Logicians and philosophers have had a good 120 years to get used to the idea that not every condition defines a set. One popular coping strategy is to maintain that each instantiated condition does at least determine a ‘plurality’ (i.e., one or more items). This is to say that friends of traditional plural logic accept—often as a trivial or evident or logical truth—each instance of plural comprehension: Unless nothing is φ, some things include everything that is φ, and nothing else. Set-theoretic paradoxes are avoided by recognizing a type distinction between singular quantifiers (‘something’) and plural ones (‘some things’).This book defends a heterodox version of plural logic. Salvatore Florio and Øystein Linnebo advocate a set theory based on a ‘critical plural logic’ that refutes many instances of plural comprehension. In particular, they deny that there are one or more things that include everything. Instead, they argue, when it comes to resolving the paradoxes, a ‘package deal’ that restricts plural comprehension to ‘extensionally definite’ conditions is more attractive than its competitors that either limit the range of our quantifiers (‘generality relativism’) or constrain ‘singularization.’ Florio and Linnebo’s rejection of traditional plural logic permits them to combine two otherwise incompatible views: (i) ‘the set of’ operation is a universal singularization, so that it injectively maps each plurality to an object (namely, its set), and (ii) the domain of ‘everything’ may contain absolutely everything, so that it cannot be surpassed by singularization.The argument for adopting critical plural logic in preference to traditional plural logic comes in the fourth and final part of the book. The first three parts make the authors’ case for taking plural resources seriously in the first place. Part I reappraises the debate between pluralism, ‘which takes plural resources at face value’ (2), and singularism, which takes the opposite view. Part II compares ‘four different ways to talk about many objects simultaneously’ (119), including second-order quantification, and the use of ‘individual sums’, in addition to sets and pluralities. Part III focuses on philosophical applications of plural logic. Along the way, the book tackles many other topics of interest, including whether plural logic counts as ‘pure logic’ (168), or carries distinctive ontological commitments (chap. 8), how plural resources interact with modality (chap. 10), and whether the pluralization operation can be iterated to obtain superplural terms the denote ‘pluralities of pluralities’ (180) (chap. 9).The Many and the One covers an impressive amount of difficult territory in an admirably clear and engaging way. Florio and Linnebo offer a fresh perspective on the pluralism debate and defend a novel response to the paradoxes. The driving force behind their arguments is usually logic, broadly construed, rather than linguistics or the philosophy of language. But Florio and Linnebo have written a book that will also be of interest and accessible to nonlogicians. Even if their opponents are not ultimately converted to the position defended in this book, open-minded readers will find much of value.Should singularists and friends of traditional plural logic be swayed by Florio and Linnebo’s arguments? There are many arguments in this book that merit careful consideration. I will consider just two in particular that give me pause for thought. The first concerns the case for pluralism as opposed to singularism. Florio and Linnebo are clear that the move to a critical plural logic undermines many of the familiar arguments for pluralism. For example, one popular style of argument, ‘the paradox of plurality,’ maintains that a singularist analysis would turn evident truths into demonstrable falsehoods (section 3.4). But this argument is no longer available, since the would-be truths are instances of plural comprehension which Florio and Linnebo reject. Another influential style of argument turns on the fact that pluralities provide a means to encode non-set-sized collections (section 4.8). But this argument is also unavailable, since their view only countenances set-sized pluralities. What reason, then, remains to take plural resources seriously?Florio and Linnebo’s main reason is that primitive plurals are needed to ‘give an account of sets’ (62) (section 4.4, chap. 12). The first half of Florio and Linnebo’s account comprises two elegant axioms that characterize the ‘singularization’ that maps each plurality to its set. These axioms are to be justified via the liberal view of definitions that Florio and Linnebo defend in section 12.3. The second half of their account consists of a critical plural logic whose axioms assert the existence of pluralities that correspond to ‘properly circumscribed’ or ‘extensionally definite’ collections. Florio and Linnebo seek to motivate some of these axioms through our intuitive grasp of these notions (which they explain in section 10.10). For example, they maintain, ‘since every single object can be circumscribed, there are singleton pluralities’ (280). In other cases, they rely on abductive considerations. One axiom permits us to obtain infinite pluralities by closing any plurality under a defined function. This axiom is justified on the grounds that taking infinite collections to be extensionally definite has been a ‘tremendous theoretical success’ (282). Combined with axioms licensing further plurality-forming operations, the end result is a set theory closely akin to the standard set theory, ZFC.Here is one reservation I have about this argument. Suppose that our grasp of ‘circumscription’ or ‘extensional definiteness’ is robust enough to vouchsafe Florio and Linnebo’s axioms. What is to stop a singularist from deploying this notion to directly motivate analogous set-forming operations in line with a first-order formulation of ZFC? The singularist may say, for example, ‘since every single object can be circumscribed, there are singleton sets.’ Moreover, given their theoretical success, she may obtain infinite sets, by permitting any set to be closed under a defined function. What would be lost by going direct from extensional definiteness to sets without the detour via pluralities?The second argument I want to pick up on targets the ‘traditional absolutist,’ who rejects generality relativism but adopts traditional plural logic (sections 11.5–11.6). Florio and Linnebo argue that this view, on its ‘most plausible development’ (261), ends up adopting a plural logic akin to their critical plural logic. First, ‘semantic considerations’ push the traditional absolutist to ascend a hierarchy that results from iterated ‘pluralization’ (256). She should countenance not just plural resources (level 1 pluralization), but also superplural resources (level 2 pluralization) and, more generally, pluralization of level n, for any finite n. Second, the infinitely many types of pluralization result in ‘expressibility problems’ (256) unless, as Florio and Linnebo recommend, the traditional absolutist takes one further step and ‘lifts the veil of type distinctions’ (261). The result is a one-sorted language whose ‘all-purpose’ variables simultaneously quantify over each individual, plurality, superplurality, or whatever, available at any level (261). Then, if she tries to ‘pluralize’ the all-purpose variables, the resulting logic does not sustain unrestricted plural comprehension. Each plurality, superplurality, and so on sits at some level in the hierarchy and only has members that belong to lower levels so there is no ‘universal plurality’ with respect to the all-purpose variable (261). The end result, Florio and Linnebo contend, is a view that has ‘much in common’ with their own (261).A traditional absolutist who is reluctant to ascend, or subsequently transcend, the pluralization hierarchy may well want to scrutinize Florio and Linnebo’s assumptions. The semantic considerations relate to Florio and Linnebo’s desire to give an ‘intensionally correct’ Tarski-style account of logical consequence (253), which generalizes not just over the set-based interpretations supplied by standard model theory but over every possible interpretation of the object language. The expressibility problems center on the inability of the infinitely typed language to articulate facts about the whole hierarchy. Even if a traditional absolutist is willing to follow Florio and Linnebo’s argument to its end point, however, I doubt that the resulting position is as similar to their view as they suggest.For one thing, the argument puts no pressure on the traditional absolutist’s contention that some things include everything. The would-be universal ‘plurality’ that Florio and Linnebo argue she should renounce is really—what to call it?—a ‘hyperplurality’ comprising every individual, plurality, superplural, or whatever, available at any level of the pluralization hierarchy. Rejecting this ‘hyperplurality’ is perfectly compatible with accepting an ordinary, level 1 plurality comprising everything. More generally—and dropping the loose ‘plurality’ talk for a moment—the mooted restrictions to plural comprehension arise only on an unintended interpretation, which gives ‘singular’ and ‘plural’ quantifiers meanings far removed from the ordinary ones. A traditional absolutist who accepts these restrictions may still maintain that plural comprehension is subject to no restriction under its intended interpretation in which singular and plural quantifiers express ordinary singular and plural quantification.The importance of this difference comes out when we set aside the higher levels of the pluralization hierarchy and focus on the plural resources available in natural languages. One unusual feature of Florio and Linnebo’s pluralism is that pluralities appear to play no essential role in the semantics of natural language plural terms. Consider, for example, a sentence such as ‘Most things are nonconcrete things.’ As Florio and Linnebo point out, the standard account of determiners like ‘most’ relies on the assumption that the underlying domain of discourse is a set (90). In these cases, they argue, sets or individual sums would serve just as well as, or perhaps better than, pluralities in the semantic analysis of plural terms (85–88, 295).What should we make of these set-based semantic theories? It is open to a generality relativist to take such a theory at face value. On this view, any universe of discourse available in natural language may be encoded as a set in a suitable metalanguage. But as Florio and Linnebo acknowledge, the same option is not available to someone who rejects generality relativism in a case when she takes the universe of discourse to comprise absolutely everything (295). Traditional absolutists have a fallback option. In cases where set-based semantic values are no longer available, a traditional absolutist may hope to salvage the linguistic core of the set-based semantic theory using plural resources. A universe comprising every individual, for example, may be encoded using the corresponding plurality. But this option is not available for an advocate of critical plural logic. Let me close then by raising what seems to me an important future task for Florio and Linnebo: if the semantics of natural language plural terms cannot always be understood in the standard way in terms of either individual sums or sets or pluralities, how is it to be understood?

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多与一：多元逻辑的哲学研究

逻辑学家和哲学家已经有 120 年的时间来习惯并非每个条件都定义一个集合的想法。一种流行的应对策略是保持每个实例化条件至少确定一个“多个”（即一个或多个项目）。这就是说，传统复数逻辑的朋友接受复数理解的每个实例——通常作为微不足道的或明显的或逻辑的真理：除非没有什么是 φ，否则有些东西包括 φ 的一切，没有别的。通过识别单数量词（'something'）和复数量词（'some things'）之间的类型区别，可以避免集合论悖论。本书为复数逻辑的异端版本辩护。Salvatore Florio 和 Øystein Linnebo 提倡一种基于“批判性复数逻辑”的集合论，该理论驳斥了许多复数理解的例子。尤其，他们否认有一个或多个事物包含一切。相反，他们争辩说，在解决悖论时，将复数理解限制在“外延确定”条件下的“一揽子交易”比其限制我们量词范围（“广义相对论”）或约束的竞争对手更具吸引力“单一化。” Florio 和 Linnebo 对传统复数逻辑的拒绝使他们能够结合两种本来不相容的观点：（i）“集合”运算是一种普遍的单数化，因此它将每个复数单射映射到一个对象（即它的集合），并且（ ii) “一切”的领域可能包含绝对的一切，因此单数化无法超越它。本书的第四部分也是最后一部分提出了采用批判性多元逻辑而不是传统多元逻辑的论点。前三部分首先说明了作者认真对待多元资源的理由。第一部分重新评估多元主义和单一主义之间的争论，多元主义“以表面价值看待多元资源”(2)，而单一主义则持相反观点。第二部分比较了“同时谈论多个对象的四种不同方式”(119)，包括二阶量化，以及“单数”的使用，以及集合和复数。第三部分侧重于多元逻辑的哲学应用。在此过程中，这本书解决了许多其他有趣的话题，包括复数逻辑是否算作“纯逻辑”（168），或者是否具有独特的本体论承诺（第 8 章），复数资源如何与模态相互作用（第 10 章），以及是否可以迭代复数运算以获得超复数项，表示“复数的复数”（180）（第 9 章）。以令人钦佩的清晰和引人入胜的方式讲述困难的领域。Florio 和 Linnebo 为多元主义辩论提供了全新的视角，并捍卫了对悖论的新颖回应。他们论点背后的驱动力通常是广义上的逻辑，而不是语言学或语言哲学。但是 Florio 和 Linnebo 写了一本书，非逻辑学家也会感兴趣并且可以阅读。即使他们的对手最终没有转变为本书所捍卫的立场，思想开放的读者也会发现很多价值。单一主义者和传统多元逻辑的朋友是否应该被弗洛里奥和林尼博的论点所左右？本书中有许多论点值得仔细考虑。我将特别考虑两个让我停下来思考的问题。第一个问题涉及多元主义相对于单一主义的情况。Florio 和 Linnebo 很清楚，转向批判的多元逻辑会破坏许多熟悉的多元主义论点。例如，一种流行的论证风格“多元悖论”认为，单一主义分析会将明显的真理变成可证明的谬误（第 3.4 节）。但是这个论点不再可用，因为可能的真理是弗洛里奥和林尼博拒绝的复数理解的例子。另一种有影响力的论证风格基于这样一个事实，即复数提供了一种对非集合大小的集合进行编码的方法（第 4.8 节）。但是这个论点也是不可用的，因为他们的观点只支持固定大小的多元化。那么，还有什么理由认真对待复数资源呢？Florio 和 Linnebo 的主要原因是需要原始复数来“给出集合的说明”（62）（第 4.4 节，第 12 章）。Florio 和 Linnebo 的前半部分包含两个优雅的公理，它们描述了将每个复数映射到其集合的“单一化”。这些公理将通过 Florio 和 Linnebo 在第 12.3 节中捍卫的自由主义定义观点得到证明。他们解释的后半部分包含一个批判性的复数逻辑，其公理断言复数的存在对应于“适当限定的”或“外延确定的”集合。Florio 和 Linnebo 试图通过我们对这些概念的直觉理解（他们在 10.10 节中解释）来激发其中一些公理。例如，他们坚持认为，“由于每个单独的对象都可以被限定，因此存在单一的复数”（280）。在其他情况下，他们依赖于溯因考虑。一个公理允许我们通过在定义的函数下关闭任何复数来获得无限复数。这个公理是合理的，因为将无限集合视为外延确定已经是一个“巨大的理论成功”（282）。结合公理许可进一步形成复数运算，最终结果是一个与标准集合论 ZFC 非常相似的集合论。这是我对这个论点的一个保留意见。假设我们对“限制”或“外延确定性”的把握足够稳健，足以保证弗洛里奥和林尼博的公理安全。是什么阻止奇点论者根据 ZFC 的一阶公式部署这个概念来直接激发类似的集合形成操作？例如，单一论者可能会说，“由于每个单独的对象都可以被限制，因此存在单一集合”。此外，考虑到他们在理论上的成功，她可以通过允许任何集合在定义的函数下闭合来获得无限集合。如果不绕道复数而直接从外延确定性转向集合，会失去什么？第二个论点我想针对“传统绝对主义者，' 他拒绝普遍相对主义，但采用传统的多元逻辑（第 11.5-11.6 节）。Florio 和 Linnebo 认为，这种观点在其“最合理的发展”（261）上最终采用了类似于他们批判的多元逻辑的多元逻辑。首先，“语义考虑”推动传统的专制主义者上升到一个由重复的“多元化”产生的等级制度 (256)。她不仅应该支持复数资源（第 1 级复数），而且应该支持超复数资源（第 2 级复数），更一般地说，对于任何有限的 n，都应该支持第 n 级的复数。其次，无限多种类型的多元化导致“可表达性问题”（256），除非像弗洛里奥和林尼博所建议的那样，传统的绝对主义者更进一步，“揭开类型区分的面纱”（261）。结果是一种单分类语言，其“通用”变量同时量化每个个体、复数、超复数或任何级别可用的任何东西 (261)。然后，如果她试图将通用变量“复数化”，那么由此产生的逻辑将无法支持不受限制的复数理解。每个复数、超复数等都位于层次结构中的某个级别，并且只有属于较低级别的成员，因此对于通用变量 (261) 没有“普遍复数”。Florio 和 Linnebo 争辩说，最终结果是一个与他们自己的观点有“很多共同点”的观点 (261)。一个不愿提升或随后超越多元等级制度的传统专制主义者可能很想仔细审查 Florio 和 Linnebo 的观点假设。语义方面的考虑与 Florio 和 Linnebo 对逻辑结果给出“内涵上正确”的 Tarski 式解释的愿望有关 (253)，它不仅概括了标准模型理论提供的基于集合的解释，而且概括了对象的所有可能解释语言。可表达性问题集中在无限类型语言无法阐明有关整个层次结构的事实。然而，即使传统的绝对主义者愿意将弗洛里奥和林尼博的论点推向终点，我怀疑最终的立场是否与他们所暗示的观点相似。一方面，该论点没有对传统绝对主义者的论点施加压力有些东西包括一切。Florio 和 Linnebo 认为她应该放弃的可能的普遍“多元性”实际上是——怎么称呼它？——一种“超多元性”，包括在多元化层次结构的任何级别上可用的每个个体、多元性、超复数或其他任何东西。拒绝这种“超多元性”与接受包含一切的普通的、第一层的多元性是完全兼容的。更一般地说——暂时放弃松散的“复数”话题——对复数理解提出的限制只出现在一种无意的解释上，它赋予“单数”和“复数”量词与普通量词相去甚远的含义。接受这些限制的传统绝对主义者可能仍然认为复数理解不受其预期解释的限制，其中单数和复数量词表示普通的单数和复数量词。当我们搁置更高层次的多元化层次结构并关注自然语言中可用的多元化资源。Florio 和 Linnebo 的多元论的一个不同寻常的特征是，复数似乎在自然语言复数术语的语义中没有发挥重要作用。例如，考虑“大多数事物都是非具体事物”这样的句子。正如 Florio 和 Linnebo 所指出的，对像“most”这样的限定词的标准解释依赖于以下假设：话语的基础领域是一个集合 (90)。在这些情况下，他们争辩说，在复数术语的语义分析中，集合或单个总和的作用与复数一样好，或者可能比复数更好 (85–88, 295)。我们应该如何看待这些基于集合的语义理论？广义相对论者可以从表面上接受这样的理论。按照这种观点，任何自然语言可用的话语域都可以用合适的元语言编码为一组。但正如 Florio 和 Linnebo 所承认的那样，当她认为话语域绝对包含一切时，对于拒绝广义相对论的人来说，同样的选择是不可用的 (295)。传统的绝对主义者有一个后备选项。在基于集合的语义值不再可用的情况下，传统的绝对主义者可能希望使用多元资源来挽救基于集合的语义理论的语言核心。例如，包括每个个体的宇宙可以使用相应的复数来编码。但是对于批判多元逻辑的倡导者来说，这个选项是不可用的。最后，让我提出在我看来 Florio 和 Linnebo 的一项重要未来任务：如果自然语言复数术语的语义不能总是以标准方式根据单个和或集合或复数来理解，那么如何被理解？