The paper’s novelty is in combining two comparatively new fields of research: non-transitive logic and the proof method of correspondence analysis. To be more detailed, in this paper the latter is adapted to Weir’s non-transitive trivalent logic \({\mathbf{NC}}_{\mathbf{3}}\). As a result, for each binary extension of \({\mathbf{NC}}_{\mathbf{3}}\), we present a sound and complete Lemmon-style natural deduction system. Last, but not least, we stress the fact that Avron and his co-authors’ general method of obtaining _n_-sequent proof systems for any _n_-valent logic (...) with deterministic or non-deterministic matrices is not applicable to \({\mathbf{NC}}_{\mathbf{3}}\) and its binary extensions. (shrink)

A new type of formalization of classical first-order logic with equality is introduced on the basis of the sequent calculus. It serves to justify the claim that equality is a logical constant characterised by well-behaved rules satisfying properties usually regarded as essential. The main feature of this approach is the application of sequents built not only from formulae but also from terms. Two variants of sequent calculus are examined, a structural and a logical one. The former is defined in accordance (...) with Dos̆en’s criteria for logical constants. The latter is a standard Gentzen’s sequent calculus and satisfies Hacking’s criteria for logicality, including cut elimination. It is also shown that provided rules are harmonious in the sense advocated by Gratzl and Orlandelli. (shrink)