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.”. (shrink)

Bob Hale’s distinguished record of research places him among the most important and influential contemporary analytic metaphysicians. In his deep, wide ranging, yet highly readable book Necessary Beings, Hale draws upon, but substantially integrates and extends, a good deal his past research to produce a sustained and richly textured essay on — as promised in the subtitle — ontology, modality, and the relations between them. I’ve set myself two tasks in this review: first, to provide a reasonably thorough (if not (...) exactly comprehensive) overview of the structure and content of Hale’s book and, second, to a limited extent, to engage Hale’s book philosophically. I approach these tasks more or less sequentially: Parts I and 2 of the review are primarily expository; in Part 3 I adopt a somewhat more critical stance and raise several issues concerning one of the central elements of Hale’s account, his essentialist theory of modality. (shrink)

This article presents an overview of the basic philosophical motivations for, and some recent work in, modal set theory.

To understand the thesis of actualism, consider the following example. Imagine a race of beings — call them ‘Aliens’ — that is very different from any life-form that exists anywhere in the universe; different enough, in fact, that no actually existing thing could have been an Alien, any more than a given gorilla could have been a fruitfly. Now, even though there are no Aliens, it seems intuitively the case that there could have been such things. After all, life might (...) have evolved very differently than the way it did in fact. So in virtue of what is it true that there could have been Aliens when in fact there are none, and when, moreover, nothing that exists in fact could have been an Alien? So-called "possibilists" offer the following answer: ‘It is possible that there are Aliens’ is true because there are in fact individuals that could have been Aliens. At first blush, this might appear directly to contradict the premise that no existing thing could possibly have been an Alien. The possibilist's thesis, however, is that existence, or actuality, encompasses only a subset of the things that, in the broadest sense, are. So for the possibilist, ‘It is possible that there are Aliens’ is true simply in virtue of the fact that there are possible-but-nonactual Aliens, i.e., things that could have existed (but do not) and which would have been Aliens if they had. Actualists reject this answer; they deny that there are any nonactual individuals. Thus, actualism is the philosophical position that everything there is — everything that can in any sense be said to be — exists, or is actual. (shrink)

According to traditional theism, God alone exists a se, independent of all other things, and all other things exist ab alio, i.e., God both creates them and sustains them in existence. On the face of it, divine "aseity" is inconsistent with classical Platonism, i.e., the view that there are objectively existing, abstract objects. For according to the classical Platonist, at least some abstract entities are wholly uncreated, necessary beings and, hence, as such, they also exist a se. The thesis of (...) theistic activism purports to reconcile divine aseity with a robust Platonism. Specifically, the activist holds that God creates the abstract objects no less than the contingent concrete objects of the physical universe and hence that, like all created things, they exist ab alio after all, their necessity notwithstanding. But many philosophers believe a severe roadblock for activism remains — a problem known as the bootstrapping objection. Despite widespread faith in the deliverances of this argument, in this paper I show that the bootstrapping objection is open to significant objections on several fronts. (shrink)

Bringsjord (1985) argues that the definition W of possible worlds as maximal possible sets of propositions is incoherent. Menzel (1986a) notes that Bringsjord’s argument depends on the Powerset axiom and that the axiom can be reasonably denied. Grim (1986) counters that W can be proved to be incoherent without Powerset. Grim was right. However, the argument he provided is deeply flawed. The purpose of this note is to detail the problems with Grim’s argument and to present a sound alternative argument (...) for his conclusion – basically the argument Russell gave to establish a well-known paradox in The Principles of Mathematics. (shrink)

This article includes a basic overview of possible world semantics and a relatively comprehensive overview of three central philosophical conceptions of possible worlds: Concretism (represented chiefly by Lewis), Abstractionism (represented chiefly by Plantinga), and Combinatorialism (represented chiefly by Armstrong).

As the story goes, the source of the paradoxes of naive set theory lies in a conflation of two distinct conceptions of set: the so-called iterative, or mathematical, conception, and the Fregean, or logical, conception. While the latter conception is provably inconsistent, the former, as Godel notes, "has never led to any antinomy whatsoever". More important, the iterative conception explains the paradoxes by showing precisely where the Fregean conception goes wrong by enabling us to distinguish between sets and proper classes, (...) collections that are "too big" to be sets. While I agree wholeheartedly with this distinction, in this paper I argue first that the iterative conception does not provide an explanation of all of the set theoretic paradoxes. I then argue that we need to reconsider the distinction between sets and proper classes rather more carefully. The result will be that ZFC does not capture the iterative conception in its full generality. I close by offering a more general theory that, arguably, does. (shrink)

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. (shrink)

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. (shrink)

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. (shrink)

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. (shrink)

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. (shrink)

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. (shrink)

The authors provide an object-theoretic analysis of two paradoxes in the theory of possible worlds and propositions stemming from Russell and Kaplan. After laying out the paradoxes, the authors provide a brief overview of object theory and point out how syntactic restrictions that prevent object-theoretic versions of the classical paradoxes are justified philosophically. The authors then trace the origins of the Russell paradox to a problematic application of set theory in the definition of worlds. Next the authors show that an (...) object-theoretic analysis of the Kaplan paradox reveals that there is no genuine paradox at all, as the central premise of the paradox is simply a logical falsehood and hence can be rejected on the strongest possible grounds—not only in object theory but for the very framework of propositional modal logic in which Kaplan frames his argument. The authors close by fending off a possible objection that object theory avoids the Russell paradox only by refusing to incorporate set theory and, hence, that the object-theoretic solution is only a consequence of the theory’s weakness. (shrink)

A central stream running through the history of philosophy has been the attempt to gather a wide range of ostensibly disparate intuitive phenomena under a small, integrated set of concepts. Edward Zalta’s work is a sustained celebration of this tradition. This paper — part of a symposium on Zalta's work — is a friendly, but critical examination of Zalta's commitment to possibilism and the roles they play in his theory.

Suppose we believe that God created the world. Then surely we want it to be the case that he intended, in some sense at least, to create THIS world. Moreover, most theists want to hold that God didn't just guess or hope that the world would take one course or another; rather, he KNEW precisely what was going to take place in the world he planned to create. In particular, of each person P, God knew that P was to exist. (...) Call this the "standard" conception. Most theists find the standard conception appealing. Unfortunately, the view seems to conflict with the equally appealing idea — call it "temporal actualism" — that there are no "future" individuals beyond those that already exist in the present moment. For, on this view, for any historical person P, prior to creation, there was no such person as P and hence nothing about P for God to know. Hence, in particular, God couldn't have known that P was to exist. This is of course not a new problem. But past solutions to it are highly problematic. In this paper, after canvassing previous approaches, I will propose a solution that seems to preserve both temporal actualism and a suitably robust form of the standard conception while avoiding the pitfalls of the past. (shrink)

The use of highly abstract mathematical frameworks is essential for building the sort of theoretical foundation for semantic integration needed to bring it to the level of a genuine engineering discipline. At the same time, much of the work that has been done by means of these frameworks assumes a certain amount of background knowledge in mathematics that a lot of people working in ontology, even at a fairly high theoretical level, lack. The major purpose of this short paper is (...) provide a (comparatively) simple model of semantic integration that remains within the friendlier confines of first-order languages and their usual classical semantics and logic. (shrink)

In a previous paper, Thomas V. Morris and I sketched a view on which abstract objects, in particular, properties, relations, and propositions , are created by God no less than contingent, concrete objects. In this paper r suggest a way of extending this account to cover mathematical objects as well. Drawing on some recent work in logic and metaphysics, I also develop a more detailed account of the structure of PRPs in answer to the paradoxes that arise on a naive (...) understanding of the structure ofthe abstract universe. (shrink)

According to many actualists, propositions, singular propositions in particular, are structurally complex, that is, roughly, (i) they have, in some sense, an internal structure that corresponds rather directly to the syntactic structure of the sentences that express them, and (ii) the metaphysical components, or constituents, of that structure are the semantic values — the meanings — of the corresponding syntactic components of those sentences. Given that reference is "direct", i.e., that the meaning of a name is its denotation, an apparent (...) consequence of this view is that any proposition expressed by a sentence containing a name that denotes a contingent being S is itself contingent — notably, the proposition [S does not exist]. Assuming that an entity must exist to have a property, necessarily, [S does not exist] must exist in order to be true. It seems to follow that, necessarily, [S does not exist] is not true and, hence, that S is not contingent after all. Past approaches to the problem — notably, those of Prior and Adams — lead to highly undesirable consequences for quantified modal logic. In this paper, several solutions to this puzzle are developed that preserve actualism, the structured view of propositions, the direct theory of reference, and the intuition that [S does not exist] is indeed possible without the adverse consequences for QML of previous solutions. (shrink)

Recent work in the philosophy of mathematics has suggested that mathematical structuralism is not committed to a strong form of the Identity of Indiscernibles (II). José Bermúdez demurs, and argues that a strong form of II can be warranted on structuralist grounds by countenancing identity properties, or haecceities, as legitimately structural. Typically, structuralists dismiss such properties as obviously non-structural. I will argue to the contrary that haecceities can be viewed as structural but that this concession does not warrant Bermdez’s version (...) of II but, rather, another easily falsified version. I close with some reflections on reference vis-à-vis structurally indiscernible objects. (shrink)

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.

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. (shrink)

In this paper, an argument of Alvin Plantinga's for the existence of abstract possible worlds is shown to be unsound. The argument is based on a principle Plantinga calls "Quasicompactness", due to its structural similarity to the notion of compactness in first-order logic. The principle is shown to be false.

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. (shrink)

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. (shrink)

In this report I motivate and develop a type-free logic with predicate quantifiers within the general ontological framework of properties, relations, and propositions. In Part I, I present the major ideas of the system informally and discuss its philosophical significance, especially with regard to Russell's paradox. In Part II, I prove the soundness, consistency, and completeness of the logic.

In this paper, an objective conception of contexts based loosely upon situation theory is developed and formalized. Unlike subjective conceptions, which take contexts to be something like sets of beliefs, contexts on the objective conception are taken to be complex, structured pieces of the world that (in general) contain individuals, other contexts, and propositions about them. An extended first-order language for this account is developed. The language contains complex terms for propositions, and the standard predicate "ist" that expresses the relation (...) that holds between a context and a proposition just in case the latter is true in the former. The logic for the objective conception features a global classical predicate calculus, a local logic for reasoning within contexts, and axioms for propositions. The specter of paradox is banished from the logic by allowing "ist" to be nonbivalent in problematic cases: it is not in general the case, for any context c and proposition p, that either ist(c,p) or ist(c, ¬p). An important representational capability of the logic is illustrated by proving an appropriately modified version of an illustrative theorem from McCarthy's classic Blocks World example. (shrink)

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. (shrink)

I address Grosholz's critique of Resnik's mathematical structuralism and suggest that although Resnik's structuralism is not without its difficulties it survives Grosholz's attacks.