 One snag for such a suggestion is that examples can be found involving two connectives with associated sets of rules each of which sets of rules yields a conservative extension of an initial proof system, apparently requiring the answer “yes, there does exist such a connective” in each case from the perspective of that original system, while extending the initial proof system with both sets of rules at once is non-conservative—undermining this double endorsement.[21] We may conclude that while yielding a non-conservative extension of a favoured proof system is sufficient to denying the existence or intelligibility of a connective with such-and-such logical properties, there may be other grounds for returning such a verdict. Boolean negation comes under just such a critique in Chapter 5 of Priest (2006), where such problems for Belnap’s account are raised and references to the technical literature establishing the conservativity of extending certain logics in the relevance tradition with suitable principles governing this connection can also be found. (For an earlier presentation of this point, see also Belnap and Dunn 1981, p. 344, and for a related discussion, Restall 1993, esp. §5.) It should be noted also that the conservativity test raises issues of its own, independently of the question of whether passing it is necessary and sufficient (and what we have been querying is the sufficiency half of this package) for acknowledging the existence of a connective. For example, there are sentential connectives which conservatively extend intuitionistic propositional logic in any of its conventional presentations as a proof system, but non-conservatively extend intuitionistic predicate logic. The best known is dual intuitionistic implication (see López-Escobar  1985).[22]  Returning to the conservative extension criterion of existence, let us note that a related connective, dual intuitionistic negation, can be added conservatively to intuitionistic predicate (and a fortiori propositional logic) but already at the propositional level produces violations of the disjunction property—that ≻ φ ∨ ψ should only be provable if at least one of ≻ φ, ≻ ψ, is provable—which on some accounts (e.g., Gabbay 1977 and 1981) renders it illegitimate from an intuitionistic perspective. Finally, it should be observed that an adherent of intuitionistic logic might be expected to take a dim view of a proposed new connective with rules conservatively extending that logic but not conserving synonymies, in the sense that formulas in the original language which are synonymous according to ⊢I L are no longer so according to the consequence relation associated with the extended proof system. This happens in at least the most straightforward extension of intuitionistic logic by what is called strong  negation.[23]