Abstract
For an arbitrary fixed element $\beta$ in $\{1; 2; 3; ...; \omega\}$ both a sequent calculus and a natural deduction calculus which axiomatise simple paracomplete logic $I_{2;\beta}$ are built. Additionally, a valuation semantic which is adequate to logic $I_{2;\beta}$ is constructed. For an arbitrary fixed element $\gamma$ in $\{1; 2; 3;...\}$ a cortege semantic which is adequate to logic $I_{2;\gamma}$ is described. A number of results obtainable with the axiomatisations and semantics in question are formulated.
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