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Unified Deductive Systems: An Outline

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Abstract

Our goal is to develop a syntactical apparatus for propositional logics in which the accepted and rejected propositions have the same status and obeying treated in the same way. The suggested approach is based on the ideas of Łukasiewicz used for the classical logic and in addition, it includes the use of multiple conclusion rules. More precisely, a consequence relation is defined on a set of statements of forms “proposition A is accepted” and “proposition A is rejected”, where A is a proposition,—a unified consequence relation. Accordingly, the rules defining a unified consequence relation,—the unified rules, have statements as premises and as conclusions. A special attention is paid to the logics in which each proposition is either accepted or rejected. If we express this property via unified rules and add them to a unified deductive system, such a unified deductive system defines a reversible unified consequence: a statement “proposition B is accepted” is derived from the statement “proposition A is accepted” if and only if a statement “proposition A is rejected” is derived from the statement “proposition B is rejected”.

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Notes

  1. Sometimes logics with rejections are called “hybrid” (e.g. [10]); we prefer to call such logics “unified”, because the term “hybrid logic” is often used in a different sense (e.g. [31]).

  2. More details about different approaches to refutation systems the reader can find in [5, 9, 10, 17, 33, 34].

  3. The similar notations are used in [24] and [29].

  4. Note that we define a protoinference from \(\langle \Gamma ;\textsf{R}\rangle \) without clarifying what we are deriving.

  5. Some of the directions were suggested by anonymous referee, to whom I am grateful.

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Acknowledgements

I am grateful to V. Goranko and A. Muravitsky for fruitful discussions. I am in debt to the anonymous referee for insightful comments and suggestions.

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    Alex Citkin

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Citkin, A. Unified Deductive Systems: An Outline. Log. Univers. 17, 483–509 (2023). https://doi.org/10.1007/s11787-023-00335-x

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