A note on Grigoriev and Zaitsev’s system CNL²₄

The present article examines a system of four-valued logic recently introduced by Oleg Grigoriev and Dmitry Zaitsev. In particular, besides other interesting results, we will clarify the connection of this system to related systems developed by Paul Ruet and Norihiro Kamide. By doing so, we discuss two philosophical problems that arise from making such connections quite explicit: first, there is an issue with how to make intelligible the meaning of the connectives and the nature of the truth values involved in the many-valued setting employed — what we have called ‘the Haackian theme’. We argue that this can be done in a satisfactory way, when seen according to the classicist’s light. Second, and related to the first problem, there is a complication arising from the fact that the proof system advanced may be made sense of by advancing at least four such different and incompatible readings — a sharpening of the so-called ‘Carnap problem’. We make explicit how the problems connect with each other precisely and argue that what results is a kind of underdetermination by the deductive apparatus for the system.