An Algebraic Approach to Intuitionistic Connectives

It is show11 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 negatio~l of such a connective is equivalent to a formula of intnitionistic calculus. Thus, under the excluded third law it collapses to a classical formula. showing that this conditio~l 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 intnitionistic 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. $