We give a proof-theoretic as well as a semantic characterization of a logic in the signature with conjunction, disjunction, negation, and the universal and existential quantifiers that we suggest has a certain fundamental status. We present a Fitch-style natural deduction system for the logic that contains only the introduction and elimination rules for the logical constants. From this starting point, if one adds the rule that Fitch called Reiteration, one obtains a proof system for intuitionistic logic in the given signature; if instead of adding Reiteration, one adds the rule of Reductio ad Absurdum, one obtains a proof system for orthologic; by adding both Reiteration and Reductio, one obtains a proof system for classical logic. Arguably neither Reiteration nor Reductio is as intimately related to the meaning of the connectives as the introduction and elimination rules are, so the base logic we identify serves as a more fundamental starting point and common ground between proponents of intuitionistic logic, orthologic, and classical logic. The algebraic semantics for the logic we motivate proof-theoretically is based on bounded lattices equipped with what has been called a weak pseudocomplementation. We show that such lattice expansions are representable using a set together with a reflexive binary relation satisfying a simple first-order condition, which yields an elegant relational semantics for the logic. This builds on our previous study of representations of lattices with negations, which we extend and specialize for several types of negation in addition to weak pseudocomplementation. Finally, we discuss ways of extending these representations to lattices with a conditional or implication operation.

翻译：我们为一种包含合取、析取、否定以及全称和存在量词的签名逻辑提供了证明论和语义刻画，并认为该逻辑具有某种基础性地位。我们为该逻辑提出了一个仅包含逻辑常项引入和消去规则的菲奇式自然演绎系统。以此为起点，若添加菲奇称为“重述”的规则，则可获得给定签名下的直觉主义逻辑证明系统；若以“归谬法”规则替代重述，则可获得正交逻辑证明系统；同时添加重述与归谬法，则能得到经典逻辑证明系统。可以说，重述和归谬法与联结词意义的关联程度均不及引入和消去规则那般直接，因此我们所识别的基础逻辑可作为直觉主义逻辑、正交逻辑与经典逻辑支持者之间更根本的出发点和共同基础。我们从证明论角度论证的该逻辑代数语义基于配备所谓弱伪补的有界格。我们证明此类格扩张可通过一个集合与满足简单一阶条件的自反二元关系加以表示，从而为该逻辑建立优雅的关系语义。这建立在我们先前关于带否定格的表示研究基础上，我们对此进行了扩展并针对弱伪补以外的多种否定类型进行了专门化处理。最后，我们探讨了将这些表示方法扩展至带条件或蕴涵运算的格的途径。