Monism and the Ontology of Logic

2015

In Mathematics is megethology (Lewis, 1993) David K. Lewis proposes a structuralist reconstruction of classical set theory based on mereology. In order to formulate suitable hypotheses about the size of the universe of individuals without the help of set-theoretical notions, he uses the device of Boolos' plural quantification for treating second order logic without commitment to set-theoretical entities.

Philosophical Studies, 2019

Since Kit Fine presented his counterexamples to the standard versions of the modal view, many have been convinced that the standard versions of the modal view are not adequate. However, the scope of Fine's argument has not been fully appreciated. In this paper, I aim to carry Fine's argument to its logical conclusion and argue that once we embrace the intuition underlying his counterexamples , we have to hold that properties obtained, totally or partially, by application of logical operations are not essential to non-logical entities. I also demonstrate that most of the post-Finean versions of the modal view, which were developed to accommodate Fine's counterexamples , entail that such properties are essential to the entities, and so fail to capture the notion of essence at issue in Fine's counterexamples. Additionally, I explore the consequence of my argument for Fine's proposed logic of essence. The logic turns out to be inadequate in its present shape as it represents such properties to be essential to the entities. I conclude by developing a modification to the logic to overcome the shortcoming.

Philosophical Perspectives, 1991

1979. Identity Logics. (Co-author: Steven Ziewacz) Notre Dame Journal of Formal Logic 20, 777–84. MR0545427 (80h: 03017) In this paper we prove the completeness of three logical systems IL.1, IL2, and IL3. IL1 deals solely with identities {a = b), and its deductions are the direct deductions constructed with the three traditional rules: (T) from a = b and b = c infer a = c, (S) from a = b infer b = a and (A) infer a = a (from anything). IL2 deals solely with identities and inidentities {a ≠ b); its deductions include both the direct and the indirect deductions constructed with the three traditional rules. IL3 is a hybrid of IL1 and IL2: its deductions are all direct as in IL1 but it deals with identities and inidentities as in IL2. IL1 and IL2 have a high degree of naturalness. Although the hybrid system IL3 was constructed as an artifact useful in the mathematical study of IL1 and IL2, it nevertheless has some intrinsically interesting aspects. The main motivation for describing and studying such simple systems is pedagogical. In teaching beginning logic one would like to present a system of logic which has the following properties. First, it exemplifies the main ideas of logic: implication, deduction, non-implication, counterargument, logical truth, self-contradiction, consistency, satisfiability, etc. Second, it exemplifies the usual general meta-principles of logic: contraposition and transitivity of implication, cut laws, completeness, soundness, etc. Third, it is simple enough to be thoroughly grasped by beginners. Fourth, it is obvious enough so that its rules do not appear to be arbitrary or purely conventional. Fifth, it does not invite confusions which must be unlearned later. Sixth, it involves a minimum of presuppositions which are no longer accepted in mainstream contemporary logic. These logics are far superior to propositional logic a pedagogical catastrophe. Tarski’s INTRODUCTION TO LOGIC doesn’t start with propositional logic but with an identity logic that is more complicated that these. They are also superior to Aristotelian logic (or syllogistic), which is irrelevant to contemporary applications despite its intrinsic beauty and its undeniable historical importance. *The authors wish to thank Wendy Ebersberger (SUNY/Buffalo) and George Weaver (Bryn Mawr College) for useful criticisms and for encouragement.

2012

In this paper, I will defend the claim that there are three existence properties: the second-order property of being instantiated, a substantive first-order property (or better a group of such properties) and a formal, hence universal, first-order property. I will first try to show what these properties are and why we need all of them for ontological purposes. Moreover, I will try to show why a Meinong-like option that positively endorses both the former and the latter first-order property is the correct view in ontology. Finally, I will add some methodological remarks as to why this debate has to be articulated from the point of view of reality, i.e., by speaking of properties, rather than from the point of view of language, i.e., by speaking of predicates (for such properties).

2011

The objective of this paper is a defense of a particular answer to van Inwagen's Special Composition Question: when is it the case that some objects together compose some additional object? The answer is the conjunction of two claims. The first claim, compositional nihilism says that, necessarily, there is never an instance of material composition, and therefore all material objects that do exist are simple, or without proper parts. The second claim, existence monism, says that there exists a material object, and that all other material objects are identical with this object. In other words, there is just one material object that extends throughout the entirety of the material world. These claims are formalized as follows, where (N) represents compositional nihilism and (M) represents existence monism: (N) [∀x: x ∈ M] ~∃y(Pxy ^ x ≠ y) (M) [∃x: x ∈ M] ∀y[(y ∈ M) ⊃ (x = y)] Other claims will be argued for. While I do believe these additional claims are true, I am not committed to them as strongly as I am to compositional nihilism and existence monism. These other claims serve mostly compliment the primary two claims. The dialectic of the paper is essentially that of an argument to the best explanation. Alternatives to compositional nihilismuniversalism and compatibilism-are eliminated on various grounds. Alternatives to existence monism-versions of pluralist nihilism-are also argued against. The idea is that the two views are the only strong candidates for an ontologically sound theory. One last task of the paper is to disarm various objections to the two primary claims. This is done by demonstrating that what was previously seen as objectionable consequences of the views are, in fact, unproblematic. In at least one instance, a previously objectionable consequence is shown to be, in fact, a potential benefit of the views. In Defense of Existence Monism Following the seminal work of figures like David Lewis and Peter van Inwagen, there has been a marked increase of interest in material composition. Theories that would have previously been dismissed as patently absurd are now given more careful consideration. Much work has already been done in ontology, semantics, and logic to make sense of these views. In this paper I hope to present a relatively broad overview of the issues at play. I will then argue for a particular view, existence monism, in light of these considerations. The first aim of this paper is to provide a critical survey of various answers to the following question: 'Under what circumstances does material composition occur?' A (perhaps artificial) dialectic will be established to assist in navigating the plethora of issues involved with establishing a coherent answer to this question. The second aim of this paper is a defense of a particular answer to this question. Two central claims will be defended. The first claim, (N), is a response to van Inwagen's Special Composition Question. It says that, necessarily, there is never an instance of material composition, and therefore all material objects that do exist are simple, or without proper parts. Call this view compositional nihilism. Note that compositional nihilism does not specify how many simple objects exist. There are three types of ontology that include (N). They differ in the number and nature of the simples in the world. The first type, call it point nihilism, says that the simples that exist are as small as is physically (or

Synthese, 2013

In a first part, I defend that formal semantics can be used as a guide to ontological commitment. Thus, if one endorses an ontological view O and wants to interpret a formal language L, a thorough understanding of the relation between semantics and ontology will help us to construct a semantics for L in such a way that its ontological commitment will be in perfect accordance with O. Basically, that is what I call constructing formal semantics from an ontological perspective. In the rest of the paper, I develop rigorously and put into practice such a method, especially concerning the interpretation of second-order quantification. I will define the notion of ontological framework: it is a set-theoretical structure from which one can construct semantics whose ontological commitments correspond exactly to a given ontological view. I will define five ontological frameworks corresponding respectively to: (i) predicate nominalism, (ii) resemblance nominalism, (iii) armstrongian realism, (iv) platonic realism, and (v) tropism. In those different frameworks, I will construct different semantics for first-order and second-order languages. Notably I will present different kinds of nominalist semantics for second-order languages, thus showing how we can quantify over properties and relations while being ontologically committed only to individuals. More generally I will show in what extent those semantics differ from each other; it will make clear how the disagreements between the ontological views extend from ontology to logic, and how metaphysical questions can be correctly treated, in those semantics, as simple questions of logic.

It is argued here that criteria of identity do not have the form of predicate-logical formulae. The conclusion is drawn that Hume's Principle cannot serve as a criterion of identity for the concept of cardinal number. The way criteria of identity are formulated in Martin-Löf's type theory is presented as an alternative that is not affected by the argument. This paper is forthcoming in the Logica Yearbook 2017.

2016

This paper presents an account of what it is for a property or relation (or ‘attribute’ for short) to be logically simple. Based on this account, it is shown, among other things, that the logically simple attributes are in at least one important way sparse. This in turn lends support to the view that the concept of a logically simple attribute can be regarded as a promising substitute for Lewis’s concept of a perfectly natural attribute. At least in part, the advantage of using the former concept lies in the fact that it is amenable to analysis, where that analysis—i.e., the account put forward in this paper—requires the adoption neither of an Armstrongian theory of universals nor of a primitive notion of naturalness, fundamentality, or grounding.

Bulletin of Symbolic Logic, 2001

We discuss the differences between first-order set theory and secondorder logic as a foundation for mathematics. We analyse these languages in terms of two levels of formalization. The analysis shows that if second-order logic is understood in its full semantics capable of characterizing categorically central mathematical concepts, it relies entirely on informal reasoning. On the other hand, if it is given a weak semantics, it loses its power in expressing concepts categorically. First-order set theory and second-order logic are not radically different: the latter is a major fragment of the former. * I am grateful to Juliette Kennedy for many helpful discussions while developing the ideas of this paper. † Research partially supported by grant 40734 of the Academy of Finland.

Mind, 2003

There is a well-known argument from Leibniz's Law for the view that coincident material things may be distinct. For given that they differ in their properties, then how can they be the same? However, many philosophers have suggested that this apparent difference in properties is the product of a linguistic illusion; there is just one thing out there, but different sorts or guises under which it may be described. I attempt to show that this 'opacity' defence has intolerable consequences for the functioning of our language and that the original argument should therefore be allowed to stand. 7 Extreme monists include Gupta (), Burke (, , ), van Inwagen (), Sider (, ). Strictly moderate monists include Lewis (), Gibbard ( in Rea  Ch. ), Robinson (a, b), Heller (, ). Opponents of moderate monism include Doepke , Wiggins (, ), Sosa (, ), Johnston (), Baker ().

Synthese, December 2013, DOI : 10.1007/s11229-013-0387-9

In a first part, I defend that formal semantics can be used as a guide to ontological commitment. Thus, if one endorses an ontological view O and wants to interpret a formal language L, a thorough understanding of the relation between semantics and ontology will help us to construct a semantics for L in such a way that its ontological commitment will be in perfect accordance with O. Basically, that is what I call constructing formal semantics from an ontological perspective. In the rest of the paper, I develop rigorously and put into practice such a method, especially concerning the interpretation of second-order quantification. I will define the notion of ontological framework: it is a set-theoretical structure from which one can construct semantics whose ontological commitments correspond exactly to a given ontological view. I will define five ontological frameworks corresponding respectively to: (i) predicate nominalism, (ii) resemblance nominalism, (iii) armstrongian realism, (iv) platonic realism, and (v) tropism. In those different frameworks, I will construct different semantics for first-order and second-order languages. Notably I will present different kinds of nominalist semantics for second-order languages, thus showing how we can quantify over properties and relations while being ontologically committed only to individuals. More generally I will show in what extent those semantics differ from each other; it will make clear how the disagreements between the ontological views extend from ontology to logic, and how metaphysical questions can be correctly treated, in those semantics, as simple questions of logic.

2014

In this paper I propose a semantic argument against the existence of universally held real properties. A semantic argument is a deductive argument from one or more premises about meaning. Real properties are properties that add something to (or modify) their bearers, such as being red, being triangular or knowing that 1+1=2. They are typically contrasted with Cambridge properties. A property is universally held if and only if everything that exists has it. The semantic argument is set within a neo-Fregean linguistic framework that distinguishes meaning from reference. Although, as Frege argued, meaning and reference do not coincide, they are quite closely related. The argument is premised on an identity criterion for (definite) meanings in terms of their reference characteristics. According to this criterion two meanings are identical if and only if their reference sets coincide. The notion of a reference set of a meaning will be made more precise in the paper.

The Review of Symbolic Logic, 2009

In Parts of Classes (1991) and Mathematics Is Megethology (1993) David Lewis defends both the innocence of plural quantification and of mereology. However, he himself claims that the innocence of mereology is different from that of plural reference, where reference to some objects does not require the existence of a single entity picking them out as a whole. In the case of plural quantification "we have many things, in no way do we mention one thing that is the many taken together". Instead, in the mereological case: "we have many things, we do mention one thing that is the many taken together, but this one thing is nothing different from the many" (Lewis, 1991, p. 87). The aim of the paper is to argue that-for a certain use of mereology, weaker than Lewis' one-an innocence thesis similar to that of plural reference is defensible. To give a precise account of plural reference, we use the idea of plural choice. We then propose a virtual theory of mereology in which the role of individuals is played by plural choices of atoms. §1. Introduction. In Parts of Classes (1991) David Lewis argues that, like logic, but unlike set theory, mereology is "ontologically innocent". Prima facie, Lewis' innocence thesis seems to be ambiguous. On the one hand, he seems to argue that, given certain objects X s, referring to their sum is ontologically innocent because there is no new entity as the referent of the expression "the sum of the X s". So, talking of the sum of the X s would simply be a different way of talking of the X s, looking at them as a whole. However, on the other hand, Lewis' innocence is not understood as a mere linguistic use, where sums are not reified. He himself claims that the innocence of mereology is different from that of plural reference, where reference to some objects does not require the existence of a single entity picking them out in a whole. In the case of plural quantification "we have many things, in no way do we mention one thing that is the many taken together". Instead, in the mereological case: "we have many things, we do mention one thing that is the many taken together, but this one thing is nothing different from the many" (Lewis, 1991, p. 87).