论文标题
非取消逻辑，悖论和乘法量词
Non-contractive logics, Paradoxes, and Multiplicative Quantifiers
论文作者
论文摘要
该论文从证据理论的角度研究了各种非缩合逻辑系统规避逻辑和语义悖论。直到最近，此类系统仅显示添加剂量词（Grišin，Cantini）。在2010年代（Zardini）也提出了具有乘法量化器的系统，但事实证明它们与真理或理解的天真规则不一致。我们首先使用添加剂量词提出了一个针对失调真理的一阶系统，然后将其与GrišinSet理论进行了比较。然后，我们分析影响乘法量化量的不一致现象背后的原因：将仿射逻辑中的指数解释为空置量词后，我们展示了如何在具有多重量化器的系统的无真实碎片中模拟这种逻辑。最后，我们证明这些乘法量化词的逻辑（但没有失调的真理）是一致的，它表明可以消除剪切规则的无限版本。这为句法方法铺平了道理，以无限顺序为无限逻辑的证明理论。
The paper investigates from a proof-theoretic perspective various non-contractive logical systems circumventing logical and semantic paradoxes. Until recently, such systems only displayed additive quantifiers (Grišin, Cantini). Systems with multiplicative quantifers have also been proposed in the 2010s (Zardini), but they turned out to be inconsistent with the naive rules for truth or comprehension. We start by presenting a first-order system for disquotational truth with additive quantifiers and we compare it with Grišin set theory. We then analyze the reasons behind the inconsistency phenomenon affecting multiplicative quantifers: after interpreting the exponentials in affine logic as vacuous quantifiers, we show how such a logic can be simulated within a truth-free fragment of a system with multiplicative quantifiers. Finally, we prove that the logic of these multiplicative quantifiers (but without disquotational truth) is consistent, by showing that an infinitary version of the cut rule can be eliminated. This paves the way to a syntactic approach to the proof theory of infinitary logic with infinite sequents.
