On the Innocence and Determinacy of Plural Quantification

Dialectica, 2005

George Boolos's employment of plurals to give an ontologically innocent interpretation of monadic higher-order quantification continues and extends a minority tradition in thinking about quantification and ontological commitment. An especially prominent member of that tradition is Stanislaw Leiniewski, and shall first draw attention to this work and its relation to that of Boolos. Secondly I shall stand up briefly for plurals as logically respectable expressions, while noting their limitations in offering ontologically deflationary accounts of higher-order quantification. Thirdly I shall focus on the key idea of ontological commitment and investigate its connection with the idea of truth-making. Fourthly I shall consider how different interpretations of quantification may sideline Boolos's work, but finally I shall largely support his analysis of quantification involving nominal expressions, while arguing, in the spirit of Arthur Prior, that non-nominal quantification is non-committing. To the memory of George Boolos, logician and gentleman.

Acta Analytica, 2017

In To be is to be the object of a possible act of choice (6) the authors defended Boolos' thesis that plural quantification is part of logic. To this purpose, plural quantification was explained in terms of plural reference, and a semantics of plural acts of choice, performed by an ideal team of agents, was introduced. In this paper, following that approach, we develop a theory of concepts that-in a sense to be explained-can be labelled as a theory of logical concepts. Within this theory we propose a new logicist approach to natural numbers. Then, we compare our logicism with Frege's traditional logicism.

A distinction is introduced between itemized and non-itemized plural predication. It is argued that a full-fledged system of plural logic is not necessary in order to account for the validity of inferences concerning itemized collective predication. Instead, it is shown how this type of inferences can be adequately dealt with in a first-order logic system, after small modifications on the standard treatment. The proposed system, unlike plural logic, has the advantage of preserving completeness. And as a result, inferences such as 'Dick and Tony emptied the bottle, hence Tony and Dick emptied the bottle' are shown to be first-order.

Mereological universalists and nihilists disagree on the conditions for composition. In this paper, we show how this debate is a function of one's chosen semantics for plural quantifiers. Debating mereologists have failed to appreciate this point because of the complexity of the debate and extraneous theoretical commitments. We eliminate this by framing the debate between universalists and nihilists in a formal model where these two theses about composition are contradictory. The examination of the two theories in the model brings clarity to a debate in which opponents frequently talk past one another. With the two views stated precisely, our investigation reveals the dependence of the mereologists' ontological commitments on the semantics of plural quantifiers. Though we discuss the debate with respect to a simplified and idealized model, the insights provided will make more complex debates on composition more productive and deflationist criticisms of the debate less substantial.

In To be is to be the object of a possible act of choice (6) the authors defended Boolos' thesis that plural quantification is part of logic. To this purpose, plural quantification was explained in terms of plural reference, and a semantics of plural acts of choice, performed by an ideal team of agents, was introduced. In this paper, following that approach, we develop a theory of concepts that – in a sense to be explained – can be labelled as a theory of logical concepts. Within this theory we propose a new logicist approach to natural numbers. Then, we compare our logicism with Frege's traditional logicism.

Proof-theoretic semantics is a well-established inferentialist theory of meaning that develops ideas proposed by Prawitz and Dummett. The main aim of this theory is to find a foundation of logic based on some aspects of the linguistic use of the logical terms, as opposed to the regular foundation offered by a model-theoretic approach à la Tarski, in which the denotation of non-linguistic entities is central. Traditionally, intuitionistic logic is considered justified in proof-theoretic semantics (although some doubts are sometimes raised regarding ex falso quodlibet). Even though this approach to semantics has greatly progressed in the last decades, it remains nonetheless controversial the existence of a justification of classical logic that suits its restraints. In this thesis I examine various proposals that try to give such a justification and propose a new one greatly inspired by one of Peter Milne’s papers. The conclusion is, to some extent, open since a reformulation of some notions of proof-theoretic semantics is needed in order to justify classical logic. I conclude the thesis with a general defence of logical pluralism and a description of the kind of pluralism that can be applied to our reformulation of proof-theoretic semantics.

The Many and the One, 2021

While plural logic can be interpreted in monadic second-order logic, the full system of second-order logic cannot be interpreted in plural logic. This means it is formally possible to eliminate plural logic in favor of monadic second-order logic. However, a number of philosophical considerations militate against such an elimination. The conclusion of this chapter echoes that of the preceding ones: although the two systems can occasionally be used for similar purposes, the notions they represent are different and must be kept apart.

2014

In Mathematics is megethology (Lewis, Philos Math 1:3-23, 1993) Lewis reconstructs set theory combining mereology with plural quantification. He introduces megethology, a powerful framework in which one can formulate strong assumptions about the size of the universe of individuals. Within this framework, Lewis develops a structuralist class theory, in which the role of classes is played by individuals. Thus, if mereology and plural quantification are ontologically innocent, as Lewis maintains, he achieves an ontological reduction of classes to individuals. Lewis'work is very attractive. However, the alleged innocence of mereology and plural quantification is highly controversial and has been criticized by several authors. In the present paper we propose a new approach to megethology based on the theory of plural reference developed in To be is to be the object of a possible act of choice (Carrara, Stud Log 96: 289-313, 2010). Our approach shows how megethology can be grounded on plural reference without the help of mereology.

Kris McDaniel has recently defended a criterion of what it is to believe in ways of being that classifies the quantifier variantist as an ontological pluralist. In this paper, I argue that this is a mistake. There is an important difference between the two views, which is often obscured by a certain approach to naturalness or fundamentality. On the atomistic approach, individual expressions are the primary bearers of naturalness, and languages are more or less natural or fundamental only derivatively. On the holistic approach, it is the other way around. I argue that whereas the atomistic approach struggles to distinguish quantifier variance from ontological pluralism, the holistic approach can do so quite easily. I then propose a criterion for believing in ways of being that does not classify the quantifier variantist as an ontological pluralist. Finally, I discuss various additional advantages of the holistic approach.