Critical Plural Logic†

Interchange, 1990

Students (and not only students) usually think of mathematics as monolithic and fixed for all time, and of mathematicians (if they give any thought to the creators of the subject they are dealing with) as in total agreement on mathematical means and ends. How many of them realize that not only the methods and results of mathematics but also its basic concepts are tentative rather than final, in flux rather than eternal? That the ideas of number, of function, of continuity, even of proof have ever been different from what they are today? We focus in this essay on the concept of proof.

Journal of Indian Council of Philosophical Research, 2017

Introduction Scientific pluralism is generally understood in the backdrop of scientific monism. So is mathematical pluralism. Though there are many culture-dependent mathematical practices, mathematical concepts and theories are generally taken to be culture invariant. We would like to explore in this paper whether mathematical pluralism is admissible or not. Materials and methods Mathematical pluralism may be approached at least from five different perspectives. 1. Foundational: The view would claim that different issues within mathematics need support of different foundations, apparently incompatible with one another. 2. Ontological: The world itself is dappled-the mathematical counterpart of which can be traced to the admission of non-Euclidean spaces and also in simultaneous acceptance of set-theoretic and category theoretic entities in the ontology of mathematics. 3. The third route is epistemological. It follows from the view that the nature of reality is so complex that different aspects of it require alternative modes of representation and explanation sometimes severally, sometimes simultaneously, e.g., classical, constructivist, computer-aided and different finitist mathematics can be used depending on the knowledge situation. 4. The fourth route is semantic. Fuzzy mathematics, we know, gives up bivalence in theory of truth, and ascribes truth to mathematical propositions in degrees. 5. The last one is Programmatic: for some, mathematical pluralism is a stance, a programme. It provides a framework which strikes at the root of cultural hegemony and nourishes a climate of intellectual humility and tolerance. It is not my intention to hold the brief of any of these versions. I just want to present to the readers a more or less complete picture of the scenario, though not an exhaustive one. Conclusion Unlike monism, mathematical pluralism does not rule out any possibilityeven the possibility of having a unitary over-arching framework of interpretation remains as an open option. This paper highlights the fact that motivations for upholding & Amita Chatterjee

Nous

Plural logic is widely assumed to have two important virtues: ontological innocence and determinacy. It is claimed to be innocent in the sense that it incurs no ontological commitments beyond those already incurred by the first-order quantifiers. It is claimed to be determinate in the sense that it is immune to the threat of non-standard (Henkin) interpretations that confronts higher-order logics on their more traditional, set-based semantics. We challenge both claims. Our challenge is based on a Henkin-style semantics for plural logic that does not resort to sets or set-like objects to interpret plural variables, but adopts the view that a plural variable has many objects as its values. Using this semantics, we also articulate a generalized notion of ontological commitment which enables us to develop some ideas of earlier critics of the alleged ontological innocence of plural logic.

Journal of Indian Council of Philosophical Research, 2017

Realist philosophers of mathematics have accounted for the objectivity and robustness of mathematics by recourse to a foundational theory of mathematics that ultimately determines the ontology and truth of mathematics. The methodology for establishing these truths and discovering the ontology was set by the foundational theory. Other traditional philosophers of mathematics, but this time those who are not realists, account for the objectivity of mathematics by fastening on to: an objective account of: epistemology, ontology, truth, epistemology or methodology. One of these has to stay stable. Otherwise, it is traditionally thought, we have a rampant relativism where 'anything goes'. Pluralism is a relatively new family of positions. The pluralist in mathematics who is pluralist in: epistemology, foundations, methodology, ontology and truth cannot account for the objectivity of mathematics in either the realist or in the other traditional ways. But such a pluralist is not a rampant relativist. In the paper, I look at what it is to be a pluralist in: epistemology, foundations, methodology, ontology and truth. I then give an account of the objectivity and robustness of mathematics in terms of rigour, borrowings, crosschecking and fixtures-all technical terms defined in the paper. This account is an alternative to the realist and traditional accounts of objectivity in mathematics.

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.

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.

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.

Purpose This paper aims to establish that a certain sort of mathematical pluralism is true. Methods The paper proceeds by arguing that that the best versions of mathematical Platonism and anti-Platonism both entail the relevant sort of mathematical pluralism.

Linguistic Inquiry, 2018

I argue that an account of both inclusive plurals and the crosslinguistic typology of grammatical number requires postulating a [−atomic] feature (or something very much like it) in the structure of exclusive-plural DPs. When combined with the only theory we currently have that accounts for the crosslinguistic typology of number ( Harbour 2014 ), theories in which the exclusive-plural DPs of a language with inclusive plurals are [−atomic]-less under- or overgenerate with respect to that typology. These problems disappear as soon as the structure of exclusive plural DPs contains a component that generates exclusive-plural interpretations, either Harbour’s [−atomic] feature (added to a system with a second, [−atomic]-less structure, a proposal compatible with, e.g., Farkas and de Swart 2010 ) or a predicate-level exhaustivity operator (from Mayr 2015 ).

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.