[FRIAM] Was: Abduction; Is Now: Dionysian and Apollonian Lives

[lrudolph at meganet.net]

I'm not sure what you're buying with your move to "continuous" rather than
(merely) "infinite-valued".  I mean, though your discretized values {0..n}
are integers, they are (in my small experience of many-valued logics,
which does not include any actually *working* with them as logics) merely
nominal labels--the order, and the arithmetic for that matter, are
irrelevant semantically: the flavors 1, 2, 3 of not-true aren't such that
2 is more not-true than 1 but less not-true than 3, and certainly aren't
such that 2 is exactly half-way from 1 to 3 in not-trueness.

And, from another point of view, contrary to most people's "intuition" (as
formed by what turns out to be bad pedagogy, not anything in the
foundation of either physics or mathematics), "continuity" doesn't require
infinitude.  Way back in the early 1960 a couple of mathematicians
independently (Bob Stong was one of them, I forget the other) noticed that
all the algebraic topology that can be done with (finite) "simplicial
complexes" (e.g., polyhedra) in Euclidean space  (so, in particular, all
the algebraic topology of compact differentiable manifolds) can be
faithfully rephrased in terms of *finite* topological spaces (I mean,
literally finite: only finitely many points, where in particular a
one-element set does not have to be closed), if you don't insist that the
topology be Hausdorff (but do impose one very weak "separation property"
which I'm currently blanking on).  Much more recently, a pair of
Argentinians, J. Barmak & E. Minian, have published a series of papers
(all available at the arXiv) extending and clarifying that.  Logics with
*that* kind of a continuum of values has, I think, already be done (the
finite topological spaces in question can be reinterpreted as finite
posets / finite lattices / etc., and at least "lattice-valued logics" has
a familiar sound to me; but, again, I'm blanking on any details).


> Since one of my dead horses is artificial discretization, I've always
> wondered what it's like to work in many-valued logics.  So, proof by
> contradiction would change from [not-true => false] to [not-0 =>
> {1,2,..,n}], assuming a discretized set of values {0..n}.  But is there a
> continuous "many valued" logic, where any proposition can be evaluated to
> take on some sub-region of a continuous set?  So, proof by contradiction
> would become something like [not∈{-∞,0} => ∈{0+ε,∞}]?