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

[Nick Thompson]

Lee, I think you got your threads seriously tangled. 

N

Nicholas S. Thompson
Emeritus Professor of Psychology and Biology
Clark University
http://home.earthlink.net/~nickthompson/naturaldesigns/


-----Original Message-----
From: Friam [mailto:friam-bounces at redfish.com] On Behalf Of lrudolph at meganet.net
Sent: Wednesday, January 02, 2019 2:14 PM
To: friam at redfish.com
Subject: Re: [FRIAM] Was: Abduction; Is Now: Dionysian and Apollonian Lives

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,∞}]?



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