An algebraic approach to intuitionistic connectives

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

What is a Relevant Connective?Shawn Standefer - 2022 - Journal of Philosophical Logic 51 (4):919-950.
Implicit connectives of algebraizable logics.Xavier Caicedo - 2004 - Studia Logica 78 (1-2):155 - 170.
Implicit connectives of algebraizable logics.Xavier Caicedo - 2004 - Studia Logica 78 (1-2):155-170.

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