Tennenbaum's Theorem yields an elegant characterisation of the standard model of arithmetic. Several authors have recently claimed that this result has important philosophical consequences: in particular, it offers us a way of responding to model-theoretic worries about how we manage to grasp the standard model. We disagree. If there ever was such a problem about how we come to grasp the standard model, then Tennenbaum's Theorem does not help. We show this by examining a parallel argument, from a simpler model-theoretic (...) result. (shrink)

We argue against the halving response to Sleeping Beauty. First, we outline an appealing constraint on probability assignments: the Principle of Irrelevant Information. Roughly, this says: if you don’t know whether C, but you would assign probability p to X regardless of whether C or not-C, then you should assign p to X. This Principle is deeply plausible, but we show that it contradicts halving. Second, we show that halving either violates solid statistical reasoning or draws absurd distinctions.

Putnam’s most famous contribution to mathematical logic was his role in investigating Hilbert’s Tenth Problem; Putnam is the ‘P’ in the MRDP Theorem. This volume, though, focusses mostly on Putnam’s work on the philosophy of logic and mathematics. It is a somewhat bumpy ride. Of the twelve papers, two scarcely mention Putnam. Three others focus primarily on Putnam’s ‘Mathematics without foundations’ (1967), but with no interplay between them. The remaining seven papers apparently tackle unrelated themes. Some of this disjointedness would (...) doubtless have been addressed, if Putnam had been able to compose his replies to these papers; sadly, he died before this was possible. In this review, I do my best to tease out some connections between the paper; and there are some really interesting connections to be made. (shrink)

Ontology after Carnap focusses on metaontology in the light of recent interest in Carnap’s ‘Empiricism, Semantics and Ontology’. That paper is at the centre of things, as it is where Carnap formulates his internal/external dichotomy. If you haven’t already encountered the dichotomy, then neither Ontology after Carnap, nor this review, is for you. My aim in this review is to try to tease out some of the book’s themes, thereby giving some sense of contemporary neo-Carnapianism.

Clicking the like button following a post on social media has become a common means of expressing and gathering social support online. Little is known about how liking expression is linked and regulated by personality traits and communication motives. Following a preliminary survey about the usage of the like function on WeChat, a Chinese social media platform, we conducted an online study to map the Big-Five personality traits and five communication motives to the frequency of liking expression. The results (...) showed that each user had, on average, 385 WeChat friends and spent 2.2 hours and used the liking function 1.1 times each day on WeChat. The personality trait conscientiousness was negatively related to the liking expression. In contrast, agreeableness promoted the expression of liking directly and indirectly via two communication motives, enjoyment and passing time. The liking expression may serve as a simple index for understanding dispositional underpinnings of social media networking. (shrink)

A review of The Limits of Realism, by Tim Button. Oxford University Press, 2013.

The internet is widely used for health information and support, often by vulnerable people. Internet-based research raises both familiar and new ethical problems for researchers and ethics committees. While guidelines for internet-based research are available, it is unclear to what extent ethics committees use these. Experience of gaining research ethics approval for a UK study (SharpTalk), involving internet-based discussion groups with young people who self-harm and health professionals is described. During ethical review, unsurprisingly, concerns were raised about the vulnerability of (...) potential participants. These were dominated by the issue of anonymity, which also affected participant safety and consent. These ethical problems are discussed, and our solutions, which included: participant usernames specific to the study, a closed website, private messaging facilities, a direct contact email to researchers, information about forum rules displayed on the website, a ‘report’ button for participants, links to online support, and a discussion room for forum moderators. This experience with SharpTalk suggests that an approach to ethics, which recognises the relational aspects of research with vulnerable people, is particularly useful for internet-based health research. The solutions presented here can act as guidance for researchers developing proposals and for ethics committees reviewing them. (shrink)

Philosophy and model theory frequently meet one another. Philosophy and Model Theory aims to understand their interactions -/- Model theory is used in every ‘theoretical’ branch of analytic philosophy: in philosophy of mathematics, in philosophy of science, in philosophy of language, in philosophical logic, and in metaphysics. But these wide-ranging appeals to model theory have created a highly fragmented literature. On the one hand, many philosophically significant mathematical results are found only in mathematics textbooks: these are aimed squarely at mathematicians; (...) they typically presuppose that the reader has a significant background in mathematics; and little clue is given as to their philosophical significance. On the other hand, the philosophical applications of these results are then scattered across disconnected pockets of papers, separated by decades or more. -/- The first aim of Philosophy and Model Theory, then, is to consider the philosophical uses of model theory. On a technical level, we try to show how philosophically significant results connect to one another, and also to state the best version of a result for philosophical purposes. On a philosophical level, we show how similar dialectical situations arise repeatedly, across fragmented debates in varied philosophical areas. -/- The second aim of Philosophy and Model Theory, though, is to consider the philosophy of model theory. Model theory itself is rarely taken as the subject matter of philosophising (contrast this, say, with the philosophy of biology, or the philosophy of set theory). But model theory is a beautiful part of pure mathematics, and worthy of philosophical study in its own right. (shrink)

In “Models and Reality” (1980), Putnam sketched a version of his internal realism as it might arise in the philosophy of mathematics. Here, I will develop that sketch. By combining Putnam’s model-theoretic arguments with Dummett’s reflections on Gödelian incompleteness, we arrive at (what I call) the Skolem-Gödel Antinomy. In brief: our mathematical concepts are perfectly precise; however, these perfectly precise mathematical concepts are manifested and acquired via a formal theory, which is understood in terms of a computable system of proof, (...) and hence is incomplete. Whilst this might initially seem strange, I show how internal categoricity results for arithmetic and set theory allow us to face up to this Antinomy. This also allows us to understand why “Models are not lost noumenal waifs looking for someone to name them,” but “constructions within our theory itself,” with “names from birth.”. (shrink)

Tim Button explores the relationship between words and world; between semantics and scepticism. -/- A certain kind of philosopher – the external realist – worries that appearances might be radically deceptive. For example, she allows that we might all be brains in vats, stimulated by an infernal machine. But anyone who entertains the possibility of radical deception must also entertain a further worry: that all of our thoughts are totally contentless. That worry is just incoherent. -/- We cannot, then, (...) be external realists, who worry about the possibility of radical deception. Equally, however, we cannot be internal realists, who reject all possibility of deception. We must position ourselves somewhere between internal realism and external realism, but we cannot hope to say exactly where. We must be realists, for what that is worth, and realists within limits. -/- In establishing these claims, The Limits of Realism critically explores and develops several themes from Hilary Putnam’s work: the model-theoretic arguments; the connection between truth and justification; the brain-in-vat argument; semantic externalism; and conceptual relativity. The book establishes the continued significance of these topics for all philosophers interested in mind, logic, language, or the possibility of metaphysics. (shrink)

Standard Type Theory, STT, tells us that b^n(a^m) is well-formed iff n=m+1. However, Linnebo and Rayo have advocated the use of Cumulative Type Theory, CTT, has more relaxed type-restrictions: according to CTT, b^β(a^α) is well-formed iff β > α. In this paper, we set ourselves against CTT. We begin our case by arguing against Linnebo and Rayo’s claim that CTT sheds new philosophical light on set theory. We then argue that, while CTT ’s type-restrictions are unjustifiable, the type-restrictions imposed by (...) STT are justified by a Fregean semantics. What is more, this Fregean semantics provides us with a principled way to resist Linnebo and Rayo’s Semantic Argument for CTT. We end by examining an alternative approach to cumulative types due to Florio and Jones; we argue that their theory is best seen as a misleadingly formulated version of STT. (shrink)

Universals are putative objects like wisdom, morality, redness, etc. Although we believe in properties (which, we argue, are not a kind of object), we do not believe in universals. However, a number of ordinary, natural language constructions seem to commit us to their existence. In this paper, we provide a fictionalist theory of universals, which allows us to speak as if universals existed, whilst denying that any really do.

No-futurists ('growing block theorists') hold that that the past and the present are real, but that the future is not. The present moment is therefore privileged: it is the last moment of time. Craig Bourne (2002) and David Braddon-Mitchell (2004) have argued that this position is unmotivated, since the privilege of presentness comes apart from the indexicality of 'this moment'. I respond that no-futurists should treat 'x is real-as-of y' as a nonsymmetric relation. Then different moments are real-as-of different times. (...) This reunites privilege with indexicality, but entails that no-futurists must believe in ineliminably tensed facts. (shrink)

Keränen (2001) raises an argument against realistic (ante rem) structuralism: where a mathematical structure has a non-trivial automorphism, distinct indiscernible positions within the structure cannot be shown to be non-identical using only the properties and relations of that structure. Ladyman (2005) responds by allowing our identity criterion to include 'irreflexive two-place relations'. I note that this does not solve the problem for structures with indistinguishable positions, i.e. positions that have all the same properties as each other and exactly the same (...) relations to all objects (including themselves). I conclude that realistic structuralists must compromise and treat some structures eliminativistically. (shrink)

This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent ‘internal’ renditions of the famous categoricity arguments for arithmetic and set theory.

Amie Thomasson and Eli Hirsch have both attempted to deflate metaphysics, by combining Carnapian ideas with an appeal to ordinary language. My main aim in this paper is to critique such deflationary appeals to ordinary language. Focussing on Thomasson, I draw two very general conclusions. First: ordinary language is a wildly complicated phenomenon. Its implicit ontological commitments can only be tackled by invoking a context principle; but this will mean that ordinary language ontology is not a trivial enterprise. Second: ordinary (...) language often points in different directions simultaneously, so that a wide variety of existence questions cannot be deflated merely by appealing to ordinary language. (shrink)

The following bare-bones story introduces the idea of a cumulative hierarchy of pure sets: 'Sets are arranged in stages. Every set is found at some stage. At any stage S: for any sets found before S, we find a set whose members are exactly those sets. We find nothing else at S.' Surprisingly, this story already guarantees that the sets are arranged in well-ordered levels, and suffices for quasi-categoricity. I show this by presenting Level Theory, a simplification of set theories (...) due to Scott, Montague, Derrick, and Potter. (shrink)

Tallant (2007) has challenged my recent defence of no-futurism (Button 2006), but he does not discuss the key to that defence: that no-futurism's primitive relation 'x is real-as-of y' is not symmetric. I therefore answer Tallant's challenge in the same way as I originally defended no-futurism. I also clarify no-futurism by rejecting a common mis-characterisation of the growing-block theorist. By supplying a semantics for no-futurists, I demonstrate that no-futurism faces no sceptical challenges. I conclude by considering the problem of (...) how to interpret the relation 'x is real-as-of y'. -/- (NB: A correction to this article appears in Analysis 68.1.). (shrink)

This book provides a sustained and penetrating critique of a wide range of views in modern cognitive science and philosophy of the mind, from Turing's famous test for intelligence in machines to recent work in computational linguistic theory. While discussing many of the key arguments and topics, the authors also develop a distinctive analytic approach. Drawing on the methods of conceptual analysis first elaborated by Wittgenstein and Ryle, the authors seek to show that these methods still have a great deal (...) to offer in the field of the cognitive theory and the philosophy of mind, providing a powerful alternative to many of the positions put forward in the contemporary literature. Amoung the many issues discussed in the book are the following: the Cartesian roots of modern conceptions of mind; Searle's 'Chinese Room' thought experiment; Fodor's 'language of thought' hypothesis; the place of 'folk psychology' in cognitivist thought; and the question of whether any machine may be said to 'think' or 'understand' in the ordinary senses of these words. Wide ranging, up-to-date and forcefully argued, this book represents a major intervention in contemporary debates about the status of cognitive science an the nature of mind. It will be of particular interest to students and scholars in philosophy, psychology, linguistics and computing sciences. (shrink)

Consider a variant of the usual story about the iterative conception of sets. As usual, at every stage, you find all the (bland) sets of objects which you found earlier. But you also find the result of tapping any earlier-found object with any magic wand (from a given stock of magic wands). -/- By varying the number and behaviour of the wands, we can flesh out this idea in many different ways. This paper's main Theorem is that any loosely constructive (...) way of fleshing out this idea is synonymous with a ZF-like theory. -/- This Theorem has rich applications; it realizes John Conway's (1976) Mathematicians' Liberation Movement; and it connects with a lovely idea due to Alonzo Church (1974). (shrink)

Hilary Putnam’s BIV argument first occurred to him when ‘thinking about a theorem in modern logic, the “Skolem–Löwenheim Theorem”’ (Putnam 1981: 7). One of my aims in this paper is to explore the connection between the argument and the Theorem. But I also want to draw some further connections. In particular, I think that Putnam’s BIV argument provides us with an impressively versatile template for dealing with sceptical challenges. Indeed, this template allows us to unify some of Putnam’s most enduring (...) contributions to the realism/antirealism debate: his discussions of brains-in-vats, of Skolem’s Paradox, and of permutations. In all three cases, we have an argument which does not merely defeat the sceptic; it also shows us that we must reject some prima facie plausible philosophical picture. (shrink)

Recent work on hypercomputation has raised new objections against the Church–Turing Thesis. In this paper, I focus on the challenge posed by a particular kind of hypercomputer, namely, SAD computers. I first consider deterministic and probabilistic barriers to the physical possibility of SAD computation. These suggest several ways to defend a Physical version of the Church–Turing Thesis. I then argue against Hogarth's analogy between non-Turing computability and non-Euclidean geometry, showing that it is a non-sequitur. I conclude that the Effective version (...) of the Church–Turing Thesis is unaffected by SAD computation. (shrink)

Prior’s Tonk is a famously horrible connective. It is defined by its inference rules. My aim in this article is to compare Tonk with some hitherto unnoticed nasty connectives, which are defined in semantic terms. I first use many-valued truth-tables for classical sentential logic to define a nasty connective, Knot. I then argue that we should refuse to add Knot to our language. And I show that this reverses the standard dialectic surrounding Tonk, and yields a novel solution to the (...) problem of many-valued truth-tables for classical sentential logic. I close by outlining the technicalities surrounding nasty connectives on many-valued truth-tables. (shrink)

There are several relations which may fall short of genuine identity, but which behave like identity in important respects. Such grades of discrimination have recently been the subject of much philosophical and technical discussion. This paper aims to complete their technical investigation. Grades of indiscernibility are defined in terms of satisfaction of certain first-order formulas. Grades of symmetry are defined in terms of symmetries on a structure. Both of these families of grades of discrimination have been studied in some detail. (...) However, this paper also introduces grades of relativity, defined in terms of relativeness correspondences. This paper explores the relationships between all the grades of discrimination, exhaustively answering several natural questions that have so far received only partial answers. It also establishes which grades can be captured in terms of satisfaction of object-language formulas and draws connections with definability theory. (shrink)

Can we quantify over everything: absolutely, positively, definitely, totally, every thing? Some philosophers have claimed that we must be able to do so, since the doctrine that we cannot is self-stultifying. But this treats restrictivism as a positive doctrine. Restrictivism is much better viewed as a kind of militant quietism, which I call dadaism. Dadaists advance a hostile challenge, with the aim of silencing everyone who holds a positive position about ‘absolute generality’.

On a very natural conception of sets, every set has an absolute complement. The ordinary cumulative hierarchy dismisses this idea outright. But we can rectify this, whilst retaining classical logic. Indeed, we can develop a boolean algebra of sets arranged in well-ordered levels. I show this by presenting Boolean Level Theory, which fuses ordinary Level Theory (from Part 1) with ideas due to Thomas Forster, Alonzo Church, and Urs Oswald. BLT neatly implement Conway’s games and surreal numbers; and a natural (...) extension of BLT is definitionally equivalent with ZF. (shrink)

Over the past several decades, scholars of liberal and democratic theory have shown a heightened interest in the role that various virtues might play in promoting the good/free society. Yet within this recent "return" to the virtues, one quality that has been almost entirely left out of the discussion is humility. In this essay, I critically address this lacuna and offer a defense of a particular form of humility, what I call democratic humility. After considering a range of moral and (...) political objections to the concept of humility, Iprovide an account that seeks to address some of these long-standing difficulties while arguing that the idea of humility, recuperated as an ethos of civic attentiveness, may be one of the most important virtues for late-modern societies marked by ethical and cultural pluralism. (shrink)

Hilary Putnam once suggested that “the actual existence of sets as ‘intangible objects’ suffers… from a generalization of a problem first pointed out by Paul Benacerraf… are sets a kind of function or are functions a sort of set?” Sadly, he did not elaborate; my aim, here, is to do so on his behalf. There are well-known methods for treating sets as functions and functions as sets. But these do not raise any obvious philosophical or foundational puzzles. For that, we (...) first need to provide a full-fledged function theory. I supply such a theory: it axiomatizes the iterative notion of function in exactly the same sense that ZF axiomatizes the iterative notion of set. Indeed, this function theory is synonymous with ZF. It might seem that set theory and function theory present us with rival foundations for mathematics, since they postulate different ontologies. But appearances are deceptive. Set theory and function theory provide the very same judicial foundation for mathematics. They do not supply rival metaphysical foundations; indeed, if they supply metaphysical foundations at all, then they supply the very same metaphysical foundations. (shrink)

Putnam famously attempted to use model theory to draw metaphysical conclusions. His Skolemisation argument sought to show metaphysical realists that their favourite theories have countable models. His permutation argument sought to show that they have permuted models. His constructivisation argument sought to show that any empirical evidence is compatible with the Axiom of Constructibility. Here, I examine the metamathematics of all three model-theoretic arguments, and I argue against Bays (2001, 2007) that Putnam is largely immune to metamathematical challenges.

Minimalists, such as Paul Horwich, claim that the notions of truth, reference and satisfaction are exhausted by some very simple schemes. Unfortunately, there are subtle difficulties with treating these as schemes, in the ordinary sense. So instead, minimalists regard them as illustrating one-place functions, into which we can input propositions (when considering truth) or propositional constituents (when considering reference and satisfaction). However, Bertrand Russell's Gray's Elegy argument teaches us some important lessons about propositions and propositional constituents. When applied to minimalism, (...) these lessons show us why we should abandon it. (shrink)

In 1919 the Animal Breeding Research Department was established in Edinburgh. This Department, later renamed the Institute of Animal Genetics, forged an international reputation, eventually becoming the centrepiece of a cluster of new genetics research units and institutions in Edinburgh after the Second World War. Yet despite its significance for institutionalising animal genetics research in the UK, the origins and development of the Department have not received as much scholarly attention as its importance warrants. This paper sheds new light on (...) Edinburgh’s place in early British genetics by drawing upon recently catalogued archival sources including the papers of James Cossar Ewart, Regius Professor of Natural History at the University of Edinburgh between 1882 and 1927. Although presently a marginal figure in genetics historiography, Ewart established two sites for experimental animal breeding work between 1895 and 1911 and played a central role in the founding of Britain’s first genetics lectureship, also in 1911. These early efforts helped to secure government funding in 1913. However, a combination of the First World War, bureaucratic problems and Ewart’s personal ambitions delayed the creation of the Department and the appointment of its director by another six years. This paper charts the institutionalisation of animal breeding and genetics research in Edinburgh within the wider contexts of British genetics and agriculture in the early twentieth century. (shrink)

It is a metaphysical orthodoxy that interesting non-symmetric relations cannot be reduced to symmetric ones. This orthodoxy is wrong. I show this by exploring the expressive power of symmetric theories, i.e. theories which use only symmetric predicates. Such theories are powerful enough to raise the possibility of Pythagrapheanism, i.e. the possibility that the world is just a vast, unlabelled, undirected graph.

A judgement analysis of people's social inferences of attitudes and ability was conducted. University students were asked to infer the liberalness ( N = 60; Study 1) or intelligence ( N = 40; Study 2) of targets seen in pictures. Multiple regression analyses revealed that attractiveness was the most important cue for predicting inferences of liberalness, while an ethnic cue (i.e., being Asian) was the most important cue for judgements about intelligence. Results also showed that a single-cue model was less (...) susceptible to overfitting, but significantly less accurate than a multiple-cue model in predicting participant's intelligence judgements. Although the multiple regression models suffered a degree of overfitting, cross validation showed that they continued to have significant predictive value when applied to new data. Furthermore, a “random partner” method (comparing each participant's own regression equation with that of another, randomly selected, participant) provided evidence of significant idiosyncratic variation in the way intelligence judgements were made. (shrink)

Potentialists think that the concept of set is importantly modal. Using tensed language as an heuristic, the following bar-bones story introduces the idea of a potential hierarchy of sets: 'Always: for any sets that existed, there is a set whose members are exactly those sets; there are no other sets.' Surprisingly, this story already guarantees well-foundedness and persistence. Moreover, if we assume that time is linear, the ensuing modal set theory is almost definitionally equivalent with non-modal set theories; specifically, with (...) Level Theory, as developed in Part 1. (shrink)

Do men and women perceive cybercrime types differently? This article draws on the distinction between socio-economic and psychosocial cybercrime proposed by Lazarus (2019) to investigate whether men and women hold different perceptions of digital crimes across these two dimensions. Informed by the synergy between feminist theory and the Tripartite Cybercrime Framework (TCF), our survey examined respondents’ differential perceptions of socio-economic cybercrime (online fraud) and psychosocial cybercrime (cyberbullying, revenge porn, cyberstalking, online harassment) among men and women in the United Kingdom. The (...) results revealed that women considered psychosocial cybercrime worse than men. Conversely, we found no differences between men and women with regard to socio-economic cybercrime. The article concludes that psychosocial cybercrimes are more gendered than socio-economic cybercrime, suggesting problems with the meaning of ‘cyber-enabled crimes’, and substantiating the synergy between the TCF and feminist perspectives. (shrink)

"Explores the concept of the social contract and how it shapes citizenship. Argues that the modern social contract is an account of the ethical and cultural conditions upon which modern citizenship depends"--Provided by publisher.

How do tweets reflect the long-standing disparities between the northern and southern regions of Nigeria? This study presents a qualitative analysis of Twitter users' responses (n = 101,518) to the tweets of the Economic and Financial Crimes Commission (EFCC) regarding the production and prosecution of cybercrime. The article uses postcolonial perspectives to shed light on the legacies of British colonial efforts in Nigeria, such as the amalgamation of the northern and southern protectorates in 1914. The results revealed significant discrepancies between (...) Nigeria's northern and southern regions regarding cybercriminal arrest, conviction, and sentencing. Specifically, the results showed that the EFCC's criminalization of Southerners differs substantially from that of Northerners. The contemporary manifestation of inequalities concerning the production and prosecution of cybercrime on Twitter reflects long-standing contestations (e.g., economic, political, cultural, geological) between the northern and southern parts of Nigerian society. Therefore, since the North-South divide in present-day Nigeria originated from British colonization, colonialism is the base that shaped the superstructure comprising political, religious, historical, geological (e.g., crude oil), and economic factors. In turn, the article spotlights that regional differences in educational attainment, originating from differing experiences of Christianization and colonization, interact with regional disparities in the production of cybercrime. More research is required to better understand how these contextual disparities in society interact with the production and prosecution of cybercrime, given that Nigerian cybercriminals defraud victims all over the world. (shrink)

This article is about the manifestations of similarities between two seemingly distinct groups of Nigerians: cybercriminals and politicians. Which linguistic strategies do Twitter users use to express their opinions on cybercriminals and politicians? The study undertakes a qualitative analysis of ‘engaged’ tweets of an elite law enforcement agency in West Africa. We analyzed and coded over 100,000 ‘engaged’ tweets based on a component of mechanisms of moral disengagement (i.e., advantageous comparison), a linguistic device. The results reveal how respondents defend the (...) actions of online fraudsters (“the deviant group”) by strategically comparing them to the wrongful acts of corrupt politicians (“the respectable group”). Similarly, the results show how respondents positioned this linguistic strategy to compare “the powerless group” (online fraudsters) and “the powerful group” (politicians) in society. Indeed, tweet responses suggest that the Economic and Financial Crimes Commission (EFCC) generally looks downwards for culprits (i.e., online fraudsters) while ignoring fraudulent politicians. We conclude that the process by which some actions are interpreted as a crime compared to others is a moral enterprise. (shrink)

In 1907–8, Russell and Stout presented an objection against James and Schiller, to which both James and Schiller replied. In this paper, I shall revisit their transatlantic exchange. Doing so will yield a better understanding of Schiller’s relationship to a worryingly solipsistic brand of phenomenalism. It will also allow us to appreciate a crucial difference between Schiller and James; a difference which James explicitly downplayed.

Wittgenstein’s atomist picture, as embodied in his Tractatus, is initially very appealing. However, it faces the famous colour-exclusion problem. In this paper, I shall explain when the atomist picture can be defended in the face of that problem; and, in the light of this, why the atomist picture should be rejected. I outline the atomist picture in Section 1. In Section 2, I present a very simple necessary and sufficient condition for the tenability of the atomist picture. The condition is: (...) logical space is a power of two. In Sections 3 and 4, I outline the colour-exclusion problem, and then show how the cardinality-condition supplies a response to exclusion problems. In Section 5, I explain how this amounts to a distillation of a proposal due to Moss, which goes back to Carruthers. And in Section 6, I show how all this vindicates Wittgenstein’s ultimate rejection of the atomist picture. The brief reason is that we have no guarantee that there are any solutions to a given exclusion problem but, if there are any, then there are far too many. (shrink)