### Natural Deduction for Four-Valued both Regular and Monotonic Logics

Yaroslav Petrukhin

DOI: http://dx.doi.org/10.12775/LLP.2017.001

#### Abstract

The development of recursion theory motivated Kleene to create regular three-valued logics. Remove it taking his inspiration from the computer science, Fitting later continued to investigate regular three-valued logics and defined them as monotonic ones. Afterwards, Komendantskaya proved that there are four regular three-valued logics and in the three-valued case the set of regular logics coincides with the set of monotonic logics. Next, Tomova showed that in the four-valued case regularity and monotonicity do not coincide. She counted that there are 6400 four-valued regular logics, but only six of them are monotonic. The purpose of this paper is to create natural deduction systems for them. We also describe some functional properties of these logics.

#### Keywords

natural deduction; four-valued logic; regular logic; monotonic logic; Kleene’s logics; Belnap–Dunn’s logic

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