Gabriel Uzquiano

The Russell–Myhill theorem threatens a familiar structured conception of propositions according to which two sentences express the same proposition only if they share the same syntactic structure and their corresponding syntactic constituents share the same semantic value. Given the role of the principle of universal instantiation in the derivation of the theorem in simple type theory, one may hope to rehabilitate the core of the structured view of propositions in ramified type theory, where the principle is systematically restricted. We suggest otherwise. The ramified core of the structured theory of propositions remains inconsistent in ramified type theory augmented with axioms of reducibility. This is significant because reducibility has been thought to be perfectly consistent with the ramified approach to the intensional antinomies. Nor is the addition of reducibility to ramified type theory sufficient to restore other intensional puzzles such as Prior’s paradox or Kripke’s puzzle about time and thought.

According to the structured theory of propositions, if two sentences express the same proposition, then they have the same syntactic structure, with corresponding syntactic constituents expressing the same entities. A number of philosophers have recently focused attention on a powerful argument against this theory, based on a result by Bertrand Russell, which shows that the theory of structured propositions is inconsistent in higher order-logic. This paper explores a response to this argument, which involves restricting the scope of the claim that propositions are structured, so that it does not hold for all propositions whatsoever, but only for those which are expressible using closed sentences of a given formal language. We call this restricted principle Closed Structure, and show that it is consistent in classical higher-order logic. As a schematic principle, the strength of Closed Structure is dependent on the chosen language. For its consistency to be philosophically significant, it also needs to be consistent in every extension of the language which the theorist of structured propositions is apt to accept. But, we go on to show, Closed Structure is in fact inconsistent in a very natural extension of the standard language of higher-order logic, which adds resources for plural talk of propositions. We conclude that this particular strategy of restricting the scope of the claim that propositions are structured is not a compelling response to the argument based on Russell’s result, though we note that for some applications, for instance to propositional attitudes, a restricted thesis in the vicinity may hold some promise.

David Kaplan observed in Kaplan that the principle \\) cannot be verified at a world in a standard possible worlds model for a quantified bimodal propositional language. This raises a puzzle for certain interpretations of the operator Q: it seems that some proposition p is such that is not possible to query p, and p alone. On the other hand, Arthur Prior had observed in Prior that on pain of contradiction, ∀p is Q only if one true proposition is Q and one false proposition is Q. The two observations are related: ∀p is elusive in that it is not possible for the proposition to be uniquely Q. Kaplan based his model-theoretic observation on Cantor’s theorem, but there is a less well-known link between this simple set-theoretic observation and Prior’s remark. We generalize the link to develop a heuristic designed to move from Cantor’s theorem to the observation that a variety of sentences of the bimodal language express propositions that cannot be Q uniquely. We highlight the analogy between some of these results and some set-theoretic antinomies and suggest that the phenomenon is richer than one may have anticipated.

This chapter takes a close look at the thought that mereological relations on material objects mirror, and are mirrored by, parallel mereological relations on their exact locations. This hypothesis is made more precise by means of a battery of principles from which more substantive consequences are derived. Mereological harmony turns out to entail, for example, that atomistic space is an inhospitable environment for material gunk or that Whiteheadian space is not a hospitable environment for unextended material atoms.

The purpose of this article is to assess the prospects for a Scottish neo-logicist foundation for a set theory. We show how to reformulate a key aspect of our set theory as a neo-logicist abstraction principle. That puts the enterprise on the neo-logicist map, and allows us to assess its prospects, both as a mathematical theory in its own right and in terms of the foundational role that has been advertised for set theory. On the positive side, we show that our abstraction based theory can be modified to yield much of ordinary mathematics, indeed everything needed to recapture all branches of mathematics short of set theory itself. However, our conclusions are mostly negative. The theory will fall far short of the power of ordinary Zermelo-Fraenkel set theory. It is consistent that our set theory has models that are relatively small, smaller than the first cardinal with an uncountable index. More important, there is a strong tension between the idea that the iterative hierarchy is somehow ineffable, or indefinitely extensible, and the neo-logicist theme of capturing mathematical theories with abstraction principles.

Groups are ubiquitous in our lives. But while some of them are highly structured and appear to support a shared intentionality and even a shared agency, others are much less cohesive and do not seem to demand much of their individual members. Queues, for example, seem to be, at a given time, nothing over and above some individuals as they exemplify a certain spatial arrangement. Indeed, the main aim of this paper is to develop the more general thought that at a given time, a group is nothing over and above some individual members as they exemplify a certain complex condition. The general conception of groups that emerges is able to accommodate a variety of constraints on a reasonable answer to the question of what are groups.

I present a puzzle for absolutely unrestricted quantification. One important advantage of absolutely unrestricted quantification is that it allows us to entertain perfectly general theories. Whereas most of our theories restrict attention to one or another parcel of reality, other theories are genuinely comprehensive taking absolutely all objects into their domain. The puzzle arises when we notice that absolutely unrestricted theories sometimes impose incompatible constraints on the size of the universe.

Thomas Hofweber has written a very rich book. In line with the conviction that ontology should be informed by linguistic considerations, he develops a systematic approach to central ontological questions as they arise in different regions of discourse. More generally, the book seeks to cast light upon the nature of ontology and its proper place in enquiry. His preferred methodology is not without consequence: it promises, for example, to solve what otherwise look like intractable philosophical puzzles raised by arithmetical practice and numerical discourse, and, likewise, his treatment of propositional discourse has ramifications for larger questions to do with the prospects of metaphysical enquiry generally.

The aim of the present paper is twofold. One task is to argue that our use of the numerical vocabulary in theory and applications determines the reference of the numerical terms more precisely than up to isomorphism. In particular our use of the numerical vocabulary in modal and counterfactual contexts of application excludes contingent existents as candidate referents for the numerical terms. The second task is to explore the impact of this conclusion on what I call semantic nominalism, which is the view that ordinary physical objects and other contingent existents are eligible to be the referents of numerical terms. I suggest that for semantic nominalism to have a chance we must reform our usage of the numerical vocabulary in modal and counterfactual contexts. I tentatively explore some ways in which this might be done and call attention to some of their costs.

The aim of the present paper is twofold. One task is to argue that our use of the numerical vocabulary in theory and applications determines the reference of the numerical terms more precisely than up to isomorphism. In particular our use of the numerical vocabulary in modal and counterfactual contexts of application excludes contingent existents as candidate referents for the numerical terms. The second task is to explore the impact of this conclusion on what I call semantic nominalism, which is the view that ordinary physical objects and other contingent existents are eligible to be the referents of numerical terms. I suggest that for semantic nominalism to have a chance we must reform our usage of the numerical vocabulary in modal and counterfactual contexts. I tentatively explore some ways in which this might be done and call attention to some of their costs.

The paper is concerned with the bad company problem as an instance of a more general difficulty in the philosophy of mathematics. The paper focuses on the prospects of stability as a necessary condition on acceptability. However, the conclusion of the paper is largely negative. As a solution to the bad company problem, stability would undermine the prospects of a neo-Fregean foundation for set theory, and, as a solution to the more general difficulty, it would impose an unreasonable constraint on mathematical practice.

We argue that certain modal questions raise serious problems for a modal metaphysics on which we are permitted to quantify unrestrictedly over all possibilia. In particular, we argue that, on reasonable assumptions, both David Lewis's modal realism and Timothy Williamson's necessitism are saddled with the remarkable conclusion that there is some cardinal number of the form ℵα such that there could not be more than ℵα-many angels in existence. In the last section, we make use of similar ideas to draw a moral for a recent debate in meta-ontology

Do mereological fusions have their parts necessarily? None of the axioms of non-modal formulations of classical mereology appear to speak directly to this question. And yet a great many philosophers who take the part-whole relation to be governed by classical mereology seem to assume that they do. In addition to this, many philosophers who make allowance for the part-whole relation to obtain merely contingently between a part and a mereological fusion tend to depart from non-modal formulations of classical mereology at least when it comes to the axiom of Unique Fusion, which states that no two different mereological fusions ever fuse exactly the same objects. This is no coincidence. There are reasons of principle why one’s adherence to classical mereology should exert some pull towards the view that mereological fusions have their parts necessarily. There is, however, no direct route from the combination of classical mereology and propositional modal logic to the hypothesis that the part-whole relation obtains necessarily between a part and a mereological fusion. In order to bridge between a modal formulation of classical mereology and the hypothesis that fusions have their parts necessarily, one needs to strengthen the axiom of Unrestricted Fusion in a way that is agreeable to many philosophers on both sides of the debate.

"Ontology and the Foundations of Mathematics" consists of three papers concerned with ontological issues in the foundations of mathematics. Chapter 1, "Numbers and Persons," confronts the problem of the inscrutability of numerical reference and argues that, even if inscrutable, the reference of the numerals, as we ordinarily use them, is determined much more precisely than up to isomorphism. We argue that the truth conditions of a variety of numerical modal and counterfactual sentences place serious constraints on the sorts of items to which numerals, as we ordinarily use them, can be taken to refer: Numerals cannot be taken to refer to objects that exist contingently such as people, mountains, or rivers, but rather must be taken to refer to objects that exist necessarily such as abstracta. ;Chapter 2, "Modern Set Theory and Replacement," takes up a challenge to explain the reasons one should accept the axiom of replacement of Zermelo-Fraenkel set theory, when its applications within ordinary mathematics and the rest of science are often described as rare and recondite. We argue that this is not a question one should be interested in; replacement is required to ensure that the element-set relation is well-founded as well as to ensure that the cumulation of sets described by set theory reaches and proceeds beyond the level o of the cumulative hierarchy. A more interesting question is whether we should accept instances of replacement on uncountable sets, for these are indeed rarely used outside higher set theory. We argue that the best case for replacement comes not from direct, intuitive considerations, but from the role replacement plays in the formulation of transfinite recursion and the theory of ordinals, and from the fact that it permits us to express and assert the content of the modern cumulative view of the set-theoretic universe as arrayed in a cumulative hierarchy of levels. ;Chapter 3, "A No-Class Theory of Classes," makes use of the apparatus of plural quantification to construe talk of classes as plural talk about sets, and thus provide an interpretation of both one- and two-sorted versions of first-order Morse-Kelley set theory, an impredicative theory of classes. We argue that the plural interpretation of impredicative theories of classes has a number of advantages over more traditional interpretations of the language of classes as involving singular reference to gigantic set-like entities, only too encompassing to be sets, the most important of these being perhaps that it makes the machinery of classes available for the formalization of much recent and very interesting work in set theory without threatening the universality of the theory as the most comprehensive theory of collections, when these are understood as objects

George Boolos has described an interpretation of a fragment of ZFC in a consistent second-order theory whose only axiom is a modification of Frege's inconsistent Axiom V. We build on Boolos's interpretation and study the models of a variety of such theories obtained by amending Axiom V in the spirit of a limitation of size principle. After providing a complete structural description of all well-founded models, we turn to the non-well-founded ones. We show how to build models in which foundation fails in prescribed ways. In particular, we obtain models in which every relation is isomorphic to the membership relation on some set as well as models of Aczel's anti-foundation axiom (AFA). We suggest that Fregean extensions provide a natural way to envisage non-well-founded membership.

The principle of universal instantiation plays a pivotal role both in the derivation of intensional paradoxes such as Prior’s paradox and Kaplan’s paradox and the debate between necessitism and contingentism. We outline a distinctively free logical approach to the intensional paradoxes and note how the free logical outlook allows one to distinguish two different, though allied themes in higher-order necessitism. We examine the costs of this solution and compare it with the more familiar ramificationist approaches to higher-order logic. Our assessment of both approaches is largely pessimistic, and we remain reluctantly inclined to take Prior’s and Kaplan’s derivations at face value.