Stewart Shapiro

Smooth Infinitesimal Analysis (SIA) is a remarkable late twentieth-century theory of analysis. It is based on nilsquare infinitesimals, and does not rely on limits. SIA poses a challenge of motivating its use of intuitionistic logic beyond merely avoiding inconsistency. The classical-modal account(s) provided here attempt to do just that. The key is to treat the identity of an arbitrary nilsquare, e, in relation to 0 or any other nilsquare, as objectually vague or indeterminate—pace a famous argument of Evans [10]. Thus, we interpret the necessity operator of classical modal logic as “determinateness” in truth-value, naturally understood to satisfy the modal system, S4 (the accessibility relation on worlds being reflexive and transitive). Then, appealing to the translation due to Gödel et al., and its proof-theoretic faithfulness (“mirroring theorem”), we obtain a core classical-modal interpretation of SIA. Next we observe a close connection with Kripke semantics for intuitionistic logic. However, to avoid contradicting SIA’s non-classical treatment of identity relating nilsquares, we translate “=” with a non-logical surrogate, ‘E,’ with requisite properties. We then take up the interesting challenge of adding new axioms to the core CM interpretation. Two mutually incompatible ones are considered: one being the positive stability of identity and the other being a kind of necessity of indeterminate identity (among nilsquares). Consistency of the former is immediate, but the proof of consistency of the latter is a new result. Finally, we consider moving from CM to a three-valued, semi-classical framework, SCM, based on the strong Kleene axioms. This provides a way of expressing “indeterminacy” in the semantics of the logic, arguably improving on our CM. SCM is also proof-theoretically faithful, and the extensions by either of the new axioms are consistent.

Hofweber (Ontology and the ambitions of metaphysics, Oxford University Press, 2016) argues for a thesis he calls “internalism” with respect to natural number discourse: no expressions purporting to refer to natural numbers in fact refer, and no apparent quantification over natural numbers actually involves quantification over natural numbers as objects. He argues that while internalism leaves open the question of whether other kinds of abstracta exist, it precludes the existence of natural numbers, thus establishing what he calls “restricted nominalism” about natural numbers. We argue that Hofweber’s internalism fails to establish restricted nominalism. Not only is his primary argument for restricted nominalism invalid, the analysis of quantification proposed threatens to collapse internalism into either a traditional form of error theory or realism.

MEDIEVAL AND RENAISSANCE LOGICSMARK D. JOHNSTON, The spiritual logic of Ramon Llull. Oxford: Clarendon Press,1987. xi + 336 pp. £35.00E. J. ASHWORTH, Thomas Bricot: Tractatus Znsolubilium. Nijmegen: Ingenium, 1986. xxiii+ 155 pp. 44 Dfl.CYPRIANI REGNERI, Demonstratio logicae verae iuridica. Edited by G. Kalinowski. Bologna: Cooperativa Libraria Universitaria Editrice Bologna, 1986. xxviii + 167 pp. No price stated.GIROLAMO SACCHERI, Euclides vindicatus. Edited and translated by George Bruce Halsted: second, expanded, edition. New York: Chelsea Publishing Company 1986. xxx + 255 pp. $16.95.THE NINETEENTH CENTURYT. HAILPERIN, Boole's logic andprobability. Second edition. Amsterdam, New York,Oxford and Tokyo: North-Holland, 1986. xii + 428 pp. $76.00.WALTER PURKERT and HANS JOACHIM ILGAUDS, Georg Cantor 1845-1918. Basel/Boston/Stuttgart: Birkhauser Verlag, 1987. 262 pp., 26 plts. SFr. 48.MAX H. FISCH, Peirce, semeiotic, andpragmatism: essays by Max H. Fisch. Edited by Kenneth Laine Ketner and Christian J. W. Kloesel. Bloomington: Indiana University Press, 1986. xiv + 464 pp. $45.00.LOGIC AND LANGUAGEJOËLLE PROUST, Qusesting de forme. logique et proposition analtique de kant á Carnap. Paris Fayard, 1986. xxviii + 504 pp. FF 160H. PALMER, Presupposition and transcendental inference. London and Sydney: Croom Helm, 1985. 208 pp. £15.95EDITIONSALFRED TARSKI, Collected papers. Edited by Steven R. Givant and Ralph N. McKenzie. Vol. I, 1921-1934, 659 pp. Vol. 11, 1935-1944, 699 pp. Vol. 111, 1945-1957, 682 pp. Vol. IV, 1958-1979, 757 pp. Basel, Boston, Stuttgart: Birkhäuser,1986. SF 1280KURT GöDEL, Collected works, Volume 1. Edited by Solomon Feferman, John W. Dawson Jr., Stephen Kleene, Gregory H. Moore, Robert M. Solovay and Jean van Heijenoort. New York: Oxford University Press; Oxford: Clarendon Press; 1986. xviii + 474 pp. £25.00.MISCELLANEOUSJ. MACNAMARA, A border dispute. The place of logic in psychology. Cambridge, Mass. and London: M. I. T. Press. 1986. xviii + 212 pp. £22.50.G. LOLLI, La macchina e le dimostrazioni. Bologna: il Mulino, 1987. 157 pp. L. 15.000.

While sets are combinatorial collections, defined by their elements, classes are logical collections, defined by their membership conditions. We develop, in a potentialist setting, a predicative approach to (logical) classes of (combinatorial) sets. Some reasons emerge to adopt a stricter form of potentialism, which insists, not only that each object is generated at some stage of an incompletable process, but also that each truth is “made true” at some such stage. The natural logic of this strict form of potentialism is semi-intuitionistic: where each set-sized domain is classical, the domain of all sets or all classes is intuitionistic.

Aristotle argued that paradoxes of the infinite can be avoided only by insisting that all infinities are potential, not actual. There is a long tradition of thinking that a Judeo-Christian God would collapse potential infinities to actual ones, thus removing the Aristotelian guard-rail against paradox. After all, does not God know all numbers, regardless of whether they are actual or merely potential? We analyze the Aristotelian guard-rail of potentiality, as well as challenges to it due to Augustine, Burley, Scotus, and Cantor. At the heart of the debate we find the metaphysical question of how the temporal or modal “dimension” that harbors potential infinities compares to ordinary spatial dimensions. Certain theological assumptions underwrite a strong analogy. The best Aristotelian response, we suggest, is to reject the analogy by refusing to analyze modality extensionally as a point-space of possible worlds. Our analysis thus reveals lessons for atheists as well as theists.

In previous work, Hellman and Shapiro present a regions-based account of a one-dimensional continuum. This paper produces a more Aristotelian theory, eschewing the existence of points and the use of infinite sets or pluralities. We first show how to modify the original theory. There are a number of theorems that have to be added as axioms. Building on some work by Linnebo, we then show how to take the ‘potential’ nature of the usual operations seriously, by using a modal language, and we show that the two approaches are equivalent.

Modal logic has been used to analyze potential infinity and potentialism more generally. However, the standard analysis breaks down in cases of divergent possibilities, where there are two or more possibilities that can be individually realized but which are jointly incompatible. This paper has three aims. First, using the intuitionistic theory of choice sequences, we motivate the need for a modal analysis of divergent potentialism and explain the challenges this involves. Then, using Beth–Kripke semantics for intuitionistic logic, we overcome those challenges. Finally, we apply our modal analysis of divergent potentialism to make choice sequences comprehensible in classical terms.

A core commitment of Bob Hale and Crispin Wright’s neologicism is their invocation of Frege’s Constraint—roughly, the requirement that the core empirical applications for a class of numbers be “built directly into” their formal characterization. According to these neologicists, if legitimate, Frege’s Constraint adjudicates in favor of their preferred foundation—Hume’s Principle—and against alternatives, such as the Dedekind–Peano axioms. In this paper, we consider a recent argument for legitimating Frege’s Constraint due to Hale, according to which the primary empirical application of the naturals is transitive counting, or answering ‘how many’-questions using numerals. We make two claims regarding Hale’s argument. First, it fails to legitimate Frege’s Constraint in virtue of resting on unsupported and highly contentious assumptions. Secondly, even if sound, Hale’s argument would vindicate a version of Frege’s Constraint which fails to adjudicate in favor of Hume’s Principle over alternative characterizations of the naturals.

This article examines a number of issues and problems that motivate at least much of the literature in the philosophy of mathematics. It first considers how the philosophy of mathematics is related to metaphysics, epistemology, and semantics. In particular, it reviews several views that account for the metaphysical nature of mathematical objects and how they compare to other sorts of objects, including realism in ontology and nominalism. It then discusses a common claim, attributed to Georg Kreisel that the important issues in the philosophy of mathematics do not concern the nature of mathematical objects, but rather the objectivity of mathematical discourse. It also explores irrealism in truth-value, the dilemma posed by Paul Benacerraf, epistemological issues in ontological realism, ontological irrealism, and the connection between naturalism and mathematics.

It has long been known that (classical) Peano arithmetic is, in some strong sense, “equivalent” to the variant of (classical) Zermelo–Fraenkel set theory (including choice) in which the axiom of infinity is replaced by its negation. The intended model of the latter is the set of hereditarily finite sets. The connection between the theories is so tight that they may be taken as notational variants of each other. Our purpose here is to develop and establish a constructive version of this. We present an intuitionistic theory of the hereditarily finite sets, and show that it is definitionally equivalent to Heyting Arithmetic, in a sense to be made precise. Our main target theory, the intuitionistic small set theory is remarkably simple, and intuitive. It has just one non-logical primitive, for membership, and three straightforward axioms plus one axiom scheme. We locate our theory within intuitionistic mathematics generally.

After a brief account of the problem of higher-order vagueness, and its seeming intractability, I explore what comes of the issue on a linguistic, contextualist account of vagueness. On the view in question, predicates like 'borderline red' and 'determinately red' are, or at least can be, vague, but they are different in kind from 'red'. In particular, 'borderline red' and 'determinately red' are not colours. These predicates have linguistic components, and invoke notions like 'competent user of the language'. On my view, so-called 'higher-order vagueness' is actually ordinary, first-order vagueness in different predicates. I explore the possibility that, nevertheless, a pernicious regress ensues.

The goal of this book is to defend a supervaluationist theory of vagueness. Keefe begins by laying out a series of desiderata for an adequate theory of vagueness generally: among other things, such a theory will need to solve the sorites paradox, provide a plausible analysis of borderline cases, preserve so-called penumbral connections among borderline predications, accommodate the phenomenon of higher-order vagueness, and comport with as many of our ordinary linguistic intuitions as possible. She then proceeds to evaluate in turn each of the main competitors to a supervaluationist theory—including epistemic, many-valued, degree-theoretic, and pragmatic accounts—in light of these desiderata, and finds each seriously lacking. Having thus cleared the ground, she advances a version of the supervaluationist approach and replies to a number of potential objections.

This paper discusses the neo-logicist approach to the foundations of mathematics by highlighting an issue that arises from looking at the Bad Company objection from an epistemological perspective. For the most part, our issue is independent of the details of any resolution of the Bad Company objection and, as we will show, it concerns other foundational approaches in the philosophy of mathematics. In the first two sections, we give a brief overview of the "Scottish" neo-logicist school, present a generic form of the Bad Company objection and introduce an epistemic issue connected to this general problem that will be the focus of the rest of the paper. In the third section, we present an alternative approach within philosophy of mathematics, a view that emerges from Hilbert's Grundlagen der Geometrie (1899, Leipzig: Teubner; Foundations of geometry (trans.: Townsend, E.). La Salle, Illinois: Open Court, 1959.). We will argue that Bad Company-style worries, and our concomitant epistemic issue, also affects this conception and other foundationalist approaches. In the following sections, we then offer various ways to address our epistemic concern, arguing, in the end, that none resolves the issue. The final section offers our own resolution which, however, runs against the foundationalist spirit of the Scottish neo-logicist program

One of the more distinctive features of Bob Hale and Crispin Wright’s neologicism about arithmetic is their invocation of Frege’s Constraint – roughly, the requirement that the core empirical applications for a class of numbers be “built directly into” their formal characterization. In particular, they maintain that, if adopted, Frege’s Constraint adjudicates in favor of their preferred foundation – Hume’s Principle – and against alternatives, such as the Dedekind-Peano axioms. In what follows we establish two main claims. First, we show that, if sound, Hale and Wright’s arguments for Frege’s Constraint at most establish a version on which the relevant application of the naturals is transitive counting – roughly, the counting procedure by which numerals are used to answer “how many”-questions. Second, we show that this version of Frege’s Constraint fails to adjudicate in favor of Hume’s Principle. If this is the version of Frege’s Constraint that a foundation for arithmetic must respect, then Hume’s Principle no more – and no less – meets the requirement than the Dedekind-Peano axioms do.

There are multiple formal characterizations of the natural numbers available. Despite being inter-derivable, they plausibly codify different possible applications of the naturals – doing basic arithmetic, counting, and ordering – as well as different philosophical conceptions of those numbers: structuralist, cardinal, and ordinal. Some influential philosophers of mathematics have argued for a non-egalitarian attitude according to which one of those characterizations is ‘more basic’ or ‘more fundamental’ than the others. This paper addresses two related issues. First, we review some of these non-egalitarian arguments, lay out a laundry list of different, legitimate, notions of relative priority, and suggest that these arguments plausibly employ different such notions. Secondly, we argue that given a metaphysical-cum-epistemological gloss suggested by Frege's foundationalist epistemology, the ordinals are plausibly more basic than the cardinals. This is just one orientation to relative priority one could take, however. Ultimately, we subscribe to an egalitarian attitude towards these formal characterizations: they are, in some sense, equally ‘legitimate’.