Roy T. Cook

I propose a novel way of viewing the connection between mathematical discourse and the mathematical logician's formalizations of it. We should abandon the idea that formalizations are accurate descriptions of mathematical activity. Instead, logicians are in the business of supplying models in much the same way that a mathematical physicist formulates models of physical phenomena or the hobbyist constructs models of ships. ;I first examine problems with the traditional view, and I survey some prior work in the spirit of the logic-as-model, including that of Gerhard Gentzen, Georg Kreisel, and Imre Lakatos. The utility of the present framework is then demonstrated in three case studies. ;First, this approach solves a purported problem with precise semantics for vagueness. Precise semantics supposedly err by replacing the defining characteristic of vagueness with precise cut-offs, but on the logic-as-model approach this precision can be seen as an artifact serving to simplify the model but corresponding to nothing actually found in vague languages. ;Second, on the logic-as-model view the advantages claimed for branching quantifiers, i.e. providing the expressiveness of second-order languages while avoiding their semantic intractability, are illusory. On the one hand, second-order formulations of mathematical concepts are usually far more natural than branching quantifier formulations; on the other hand, branching quantifiers are no more tractable than second-order languages. ;Finally, I examine the claim that Gottfried Leibniz's infinitesimal calculus is nothing other than Abraham Robinson's non-standard analysis. Although non-standard analysis is a useful model, there are other formalizations of Leibniz's mathematics as fruitful as, yet incompatible with, the non-standard approach, a fact hard to reconcile with the idea that logic provides accurate descriptions. ;After the case studies, I examine the objectivity of logic from the logic-as-model framework. This issue is difficult enough on traditional accounts of logic, and would seem to become even more difficult from the present point of view. Nevertheless, a rich picture of the objectivity of logic is worked out with interesting new sorts of indeterminacy arising from the idea that logic provides models and not descriptions.

Es bestehen tiefgreifende Zusammenhänge zwischen Leibniz' Mathematik und seiner Metaphysik. Dieser Aufsatz hat das Ziel, das Verständnis für diese beiden Bereiche zu erweitern, indem er Leibniz' Mereologie (die Theorie der Teile und des Ganzen) näher untersucht. Zunachst wird Leibniz' Mereologie primär anhand seiner Schrift “Initia rerum mathematicarum metaphysica" rekonstruiert. Dieses ehrgeizige Programm beginnt mit dem einfachen Begriff der Kompräsenz, geht dann iiber zu komplexeren Begriffen wie Gleichheit, Ähnlichkeit und Homogenität und kulminiert letztlich in der Leibnizschen Definition der Begriffe Teil, Ganzes und Komposition. Im Verlauf des Aufsatzes werden auch weitere Erkenntnisse in anderen Bereichen gewonnen, so z. B. in bezug auf die Identität des Ununterscheidbaren. Schließlich werde ich im Rahmen der vorgelegten Theorie versuchen, ein Mißverständnis bezüglich Leibniz' Analogie zwischen Monaden und mathematischen Punkten, welche sich auf die Räumlichkeit der Monaden bezieht, auszuräumen.

This dictionary introduces undergraduate and post-graduate students in philosophy, mathematics, and computer science to the main problems and positions in philosophical logic. Coverage includes not only key figures, positions, terminology, and debates within philosophical logic itself, but issues in related, overlapping disciplines such as set theory and the philosophy of mathematics as well. Entries are extensively cross-referenced, so that each entry can be easily located within the context of wider debates, thereby providing a valuable reference both for tracking the connections between concepts within logic and for examining the manner in which these concepts are applied in other philosophical disciplines. Roy T. Cook is Assistant Professor in the Department of Philosophy at the University of Minnesota and an Associate Fellow at Arché, the Philosophical Research Centre for Logic, Language, Metaphysics and Epistemology at the University of St Andrews. He works primarily in the philosophy of logic, language, and mathematics, and has also published papers on seventeenth century philosophy.

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.

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

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.

We begin with a prepositional languageLpcontaining conjunction (Λ), a class of sentence names {Sα}αϵA, and a falsity predicateF. We (only) allow unrestricted infinite conjunctions, i.e., given any non-empty class of sentence names {Sβ}βϵB,is a well-formed formula (we will useWFFto denote the set of well-formed formulae).The language, as it stands, is unproblematic. Whether various paradoxes are produced depends on which names are assigned to which sentences. What is needed is a denotation function:For example, theLPsentence “F(S1)” (i.e.,Λ{F(S1)}), combined with a denotation functionδsuch thatδ(S1)“F(S1)”, provides the (or, in this context, a)Liar Paradox.To give a more interesting example,Yablo's Paradox[4] can be reconstructed within this framework.Yablo's Paradoxconsists of an ω-sequence of sentences {Sk}kϵωwhere, for eachnϵω:WithinLPan equivalent construction can be obtained using infinite conjunction in place of universal quantification - the sentence names are {Si}iϵωand the denotation function is given by:We can express this in more familiar terms as:etc.

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)

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.

On the Dummettian understanding, anti-realism regarding a particular discourse amounts to (or at the very least, involves) a refusal to accept the determinacy of the subject matter of that discourse and a corresponding refusal to assert at least some instances of excluded middle (which can be understood as expressing this determinacy of subject matter). In short: one is an anti-realist about a discourse if and only if one accepts intuitionistic logic as correct for that discourse. On careful examination, the strongest Dummettian arguments for anti-realism of this sort fail to secure intuitionistic logic as the single, correct logic for anti-realist discourses. Instead, antirealists are placed in a situation where they fail to be justified in asserting monism (that intuitionistic logic is the unique correct logic). Thus, antirealists seem forced either to accept pluralism (i.e. one or more intermediate logic is at least as `correct’ as intuitionistic logic–an option I take to be unattractive from the anti-realist perspective), or they must be anti-realists about the realism/anti-realism debate (and, in particular, must refuse to assert the instance of excluded middle equivalent to logical monism or logical pluralism)

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.