Christopher Menzel

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.

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

Textual and historical subtleties aside, let's call the idea that numbers are properties of equinumerous sets ‘the Fregean thesis.’ In a recent paper, Palle Yourgrau claims to have found a decisive refutation of this thesis. More surprising still, he claims in addition that the essence of this refutation is found in the Grundlagen itself – the very masterpiece in which Frege first proffered his thesis. My intention in this note is to evaluate these claims, and along the way to shed some light on relevant passages of the Grundlagen. I will argue that Yourgrau does not make his case.The arguments with which we are concerned are found in the last three sections of Yourgrau's paper. A pervasive difficulty in these sections is that it is not clear exactly what Yourgrau is arguing against. The stated object of his attack is the Fregean thesis, a thesis about what numbers are; however, instead of a frontal assault, his strategy is to embark on a foray into the ill-defined issue of what it is that numbers number, where, roughly speaking, a number n numbers an object x just in case n can be legitimately assigned to x. The reason for this shift in emphasis appears to be rooted in a misconception. As we’ll see in more detail shortly, Yourgrau's argument against the Fregean thesis is based on an extension of a well known argument of Frege's found in §§22-3 of the Grundlagen, which Glenn Kessler has tagged the ‘relativity argument,’. According to Yourgrau, this is an argument ‘to the effect that what is literally numbered cannot simply be concrete objects’. This is incorrect. Frege himself clarifies the point of the argument in §21 with the following preface:In language, numbers (i.e., numerals] most commonly appear in adjectival form and attributive construction in the same sort of way as the words "hard" or "heavy" or "red," which have for their meanings properties of external things. It is natural to ask whether we must think of the individual numbers too as such properties, and whether, accordingly, the concept of number can be classed along with that, say, of color.

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.

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.

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.

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

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.

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.

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 Bermúdez’s version of II but, rather, another easily falsified version. I close with some reflections on reference vis-à-vis structurally indiscernible objects.

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.