Abstract
The present paper deals with natural intuitionistic semantics for intuitionistic logic within an intuitionistic metamathematics. We show how strong completeness of full first order logic fails. We then consider a negationless semantics à la Henkin for second order intuitionistic logic. By using the theory of lawless sequences we prove that, for such semantics, strong completeness is restorable. We argue that lawless negationless semantics is a suitable framework for a constructive structuralist interpretation of any second order formalizable theory (classical or intuitionistic, contradictory or not).
Similar content being viewed by others
REFERENCES
Dummett, M. (1977): Elements of Intuitionism. Oxford.
Franchella, M. (1992): The Griss- Brouwer Debate on Negation, CUEM preprint, Univ. di Milano.
Griss, G. F. C. (1946): Negationless Intuitionistic Mathematics, Indag. Math. 8: 675- 681.
Heyting, A. (1976): Intuitionism: an Introduction, North-Holland, Amsterdam.
Leblanc, H. (1975): That Principia Mathematica, first edition, has a predicative interpretation after all, Journal of Philosophical Logic 4: 67- 70.
McCarty, D. C. (1991): Incompleteness in intuitionistic metamathematics, Notre Dame J. of Formal Logic 32(3): 323- 358.
Martino, E. (1988): An intuitionistic notion of hypothetical truth for which strong completeness intuitionistically holds, Teoria viii(2): 131- 144.
Russell, B. (1906): The Theory of Implication, American Journal of Mathematics 28: 159- 202.
De Swart, H. (1976): Another intuitionistic completeness proof, Journal of Symbolic Logic 41: 644- 662.
Troelstra, A. S. and van Dalen, D. (1988): Constructivism in Mathematics, vol. I, vol. II, North-Holland.
Veldman, W. (1976): An intuitionistic completeness theorem for intuitionistic predicate logic, Journal of Symbolic Logic 41: 159- 176.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Martino, E. Negationless intuitionism. Journal of Philosophical Logic 27, 165–177 (1998). https://doi.org/10.1023/A:1004278211254
Issue Date:
DOI: https://doi.org/10.1023/A:1004278211254