4 ms·

Wow. Replying to myself. What is brought into question is the status of Proof by Contradiction, not the status of the Law of Noncontradiction. Russell seems to

Wow. Replying to myself. What is brought into question is the status of Proof by Contradiction, not the status of the Law of Noncontradiction. Russell seems to have confused these two things. And I don't blame him. It was 1940 after all, and there was a war on. Another interesting fact. The book is built up from from accumulated material finally delivered over a course of lectures at Harvard University in a series called the William James Lectures in 1940 when England was at war but the USA was not.

noreasonw 11y ago
Just reading your comment I was thinking about the closed world assumption in logic (prolog), perhaps there are some middle concept between the Proof by Contradiction and the closed world assumption. In the real world we learn concepts (think about relativity or a non planar world) and then the old question is no longer a two value alternative. Perhaps something like: In this field a new conception could appear that would invalidate what we consider today to be only two alternatives. Is not about many worlds reality or quantum computation, (3 minutes thinking time out).
- igravious 11y ago
  This is out of my depth but the work of the physicists David Bohm and Basil Hiley (an article with him was recently linked to HN) seems to point in this direction? See, for instance, http://www.amazon.com/The-Undivided-Universe-Ontological-Interpretation/dp/0415065887 http://www.amazon.com/The-Undivided-Universe-Ontological-Int... But like I say, out of my depth! :)
- gallabytes 11y ago
  We have something between the two in HoTT - universes of types is stratified by homotopy levels, corresponding to how many dimensions of structure a type has. A space with only points is thus a 0-type, a space with at most 1 point is a -1-type, and a space with only one is a -2-type. The catch is that univalence is inconsistent with LEM at h-levels greater than -1, but assuming it is perfectly consistent for -1 types, which can be thought of as the "at most true" propositions of classical logic.

noreasonw 11y ago

Just reading your comment I was thinking about the closed world assumption in logic (prolog), perhaps there are some middle concept between the Proof by Contradiction and the closed world assumption. In the real world we learn concepts (think about relativity or a non planar world) and then the old question is no longer a two value alternative. Perhaps something like: In this field a new conception could appear that would invalidate what we consider today to be only two alternatives. Is not about many worlds reality or quantum computation, (3 minutes thinking time out).

igravious 11y ago
This is out of my depth but the work of the physicists David Bohm and Basil Hiley (an article with him was recently linked to HN) seems to point in this direction? See, for instance, http://www.amazon.com/The-Undivided-Universe-Ontological-Interpretation/dp/0415065887 http://www.amazon.com/The-Undivided-Universe-Ontological-Int... But like I say, out of my depth! :)
gallabytes 11y ago
We have something between the two in HoTT - universes of types is stratified by homotopy levels, corresponding to how many dimensions of structure a type has. A space with only points is thus a 0-type, a space with at most 1 point is a -1-type, and a space with only one is a -2-type. The catch is that univalence is inconsistent with LEM at h-levels greater than -1, but assuming it is perfectly consistent for -1 types, which can be thought of as the "at most true" propositions of classical logic.