Kaj Börge Hansen

Short presentation

This homepage contains information about my education, teaching “career”, research, and publications. There are links to some of the works.

Here is my Curriculum Vitae in English.

Här är min Meritförteckning på svenska.



I was born 1946 on the island of Fyn (Funen) in Denmark and grew up in the village of Korinth. Primary school in Korinth. Secondary school in Svendborg. During the last year in secondary school, I became convinced that research on the foundations of mathematics was my mission in life: (1) What is mathematics? (2) How can mathematical proofs be understood? (3) Where do the fundamental assumptions of mathematics come from? (4) How can the applications of mathematics to physics be understood? I started my university studies at ÅrhusUniversity. To get the possibility of studying logic and the foundations of mathematics, I moved in 1974 to the Department of Philosophy at Uppsala University in Sweden from which university I have most of my post-secondary education; it covers Philosophy, Logic, Mathematics, and Physics. In the mid‑1980s, I realized that it is impossible to understand the foundations of mathematics without understanding the applications of mathematics, especially to physics. This made me broaden my research interests to include also the foundations of physics. In 1996, I earned a Fil Dr/PhD degree in Theoretical Philosophy at Uppsala with a dissertation on the foundations of quantum mechanics; and the same year, I was appointed Docent in Theoretical Philosophy and Logic, also at Uppsala University. I have never had an academic position. Over the years, I have mostly made my living as a teaching assistant (“timlärare”) in Århus in Denmark, in Uppsala, Stockholm, and Luleå in Sweden, in Mariehamn on the Åland Islands in Finland, and in Bahía Blanca in Argentina. Since 1997, I hold a part time research grant,”Burmans Docentstipendium”, at Uppsala University.


Educated at Århus, Uppsala, Fairfax-Baton Rouge, and Mälardalen, I hold seven academic degrees, including an Exam. Art./AA in Philosophy, a Fil Kand/BA in Philosophy and Mathematics, and a Fil Kand/BSc in Mathematics and Physics as well as three Master degrees in Logic (MS), in Mathematics and Physics (Fil Mag/MSc), and in Theoretical Philosophy (Fil Lic/MA).
Fil Dr/PhD (Uppsala, 1996) in Theoretical Philosophy.
Docent/Postdoctoral education/Habilitation (Uppsala, 1996) in Theoretical Philosophy and Logic.


I have taught in the Department of Philosophy, Århus University (1971-1973); Department of Philosophy, Uppsala University (1975‑1993); Department of Computer and System Science, Stockholm University (1986); Department of Information Science, Uppsala University (1986‑2003); Department of Mathematics, Luleå University of Technology (1996‑1997); Departamento de Ciencias e Ingeniería de la Computacíon, Universidad Nacional del Sur, Bahía Blanca (1998); Åland Institute of Technology (1999‑2001). Mostly I have taught as a teaching assistant (“timlärare”).

I have published two textbooks of logic in Swedish (B2, B6). Grundläggande logik (four editions, Lund, 1992-2003) is or has been used in several educations in Sweden and Finland – in computer science, mathematics, and philosophy. The textbook covers the fundamentals of sentential logic and predicate logic, basic set theory, definition theory, and the logical foundations of logic programming. It outlines metalogic up to the completeness theorems and their immediate consequences. The book is to a large extent based on the principle of learning-by-doing in order to develop logical craftsmanship in the reader. All the basic problem types in elementary logic are defined, and methods and heuristic rules for their solution are formulated. The book contains a wealth of exercises most of which have been developed by the author for exams or for the book.

Over the years, I have had c. 1200 students. I have participated in the education of c. 900 programmers and computer technologists. Through my textbooks, I have contributed to the education of another c. 6000 programmers, computer technologists, mathematicians, and philosophers.


My main fields of research are logic, the foundations of mathematics, and the foundations of physics.

In logic, I have published on the foundations of logic (B4, A21), mathematical logic (B4, A6, A9, A10, A19), set theory (A6, A9, A20), and the logical foundations of logic programming (B4, A12, A14).

Though a considerable part of my research efforts has been devoted to the foundations of mathematics, only little has been published so far. Exceptions occur in (B4, A4, A9, A10, A18, A20). Work over more than 40 years has resulted in a new program for the foundations of mathematics which is, however, still unpublished.

My work on the foundations of quantum mechanics can be found in (B1, B3, B5, A1, A2, A3, A8). Here the main publication is my PhD thesis, Logical Physics: Quantum Reality Theory, from 1996. According to the established interpretation of Bell’s theorem, it shows that quantum mechanics is incompatible with the conjunction of realism and locality. At least one of them must be abandoned, it is claimed. I show that that there is an alternative interpretation of the theorem which allows for both locality and realism, and I begin to outline new foundations for quantum mechanics which are both local, realistic, and complete in the sense of Einstein-Podolski, Rosen — in accordance with the guidelines which can be inferred from the new interpretation of Bell’s theorem. This demands a quantum ontology where functions are fundamental rather than space, time, particles, and fields, where reality is operationally defined, and where logic is based on this operational ontology. In Chapter 2 of (B5), “Realism and Causality in Quantum Mechanics” I prove the inconsistency of Bohm’s Quantum Potentiality Theory. In (A11), I use my ideas on the foundations of quantum mechanics to solve the ontological problem of free will, something which cannot be achieved in any other approach to quantum mechanics..

Between 1998 and 2007, I worked mostly on the foundations of relativity theory (B8, A15). The work is still in progress. I show that there are two forms of the Special Theory of Relativity, a weak form, MIN‑STR, and a strong form, MAX-STR. Though they are not equivalent, physicists slip back and forth between them as if they were. MIN-STR is consistent and well verified; and it is sufficient for most of the important results in the Special Theory of Relativity, STR. MAX-STR is inconsistent which can be shown by the Sagnac effect, by the Real Twin Paradox (A15), and in at least seven other ways. The General Theory of Relativity, GTR, implies MAX-STR and is therefore also inconsistent. I formulate a consistent alternative to MAX-STR, Transformation Theory (TT), which is consistent and well verified. It can be extended to a General Transformation Theory (GTT) which replaces GTR. In contrast to MAX-STR and GTR, TT and GTT give consistent theories of space and time and they are superior to STR and GTR in the analysis of rotation. In TT and GTT, space and time are entangled; but the idea of a spacetime so dear to relativity theorists has to be given up.

In a recent, still unpublished, work in progress with the working title “Logical Analysis of the Dark Energy Hypothesis”, I try to show that the “proof” given by astrophysicists of the accelerating expansion of the universe is incomplete and that there, from a physicist’s point of view, are good reasons to believe that though the universe is expanding, the expansion does not accelerate.

Occasionally I have published articles in general philosophy: Metaphilosophy (B4, A18), philosophical method (B4, A18), synthetic apriori (B4), Gödel’s philosophy of mathematics (A4), free will (A11), Wittgenstein’s philosophy (A16, A17, A18), philosophy of mind (A16, A17), philosophy of language (A17, A19, A21), semantics (A17, A19, A21), truth (A7, A19, A21), Tarski’s theorem (A19), Ross’s paradox (A7, A21), the Liar paradox (A7, A19), Grelling’s paradox (A19), reference and existence (A21), ontological commitment (A21).


B1. Ideas on Bell's Theorem. Uppsala Philosophical Studies 41, Uppsala, 1989. (pdf)
B2. Kompendium i logik [Compendium on Logic]. Uppsala Philosophical Studies, Uppsala, 1989.
B3. Logical Physics: Quantum Reality Theory. Library of Theoria, Thales, Stockholm, 1996. (pdf)
B4. Applied Logic. Acta Universitatis Upsaliensis, Uppsala, 1996. (pdf)
B5. Essays 2002: Logic and Philosophy. Department of Philosophy, Uppsala University, Uppsala, 2002. (pdf)
B6. Grundläggande logik [Fundamentals of Logic]. Fourth edition. Studentlitteratur, Lund, 2003.
B7. Grundläggande logik: Lösningsdel [Fundamentals of Logic: Solutions to Exercises]. Studentlitteratur, Lund, 2003.
B8. Transformation Theory: Relativity Revised. Department of Philosophy, Uppsala Univer-sity, Uppsala, 2007. (pdf)

Articles (selection)

A1. "Some Derivations of Bell's Inequality." Danish Yearbook of Philosophy, 1994. See Item B3.
A2. "An Inverse of Bell's Theorem." Journal for General Philosophy of Science, 1995. See Item B3.
A3. “An Analysis of the EPR Argument.” Department of Philosophy, Uppsala University, Uppsala, 1995. See Item B3.
A4. "Gödel's Philosophy of Mathematics." Lychnos, 1997. See Item B5.
A5. "Stig Kanger as an Educator and as a Thinker." In G. Holmström-Hintikka, S. Lindström & R. Sliwinski (eds.), Collected Papers of Stig Kanger, Vol. 2. Springer-Kluwer, Dordrecht 2001. See Item B5.
A6. "Kanger's Ideas on Non-Well-Founded Sets: Some Remarks." In G. Holmström-Hintikka, S. Lindström & R. Sliwinski (eds.), Collected Papers of Stig Kanger, Vol. 2. Springer-Kluwer, Dordrecht 2001. See Item B5.
A7. “Two Paradoxes Revisited.” In E. Carlson and R. Sliwinski (eds.), Omnium-Gatherum. Uppsala Philosophical Studies 50, Uppsala, 2001. See Item B5.
A8. “Interpreting the Quantum World.” Lychnos, 2001. See Item B5.
A9. “A Study of König’s Lemma.” Department of Philosophy, Uppsala University, 2002. (pdf)
A10. “New Definitions of the Recursive Functions.” Department of Philosophy, Uppsala University, 2002. (pdf)
A11. “Free Will and Operationalism.” In K. Segerberg and R. Sliwinski (eds.), A Philosophi-cal Smorgasbord: Essays on Truth, Action, and other Things. Uppsala Philosophical Stud-ies 52, Uppsala, 2003. (pdf)
A12. “Applied Logic: Logical Foundations of Logic Programming.” Baton Rouge, 2004. (pdf)
A13. Articles on “Allais’ paradox”, “Epistemic logic”, “Hintikka, Jaakko”, and “Kanger, Stig” in Thomas Mautner (ed.), Dictionary of Philosophy, second edition. Penguin Books, London, 2005.
A14. “Analysis of Fast Unification: An Exercise in Applied Logic.” In H. Lagerlund, S. Lindström, and R. Sliwinski (eds.), Modality Matters. Uppsala Philosophical Studies, Upp-sala, 2006. (pdf)
A15. “The Real Twin Paradox.” Department of Philosophy, Uppsala University, 2007. (pdf)
A16. “Remarks on the Private Language Argument.” In N. Forsberg, S. Rider, and P. Segerdahl (eds.), Tankar tillägnade Sören Stenlund. Uppsala Philosophical Studies 54, Uppsala, 2008. (pdf)
A17. “Remarks on Wittgenstein's Philosophy: Private Language and Meaning.” Danish Yearbook of Philosophy, Vol. 42, 2007. (pdf)
A18. “Remarks on Wittgenstein's Philosophy: Philosophical Method and Contradictions.” Danish Yearbook of Philosophy, Vol. 43, 2008. (pdf)
A19. “What the Liar Said to Grelling.” In L G. Johansson, J. Österberg, and R. Sliwin-ski (eds.), Logic, Ethics, and All That Jazz. Uppsala Philosophical Studies 57, Uppsala, 2009. (pdf)
A20. “Conceptual Foundations of Operational Set Theory.” Department of Philosophy, Upp-sala University, Uppsala, 2010. Forthcoming in Danish Yearbook of Philosophy. (pdf)
A21. “Formal Logic, Models, Reality.” In R. Sliwinski and F. Svensson (eds.), Neither/Nor. Uppsala Philosophical Studies 58, Uppsala, 2011. (pdf)

Kaj Börge Hansen