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09030026

We know the mathematical laws of quantum mechanics, but as yet we are not so sure why those laws should be inevitable. In the simpler but related environment of classical inference, we also know the laws (of probability). With better understanding of quantum mechanics as the eventual goal, Kevin Knuth and I have been probing the foundations of inference. The world we wish to infer is a partially-ordered set ('poset') of states, which may as often supposed be exclusive, but need not be (e.g. A might be a requirement for B). In inference, a state of mind about the world degrades from perfect knowledge through logical OR, which allows for uncertain alternatives. We don't need AND, and we don't need NOT; we just need OR. This display of acceptable states of mind is [close to] a mathematical 'lattice'. We find that the OR structure by itself (!) forces the ordinary rules of probability calculus. No other rules are compatible with the structure of a lattice, so the ordinary rules are inevitable. The standard Shannon information/entropy is likewise inevitable. Taking this idea further, the OR of states of mind gives a lattice of 'Questions' that might be useful for automated learning. Disconcertingly, this lattice is very much larger (in class aleph-2), and the natural valuations on it exhibit large range. I will present this extension, and ask whether we can rationally foresee its use in practical application.

©2012 Perimeter Institute for Theoretical Physics