Christopher Menzel

Professor Grosholz packs a lot into her interesting and suggestive paper “Formal Unities and Real Individuals” (Grosholz 1990b). In the limited space available I can comment briefly on its several parts, or direct more substantive comments at a single issue. I will opt for the latter; specifically, I want to address her critique of mathematical structuralism, as found especially in the writings of Michael Resnik.I begin with a brief, hence necessarily caricatured, summary of Resnik’s influential view. According to structuralism, the subject matter of a mathematical theory is a given pattern, or structure, and the objects of the theory are intrinsically unstructured points, or positions, within that pattern. Mathematical objects thus have no identity, and no intrinsic features, outside of the patterns in which they occur. Hence, they cannot be given in isolation but only in their role within an antecedently given pattern, and are distinguishable from one another only in virtue of the relations they bear to one another in the pattern (see, e.g., Resnik (1981)).

Epistemologists often offer theories of justification without paying much attention to the variety and diversity of locutions in which the notion of justification appears. For example, consider the following claims which contain some notion of justification: B is a justified belief, S's belief that p is justified, p is justified for S, S is justified in believing that p, S justifiably believes that p, S's believing p is justified, there is justification for S to believe that p, there is justification for S's believing p, and S has a justification for believing that p. In addition to these passive uses of the notion of justification, there are active uses as well: S justified his belief in p, believing e justifies believing p, etc. The syntactic variety involves semantic difference as well. For example, the proposition S has a justification for believing that p does not entail that S believes p, whereas the proposition S justifiably believes that p does entail that S believes p. Our ultimate goal is to show that this diversity is only superficial by arguing that there is a basic kind of justification. On the way, however, we shall argue that there are three central uses of a notion of justifica- tion in the above list: propositional justification (as in p is justified for S), personal justification (as in S is justified in believing that p) and doxastic justification (as in S's believing p is justified). Our preliminary argument will be that the multiplicity above can be explained in terms of these three locutions, and the substance of our argument will be to show that one of these three is the basic kind of justification. Success in this task will thereby justify, at least in part, the practice of contem- porary epistemologists. Our conclusions, however, shall not be of much comfort to contemporary epistemology, for the way in which the apparent diversity in the uses of the notion of justification is eliminated undermines much of recent epistemology.

W. V. Quine famously defended two theses that have fallen rather dramatically out of fashion. The first is that intensions are “creatures of darkness” that ultimately have no place in respectable philosophical circles, owing primarily to their lack of rigorous identity conditions. However, although he was thoroughly familiar with Carnap’s foundational studies in what would become known as possible world semantics, it likely wouldn’t yet have been apparent to Quine that he was fighting a losing battle against intensions, due in large measure to developments stemming from Carnap’s studies and culminating in the work of Kripke, Hintikka, and Bayart. These developments undermined Quine’s crusade against intensions on two fronts. First, in the context of possible world semantics, intensions could after all be given rigorous identity conditions by defining them (in the simplest case) as functions from worlds to appropriate extensions, a fact exploited to powerful and influential effect in logic and linguistics by the likes of Kaplan, Montague, Lewis, and Cresswell. Second, the rise of possible world semantics fueled a strong resurgence of metaphysics in contemporary analytic philosophy that saw properties and propositions widely, fruitfully, and unabashedly adopted as ontological primitives in their own right. This resurgence — happily, in my view — continues into the present day. For a time, at any rate, Quine experienced somewhat better success with his second thesis: that higher-order logic is, at worst, confused and, at best, a quirky notational alternative to standard first-order logic. However, Quine notwithstanding, a great deal of recent work in formal metaphysics transpires in a higher-order logical framework in which properties and propositions fall into an infinite hierarchy of types of (at least) every finite order. Initially, the most philosophically compelling reason for embracing such a framework since Russell first proposed his simple theory of types was simply that it provides a relatively natural explanation of the paradoxes. However, since the seminal work of Prior there has been a growing trend to consider higher-order logic to be the most philosophically natural framework for metaphysical inquiry, many of the contributors to this volume being among the most important and influential advocates of this view. Indeed, this is now quite arguably the dominant view among formal metaphysicians. In this paper, and against the current tide, I will argue in §1 that there are still good reasons to think that Quine’s second battle is not yet lost and that the correct framework for logic is first-order and type-free — properties and propositions, logically speaking, are just individuals among others in a single domain of quantification — and that it arises naturally out of our most basic logical and semantical intuitions. The data I will draw upon are not new and are well-known to contemporary higher-order metaphysicians. However, I will try to defend my thesis in what I believe is a novel way by suggesting that these basic intuitions ground a reasonable distinction between “pure” logic and non-logical theory, and that Russell-style semantic paradoxes of truth and exemplification arise only when we move beyond the purely logical and, hence, do not of themselves provide any strong objection to a type-free conception of properties and propositions. Most of my arguments in §1 are largely independent of any specific account of the nature of properties and propositions beyond their type-freedom. However, I will in addition argue that there are good reasons to take propositions, at least, to be very fine-grained. My arguments are thus bolstered significantly if it can be shown that there are in fact well-defined examples of logics that are not only type-free but which comport with such a conception of propositions. It is the purpose of §2 to lay out a logic of this sort in some detail, drawing especially upon work by George Bealer and related work of my own. With the logic in place, it will be possible to generalize the line of argument noted above regarding Russell-style paradoxes and, in §3, apply it to two propositional paradoxes — the Prior-Kaplan paradox and the Russell-Myhill paradox — that are often taken to threaten the sort of account developed here.

Actualism is a widely-held view in the metaphysics of modality that arises in response to the thesis of possibilism, the doctrine that, in addition to the things that actually exist — in particular, things that exist alongside us in the causal order — there are merely possible things as well, things that, in fact, fail to be actual but which could have been. The central motivation for possibilism is to explain what it is about reality that grounds such intuitively true propositions as that Wittgenstein (who was childless) could have had children. In answer, possibilists argue that we must simply broaden our understanding of reality, of what there is in the broadest sense, beyond the actual, beyond what actually exists, so that it also includes the merely possible. In particular, says the possibilist, there are merely possible people, things that are not, in fact, people but which could have been. So, for the possibilist, the proposition that Wittgenstein could have had children is grounded in the fact that, among the possibilia, there are those that could have been his children. Actualism is (at the least) the denial of possibilism; to be an actualist is to deny that there are any possibilia. Put another way, for the actualist, there is no realm of reality, or being, beyond actual existence; to be is to exist, and to exist is to be actual. This article investigates the origins and nature of the debate between possibilists and actualists, with a particular focus on the implications of the debate for quantified modal logic.

Contains the following contributions: Ingvar Johansson: Ontologies and Concepts. Two Proposals Christopher Menzel: Reference Ontologies - Application Ontologies: Either/Or or Both/And? Luc Schneider: Foundational Ontologies and the Realist Bias Guenther Goerz, Kerstin Buecher, Bernd Ludwig, Frank-Peter Schweinberger, and Iman Thabet: Combining a Lexical Taxonomy with Domain Ontology in the Erlangen Dialogue System Vim Vandenberghe, Burkhard Schafer, John Kingston: Ontology Modelling in the Legal Domain - Realism Without Revisionism A Proposed Methodology for the Development of Application-Based Formal Ontologies Eric Little

In Modal Logic as Metaphysics, Timothy Williamson claims that the possibilism-actualism (P-A) distinction is badly muddled. In its place, he introduces a necessitism-contingentism (N-C) distinction that he claims is free of the confusions that purportedly plague the P-A distinction. In this paper I argue first that the P-A distinction, properly understood, is historically well-grounded and entirely coherent. I then look at the two arguments Williamson levels at the P-A distinction and find them wanting and show, moreover, that, when the N-C distinction is broadened (as per Williamson himself) so as to enable necessitists to fend off contingentist objections, the P-A distinction can be faithfully reconstructed in terms of the N-C distinction. However, Williamson’s critique does point to a genuine shortcoming in the common formulation of the P-A distinction. I propose a new definition of the distinction in terms of essential properties that avoids this shortcoming.

Ontology today is in many ways in a state similar to that of analysis in the late 18th century prior to arithmetization: it lacks the sort rigorous theoretical foundations needed to elevate ontology to the level of a genuine scientific discipline. This paper attempts to make some first steps toward the development of such foundations. Specifically, starting with some basic intuitions about ontologies and their content, I develop an expressively rich framework capable of treating ontologies as theoretical objects whose properties and logical inter- connections — notably, potential for integration — we can clearly define and study.

The distinction between reference ontologies and application ontologies crept rather unobtrusively into the recent literature on knowledge engineering. A lot of the discourse surrounding this distinction – notably, the one framing the workshop generating this collection of papers – suggests the two types of ontologies are in some sort of opposition to one another. Thus, Borge et al. [3] characterize reference ontologies (more recently, foundational ontologies) as rich, axiomatic theories whose focus is to clarify the intended meanings of terms used in specific domains. Application ontologies, by contrast, provide a minimal terminological structure to fit the needs of a specific community. Reflecting their minimal nature, Masolo et al. [7] refer to such ontologies as “lightweight” ontologies. An application ontology can be lightweight in a second respect as well, namely, that it may not necessarily take the form of fully-fledged axiomatic theory. Rather, it might only be a taxonomy of the relevant domain, a division of the domain into a salient collection of classes, perhaps ordered by the subclass relation. Importantly, though, for an application ontology to “fit the needs of a specific community” needn’t require representational accuracy. In the “worst” case (from a reference ontology perspective), to fit the needs of a community is just to represent uncritically what people in that community think about the ontology’s domain.

Providing a means of translating RDF, RDF-S, and DAML+OIL descriptions into a first-order predicate calculus logical theory not only specifies the intended meaning of the descriptions, but also produces a representation of the descriptions from which inferences can automatically be made using traditional automatic theorem provers and problem solvers. For example, the DAML+OIL axioms enable a reasoner to infer from the two statements “Class Male and class Female are disjointWith.” and “John is type Male.” that the statement “John is type Female.” is false.

Summary. The purpose of this article is to serve as a clear introduction to the modeling languages of the three most widely used IDEF methods: IDEF0, IDEF1X, and IDEF3. Each language is presented in turn, beginning with a discussion of the underlying “ontology” the language purports to describe, followed by presentations of the syntax of the language — particularly the notion of a model for the language — and the semantical rules that determine how models are to be interpreted. The level of detail should be sufficient to enable the reader both to understand the intended areas of application of the languages and to read and construct simple models of each of the three types.

Set theoretic platonism is the view that there exist objective, mind-independent abstract sets, and that set theory is the science of these entities. For the realist, this view offers the most natural semantical account of set theoretic discourse. Nonetheless, set theoretic platonism is beset by a number of serious difficulties. Chief among these, it turns out, is that it must deny the fundamental set theoretic intuition that any available objects can be collected into a further object. After a brief historical overview, I discuss this difficulty at length in Part I of my dissertation, and argue that three responses to it proffered respectively by Lear, Parsons, and Maddy are unsatisfactory. A comprehensive solution to this problem, I suggest, would have to consist in the development of a metaphysically and mathematically adequate alternative ontological framework. Necessary conditions of such a framework's metaphysical adequacy would be that it be rich enough to supplant the platonistic ontology of sets, and that it avoid the problem above. To be mathematically adequate it would have to preserve, in some strong sense, classical set theory . In Part II I develop just such an alternative framework. Beginning with Hao Wang's suggestive elaborations of some of Cantor's remarks, I develop a detailed account according to which set theory is based upon an idealization of our ability to collect objects together mentally. I then go on to show that this account is both metaphysically and mathematically adequate. An appendix is included in which the work of Cantor is discussed

It is almost universally acknowledged that first-order logic (FOL), with its clean, well-understood syntax and semantics, allows for the clear expression of philosophical arguments and ideas. Indeed, an argument or philosophical theory rendered in FOL is perhaps the cleanest example there is of “representing philosophy”. A number of prominent syntactic and semantic properties of FOL reflect metaphysical presuppositions that stem from its Fregean origins, particularly the idea of an inviolable divide between concept and object. These presuppositions, taken at face value, reflect a significant metaphysical viewpoint, one that can in fact hinder or prejudice the representation of philosophical ideas and arguments. Philosophers have of course noticed this and have, accordingly, sought to alter or extend traditional FOL in novel ways to reflect a more flexible and egalitarian metaphysical standpoint. The purpose of this paper, however, is to document and discuss how similar “adaptations” to FOL—culminating in a standardized framework known as Common Logic —have evolved out of the more practical and applied encounter of FOL with the problem of representing, sharing, and reasoning upon information on the World Wide Web.

Process modeling is ubiquitous in business and industry. While a great deal of effort has been devoted to the formal and philosophical investigation of processes, surprisingly little research connects this work to real world process modeling. The purpose of this paper is to begin making such a connection. To do so, we first develop a simple mathematical model of activities and their instances based upon the model theory for the NIST Process Specification Language (PSL), a simple language for describing these entities, and a semantics for the latter in terms of the former, and a set of axioms for the semantics based upon the NIST Process Specification Language (PSL). On the basis of this foundation, we then develop a general notion of a process model, and an account of what it is for such a model to be realized by a collection of events.

Very broadly, an argument from collections is an argument that purports to show that our beliefs about sets imply — in some sense — the existence of God. Plantinga (2007) first sketched such an argument in “Two Dozen” and filled it out somewhat in his 2011 monograph Where the Conflict Really Lies: Religion, Science, and Naturalism. In this paper I reconstruct what strikes me as the most plausible version of Plantinga’s argument. While it is a good argument in at least a fairly weak sense, it doesn’t initially appear to have any explanatory advantages over a non-theistic understanding of sets — what I call set theoretic realism. However, I go on to argue that the theist can avoid an important dilemma faced by the realist and, hence, that Plantinga’s argument from collections has explanatory advantages that realism does not have.

In this paper I rehearse two central failings of traditional possible world semantics. I then present a much more robust framework for intensional logic and semantics based liberally on the work of George Bealer in his book Quality and Concept. Certain expressive limitations of Bealer's approach, however, lead me to extend the framework in a particularly natural and useful way. This extension, in turn, brings to light associated limitations of Bealer's account of predication. In response, I develop a more general and intuitively more adequate account of the logical form of predication.

The Knowledge Interchange Format (KIF) [2] is an ASCII- based framework for use in exchanging of declarative knowledge among disparate computer systems. KIF has been widely used in the fields of knowledge engineering and artificial intelligence. Due to its growing importance, there arose a renewed push to make KIF an offi- cial international standard. A central motivation behind KIF standardization is the wide variation in quality, style, and content — of logic-based frameworks being used for knowledge representation. Variations of all three types, of course, hinder the possibility of semantic integration. A well-crafted logic standard for the representation of declarative knowledge would impose some greatly needed syntactic and semantic uniformity on the current somewhat chaotic situation, uniformity that would in turn greatly enhance the capacity for semantic integration. For all its potential advantages, however, the idea of a logic standard is problematic for at least two reasons.

Consider the following argument: All men are mortal; Socrates is a man; therefore, Socrates is mortal. Intuitively, what makes this a valid argument has nothing to do with Socrates, men, or mortality. Rather, each sentence in the argument exhibits a certain logical form, which, together with the forms of the other two, constitute a pattern that, of itself, guarantees the truth of the conclusion given the truth of the premises. More generally, then, the logical form of a sentence of natural language is what determines both its logical properties and its logical relations to other sentences. The logical form of a sentence of natural language is typically represented in a theory of logical form by a well-formed formula in a ‘logically pure’ language whose only meaningful symbols are expressions with fixed, distinctly logical meanings (e.g., quantifiers). Thus, the logical forms of the sentences in the above argument would be represented in a theory based on pure predicate logic by the formulas ‘∀x(Fx ⊃ Gx)’, ‘Fy’, and ‘Gy’, respectively, where ‘F’, ‘G’, and ‘y’ are all free variables. The argument’s intuitive validity is then explained in virtue of the fact that the logical forms of the premises formally entail the logical form of the conclusion. The primary goal of a theory of logical form is to explain as broad a range of such intuitive logical phenomena as possible in terms of the logical forms that it assigns to sentences of natural language.

In studying the early history of mathematical logic and set theory one typically reads that Georg Cantor discovered the so-called Burali-Forti (BF) paradox sometime in 1895, and that he offered his solution to it in his famous 1899 letter to Dedekind. This account, however, leaves it something of a mystery why Cantor never discussed the paradox in his writings. Far from regarding the foundations of set theory to be shaken, he showed no apparent concern over the paradox and its implications whatever. Against this account, I will argue here that in fact Cantor never saw any paradox at all, but that his conception of set at that time, and already as far back as 1883, was one in which the paradoxes cannot arise.

The Process Specification Language (PSL) has been designed to facilitate correct and complete exchange of process information among manufacturing systems, such as scheduling, process modeling, process planning, production planning, simulation, project management, work flow, and business process reengineering. We given an overview of the theories with the PSL ontology, discuss some of the design principles for the ontology, and finish with examples of process specifications that are based on the ontology.

The fundamental principle of the theory of possible worlds is that a proposition p is possible if and only if there is a possible world at which p is true. In this paper we present a valid derivation of this principle from a more general theory in which possible worlds are defined rather than taken as primitive. The general theory uses a primitive modality and axiomatizes abstract objects, properties, and propositions. We then show that this general theory has very small models and hence that its ontological commitments—and, therefore, those of the fundamental principle of world theory—are minimal

Arthur Prior was a truly philosophical logician. Though he believed formal logic to be worthy of study in its own right, of course, the source of Prior’s great passion for logic was his faith in its capacity for clarifying philosophical issues, untangling philosophical puzzles, and solving philosophical problems. Despite the fact that he has received far less attention than he deserves, Prior has had a profound influence on the development of philosophical and formal logic over the past forty years, a fact to which the present volume bears eloquent witness. The genesis of the volume was the 1989 Arthur Prior Memorial Conference, held appropriately enough at the University of Canterbury in Christchurch, New Zealand, where Prior had his first appointment in philosophy. However, this is not a volume of proceedings. Only eight of the twenty-two essays were actually presented at the conference. The rest were solicited by the editor specially for the volume. The subtitle—Essays on the Legacy of Arthur Prior—is an appropriate one. Few of the essays devote much space to Prior’s work per se. Rather, most address issues that, if not first raised by Prior, were revisited by him and subjected to his typically keen and distinctively original analysis. The papers themselves divide pretty evenly into the technical and the less technical. In this short review I will focus on the latter.

Actualism is the doctrine that the only things there are, that have being in any sense, are the things that actually exist. In particular, actualism eschews possibilism, the doctrine that there are merely possible objects. It is widely held that one cannot both be an actualist and at the same time take possible world semantics seriously — that is, take it as the basis for a genuine theory of truth for modal languages, or look to it for insight into the modal structure of reality. For possible world semantics, it is supposed, commits one to possibilism. In this paper I take issue with this view. To the contrary, I argue that one can take possible world semantics seriously without any commitment to possible worlds or possibilism and hence remain in full compliance with actualist scruples. Moreover, one can do so without without invoking either "ersatz" worlds or haecceities.

The iterative conception of set is typically considered to provide the intuitive underpinnings for ZFCU (ZFC+Urelements). It is an easy theorem of ZFCU that all sets have a definite cardinality. But the iterative conception seems to be entirely consistent with the existence of “wide” sets, sets (of, in particular, urelements) that are larger than any cardinal. This paper diagnoses the source of the apparent disconnect here and proposes modifications of the Replacement and Powerset axioms so as to allow for the existence of wide sets. Drawing upon Cantor’s notion of the absolute infinite, the paper argues that the modifications are warranted and preserve a robust iterative conception of set. The resulting theory is proved consistent relative to ZFC + “there exists an inaccessible cardinal number.”

This paper traces the course of Prior’s struggles with the concepts and phenomena of modality, and the reasoning that led him to his own rather peculiar modal logic Q. I find myself in almost complete agreement with Prior’s intuitions and the arguments that rest upon them. However, I argue that those intuitions do not of themselves lead to Q, but that one must also accept a certain picture of what it is for a proposition to be possible. That picture. though, is not inevitable. Rather, implicit in Prior’s own account is an alternative picture that has already appeared in various guises, most prominently in the work of Adams, Fine, Deutsch, and Almog. I, too, will opt for this alternative, though I will spell it out rather differently than these philosophers. I will then show that, starting with the alternative picture, Prior’s intuitions can lead instead to a much happier and more standard quantified modal logic than Q. The last section of the paper is devoted to the formal development of the logic and its metatheory.