Journal of Symbolic Logic 66 (4):1620-1636 (2001)

It is shown that axiomatic extensions of intuitionistic propositional calculus defining univocally new connectives, including those proposed by Gabbay, are strongly complete with respect to valuations in Heyting algebras with additional operations. In all cases, the double negation of such a connective is equivalent to a formula of intuitionistic calculus. Thus, under the excluded third law it collapses to a classical formula, showing that this condition in Gabbay's definition is redundant. Moreover, such connectives can not be interpreted in all Heyting algebras, unless they are already equivalent to a formula of intuitionistic calculus. These facts relativize to connectives over intermediate logics. In particular, the intermediate logic with values in the chain of length n may be "completed" conservatively by adding a single unary connective, so that the expanded system does not allow further axiomatic extensions by new connectives
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DOI 10.2307/2694965
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References found in this work BETA

Intuitionistic Logic Model Theory and Forcing.F. R. Drake - 1971 - Journal of Symbolic Logic 36 (1):166-167.
Bi-Heyting Algebras, Toposes and Modalities.Gonzalo E. Reyes & Houman Zolfaghari - 1996 - Journal of Philosophical Logic 25 (1):25 - 43.
Distributive Lattices.Raymond Balbes & Philip Dwinger - 1977 - Journal of Symbolic Logic 42 (4):587-588.
Finite Limitations on Dummet's LC.Ivo Thomas - 1962 - Notre Dame Journal of Formal Logic 3 (3):170-174.

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Citations of this work BETA

Sentence Connectives in Formal Logic.Lloyd Humberstone - forthcoming - Stanford Encyclopedia of Philosophy.
Implicit Connectives of Algebraizable Logics.Xavier Caicedo - 2004 - Studia Logica 78 (1-2):155 - 170.

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