Gabriel Uzquiano

Toward a Theory of Second-Order Consequence

We develop an account of logical consequence for the second-order language of set theory in the spirit of Booloss plural interpretation of monadic second-order logic. There is little doubt that a second-order axiomatization of Zermelo-Fraenkel set the- ory plus the axiom of choice (ZFC) is desirable. One advantage of such an ax- iomatization is that it permits us to express the principles underlying the first-order schemata of separation and replacement. Another is its almost-categoricity: M is a model of second-order ZFC if and only if it is isomorphic to a model of the form � Vκ, ∈∩ (Vκ × Vκ)� ,f orκ a strongly inaccessible ordinal. We obtain similar benefits when we allow for the existence of Urelemente. The axioms of second-order ZFC with Urelemente (ZFCU) are not able to specify the structure of the universe up to isomorphism, but McGee has recently shown that, pro- vided one takes the range of its quantifiers to be unrestricted, the addition of an axiom that states that the Urelemente form a set to the axioms of ZFCU will characterize the structure of the universe of pure sets up to isomorphism. 1 In sum, there is much to be gained from the ability to employ second-order quantification in the context of set theory. What is much more controversial is that we can, with a clear conscience, develop set theory within a second-order language. The standard interpretation of second- order quantification takes second-order variables to range over the sets of individuals which first-order variables range over. This interpretation may be convenient for the development of second-order arithmetic but it will not do for the purpose of develop- ing set theory in a second-order language. The reason is not difficult to state. When we do set theory, we take our first-order variables to range over all sets. But if we take our second-order variables to range over sets of sets in the range of the first-order vari- ables, then second-order comprehension will fail. A simple instance of second-order comprehension such as ∃X∀y (Xy←→ y / ∈ y) will be false on account of Russells

Higher-order free logic and the Prior-Kaplan paradox

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.

Which Abstraction Principles are Acceptable? Some Limitative Results

Neo-Fregean logicism attempts to base mathematics on abstraction principles. Since not all abstraction principles are acceptable, the neo-Fregeans need an account of which ones are. One of the most promising accounts is in terms of the notion of stability; roughly, that an abstraction principle is acceptable just in case it is satisfiable in all domains of sufficiently large cardinality. We present two counterexamples to stability as a sufficient condition for acceptability and argue that these counterexamples can be avoided only by major departures from the existing neo-Fregean programme. 1. Introduction2. A Simple Counterexample3. A Fregean Counterexample4. The Argument 4.1. Defending step 14.2. Defending step 24.3. Defending step 34.4. Defending step 45. Concluding Remarks Introduction A Simple Counterexample A Fregean Counterexample The Argument 4.1. Defending step 14.2. Defending step 24.3. Defending step 34.4. Defending step 4 Defending step 1 Defending step 2 Defending step 3 Defending step 4 Concluding Remarks

A neglected resolution of Russell's paradox of propositions

Bertrand Russell oered an inuential paradox of propositions in Appendix B of The Principles of Mathematics, but there is little agreement as to what to conclude from it. We suggest that Russells paradox is best regarded as a limitative result on propositional granularity. Some propositions are, on pain of contradiction, unable to discriminate between classes with dierent members: whatever they predicate of one, they predicate of the other. When accepted, this remarkable fact should cast some doubt upon some of the uses to which modern descendants of Russells paradox of propositions have been put in recent literature.

Bad company generalized

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.

Quantification Without a Domain

Much recent work in the philosophy of mathematics has been concerned with indefinite extensibility and the problem of absolute generality. It is not uncommon to take the set-theoretic paradoxes to illustrate a phenomenon of indefinite extensibility whereby certain concepts, for example, set and ordinal, are indefinitely extensible. What is less clear is how to articulate this response to paradox or what it means for the prospects of absolute generality. While some philosophers take the indefinite extensibility of certain concepts to imply the unavailability of unrestricted quantification over their instances, others seem inclined to conclude that there is no comprehensive domain.1 These conclusions are importantly different: One seems to place a serious limitation on thought and language while the other might seem a subtle ontological discovery.

FREGE MEETS ZERMELO: A PERSPECTIVE ON INEFFABILITY AND REFLECTION

There are at least two heuristic motivations for the axioms of standard set theory, by which we mean, as usual, first-order Zermelo-Fraenkel set theory with the axiom of choice (ZFC): the iterative conception and limitation of size (see (Boolos, 1989)). Each strand provides a rather hospitable environment for the hypothesis that the set-theoretic universe is ineffable, which is our target in this paper, although the motivation is different in each case. Contemporary ZFC does not countenance urelements. It is a theorem that every set is a member of some Vα, where, as usual:

Some Results on the Limits of Thought

Generalizing on some arguments due to Arthur Prior and Dmitry Mirimanoff, we provide some further limitative results on what can be thought.