The Many and the One: A Philosophical Study of Plural Logic

Abstract

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?