Not all truths are on a par. The realm of truths is structured: some propositions are only true because others are. The relation that endows the realm of truths with this structure is often called grounding. Grounding has achieved much attention in 21st century metaphysics, but the topic is arguably as old as philosophy itself. -/- This becomes apparent when investigating the works of the 19th-century philosopher Bernard Bolzano, who developed what is perhaps the first comprehensive theory of grounding, drawing (...) on a rich tradition that goes back to Aristotle’s Posterior Analytics. Roski’s book provides, for the first time, a comprehensive study of Bolzano’s theory of grounding in its entirety, paying more attention than previous studies to the interaction between grounding and the consequence-relation of deducibility. -/- . (shrink)

Users of sociotechnical systems often have no way to independently verify whether the system output which they use to make decisions is correct; they are epistemically dependent on the system. We argue that this leads to problems when the system is wrong, namely to bad decisions and violations of the norm of practical reasoning. To prevent this from occurring we suggest the implementation of defeaters: information that a system is unreliable in a specific case (undercutting defeat) or independent information that (...) the output is wrong (rebutting defeat). Practically, we suggest to design defeaters based on the different ways in which a system might produce erroneous outputs, and analyse this suggestion with a case study of the risk classification algorithm used by the Dutch tax agency. (shrink)

Process reliabilist accounts claim that a belief is justified when it is the result of a reliable belief-forming process. Yet over what range of possible token processes is this reliability calculated? I argue against the idea that _all_ possible token processes (in the actual world, or some other subset of possible worlds) are to be considered using the case of a user acquiring beliefs based on the output of an AI system, which is typically reliable for a substantial local range (...) but unreliable when all possible inputs are considered. I show that existing solutions to the generality problem imply that these cases cannot be solved by a more fine-grained typing of the belief-forming process. Instead, I suggest that reliability is evaluated over a range restricted by the content of the actual belief and by the similarity of the input to the actual input. (shrink)

How do we get out knowledge of the natural numbers? Various philosophical accounts exist, but there has been comparatively little attention to psychological data on how the learning process actually takes place. I work through the psychological literature on number acquisition with the aim of characterising the acquisition stages in formal terms. In doing so, I argue that we need a combination of current neologicist accounts and accounts such as that of Parsons. In particular, I argue that we learn the (...) initial segment of the natural numbers on the basis of the Fregean definitions, but do not learn the natural number structure as a whole on the basis of Hume's principle. Therefore, we need to account for some of the consistency of our number concepts with the Dedekind-Peano axioms in other terms. (shrink)

In their paper Nothing but the Truth Andreas Pietz and Umberto Rivieccio present Exactly True Logic, an interesting variation upon the four-valued logic for first-degree entailment FDE that was given by Belnap and Dunn in the 1970s. Pietz & Rivieccio provide this logic with a Hilbert-style axiomatisation and write that finding a nice sequent calculus for the logic will presumably not be easy. But a sequent calculus can be given and in this paper we will show that a calculus for (...) the Belnap-Dunn logic we have defined earlier can in fact be reused for the purpose of characterising ETL, provided a small alteration is made—initial assignments of signs to the sentences of a sequent to be proved must be different from those used for characterising FDE. While Pietz & Rivieccio define ETL on the language of classical propositional logic we also study its consequence relation on an extension of this language that is functionally complete for the underlying four truth values. On this extension the calculus gets a multiple-tree character—two proof trees may be needed to establish one proof. (shrink)

I argue for an account of know‐how as a capacity for practical judgment—a view I derive from Gilbert Ryle. I begin by offering an interpretation of Ryle and by correcting a number of widespread misconceptions about his views in the current debate. I then identify some problems with Ryle's account and finally present my own view which, I argue, retains Ryle's insights while avoiding his mistakes.

By using the notions of exact truth and exact falsity, one can give 16 distinct definitions of classical consequence. This paper studies the class of relations that results from these definitions in settings that are paracomplete, paraconsistent or both and that are governed by the Strong Kleene schema. Besides familiar logics such as Strong Kleene logic, the Logic of Paradox and First Degree Entailment, the resulting class of all Strong Kleene generalizations of classical logic also contains a host of unfamiliar (...) logics. We first study the members of our class semantically, after which we present a uniform sequent calculus that is sound and complete with respect to all of them. Two further sequent calculi and \ calculus) will be considered, which serve the same purpose and which are obtained by applying general methods to construct sequent calculi for many-valued logics. Rules and proofs in the SK calculus are much simpler and shorter than those of the \ and the \ calculus, which is one of the reasons to prefer the SK calculus over the latter two. Besides favourably comparing the SK calculus to both the \ and the \ calculus, we also hint at its philosophical significance. (shrink)

The rise of causality and the attendant graph-theoretic modeling tools in the study of counterfactual reasoning has had resounding effects in many areas of cognitive science, but it has thus far not permeated the mainstream in linguistic theory to a comparable degree. In this study I show that a version of the predominant framework for the formal semantic analysis of conditionals, Kratzer-style premise semantics, allows for a straightforward implementation of the crucial ideas and insights of Pearl-style causal networks. I spell (...) out the details of such an implementation, focusing especially on the notions of intervention on a network and backtracking interpretations of counterfactuals. (shrink)

We present the first comprehensive theory of fairness that conceives of fairness as having two dimensions: a comparative and an absolute one. The comparative dimension of fairness has traditionally been the main interest of Broomean fairness theories. It has been analysed as satisfying competing individual claims in proportion to their respective strengths. And yet, many key contributors to Broomean fairness agree that ‘absolute’ fairness is important as well. We make this concern precise by introducing the Fairness formula and the absolute (...) priority rule and analyse their implications for comparative fairness. (shrink)

According to an increasingly popular view among philosophers of science, both causal and non-causal explanations can be accounted for by a single theory: the counterfactual theory of explanation. A kind of non-causal explanation that has gained much attention recently but that this theory seems unable to account for are grounding explanations. Reutlinger :239-256, 2017) has argued that, despite these appearances to the contrary, such explanations are covered by his version of the counterfactual theory. His idea is supported by recent work (...) on grounding by Schaffer and Wilson who claim there to be a tight connection between grounding and counterfactual dependence. The present paper evaluates the prospects of the idea. We show that there is only a weak sense in which grounding explanations convey information about counterfactual dependencies, and that this fact cannot plausibly be taken to reveal a distinctive feature that grounding explanations share with other kinds of explanations. (shrink)

The Approximate Number System (ANS) is a system that allows us to distinguish between collections based on the number of items, though only if the ratio between numbers is high enough. One of the questions that has been raised is what the representations involved in this system represent. I point to two important constraints for any account: (a) it doesn’t involve numbers, and (b) it can account for the approximate nature of the ANS. Furthermore, I argue that representations of pure (...) magnitude with vehicles that have an imprecision in the value of the unit of measurement (further clarified through a formal model from measurement theory) fit both these requirements. (shrink)

In the article ‘How to be absolutely fair, Part I: the Fairness formula’, we presented the first theory of comparative and absolute fairness. Here, we relate the implications of our Fairness formula to economic theories of fair division. Our analysis makes contributions to both philosophy and economics: to the philosophical literature, we add an axiomatic discussion of proportionality and fairness. To the economic literature, we add an appealing normative theory of absolute and comparative fairness that can be used to evaluate (...) axioms and division rules. Also, we provide a novel definition and characterization of the absolute priority rule. (shrink)

This paper is devoted to Bolzano’s theory of grounding (Abfolge) in his Wissenschaftslehre. Bolzanian grounding is an explanatory consequence relation that is frequently considered an ancestor of the notion of metaphysical grounding. The paper focuses on two principles that concern grounding in the realm of conceptual sciences and relate to traditionally widespread ideas on explanations: the principles, namely, that grounding orders conceptual truths from simple to more complex ones (Simplicity), and that it comes along with a certain theoretical economy among (...) them (Economy). Being spelled out on the basis of Bolzano’s notion of deducibility (Ableitbarkeit), these principles are revealing for the question to what extent grounding can be considered a formal relation. (shrink)

For Aristotle, a just political community has to find similarity in difference and foster habits of reciprocity. Conventionally, speech and law have been seen to fulfill this role. This article reconstructs Aristotle’s conception of currency ( nomisma) as a political institution of reciprocal justice. By placing Aristotle’s treatment of reciprocity in the context of the ancient politics of money, currency emerges not merely as a medium of economic exchange but also potentially as a bond of civic reciprocity, a measure of (...) justice, and an institution of ethical deliberation. Reconstructing this account of currency ( nomisma) in analogy to law ( nomos) recovers the hopes Aristotle placed in currency as a necessary institution particular to the polis as a self-governing political community striving for justice. If currency was a foundational institution, it was also always insufficient, likely imperfect, and possibly tragic. Turned into a tool for the accumulation of wealth for its own sake, currency becomes unjust and a serious threat to any political community. Aristotelian currency can fail precisely because it contains an important moment of ethical deliberation. This political significance of currency challenges accounts of the ancient world as bifurcated between oikos and polis and encourages contemporary political theorists to think of money as a constitutional project that can play an important role in improving reciprocity across society. (shrink)

It is widely recognized that the innate versus acquired distinction is a false dichotomy. Yet many scientists continue to describe certain traits as “innate” and take this to imply that those traits are not acquired, or “unlearned.” This article asks what cognitive role, if any, the concept of innateness should play in the psychological and behavioural sciences. I consider three arguments for eliminating innateness from scientific discourse. First, the classification of a trait as innate is thought to discourage empirical research (...) into its developmental origin. Second, this concept lumps together a number of different biological properties that ought to be treated as distinct. Third, innateness is associated with the outmoded folk biological theory of essentialism. In response to these objections, I consider two attempts to revise the concept of innateness which aim to make it more suitable for scientific explanation and research. One proposal is that innateness can be defined in terms of the biological property of environmental canalization. On this view, a trait is innate to the extent that it is developmentally buffered against a range of different environments. Another proposal is that innateness serves as an explanatory primitive for cognitive science. This view holds that there exist a sharp boundary between psychological and biological explanations and that to identify a trait as innate means that it falls into the latter explanatory domain. This essay ends with some questions for future research. (shrink)

The entropy-reduction hypothesis claims that the cognitive processing difficulty on a word in sentence context is determined by the word's effect on the uncertainty about the sentence. Here, this hypothesis is tested more thoroughly than has been done before, using a recurrent neural network for estimating entropy and self-paced reading for obtaining measures of cognitive processing load. Results show a positive relation between reading time on a word and the reduction in entropy due to processing that word, supporting the entropy-reduction (...) hypothesis. Although this effect is independent from the effect of word surprisal, we find no evidence that these two measures correspond to cognitively distinct processes. (shrink)

A measure of coherence is said to be reliability conducive if and only if a higher degree of coherence (as measured) among testimonies implies a higher probability that the witnesses are reliable. Recently, it has been proved that several coherence measures proposed in the literature are reliability conducive in scenarios of equivalent testimonies (Olsson and Schubert 2007; Schubert, to appear). My aim is to investigate which coherence measures turn out to be reliability conducive in the more general scenario where the (...) testimonies do not have to be equivalent. It is shown that four measures are reliability conducive in the present scenario, all of which are ordinally equivalent to the Shogenji measure. I take that to be an argument for the Shogenji measure being a fruitful explication of coherence. (shrink)

Within evolutionary biology, life-history theory is used to explain cross-species differences in allocation strategies regarding reproduction, maturation, and survival. Behavioral scientists have recently begun to conceptualize such strategies as a within-species individual characteristic that is predictive of behavior. Although life history theory provides an important framework for behavioral scientists, the psychometric approach to life-history strategy measurement—as operationalized by K-factors—involves conceptual entanglements. We argue that current psychometric approaches attempting to identify K-factors are based on an unwarranted conflation of functional descriptions and (...) proximate mechanisms—a conceptual mix-up that may generate unviable hypotheses and invites misinterpretation of empirical findings. The assumptions underlying generic psychometric methodology do not allow measurement of functionally defined variables; rather these methods are confined to Mayr’s proximate causal realm. We therefore conclude that K-factor scales lack validity, and that life history strategy cannot be identified with psychometrics as usual. To align theory with methodology, suggestions for alternative methods and new avenues are proposed. (shrink)

A measure of coherence is said to be reliability conducive if and only if a higher degree of coherence (asmeasured) of a set of testimonies implies a higher probability that the witnesses are reliable. Recently, it has been proved that the Shogenji measure of coherence is reliability conducive in restricted scenarios (e.g., Olsson and Schubert, Synthese, 157:297–308, 2007). In this article, I investigate whether the Shogenji measure, or any other coherence measure, is reliability conducive in general. An impossibility theorem is (...) proved to the effect that this is not the case. I conclude that coherence is not reliability conducive. (shrink)

ABSTRACT How do non-experts single out numbers for reference? Linnebo has argued that they do so using a criterion of identity based on the ordinal properties of numerals. Neo-logicists, on the other hand, claim that cardinal properties are the basis of individuation, when they invoke Hume’s Principle. I discuss empirical data from cognitive science and linguistics to answer how non-experts individuate numbers better in practice. I use those findings to develop an alternative account that mixes ordinal and cardinal properties to (...) provide a detailed answer to the question: how do we in fact semantically individuate numbers? (shrink)

We show that under the assumption of the existence of the canonical inner model with one Woodin cardinal$M_1$, there is a model of$\mathsf {ZFC}$in which$\mbox {NS}_{\omega _{1}}$is$\aleph _2$-saturated and${\Delta }_{1}$-definable with$\omega _1$as a parameter which answers a question of S. D. Friedman and L. Wu. We also show that starting from an arbitrary universe with a Woodin cardinal, there is a model with$\mbox {NS}_{\omega _{1}}$saturated and${\Delta }_{1}$-definable with a ladder system$\vec {C}$and a full Suslin treeTas parameters. Both results rely on (...) a new coding technique whose presentation is the main goal of this article. (shrink)

The recent explosion of interest in epigenetics is often portrayed as the dawning of a scientific revolution that promises to transform biomedical science along with developmental and evolutionary biology. Much of this enthusiasm surrounds what we call the epigenetic switch hypothesis, which regards certain examples of epigenetic inheritance as an adaptive organismal response to environmental change. This interpretation overlooks an alternative explanation in terms of coevolutionary dynamics between parasitic transposons and the host genome. This raises a question about whether epigenetics (...) researchers tend to overlook transposon dynamics more generally. To address this question, we surveyed a large sample of scientific publications on the topics of epigenetics and transposons over the past fifty years. We found that enthusiasm for epigenetics is often inversely related to interest in transposon dynamics across the four disciplines we examined. Most surprising was a declining interest in transposons within biomedical science and cellular and molecular biology over the past two decades. Also notable was a delayed and relatively muted enthusiasm for epigenetics within evolutionary biology. An analysis of scientific abstracts from the past twenty-five years further reveals systematic differences among disciplines in their uses of the term epigenetic, especially with respect to heritability commitments and functional interpretations. Taken together, these results paint a nuanced picture of the rise of epigenetics and the possible neglect of transposon dynamics, especially among biomedical scientists. (shrink)

This study examines the conflation of terms such as “knowledge” and “understanding” in peer-reviewed literature, and tests the hypothesis that little current research clearly distinguishes between importantly distinct epistemic states. Two sets of data are presented from papers published in the journal Public Understanding of Science. In the first set, the digital text analysis tool, Voyant, is used to analyze all papers published in 2014 for the use of epistemic success terms. In the second set of data, all papers published (...) in Public Understanding of Science from 2010–2015 are systematically analyzed to identify instances in which epistemic states are empirically measured. The results indicate that epistemic success terms are inconsistently defined, and that measurement of understanding, in particular, is rarely achieved in public understanding of science studies. We suggest that more diligent attention to measuring understanding, as opposed to mere knowledge, will increase efficacy of scientific outreach and communication efforts. (shrink)

The normative mechanics of promising seem complex. The strength and content of promissory obligations, and the residual duties they entail upon being violated, have various prima facie surprising features. We give an account to explain these features. Promises have a point. The point of a promise to φ is a promise-independent reason to φ for the promisee’s sake. A promise turns this reason into a duty. This explains the mechanics of promises. And it grounds a nuanced picture of immoral promises, (...) an argument against promissory bare wrongings, and a constraint on theories of why we have promissory obligations at all. (shrink)

Landgrebe and Smith provide an argument for the impossibility of Artificial General Intelligence based on the limits of simulating complex systems. However, their argument presupposes a very contemporary vision of artificial intelligence as a model trained on data to produce an algorithm executable in a modern digital computing system. The present contribution explores what it means to be artificial. Current artificial intelligence approaches on modern computing systems are not the only conceivable way in which artificial intelligence technology might be created. (...) If there are conceivable routes by which an artificial intelligence might be developed that are not constrained in the same way as the current generation of artificial neural networks, then these might be plausible routes towards the engineered generation of artificial general intelligence that are not precluded by the Landgrebe and Smith argument against the possibility of artificial general intelligence. (shrink)

Journal of Mathematical Logic, Volume 22, Issue 02, August 2022. We generically construct a model in which the [math]-separation property is true, i.e. every pair of disjoint [math]-sets can be separated by a [math]-definable set. This answers an old question from the problem list “Surrealist landscape with figures” by A. Mathias from 1968. We also construct a model in which the (lightface) [math]-separation property is true.

Recent examples of rapid evolution under natural selection seem to require that the disciplines of ecology and evolution become better integrated. This inference makes sense only if one’s understanding of these disciplines is based on Hutchinson’s two-speed model of the ecological theater and the evolutionary play. Instead, these disciplines are more accurately viewed as occupying distinct “epistemic niches.” When so understood, we see that rapid evolution under selection, even if it is generally true, does not imply that evolutionary explanations are (...) improved by the inclusion of ecological details. Nor are ecological explanations necessarily improved by the inclusion of information about trait variation, heritability, effective population size, or other standard evolutionary factors. To illustrate, I develop a version of Kitcher’s “gory details” argument to show that, even for some trait that is under strong directional selection, a dynamically sufficient explanation of its ecological relationships should ignore most of the information explaining why that trait is evolving. The wholesale integration of ecology and evolution looks even less appealing when empirical sufficiency, a purely practical requirement, is taken into account. As a way forward, I propose an eco-evo partitioning framework. This strategy enables researchers to estimate the empirical sufficiency of a purely ecological, a purely evolutionary, or a combined eco-evo approach. (shrink)

We investigate the issue of aggregativity in fair division problems from the perspective of cooperative game theory and Broomean theories of fairness. Paseau and Saunders proved that no non-trivial theory of fairness can be aggregative and conclude that theories of fairness are therefore problematic, or at least incomplete. We observe that there are theories of fairness, particularly those that are based on cooperative game theory, that do not face the problem of non-aggregativity. We use this observation to argue that the (...) universal claim that no non-trivial theory of fairness can guarantee aggregativity is false. Paseau and Saunders’s mistaken assertion can be understood as arising from a neglect of the games approach to fair division. Our treatment has two further pay-offs: for one, we give an accessible introduction to the games approach to fair division, whose significance has hitherto not been appreciated by philosophers working on fairness. For another, our discussion explores the issue of aggregativity in fair division problems in a comprehensive fashion. (shrink)

Most accounts of our knowledge of the successor axiom claim that this is based on the procedure of adding one. While they usually don’t claim to provide an account of how children actually acquire this knowledge, one may well think that this is how they get that knowledge. I argue that when we look at children’s responses in interviews, the time when they learn the successor axiom and the intermediate learning stages they find themselves in, that there is an empirically (...) viable alternative. I argue that they could also learn it on the basis of a method that has to do with the structure of the numeral system. Specifically, that they (1) use the syntactic structure of the numeral system and (2) attend to the leftmost digits, the one with the highest place-value. Children can learn that this is a reliable method of forming larger numbers by combining two elements. First, a grasp of the syntactic structure of the numeral system. That way they know that the leftmost digit receives the highest value. Second, an interpretation of numerals as designating cardinal values, so that they also realise that increasing or adding digits on the lefthand side of a numeral produces a larger number. There are thus two, currently equally well-supported, ways in which children might learn that there are infinitely many natural numbers. (shrink)

Why should governments or individuals invest time and resources in conserving biodiversity? A popular answer is that biodiversity has both instrumental value for humans and intrinsic value in its own right. Defending Biodiversity critically evaluates familiar arguments for these claims and finds that, at best, they provide good reasons for conserving particular species or regions. However, they fail to provide a strong justification for conserving biodiversity per se. Hence, either environmentalists must develop more compelling arguments for conserving biodiversity or else (...) they should modify their agenda. This short precis is an overview of the central findings of our book. (shrink)

In a recent article, Emanuel Rutten has presented a novel argument for the existence of God, defined as a personal being that is the first cause of reality. An interesting feature of the argument, which caused quite a stir, is that it does not fall within any of the traditional categories of arguments for God’s existence. Rutten calls his argument a modal-epistemic one, which reflects the fact that the first premise of his argument states that all possible truths are knowable. (...) The main purpose of this article is a simple one: to point out that Rutten’s modal-epistemic argument is flawed. (shrink)

In a recent article, Emanuel Rutten defends his Modal-Epistemic Argument (MEA) for the existence of God against various objections that I raised against it. In this article, I observe that Rutten’s defence fails for various reasons. Most notably though, the defence is self-undermining: the very claims that Rutten argues for in his defence yield novel counterexamples to the first premise of the MEA.

Business Ethics, the Environment &Responsibility, Volume 32, Issue 2, Page 415-418, April 2023.

This volume features a collection of essays on consciousness, which has become one of the hot topics at the crossroads between neuroscience, philosophy, and religious studies. Is consciousness something the brain produces? How can we study it? Is there just one type of consciousness or are there different states that can be discriminated? Are so called "higher states of consciousness" that some people report during meditation pointing towards a new understanding of consciousness? Meditation research is a new discipline that shows (...) new inroads into the study of consciousness. If a meditative practice changes brain structure itself this is direct proof of the causal influence of consciousness onto its substrate. If different states of consciousness can be linked with properties and states of the brain this can be used to study consciousness more directly. If the sense of self is modifiable through meditative techniques and this can be objectively shown through neuro-imaging, this has profound implications for our understanding of who we are. Can consciousness, in deep states of meditative absorption, actually access some aspect of reality which we normally don't? Meditation research can potentially foster us with a new access to the phenomenological method in general. This has even been branded with a new catch-phrase: Contemplative Science. It brings together the most modern neuroscientific approach and the most advanced phenomenological methodology of studying the mind from within, through highly skilled self-observation that has gone through many thousand hours of honing the capacity to look carefully, without distraction. This book addresses these issues by bringing together some of the leading researchers and thinkers in the field. The scope of the volume reaches from first person neuroscience to Indian philosophy, from pedagogic applications to epistemological aspects and from compassion meditation to the study of brain activity. (shrink)

Lawton’s contingency thesis states that there are no useful generalizations at the level of ecological communities because these systems are especially prone to contingent historical events. I argue that this influential thesis has been grounded on the wrong kind of evidence. CT is best understood in Woodward’s terms as a claim about the instability of certain causal dependencies across different background conditions. A recent distinction between evolution and ecology reveals what an adequate test of Lawton’s thesis would look like. To (...) date, CT remains untested. But developments in genome and molecular ecology point in a promising direction. (shrink)

The study of human-human communication and the development of computational models for human-agent communication have diverged significantly throughout the last decade. Yet, despite frequently made claims of “super-human performance” in, e.g., speech recognition or image processing, so far, no system is able to lead a half-decent coherent conversation with a human. In this paper, we argue that we must start to re-consider the hallmarks of cooperative communication and the core capabilities that we have developed for it, and which conversational agents (...) need to be equipped with: incremental joint co-construction and mentalizing. We base our argument on a vast body of work on human-human communication and its psychological processes that we reason to be relevant and necessary to take into account when modeling human-agent communication. We contrast those with current conceptualizations of human-agent interaction and formulate suggestions for the development of future systems. (shrink)

In their recent paper Bi-facial truth: a case for generalized truth values Zaitsev and Shramko [7] distinguish between an ontological and an epistemic interpretation of classical truth values. By taking the Cartesian product of the two disjoint sets of values thus obtained, they arrive at four generalized truth values and consider two “semi-classical negations” on them. The resulting semantics is used to define three novel logics which are closely related to Belnap’s well-known four valued logic. A syntactic characterization of these (...) logics is left for further work. In this paper, based on our previous work on a functionally complete extension of Belnap’s logic, we present a sound and complete tableau calculus for these logics. It crucially exploits the Cartesian nature of the four values, which is reflected in the fact that each proof consists of two tableaux. The bi-facial notion of truth of Z&S is thus augmented with a bi-facial notion of proof. We also provide translations between the logics for semi-classical negation and classical logic and show that an argument is valid in a logic for semi-classical negation just in case its translation is valid in classical logic. (shrink)

Mathematical cognition has become an interesting case study for wider theories of cognition. Menary :1–20, 2015) argues that arithmetical cognition not only shows that internalist theories of cognition are wrong, but that it also shows that the Hypothesis of Extended Cognition is right. I examine this argument in more detail, to see if arithmetical cognition can support such conclusions. Specifically, I look at how the use of numerals extends our arithmetical abilities from quantity-related innate systems to systems that can deal (...) with exact numbers of arbitrary size. I then argue that the system underlying our grasp of small numbers is an unhelpful case study for Menary; it doesn’t support an argument for externalism over internalism. The system for large numbers, on the other hand, clearly displays important interactions between public numeral systems and our cognitive processes. I argue that the large number system supports an argument against internalist theories of arithmetical cognition, but that one cannot conclude that the Hypothesis of Extended Cognition is correct. In other words, the large number case doesn’t decide between the Hypothesis of Extended Cognition and the Hypothesis of EMbedded Cognition. (shrink)