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  1. 2 Logic and Logical Philosophy

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  1. 2 Petrukhin Y.

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  1. 1 2018

  2. 1 2017

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Natural Deduction for Three-Valued Regular Logics

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Petrukhin Y.
Logic and Logical Philosophy
|
2017
|
vol. 26
|
issue 2
197–206
EN
In this paper, I consider a family of three-valued regular logics: the well-known strong and weak S.C. Kleene’s logics and two intermediate logics, where one was discovered by M. Fitting and the other one by E. Komendantskaya. All these systems were originally presented in the semantical way and based on the theory of recursion. However, the proof theory of them still is not fully developed. Thus, natural deduction systems are built only for strong Kleene’s logic both with one (A. Urquhart, G. Priest, A. Tamminga) and two designated values (G. Priest, B. Kooi, A. Tamminga). The purpose of this paper is to provide natural deduction systems for weak and intermediate regular logics both with one and two designated values.
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Natural Deduction for Four-Valued both Regular and Monotonic Logics

75%
Petrukhin Y.
Logic and Logical Philosophy
|
2018
|
vol. 27
|
issue 1
53–66
EN
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.
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