Suszko问题：混合结果与组合性
Suszko's Problem: Mixed Consequence and Compositionality
摘要：
Suszko问题是寻找语义上表征一个句法结果关系所需的最小数量的真假值的问题。Suszko证明了每一个塔斯基推论关系都可以只用两个真假值来刻画。马林诺夫斯基表明，如果放松塔斯基的一些结构约束，这个数字可以等于3。通过这样做，马林诺夫斯基引入了一个所谓的混合后果的情况，允许指定值的概念在论点的前提和结论之间变化。在本文中，我们给出了一个更系统的角度来看Suszko的问题和混合后果。首先，我们证明了一般表示定理的结构性质的后果关系，其语义解释，揭示了语义对应的替代不变，并建立（相交）混合后果是从根本上的语义对应的结构性质的单调性。我们使用这些导出最大秩的结果证明最近在不同的设置由法国和里普利，以及布拉西奥，马科斯和万辛，逻辑与各种结构属性（自反性，传递性，没有，或两者兼而有之）。我们将这些结果加强为非渗透逻辑的精确秩结果（粗略地说，就是区分前提和结论的作用的结果）。我们讨论的基本概念的排名，以及相关的减少斯科特和Suszko独立提出的。正如Suszko所强调的那样，这种归约在一般情况下不能保持组合性，这意味着所得到的语义不再是真值函数的。我们提出了一个修改的概念，减少，使我们能够证明，在紧凑的逻辑与我们所谓的定期连接词，排名结果保持，即使我们要求保存的真理功能和额外的语义属性。
英文摘要：
Suszko's problem is the problem of finding the minimal number of truth values needed to semantically characterize a syntactic consequence relation. Suszko proved that every Tarskian consequence relation can be characterized using only two truth values. Malinowski showed that this number can equal three if some of Tarski's structural constraints are relaxed. By so doing, Malinowski introduced a case of so-called mixed consequence, allowing the notion of a designated value to vary between the premises and the conclusions of an argument. In this paper we give a more systematic perspective on Suszko's problem and on mixed consequence. First, we prove general representation theorems relating structural properties of a consequence relation to their semantic interpretation, uncovering the semantic counterpart of substitution-invariance, and establishing that (intersective) mixed consequence is fundamentally the semantic counterpart of the structural property of monotonicity. We use those to derive maximum-rank results proved recently in a different setting by French and Ripley, as well as by Blasio, Marcos and Wansing, for logics with various structural properties (reflexivity, transitivity, none, or both). We strengthen these results into exact rank results for non-permeable logics (roughly, those which distinguish the role of premises and conclusions). We discuss the underlying notion of rank, and the associated reduction proposed independently by Scott and Suszko. As emphasized by Suszko, that reduction fails to preserve compositionality in general, meaning that the resulting semantics is no longer truth-functional. We propose a modification of that notion of reduction, allowing us to prove that over compact logics with what we call regular connectives, rank results are maintained even if we request the preservation of truth-functionality and additional semantic properties.
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