Roy T. Cook

One of the main problems plaguing neo-logicism is the Bad Company challenge: the need for a well-motivated account of which abstraction principles provide legitimate definitions of mathematical concepts. In this article a solution to the Bad Company challenge is provided, based on the idea that definitions ought to be conservative. Although the standard formulation of conservativeness is not sufficient for acceptability, since there are conservative but pairwise incompatible abstraction principles, a stronger conservativeness condition is sufficient: that the class of acceptable abstraction principles be strictly logically symmetrically class conservative . The article concludes with an examination of which classes of abstraction principles meet this criteria

In this paper I examine the prospects for a successful neo–logicist reconstruction of the real numbers, focusing on Bob Hale's use of a cut-abstraction principle. There is a serious problem plaguing Hale's project. Natural generalizations of this principle imply that there are far more objects than one would expect from a position that stresses its epistemological conservativeness. In other words, the sort of abstraction needed to obtain a theory of the reals is rampantly inflationary. I also indicate briefly why this problem is likely to reappear in any neo–logicist reconstruction of real analysis.

A neologicist set theory based on an abstraction principle (NewerV) codifying the iterative conception of set is investigated, and its strength is compared to Boolos's NewV. The new principle, unlike NewV, fails to imply the axiom of replacement, but does secure powerset. Like NewV, however, it also fails to entail the axiom of infinity. A set theory based on the conjunction of these two principles is then examined. It turns out that this set theory, supplemented by a principle stating that there are infinitely many nonsets, captures all (or enough) of standard second-order ZFC. Issues pertaining to the axiom of foundation are also investigated, and I conclude by arguing that this treatment provides the neologicist with the most viable reconstruction of set theory he is likely to obtain

Truth values are, properly understood, merely proxies for the various relations that can hold between language and the world. Once truth values are understood in this way, consideration of the Liar paradox and the revenge problem shows that our language is indefinitely extensible, as is the class of truth values that statements of our language can take – in short, there is a proper class of such truth values. As a result, important and unexpected connections emerge between the semantic paradoxes and the set-theoretic paradoxes.

A number of authors have argued that Peano Arithmetic supplemented with a logical validity predicate is inconsistent in much the same manner as is PA supplemented with an unrestricted truth predicate. In this paper I show that, on the contrary, there is no genuine paradox of logical validity—a completely general logical validity predicate can be coherently added to PA, and the resulting system is consistent. In addition, this observation lead to a number of novel, and important, insights into the nature of logical validity itself

Stewart Shapiro and Alan Weir have argued that a crucial part of the demonstration of Frege's Theorem (specifically, that Hume's Principle implies that there are infinitely many objects) fails if the Neo-logicist cannot assume the existence of the empty property, i.e., is restricted to so-called Aristotelian Logic. Nevertheless, even in the context of Aristotelian Logic, Hume's Principle implies much of the content of Peano Arithmetic. In addition, their results do not constitute an objection to Neo-logicism so much as a clarification regarding the view of logic that the Neo-logicist must take.

Timothy Williamson’s Modal Logic as Metaphysics is a book-length defense of necessitism about objects—roughly put, the view that, necessarily, any object that exists, exists necessarily. In more formal terms, Williamson argues for the validity of necessitism for objects (NO: ◻︎∀x◻︎∃y(x=y)). NO entails both the (first-order) Barcan formula (BF: ◇∃xΦ → ∃x◇Φ, for any formula Φ) and the (first-order) converse Barcan formula (CBF: ∃x◇Φ → ◇∃xΦ, for any formula Φ). The purpose of this essay is not to assess Williamson’s arguments either for necessitism (although discussion of these arguments will play a central role in the dialectic) or for necessitism’s two famous corollaries. Instead, the focus shall be a general principle governing abstract objects—the abstract of principle (or AOP) —instances of which seems to be at work in some of Williamson’s central arguments for necessitism. The AOP can be straightforwardly formulated and applied within the neo-logicist framework—in fact, arguably the principle is most naturally formulated in neo-logicist terms. After closely examining, and carefully formalizing, the AOP, the remainder of the paper focuses on arguments for necessitism-like claims (the exact meaning of “necessitism-like” will become clearer as the essay progresses) based on the AOP. In particular, we shall focus on the instance of the AOP that applies to the abstract objects governed by the most well-known and most fully studied abstraction principle: Hume’s Principle (HP). It turns out that, although we cannot reconstruct a valid argument for necessitism based on this numerical instance of the AOP, we can obtain valid arguments for weaker, but equally interesting conclusions. In particular, we shall show that, although HP combined with the AOP (and some additional, related assumptions) allows the contents of the domains of possible worlds to vary, the size of those domains must remain constant. The paper concludes by developing and critiquing some related arguments for necessitism based on applying relevant instances of the AOP to abstraction principles governing sets (or extensions), and to a simple objectual abstraction principle.

Fine and Antonelli introduce two generalizations of permutation invariance — internal invariance and simple/double invariance respectively. After sketching reasons why a solution to the Bad Company problem might require that abstraction principles be invariant in one or both senses, I identify the most fine-grained abstraction principle that is invariant in each sense. Hume’s Principle is the most fine-grained abstraction principle invariant in both senses. I conclude by suggesting that this partially explains the success of Hume’s Principle, and the comparative lack of success in reconstructing areas of mathematics other than arithmetic based on non-invariant abstraction principles.

It is sometimes suggested that impure sets are spatially co-located with their members (and hence are located in space). Sets, however, are in important respects like numbers. In particular, sets are connected to concepts in much the same manner as numbers are connected to concepts—in both cases, they are fundamentally abstracts of (or corresponding to) concepts. This parallel between the structure of sets and the structure of numbers suggests that the metaphysics of sets and the metaphysics of numbers should parallel each other in relevant ways. This entails, in turn, that impure sets are not co-located with their members (nor are they located in space)

The No-No Paradox consists of a pair of statements, each of which ?says? the other is false. Roy Sorensen claims that the No-No Paradox provides an example of a true statement that has no truthmaker: Given the relevant instances of the T-schema, one of the two statements comprising the ?paradox? must be true (and the other false), but symmetry constraints prevent us from determining which, and thus prevent there being a truthmaker grounding the relevant assignment of truth values. Sorensen's view is mistaken: situated within an appropriate background theory of truth, the statements comprising the No-No Paradox are genuinely paradoxical in the same sense as is the Liar (and thus, on Sorensen's view, must fail to have truth values). This result has consequences beyond Sorensen's semantic framework. In particular, the No-No Paradox, properly understood, is not only a new paradox, but also provides us with a new type of paradox, one which depends upon a general background theory of the truth predicate in a way that the Liar Paradox and similar constructions do not

A difficulty for alethic pluralism has been the idea that semantic evaluation of conjunctions whose conjuncts come from discourses with distinct truth properties requires a third notion of truth which applies to both of the original discourses. But this line of reasoning does not entail that there exists a single generic truth property that applies to all statements and all discourses, unless it is supplemented with additional, controversial, premises. So the problem of mixed conjunctions, while highlighting other aspects of alethic pluralism worth investigating further, does not constitute an effective objection to it

During the Winter of 2011 I visited SADAF and gave a series of talks based on the central chapters of my manuscript on the Yablo paradox. The following year, I visited again, and was pleased and honored to find out that Eduardo Barrio and six of his students had written ‘responses’ that addressed the claims and arguments found in the manuscript, as well as explored new directions in which to take the ideas and themes found there. These comments reflect my thoughts on these responses (also collected in this issue), as well as my thoughts on further issues that arose during the symposium that was based on the papers and during the many hours I spent talking and working with Eduardo and his students. Durante el invierno de 2011, visité SADAF y dí una serie de conferencias sobre los capítulos centrales de mi manuscrito sobre la paradoja de Yablo. El año siguiente, visité Buenos aires nuevamente y tuve el placer y el honor de descubrir que Eduardo Barrio y seis de sus estudiantes habían escrito respuestas que abordan las afirmaciones y argumentos de mi manuscrito, además de explorar nuevas direcciones en las cuales considerar las ideas y temas encontrados allí. El presente trabajo refleja mis pensamientos sobre estas respuestas (también incluidas en este volumen), así como mis ideas sobre otras cuestiones que surgieron durante el simposio en el que se presentaron los artículos y en las muchas horas que nosotros discutimos y trabajamos con Eduardo y sus estudiantes