Librationist closures of the paradoxes

Frode Bjørdal

DOI: http://dx.doi.org/10.12775/LLP.2012.016

Abstract


We present a semi-formal foundational theory of sorts, akin to sets, named librationism because of its way of dealing with paradoxes. Its semantics is related to Herzberger’s semi inductive approach, it is negation complete and free variables (noemata) name sorts. Librationism deals with paradoxes in a novel way related to paraconsistent dialetheic approaches, but we think of it as bialethic and parasistent. Classical logical theorems are retained, and none contradicted. Novel inferential principles make recourse to theoremhood and failure of theoremhood. Identity is introduced à la Leibniz-Russell, and librationism is highly non-extensional. Π11-comprehension with ordinary Bar-Induction is accounted for (to be lifted). Power sorts are generally paradoxical, and Cantor’s Theorem is blocked as a camouflaged premise is naturally discarded.


Keywords


Bialethism; Burali-Forti Paradox; Cantor’s Theorem; Curry’s Paradox; Dialetheism; Foundations of Mathematics; Liar’s Paradox; Paraconsistency; Parasistency; Paradoxes; Reverse Mathematics; Russell’s Paradox; Second Order Arithmetic; Semantical paradoxes;

Full Text:

PDF

References


Beeson, Michael, Foundations of Constructive Mathematics, Metamathematical Studies, Springer, Berlin/Heidelberg/New York 1985.

Bjørdal, Frode, “Considerations Contra Cantorianism”, in: The Logica Yearbook 2010, M. Peliš, V. Punčochář (eds.), College Publications, London, 2011.

Bjørdal, Frode, “2+2=4” er misvisande, (“2+2=4” is misleading), pages 55–65 in: Enhet i mangfold, Festskrift i anledning Johan Arnt Myrstads 60-årsdag, Anita Leirfall & Thor Sandmel (eds.), Unipub, Oslo, 2009.

Bjørdal, Frode, “Minimalistic Liberalism”, in: The Logica Yearbook 2005, M.Bílková and O.Tomala (eds.), Filosofia, Prague, 2006.

Bjørdal, Frode, “There are Only Countably Many Objects”, pages 47–58 in: The Logica Yearbook 2004, Libour Behounek & Marta Bilková (eds.) Filosofia, Prague, 2005.

Burgess, John P., Philosophical Logic, Princeton University Press, Princeton and Oxford, 2009.

Cantini, Andrea, Logical Frameworks for Truth and Abstraction, Elsevier, Amsterdam, 1996.

Friedman, Harvey, and Michael Sheard, “An Axiomatic Approach to Self-Referential Truth”, Annals of Pure and Applied Logic 33, (1987): 1–21.

Gupta, Anil, “Truth and Paradox”, Journal of Philosophical Logic XI, 1 (1982): 1–60.

Hajek, Petr, On equality and natural numbers in Cantor-Łukasiewicz set theory (forthcoming).

Herzberger, Hans, Notes on Periodicity, unpublished and circulated, 1980.

Herzberger, Hans, “Notes on Naive Semantics”, Journal of Philosophical Logic XI, 1 (1982): 61–102.

Jensen, Ronald Björn, “The Fine Structure of the Constructible Hierarchy”, Annals of Mathematical Logic 4, (1972): 229–308.

McGee, Vann, “How Truthlike Can a Predicate Be?”, Journal of Philosophical Logic 14, (1985): 399–410.

Myhill, John, Paradoxes, Synthese 60, (1984): 129–143.

Simpson, Stephen G., Subsystems of Second Order Arithmetic, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, Heidelberg, New York, 1999.

Smorynsky, Craig, “The Incompleteness Theorems”, pages 821–865 in: Handbook of Mathematical Logic, Jon Barwise (ed.), Elsevier Science Publishers B.V., Amsterdam, 1977.

Visser, Albert, Semantics and the Liar Paradox, pages 617–706 in: Handbook of Philosophical Logic. Vol. IV, Dov M. Gabbay & Franz Guenthner (eds.), Reidel, Dordrecht, 1989.

Welch, Philip D., “On Revision Operators”, Journal of Symbolic Logic 68, 3 (2003): 689–711.

Welch, Philip D., “On Gupta-Belnap revision theories of truth, Kripkean fixed-points and the next stable set”, Bulletin of Symbolic Logic 7, 3 (2001): 345–360.








Print ISSN: 1425-3305
Online ISSN: 2300-9802

Partnerzy platformy czasopism