Plural Expression and Converse Relations

Constraints on the formation of plural reference objects: The influence of role, conjunction, and type of description

English contains two sorts of object quantifiers. In addition to ordinary singular quantifiers, as in the sentence 'There is a Cheerio in the bowl', there are plural quantifiers, as in 'There are some Cheerios in the bowl'. In the 1980s, George Boolos showed how these plural quantifiers can be used to interpret monadic second-order (henceforth MSOL).' This is an indisputable technical result. However, as with most technical results, it is not obvious what its philosophical cash value is. According to Boolos, its cash value is that MSOL really is logic. In particular, the result is said to show that MSOL introduces no new ontological commitments: that no sets or classes or Fregean concepts are needed as values of the second-order variables, but that the entities already in the first-order domain suffice. Despite some isolated voices of dissent,2 this philosophical interpretation of Boolos' technical result has become enormously popular. MSOL is now widely regarded as an important part of the philosopher's logical tool kit, of great value not only in the philosophy of mathematics but also in analytic metaphysics more generally. Most of all, Boolos' interpretation of MSOL has been hailed as a way of making great ontological bargains. According to its supporters, it allows us to pay the ontological price of a mere first-order theory and get the corresponding monadic second-order theory for free.3 In this paper I will challenge this wide-spread view of the philosophical cash value of Boolos' technical result. I will reject Boolos' claim that this result shows MSOL to be pure logic. And I will express serious misgivings about his claim that MSOL in this way is shown to be ontologically innocent. To assess what is achieved by Boolos' result, we need to get clear on the logical status of the theory of plural quantification. Only if this theory deserves the honorific logic will Boolos be justified in claiming that his

The view that plural reference is reference to a set is examined in light of George Boolos's treatment of second-order quantification as plural quantification in English. I argue that monadic second-order logic does not, in Boolos's treatment, reflect the behavior of plural quantifiers under negation and claim that any sentence that properly translates a second-order formula, in accordance with his treatment, has a first-order formulation. Support for this turns on the use of certain partitive constructions to assign values to variables in a way that makes Boolos's reading of second-order variables available for a first-order language and, with it, the possibility of interpreting quantification in an unrestricted domain.

Contents: Is Second-Order Logic Logic?: Beyond first-order logic: the historical interplay between mathematical logic and axiomatic set theory, Gregory H. Moore Which logic is the right logic?, Leslie H. Tharp On second-order logic, George S. Boolos Second-order languages and mathematical practice, Stewart Shapiro What are logical notions?, Alfred Tarski A curious inference, George Boolos The rationalist conception of logic, Steven J. Wagner A critical appraisal of second-order logic, Ignacio JanA(c) Who's afraid of higher-order logic?, Peter Simons. Ontological Reduction, Intended Interpretations and the LA wenheim-Skolem Theorems: Ontological reduction, Leslie H. Tharp Intended models and the LA wenheim-Skolem theorem, Virginia Klenk Categoricity, John Corcoran Skolem's paradox and constructivism, Charles McCarty and Neil Tennant Second-order logic, foundations and rules, Stewart Shapiro. Plural Quantification: To be is to be a value of a variable (or to be some values of some variables), George Boolos Nominalist Platonism, George Boolos Second-order logic still wild, Michael D. Resnick. Philosophy of Set Theory: Kreisel, the continuum hypothesis, and second-order set theory, Thomas Weston Skolem and the LA wenheim-Skolem theorem: a case study of the philosophical significance of mathematical results, Alexander George Skolem and the skeptic, Paul Benacerraf Skolem and the skeptic, Crispin Wright Predication versus membership in the distinction between logic as language and logic as calculus, Nino B. Cocchiarella Logicism, the continuum and anti-realism, Peter Clark Name index.

Say that some things form a set just in case there is a set whose members are precisely the things in question. For instance, all the inhabitants of New York form a set. So do all the stars in the universe. And so do all the natural numbers. Under what conditions do some things form a set? Let us make the question more precise. Plural expressions such as ‘some things’ are best explained by means of plural logic. In ordinary singular first-order logic we have singular variables such as x and y, which can be bound by the existential and universal quantifiers. In plural first-order logic we have in addition plural variables such as xx and yy, which can also be bound by existential and universal quantifiers to yield so-called plural quantifiers.1 ‘∃xx’ is read as ‘there are some things xx such that . . . ’, and ‘∀xx’, as ‘given any things xx, . . . ’. These quantifiers are subject to inference rules analogous to those of the ordinary singular quantifiers. There is also a two-place logical predicate ≺, where ‘u ≺ xx’ is to be read as ‘u is one of xx’. By ‘set’ I mean set as on the standard iterative conception, according to which the sets are “formed” in stages.2 We begin at stage 0 with all the non-sets, which are known as ∗I am grateful to a large number of people for comments and discussion, in particular Daniel Isaacson, Jose Martinez, David Nicolas, Vann McGee, Richard Pettigrew, Agustin Rayo, Gabriel Uzquiano, Tim Williamson, and two anonymous referees, as well as audiences in Bristol, Buenos Aires, MIT, Munich, Oxford, Riga, and Southampton. (At some of these places an earlier version was presented under the title “Why size does not matter.”) Much of the paper was written during a period of research leave funded by the AHRC (grant AH/E003753/1), whose support I gratefully acknowledge. Plural first-order logic was made popular by George Boolos, “To Be Is to Be a Value of a Variable (or to Be Some Values of Some Variables),” this Journal, LXXXI (1984): 430-439. For an introduction, see Oystein Linnebo, “Plural Quantification,” in Edward N. Zalta, ed., Stanford Encyclopedia of Philosophy (2008). See George Boolos, “The Iterative Conception of Set,” this Journal, LXVIII (1971):215-32. It is of course controversial how the metaphor of “set formation” should be understood. I will provide a modal explication in Section 5, following Charles Parsons, “What Is the Iterative Conception of Set?,” repr. in his Mathematics in Philosophy (Ithaca: Cornell, 1983), pp. 268-297.

Cross-categorial singular and plural reference in sign language

Can Quine’s criterion for ontological commitment be comparatively applied across different logics? If so, how? Cross-logical evaluations of discourses are central to contemporary philosophy of mathematics and metaphysics. The focus here is on the influential and important arguments of George Boolos and David Lewis that second-order logic and plural quantification don’t incur additional ontological commitments over and above those incurred by first-order quantifiers. These arguments are challenged by the exhibition of a technical tool—the truncation-model construction of notational equivalents—that compares the ontological role and increased expressive strength of non-first-order ideology to first-order ideology.

A plural referring expression (‘the Fs’ or ‘Tom, Dick and Harriet’) may be used to refer either distributively, saying something which applies to each of the Fs individually, or collectively, to the Fs taken as a single totality. Predicate Logic has to analyse both uses in terms of singular reference, treating them quite differently in so doing; but we think of such an expression as functioning in basically the same way in both kinds of use. This understanding can be vindicated if we recognise that what a plural referring expression picks out is not either an aggregate simpliciter or a set, but a plurality– an aggregate taken relative to a principle for individuating its constituents; this admits of being seen either as many things or as one. In any given case, it is the nature of what is being said about the plurality which tells us whether the reference to it is to be taken as distributive, collective, or a combination of the two. Talk about pluralities is extensional. Augmenting Predicate Logic to accommodate the distinctive inference‐pattern associated with distributive plural reference is simple – and arguably necessary, to cope with cases in which distributive and collective reference are essentially combined (e.g., attributions of concerted action).

Constructionism in Logic

Metaphysicians of composite objects commonly distinguish two types of composite objects: wholes and sums. The former can survive some changes of parts, while the latter cannot. This paper investigates how the distinction between wholes and sums can be respected, while denying that a sum is an individual composite object. In particular, the view presented here identifies wholes with the fusions of (extensional) mereology – hence going against a common tendency to identify sums with mereological fusions – and identifies sums with the pluralities of plural logic. It is argued that this view better captures the ideas behind the whole/sum-distinction while also avoiding various challenges the other accounts face.

PLV (Plural Basic Law V) is a consistent second-order system which is aimed to derive second-order Peano arithmetic. It employs the notion of plural quantification and a first-order formulation of Frege's infamous Basic Law V. George Boolos' plural semantics is replaced with Enrico Martino's Acts of Choice Semantics (ACS), which is developed from the notion of arbitrary reference in mathematical reasoning. ACS provides a form of logicism which is radically alternative to Frege's and which is grounded on the existence of individuals rather than on the existence of concepts.

The curiously sweeping assumption that all reference is ‘ultimately’ singular — even in the case of plural or non-count reference — is presented and examined. In the case of plural reference, especially when associated with collective predication, the assumption takes the form of the thought that this is reference to collective entities, plural objects, or sets. Perhaps the most suggestive and profound, albeit notorious idea of this genre is Russell’s doctrine of the ‘class as many’. George Boolos’ explicitly ‘no-class’ approach to the logic of plurality is then compared favourably with ‘reductive’ approaches.

The second application of the assumption that singular reference is ‘ultimately’ exhaustive also represents non-count reference as singular — as reference to individual ‘quantities’ or ‘parcels’ of stuff. Unsurprisingly, the idea is sometimes explicitly advanced on the model of plural reference as singular. However, any such view must attempt to circumvent the difficulties posed by Russell’s analysis of the conditions, whereby descriptions count as semantically singular. It is argued that such an attempt cannot succeed.

We introduce our implementation in HOL Light of the metatheory for Gödel–Löb provability logic (GL), covering soundness and completeness w.r.t. possible world semantics and featuring a prototype of a theorem prover for GL itself. The strategy we develop here to formalise the modal completeness proof overcomes the technical difficulty due to the non-compactness of GL and is an adaptation—according to the formal language and tools at hand—of the proof given in George Boolos’ 1995 monograph. Our theorem prover for GL relies then on this formalisation, is implemented as a tactic of HOL Light that mimics the proof search in the labelled sequent calculus textsf{G3KGL}, and works as a decision algorithm for the provability logic: if the algorithm positively terminates, the tactic succeeds in producing a HOL Light theorem stating that the input formula is a theorem of GL; if the algorithm negatively terminates, the tactic extracts a model falsifying the input formula. We discuss our code for the formal proof of modal completeness and the design of our proof search algorithm. Furthermore, we propose some examples of the latter’s interactive and automated use.

Corpus evidence suggests that in contexts in which the presence of multiple antecedents might favor plural reference, the disadvantage observed for singular reference may disappear if the potential antecedents are combined in a group-like plural entity. We examined the relative salience of antecedents in conditions where the context either made a group interpretation available (i.e., mereological entity) (e.g., The engineer hooked up the engine to the boxcar …, where group = train), or not (e.g., The engineer detached the engine from the boxcar …). Results from three experiments in which participants were asked to identify referents for singular versus plural pronouns (Experiment 1), to confirm the referents of pronouns in a sentence completion task (Experiment 2), and to provide paraphrases for given texts (Experiment 3), collectively provided evidence that the creation of a group makes that entity (i) a possible referent for singular anaphoric reference and (ii) more salient than its constituents.