logic sounds kind of like the construction of the real numbers and all that from set theory. It's not really reduced to symbolic logic, but it's reduced to sets.

As far as recent developments go, this came to mind:

http://blogs.scientificamerican.com/guest-blog/2013/10/01/voevodskys-mathematical-revolution/

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HallsofIvy

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Yes, I should have said that.Any set of axioms powerful enough to do arithmetic. Proposition logic is an exception, for example.

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