On Some Extensions Of Intuitionistic Logic

We prove that many extensions of Intuitionistic Sentential Calculus ISC with new intuitionistic connectives that are known to be conservative extensions of ISC are not conservative extensions of Intuitionistic Predicate Calculus because formulas such as Kuroda's are derivable. We thus solve a problem posed by López-Escobar in 1985 and answer a question posed by Humberstone in 2001 regarding a connective called the strongest anticipator.411-21722Caicedo, X., Cignoli, R., An algebraic approach to intuitionistic connectives (2001) The Journal of Symbolic Logic, 66, pp. 1620-1636Ertola Biraben, R.C., San Martín, H.J., On some compatible operations on heyting algebras (2011) Studia Logica, 98, pp. 331-345Esakia, L., Quantification in intuitionistic logic with provability smack (1998) Bulletin of the Section of Logic, 27, pp. 26-28Esakia, L., Scattered toposes (2000) Annals of Pure and Applied Logic, 103, pp. 97-107Esakia, L., The modalized heyting calculus: A conservative modal extension of the intuitionistic logic (2006) Journal of Applied Non-Classical Logics, 16, pp. 349-366Gabbay, D.M., On some new intuitionistic propositional connectives. I (1977) Studia Logica, 36, pp. 127-139Gabbay, D.M., (1981) Semantical Investigations in Heyting's Intuitionistic Logic, , Reidel Publishing Company, DordrechtHumberstone, L., The pleasures of anticipation: Enriching intuitionistic logic (2001) Journal of Philosophical Logic, 30, pp. 395-438Humberstone, L., (2011) The Connectives, , MIT Press, Cambridge, MassKuznetsov, A., On the propositional calculus of intuitionistic provability (1985) Soviet Math. Dokl., 32, pp. 18-21López-Escobar, E.G.K., On intuitionistic sentential connectives I (1985) Revista Colombiana de Matemáticas, 19, pp. 117-130Smetanich, Y., On the completeness of a propositional calculus with a supplementary operation in one variable (1960) Tr. Mosk. Mat. Obsch., 9, pp. 357-371. , (in Russian)(1962) MR, 24, pp. A680. , reviewedSmetanich, Y., On propositional calculi with an additional connective (1961) Soviet Math. Doklady, 139, pp. 309-312. , (in Russian)(1963) MR, 26. , reviewedTroelstra, A., Van Dalen, D., (1988) Constructivism in Mathematics. An Introduction, 1. , North-Holland, AmsterdamYashin, A.D., New solutions for novikov's problem for intuitionistic connectives (1998) Journal of Logic and Computation, 8, pp. 637-66