Beall and Restall’s Logical Pluralism (2006) characterises pluralism about logical consequence in terms of the different ways cases can be selected in the analysis of logical consequence as preservation of truth over a class of cases. This is not the only way to understand or to motivate pluralism about logical consequence. Here, I will examine pluralism about logical consequence in terms of different standards of proof. We will focus on sequent derivations for classical logic, imposing two different restrictions on classical derivations to produce derivations for intuitionistic logic and for dual intuitionistic logic. The result is another way to understand the manner in which we can have different consequence relations in the same language. Furthermore, the proof-theoretic perspective gives us a different explanation of how the one concept of negation can have three different truth conditions, those in classical, intuitionistic and dual-intuitionistic models.
KeywordsLogical Consequence Classical Logic Intuitionist Logic Kripke Model Proof Theory
Thanks to audiences at the “Logical Pluralism” workshop in Tartu, the Logic Seminar at the University of Melbourne, and the Logic or Logics Workshop of the Foundations of Logical Consequence Project at Arché at the University of St Andrews for comments on this paper. I was supported by the ARC Discovery Grants DP0556827 and DP1094962, and Tonio K’s Life in the Foodchain.
- Beall, J. C., & Restall, G. (2006). Logical pluralism. Oxford University Press, Oxford.Google Scholar
- Bishop, E. (1967). Foundations of constructive analysis. New York: McGraw-Hill. Out of print. A revised and extended version of this volume has appeared (Bishop and, Bridges 1985).Google Scholar
- Bishop, E., & Bridges, D. (1985). Constructive analysis. Springer, Berlin.Google Scholar
- Bridges, D. S., & Richman, F. (1987). Varieties of constructive mathematics, volume 97 of London Mathematical Society Lecture Notes. Cambridge University Press, Cambridge.Google Scholar
- Restall, G. (1996). Truthmakers, entailment and necessity. Australasian Journal of Philosophy, 74, 331–340.Google Scholar
- Restall, G. (1997). Ways things can’t be. Notre Dame Journal of Formal Logic, 38(4), 583–596.Google Scholar
- Restall, G. (1999). Negation in relevant logics: How I stopped worrying and learned to love the Routley star. In D. Gabbay, & H. Wansing (Eds.), What is negation?, volume 13 of Applied Logic Series (pp. 53–76). Dordrecht: Kluwer Academic Publishers.Google Scholar
- Restall, G. (2001). Constructive logic, truth and warranted assertibility. Philosophical Quarterly, 51, 474–483. http://consequently.org/writing/conlogtr/.
- Restall, G. (2002). Carnap’s tolerance, meaning and logical pluralism. The Journal of Philosophy, 99, 426–443. http://consequently.org/writing/carnap/.
- Restall, G. (2005). Łukasiewicz, supervaluations and the future. Logic and Philosophy of Science, 3(1):1–10.Google Scholar
- Richman, F. (1990). Intuitionism as generalization. Philosophia Mathematica, 5, 124–128.Google Scholar
- Richman, F. (1996). Interview with a constructive mathematician. Modern Logic, 6, 247–271.Google Scholar