Non-classical logics is a (rather ample) set of logics. To-date the discussion runs as whether they are complementary or alternative to formal classical logic. NCLs shed fresh lights to nature, society and the world.
Susan Haack wrote an excellent book on this matter from the point of view of a logician and philosopher of logic, but it took me some time to appreciate it as I had been using fuzzy logic more and more for applications. Fuzzy logic, however, is one of the few non-classical logics that lends itself so readily to applications in ways many-valued logics do not (in general) and modal or tense logics certainly don't. One the other hand, quantum-logics are fascinating because they are not motivated by something like the sorites paradox or expanding beyond a the binary (Boolean algebraic) truth assignment of classical logic in order to evaluate (or perhaps not evaluate) truth values of WFFs that resist such assignments. They are adaptions of the statistical structure of quantum mechanics that, as a physical theory rather than the application of statistical mechanics due to complexity, imprecision, unknowns, etc., and as such require an ontological treatment and a suitable formal language for philosophers of science (among others). Yet such logics allow for "tertium datur" and/or ~~(A & ~A) when ~( A & ~A) as well as (A v ~ A) are cornerstones to logic and the philosophy of logic since Aristotle. Also, as modern physics has required a re-assessment of theories of probabilities, even possible world semantics and modal logics have, like counterfactuals, provided foundations for causality which both general relativity and quantum mechanics challenged independent of one another and more so in the ways in which they might be combined.
In the same way any generalization of a concept does. With each generalization step, we can create richer formal models of the reality and thus come closer to it. However they still remain idealized representations of reality. Thus far, non-classical logic (including here modal logic) captured more features of the truth and possibility than classical logic did and focused on representing formally truth and reference mainly. Still, it failed thus far to provide an acceptable account of the natural-language semantics, for instance, and language is tightly related to knowledge. Overall, classical and non-classical logic remained to serve the mathematics rather than making big steps in representing the physical world.
Two basic reasons have led to the origins of NCLs, thus: as an alternativa of the fact thta formalclassical logic (FCL) was sometimes too loose, or else because FCL was too rigorous and strict. Whence the new various modes of notation, quantification, formalization of both natural and artificial languages, among other salient features.
I believe we may need to go beyond classical logic and/or side step it in order to explain our universe. Our quantum world and spacetime continuum for example use variations of space and logic that use complex numbers involving square roots of negative numbers. These complex numbered versions of Boolean logic shadow on to classical Boolean outcomes as observations are made or questions are asked of the universe. In such systems truth values of whether Schrodiger's cat is alive or event A precedes or follows event B may be relative to different observers perspective, or how the question is asked; or the Boolean value of past events may be decided at a latter time. I have suggested an Impossible Worlds variant of Modal Logics possible worlds math/semantics may need to be invented to cope with such situations.
I completely agree with you dear Graham. Quantum logics points to that direction, and I've working on a step somehow beyond the original settings by birkhoff and von Neumann. Sound very interesting what you say about Impossible Worlds. Could you say something else, please. My own approach to the subject would be (is) via cohomology.
Here are some references in relation to impossible worlds that might be helpful:
R. Stalnaker, Ways a world may be, Oxford, New York 2003 in particular p.58. 63,
D. Lewis, Phil. Papers , Oxford ,New York, 1983, in particular Vol I p. 21, Vol. II p. 14 – 18
Excerpts of these places plus more references and commetaries in:
http://philosophy-science-humanities-controversies.com/search.php?suche3=unm%F6gliche%20welt
This lexicon is in German with automatic translations.
I found this a good introduction into fuzzy logic:
Markus Spies Unsicheres Wissen Heidelberg 1993 (German).
For fuzzy logic in relation to disjunction, belief degree, uncertainty and other concepts:
http://philosophy-science-humanities-controversies.com/search.php?suche3=Spies
Thank you Martin. Radicalizing the point, I think studying and explaining reality is important, but it's not enough. Moreover, it's rather limited. We can and should be concerned also (and I claim), mainly) with the possible - wherein, yes, the impossible is a modality. Vis-a-vis NCLs, this is the most remarkable contrast with formal classical logic.
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