Librationist Closures of the Paradoxes

• Authors: Frode Bjørdal
• Languages of publication: EN

Abstracts

EN

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

EN

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; Set Theoretic Paradoxes; Set Theory; Theory of Truth

• Publisher: Uniwersytet Mikołaja Kopernika (Nicolaus Copernicus University in Toruń)
• Journal: Logic and Logical Philosophy
• Year: 2012
• Volume: 21
• Issue: 4
• Pages: 323-361

Dates

published

2012-12-01

online

2013-07-02

Contributors

author

Frode Bjørdal

• Department of Philosophy Classics, History of Art and Ideas The University of Oslo P.O. Box 1020 Blindern 0315 Oslo, Norway http://www.hf.uio.no/ifikk/personer/vit/fbjordal/index.html

References

• [1] Beeson, Michael, Foundations of Constructive Mathematics, Metamathe- matical Studies, Springer, Berlin/Heidelberg/New York 1985.
• [2] Bjørdal, Frode, “Considerations Contra Cantorianism”, in: The LogicaYearbook 2010, M. Peliš, V. Punčochař (eds.), College Publications, Lon- don, 2011.
• [3] Bjørdal, Frode, “2+2=4” er misvisande, (“2+2=4” is misleading), pages 55-65 in: Enhet i mangfold, Festskrift i anledning Johan Arnt Myrstads60-årsdag, Anita Leirfall & Thor Sandmel (eds.), Unipub, Oslo, 2009.
• [4] Bjørdal, Frode, “Minimalistic Liberalism”, in: The Logica Yearbook 2005, M.Bilkova and O.Tomala (eds.), Filosofia, Prague, 2006.
• [5] Bjørdal, Frode, “There are Only Countably Many Objects”, pages 47-58 in: The Logica Yearbook 2004, Libour Behounek & Marta Bilkova (eds.) Filosofia, Prague, 2005.
• [6] Burgess, John P., Philosophical Logic, Princeton University Press, Prince- ton and Oxford, 2009.
• [7] Cantini, Andrea, Logical Frameworks for Truth and Abstraction, Elsevier, Amsterdam, 1996.
• [8] Friedman, Harvey, and Michael Sheard, “An Axiomatic Approach to Self- Referential Truth”, Annals of Pure and Applied Logic 33, (1987): 1-21.
• [9] Gupta, Anil, “Truth and Paradox”, Journal of Philosophical Logic XI, 1 (1982): 1-60.
• [10] Hajek, Petr, On equality and natural numbers in Cantor-Łukasiewicz settheory (forthcoming).
• [11] Herzberger, Hans, Notes on Periodicity, unpublished and circulated, 1980.
• [12] Herzberger, Hans, “Notes on Naive Semantics”, Journal of PhilosophicalLogic XI, 1 (1982): 61-102.
• [13] Jensen, Ronald Bjorn, “The Fine Structure of the Constructible Hierar- chy”, Annals of Mathematical Logic 4, (1972): 229-308.
• [14] McGee, Vann, “How Truthlike Can a Predicate Be?”, Journal of PhilosophicalLogic 14, (1985): 399-410.
• [15] Myhill, John, Paradoxes, Synthese 60, (1984): 129-143.
• [16] Simpson, Stephen G., Subsystems of Second Order Arithmetic, Perspec- tives in Mathematical Logic, Springer-Verlag, Berlin, Heidelberg, New York, 1999.
• [17] Smorynsky, Craig, “The Incompleteness Theorems”, pages 821-865 in: Handbook of Mathematical Logic, Jon Barwise (ed.), Elsevier Science Pub- lishers B.V., Amsterdam, 1977.
• [18] Visser, Albert, Semantics and the Liar Paradox, pages 617-706 in: Hand- book of Philosophical Logic. Vol. IV, Dov M. Gabbay & Franz Guenthner (eds.), Reidel, Dordrecht, 1989.
• [19] Welch, Philip D., “On Revision Operators”, Journal of Symbolic Logic 68, 3 (2003): 689-711.
• [20] 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.

Identifiers

DOI

10.12775/LLP.2012.016