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Generalized Correspondence Analysis for Three-Valued Logics

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Abstract

Correspondence analysis is Kooi and Tamminga’s universal approach which generates in one go sound and complete natural deduction systems with independent inference rules for tabular extensions of many-valued functionally incomplete logics. Originally, this method was applied to Asenjo–Priest’s paraconsistent logic of paradox LP. As a result, one has natural deduction systems for all the logics obtainable from the basic three-valued connectives of LP (which is built in the \( \{\vee ,\wedge ,\lnot \} \)-language) by the addition of unary and binary connectives. Tamminga has also applied this technique to the paracomplete analogue of LP, strong Kleene logic \( \mathbf K_3 \). In this paper, we generalize these results for the negative fragments of LP and \( \mathbf K_3 \), respectively. Thus, the method of correspondence analysis works for the logics which have the same negations as LP or \( \mathbf K_3 \), but either have different conjunctions or disjunctions or even don’t have them as well at all. Besides, we show that correspondence analyses for the negative fragments of \( \mathbf K_3 \) and LP, respectively, are also suitable without any changes for the negative fragments of Heyting’s logic \( \mathbf G_3 \) and its dual \( \mathbf DG_3 \) (which have different interpretations of negation than \( \mathbf K_3 \) and LP).

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Acknowledgements

The author’s special thanks go to Vasilyi Shangin for valuable remarks regarding the earlier version of this paper. The author also expresses his gratitude to the organizers of Vasiliev Logic Prize as well as the organizers of Universal Logic Prize.

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  1. Department of Logic, Faculty of Philosophy, Lomonosov Moscow State University, Lomonosovsky prospekt 27-4, Moscow, Russia, 119991

    Yaroslav Petrukhin

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Correspondence to Yaroslav Petrukhin.

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Presented at Unilog’18, Vichy, France as the winner of the first Vasiliev Logic Prize (Russia) and a runner-up for the Universal Logic Prize 2018.

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Petrukhin, Y. Generalized Correspondence Analysis for Three-Valued Logics. Log. Univers. 12, 423–460 (2018). https://doi.org/10.1007/s11787-018-0212-9

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