Geoffrey Hellman

In concluding their major survey, “Quantum Logics,” Greechie and Gudder (1975) listed four outstanding directions for future research, the last of which was, “Explain the meaning of the word ‘logic’ in the title of this paper.” Broadly speaking, there are two radically different points of view on this. There is the view that, taken strictly, ‘logic’ is a misnomer: quantum logic studies certain algebraic structures naturally arising from the formal, apparatus of quantum theory, structures which may afford insights into quantum physics, on a number of levels, but which do not in any way involve giving up or replacing classical logical laws. (Cf., e.g., Jauch (1968, p. 77).) In contrast, there is the view that the non-Boolean structures arising from the Hilbert space formalism should be taken as giving rise to a genuine non-classical logic, in conflict with classical logic, but appropriate to the description of quantum mechanical reality.

Neologicism (NL) invokes a “syntactic-priority thesis” (SPT) to derive existence of numbers, etc., from abstraction principles. Innumerable counterexamples to the SPT, however, are seen to arise from fiction, e.g., “Pegasus is (entirely) fictive”. Examination of possible defenses of the SPT leads to just one viable option, based on quasi-modal “in-fiction” operators. This, however, applies just as well to abstraction principles themselves, thereby undermining NL’s case for countenancing mathematical abstracta. Appeal to the linguistics principle of compositionality is seen not to help NL.

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.

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.

With developments in the 19th and early 20th centuries, structuralist ideas concerning the subject matter of mathematics have become commonplace. Yet fundamental questions concerning structures and relations themselves as well as the scope of structuralist analyses remain to be answered. The distinction between axioms as defining conditions and axioms as assertions is highlighted as is the problem of the indefinite extendability of any putatively all-embracing realm of structures. This chapter systematically compares four main versions: set-theoretic structuralism, a version taking structures as sui generis universals, structuralism based on category theory, and a quasi-nominalist modal-structuralism. While none of the approaches is problem-free, it appears that some synthesis of the category-theoretic approach with modal-structuralism can meet the challenges set out, given the notion of “logical possibility.”

It is traditional in volumes such as this – especially volumes appearing relatively soon after the death of their subject – to include a lengthy, and often rather dry and boring, intellectual biography. In the case of Putnam, such an essay would not, in fact, be boring (although, given the richness of Putnam’s life, such an essay might have to be rather lengthy!) But we wanted to do something a little different, and at any rate we knew we couldn’t do a better job at biography than what is already in the gripping intellectual autobiography Putnam wrote for his installment in the Library of Living Philosophers series (Auxier et al. 2015).

In this essay we examine the revenge problem as it arises with respect to accounts of both the set-theoretic and the semantic paradoxes. First we review revenge as it arises in the set-theoretic setting – the Burali-Forti paradox – and outline its modal-structural resolution, highlighting the roles played by the logic of plurals, modal principles, and especially the extendability of models of set theory on this account. We then we turn to the semantic paradoxes, especially the Liar, and develop an analogy between the problems of expressive incompleteness and revenge affecting proposals to resolve the semantic paradoxes, on the one hand, and, on the other, the always incomplete and extendable nature of domains of sets on the modal structural approach to set theory. We then argue for a corresponding parallelism in resolutions of the set-theoretic and the semantic paradoxes. Focusing on recent accounts stemming from the work of Martin and Woodruff (Philosophia, 5(3):213–217, 1975) and Kripke (Journal of Philosophy, 72:690–716, 1975), we formulate a modal account of the extendability of languages on the Embracing Revenge account of the semantic paradoxes (see, e.g. Cook, Embracing Revenge: On the Indefinite Extendability of Language, 2007; Schlenker, Review of Symbolic Logic, 3(3):374–414, 2010) analogous to the formulation of extendability principles for set theoretic universes on the modal structural approach. Finally, however, we examine an interesting dis-analogy via a meta-revenge version of the Liar paradox that seems to have no analogue in the set-theoretic context, and we show how the solution to this puzzle also highlights even deeper connections between the modal-structural account of set theory and the Embracing Revenge account of truth and semantics.