Using recent results in topos theory, two systems of higher-order logic are shown to be complete with respect to sheaf models over topological spaces- so -called "topological semantics." The first is classical higher-order logic, with relational quantification of finitely high type; the second system is a predicative fragment thereof with quantification over functions between types, but not over arbitrary relations. The second theorem applies to intuitionistic as well as classical logic.

Using recent results in topos theory, two systems of higher-order logic are shown to be complete with respect to sheaf models over topological spaces-so-called "topological semantics". The first is classical higher-order logic, with relational quantification of finitely high type; the second system is a predicative fragment thereof with quantification over functions between types, but not over arbitrary relations. The second theorem applies to intuitionistic as well as classical logic.

Open peer commentary on the target article “How and Why the Brain Lays the Foundations for a Conscious Self” by Martin V. Butz. Excerpt: Avoiding the term “representation” would make the article much more consistent with a radical constructivist way of thinking. It would also open up unexpected opportunities for realizing the potential of some of its most interesting ideas, such as the connection between anticipatory drive and attention (§31).

Open peer commentary on the target article “How and Why the Brain Lays the Foundations for a Conscious Self” by Martin V. Butz. Excerpt: One intriguing concept that the author introduces and uses throughout the paper is the idea of an “anticipatory drive,” which is described as explaining the systematic tendency to develop anticipatory capabilities that ultimately support goal-oriented action. Although the idea of a common mechanism that explains a multitude of capabilities can be appreciated, it is unclear if (...) the author uses the term “drive” in a literal or metaphorical sense. Should the evolutionary advantage, the selective “pressure” for anticipatory capacities and representation be considered a drive in the sense of Hull like, for example, the sex drive, or hunger? If this is true, it should be related to some bodily control structure like the other mentioned drives. Another possibility is that the anticipatory drive is an intrinsic motivation, like curiosity, which “pushes” towards certain situations, such as those that are more predictable. If this is true, it should be explained what is the exact nature of such intrinsic motivation. (shrink)

In this paper we study the logic $\mathcal{L}^\lambda_{\omega\omega}$, which is first-order logic extended by quantification over functions . We give the syntax of the logic as well as the semantics in Heyting categories with exponentials. Embedding the generic model of a theory into a Grothendieck topos yields completeness of $\mathcal{L}^\lambda_{\omega\omega}$ with respect to models in Grothendieck toposes, which can be sharpened to completeness with respect to Heyting-valued models. The logic $\mathcal{L}^\lambda_{\omega\omega}$ is the strongest for which Heyting-valued completeness is known. Finally, (...) we relate the logic to locally connected geometric morphisms between toposes. (shrink)

ID 68, the important Delian inscription found in 1910 prohibiting access to outsiders and with jurisdiction probably over the Archegesion, is shown to have been inscribed twice. This is not a matter of republication, rather of simultaneous publication within the same sanctuary. The article provides a new text and a complete bibliography for the double publication before analyzing the architectural function of the two inscriptions and their significance as sacred law. With emphasis on the palaeography and aesthetic value of the (...) letterforms as well as on the physical evidence of the stones themselves, the meaning of the prohibition is reexamined, resulting in the proposal of a specific social context and date. (shrink)

Butz, Achimova, Bilkey, and Knott provide a topic overview and discuss whether the special issue contributions may imply that event‐predictive abilities constitute a root for conceptual human thought, because they enable complex, mutually beneficial, but also intricately competitive, social interactions and language communication.

Butz, Achimova, Bilkey, and Knott provide a topic overview and discuss whether the special issue contributions may imply that event‐predictive abilities constitute a root for conceptual human thought, because they enable complex, mutually beneficial, but also intricately competitive, social interactions and language communication.

This paper proposes how various disciplinary theories of cognition may be combined into a unifying, sub-symbolic, computational theory of cognition. The following theories are considered for integration: psychological theories, including the theory of event coding, event segmentation theory, the theory of anticipatory behavioral control, and concept development; artificial intelligence and machine learning theories, including reinforcement learning and generative artificial neural networks; and theories from theoretical and computational neuroscience, including predictive coding and free energy-based inference. In the light of such a (...) potential unification, it is discussed how abstract cognitive, conceptualized knowledge and understanding may be learned from actively gathered sensorimotor experiences. The unification rests on the free energy-based inference principle, which essentially implies that the brain builds a predictive, generative model of its environment. Neural activity-oriented inference causes the continuous adaptation of the currently active predictive encodings. Neural structure-oriented inference causes the longer term adaptation of the developing generative model as a whole. Finally, active inference strives for maintaining internal homeostasis, causing goal-directed motor behavior. To learn abstract, hierarchical encodings, however, it is proposed that free energy-based inference needs to be enhanced with structural priors, which bias cognitive development towards the formation of particular, behaviorally suitable encoding structures. As a result, it is hypothesized how abstract concepts can develop from, and thus how they are structured by and grounded in, sensorimotor experiences. Moreover, it is sketched-out how symbol-like thought can be generated by a temporarily active set of predictive encodings, which constitute a distributed neural attractor in the form of an interactive free-energy minimum. The activated, interactive network attractor essentially characterizes the semantics of a concept or a concept composition, such as an actual or imagined situation in our environment. Temporal successions of attractors then encode unfolding semantics, which may be generated by a behavioral or mental interaction with an actual or imagined situation in our environment. Implications, further predictions, possible verification and falsifications, as well as potential enhancements into a fully spelled-out unified theory of cognition are discussed at the end of the paper. (shrink)

By a classifying topos for a first-order theory , we mean a topos such that, for any topos models of in correspond exactly to open geometric morphisms → . We show that not every first-order theory has a classifying topos in this sense, but we characterize those which do by an appropriate ‘smallness condition’, and we show that every Grothendieck topos arises as the classifying topos of such a theory. We also show that every first-order theory has a conservative extension (...) to one which possesses a classifying topos, and we obtain a Heyting-valued completeness theorem for infinitary first-order logic. (shrink)

Purpose: Constructivism postulates that the perceived reality is a complex construct formed during development. Depending on the particular school, these inner constructs take on different forms and structures and affect cognition in different ways. The purpose of this article is to address the questions of how and, even more importantly, why we form such inner constructs. Approach: This article proposes that brain development is controlled by an inherent anticipatory drive, which biases learning towards the formation of forward predictive structures and (...) inverse goal-oriented control structures. This drive, in combination with increasingly complex environmental interactions during cognitive development, enforces the structuring of our conscious self, which is embedded in a constructed inner reality. Essentially, the following questions are addressed: Which basic mechanisms lead us to the construction of inner realities? How are these emergent inner realities structured? How is the self represented within the inner realities? And consequently, which cognitive structures constitute the media for conscious thought and selfconsciousness? Findings: Due to the anticipatory drive, representations in the brain shape themselves predominantly purposefully or intentionally. Taking a developmental, evolutionary perspective, we show how the brain is forced to develop progressively complex and abstract representations of the self embedded in the constructed inner realities. These self representations can evoke different stages of self-consciousness. Implications: The anticipatory drive shapes brain structures and cognition during the development of progressively more complex, competent, and flexible goal-oriented bodyenvironment interactions. Self-consciousness develops because increasingly abstract, individualizing self representations are necessary to realize these progressively more challenging environmental interactions. (shrink)

“Probably Peirce’s best-known works are the first two articles in a series of six that originally were collectively entitled Illustrations of the Logic of Science and published in Popular Science Monthly from November 1877 through August 1878. The first is entitled ‘The Fixation of Belief’ and the second is entitled ‘How to Make Our Ideas Clear.’ In the first of these papers Peirce defended, in a manner consistent with not accepting naive realism, the superiority of the scientific method over other (...) methods of overcoming doubt and ‘fixing belief.’” — Robert Burch, “Charles Sanders Peirce,” entry in the Stanford Encyclopedia of Philosophy (2021 revision) -/- “Pragmatist epistemologies often explore how we can carry out inquiries in a self-controlled and fruitful way. (Where much analytic epistemology centres around the concept of knowledge, considered as an idealised end-point of human thought, pragmatist epistemology centres around the concept of inquiry, considered as the process of knowledge-seeking and how we can improve it.) So pragmatists often provide rich accounts of the capacities or virtues that we must possess in order to inquire well, and the rules or guiding principles that we should adopt. A canonical account is Peirce’s classic early paper ‘The Fixation of Belief‘. Here Peirce states that inquiry is a struggle to replace doubt with “settled belief“, and that the only method of inquiry that can make sense of the fact that at least some of us are disturbed by inconsistent beliefs, and will subsequently reflect upon which methods of fixing belief are correct is the Method of Science, which draws on the Pragmatic Maxim described above. This contrasts with three other methods of fixing belief: i) refusing to consider evidence contrary to one’s favored beliefs (the Method of Tenacity), ii) accepting an institution’s dictates (the Method of Authority), iii) developing the most rationally coherent or elegant-seeming belief-set (the A Priori Method).“ — Catherine Legg, “Pragmatism,” entry in the Stanford Encyclopedia of Philosophy (2021 revision). (shrink)

According to most accounts of trust, you can only trust other people (or groups of people). To trust is to think that another has goodwill, or something to that effect. I sketch a different form of trust: the unquestioning attitude. What it is to trust, in this sense, is to settle one’s mind about something, to stop questioning it. To trust is to rely on a resource while suspending deliberation over its reliability. Trust lowers the barrier of monitoring, challenging, checking, (...) and questioning. Trust sets up open pipelines between yourself and parts of the external world. Trust permits external resources to have a similar relationship to one as one’s internal cognitive faculties. This creates efficiency, but at the price of exquisite vulnerability. We must trust in this way because we are cognitively limited beings in a cognitively overwhelming world. Crucially, we can hold the unquestioning attitude towards objects and technologies. When I trust my climbing rope, I stop worrying about its reliability. When I trust my online calendaring system, I simply go to the events indicated, without question. But, one might worry, how could one ever hold such a normatively loaded attitude as trust towards mere objects? How could it ever make sense to feel betrayed by an object? Trust is our engine for expanding and outsourcing our agency — for binding external processes into our practical selves. Thus, we can be betrayed by our smartphones in the same way that we can be betrayed by our memory. When we trust, we try to make something a part of our agency, and we are betrayed when our part lets us down. To unquestioningly trust something is to let it in—to attempt to bring it inside one’s practical functioning. This suggests a new form of gullibility: agential gullibility, which occurs when agents too hastily and carelessly integrate external resources into their own agency. (shrink)

Value capture occurs when an agent’s values are rich and subtle; they enter a social environment that presents simplified — typically quantified — versions of those values; and those simplified articulations come to dominate their practical reasoning. Examples include becoming motivated by FitBit’s step counts, Twitter Likes and Re-tweets, citation rates, ranked lists of best schools, and Grade Point Averages. We are vulnerable to value capture because of the competitive advantage that such crisp and clear expressions of value have in (...) our private reasoning and our public justification. There is, however, a price. In value capture, we take a central component of our autonomy — our ongoing deliberation over the exact articulation of our values — and we outsource it. And the metrics to which we outsource usually engineered for the interests of some external force, like a large-scale institution’s interest in cross-contextual comprehensibility and quick aggregability. That outsourcing cuts off one of the key benefits to personal deliberation. In value capture, we no longer adjust our values and their articulations in light of own rich experience of the world. Our values should often be carefully tailored to our particular selves or our small-scale communities, but in value capture, we buy our values off the rack. In some cases – like decreasing CO2 emissions – the costs of non-tailored values are outweighed by the benefit of precise collective coordination. In other cases, like in our aesthetic lives, they are not. This suggests that we should want different values suited to different scales. We should want value federalism. Some values are perhaps best pursued at the largest-scale level, others at smaller scales. The problem occurs when we exhibit an excess preference for the largest-scale values – when we consistently let the universal metrics swamp our quieter interests. (shrink)

This study compares the relationship between the moral reasoning modes and leadership orientation of males versus females, and managers versus engineers/scientists. A questionnaire developed by Worthley (1987) was used to measure the degree of each participant's respective independence and justice, and relationships and caring moral reasoning modes. Leadership orientation values and attitudes were measured using the Fiedler and Chemers (1984) Least Preferred Coworker Scale.The results suggest that, although males differ from female in their dominant moral reasoning modes, managers are not (...) distinguishable from the engineers/scientists they manage in terms of their moral reasoning mode or Least Preferred Coworker score. (shrink)

We use the language of categorical logic to construct generic saturated models of intuitionistic theories. Our main technique is the thorough study of the filter construction on categories with finite limits, which is the completion of subobject lattices under filtered meets. When restricted to coherent or Heyting categories, classifying categories of intuitionistic first-order theories, the resulting categories are filtered meet coherent categories, coherent categories with complete subobject lattices such that both finite disjunctions and existential quantification distribute over filtered meets. Such (...) categories naturally embed into Grothendieck toposes which then contain saturated models of the theory we started with. (shrink)

We show how to interpret the language of first-order set theory in an elementary topos endowed with, as extra structure, a directed structural system of inclusions (dssi). As our main result, we obtain a complete axiomatization of the intuitionistic set theory validated by all such interpretations. Since every elementary topos is equivalent to one carrying a dssi, we thus obtain a first-order set theory whose associated categories of sets are exactly the elementary toposes. In addition, we show that the full (...) axiom of Separation is validated whenever the dssi is superdirected. This gives a uniform explanation for the known facts that cocomplete and realizability toposes provide models for Intuitionistic Zermelo—Fraenkel set theory (IZF). (shrink)

This book offers an interdisciplinary introduction to embodied cognitive science, addressing the question of how the mind comes into being while actively interacting with and learning from the environment by means of the own body. By pursuing a functional and computational perspective, concrete answers are provided about the fundamental mechanisms and developing structures that must bring the mind about, taking into account insights from biology, neuroscience, psychology, and philosophy as well as from computer science, machine learning, and artificial intelligence.

We prove a topological completeness theorem for infinitary geometric theories with respect to sheaf models. The theorem extends a classical result of Makkai and Reyes, stating that any topos with enough points has an open spatial cover. We show that one can achieve in addition that the cover is connected and locally connected.

Open peer commentary on the article “A Computational Constructivist Model as an Anticipatory Learning Mechanism for Coupled Agent–Environment Systems” by Filipo Studzinski Perotto. Upshot: The schema system presented in the target article suffers from problems that had been acknowledged more than ten years ago. The main point is that our world is neither deterministic nor symbolic. Sensory as well as motor noise is ubiquitous in our environment. Symbols do not exist a priori but need to be grounded within our continuous (...) world. In conclusion, I suggest that research on schema-learning systems should tackle small but real-world, continuous, and noisy problem domains. (shrink)

Open peer commentary on the target article “Who Conceives of Society?” by Ernst von Glasersfeld. First paragraph: Cognitive psychology, neurobiology, and cognitive systems research provide diverse clues as to how we are able to incrementally construct representations of the perceived environment and how we consequently understand other individuals and society. The construction of an individual’s reality starts with the capability to control one’s own body and to be able to predict the usual sensory effects caused by body movements. To be (...) able to infer the potential intentions of others, mirror neurons project one’s own behavioral codes onto perceived patterns that are caused by others. Equipped with representations of many other individuals, personal social realities are constructed. In this commentary, I focus on these points for the construction of social reality and the consequent existence of society as a whole. (shrink)

Pursuing a precise, focused train of thought requires cognitive effort. Even more effort is necessary when more alternatives need to be considered or when the imagined situation becomes more complex. Cognitive resources available to us limit the cognitive effort we can spend. In line with previous work, an information-theoretic, Bayesian brain approach to cognitive effort is pursued: to solve tasks in our environment, our brain needs to invest information, that is, negative entropy, to impose structure, or focus, away from a (...) uniform structure or other task-incompatible, latent structures. To get a more complete formalization of cognitive effort, a resourceful event-predictive inference model is introduced, which offers computational and algorithmic explanations about the latent structure of our generative models, the active inference dynamics that unfold within, and the cognitive effort required to steer the dynamics—to, for example, purposefully process sensory signals, decide on responses, and invoke their execution. REPI suggests that we invest cognitive resources to infer preparatory priors, activate responses, and anticipate action consequences. Due to our limited resources, though, the inference dynamics are prone to task-irrelevant distractions. For example, the task-irrelevant side of the imperative stimulus causes the Simon effect and, due to similar reasons, we fail to optimally switch between tasks. An actual model implementation simulates such task interactions and offers first estimates of the involved cognitive effort. The approach may be further studied and promises to offer deeper explanations about why we get quickly exhausted from multitasking, how we are influenced by irrelevant stimulus modalities, why we exhibit magnitude interference, and, during social interactions, why we often fail to take the perspective of others into account. (shrink)

Open peer commentary on the article “Exploration of the Functional Properties of Interaction: Computer Models and Pointers for Theory” by Etienne B. Roesch, Matthew Spencer, Slawomir J. Nasuto, Thomas Tanay & J. Mark Bishop. Upshot: Although the authors investigate a form of distributed swarm intelligence and solve some problems with it – including sorting and summing – the major goal, which is constructing cognition, cannot be achieved by this approach alone. I propose that anticipatory mechanisms have the potential to construct (...) cognition and may very well be combined with swarm intelligence principles. (shrink)

In this paper we study the logic , which is first-order logic extended by quantification over functions (but not over relations). We give the syntax of the logic as well as the semantics in Heyting categories with exponentials. Embedding the generic model of a theory into a Grothendieck topos yields completeness of with respect to models in Grothendieck toposes, which can be sharpened to completeness with respect to Heyting-valued models. The logic is the strongest for which Heyting-valued completeness is known. (...) Finally, we relate the logic to locally connected geometric morphisms between toposes. (shrink)

In this paper we study the logic $\mathcal{L}^\lambda_{\omega\omega}$, which is first-order logic extended by quantification over functions (but not over relations). We give the syntax of the logic as well as the semantics in Heyting categories with exponentials. Embedding the generic model of a theory into a Grothendieck topos yields completeness of $\mathcal{L}^\lambda_{\omega\omega}$ with respect to models in Grothendieck toposes, which can be sharpened to completeness with respect to Heyting-valued models. The logic $\mathcal{L}^\lambda_{\omega\omega}$ is the strongest for which Heyting-valued completeness (...) is known. Finally, we relate the logic to locally connected geometric morphisms between toposes. (shrink)

Recent conversation has blurred two very different social epistemic phenomena: echo chambers and epistemic bubbles. Members of epistemic bubbles merely lack exposure to relevant information and arguments. Members of echo chambers, on the other hand, have been brought to systematically distrust all outside sources. In epistemic bubbles, other voices are not heard; in echo chambers, other voices are actively undermined. It is crucial to keep these phenomena distinct. First, echo chambers can explain the post-truth phenomena in a way that epistemic (...) bubbles cannot. Second, each type of structure requires a distinct intervention. Mere exposure to evidence can shatter an epistemic bubble, but may actually reinforce an echo chamber. Finally, echo chambers are much harder to escape. Once in their grip, an agent may act with epistemic virtue, but social context will pervert those actions. Escape from an echo chamber may require a radical rebooting of one's belief system. (shrink)

In his celebrated masterpiece, Symposium, Plato imagines a high-society dinner-party in Athens in 416 BC at which the guests - including the comic poet Aristophanes and, of course, Plato's mentor Socrates - each deliver a short speech in praise of love. The sequence of dazzling speeches culminates in Socrates' famous account of the views of Diotima, a prophetess who taught him that love is our means of trying to attain goodness. And then into the party bursts the drunken Alcibiades, the (...) most popular and notorious Athenian of the time, who insists on praising Socrates himself rather than love, and gives us a brilliant sketch of this enigmatic character. The power, humour, and pathos of Plato's creation engages the reader on every page. This new translation is complemented by full explanatory notes and an illuminating introduction. (shrink)

"This edition, which replaces the original Loeb edition..., offers text, translation, and annotation that are fully current with modern scholarship"--Front flap of dust jacket, volume 1.

forthcoming in R. Cameron, B. Hale and A. Hoffmann (ed.s), The Logic, Epistemology and Metaphysics of Modality, Oxford University Press. Presents a concept-grounding account of modal knowledge.

I argue that acknowledging Hume as a doxastic naturalist about belief in a deity allows an elegant, holistic reading of his Dialogues. It supports a reading in which Hume's spokesperson is Philo throughout, and enlightens many of the interpretive difficulties of the work. In arguing this, I perform a comprehensive survey of evidence for and against Philo as Hume's voice, bringing new evidence to bear against the interpretation of Hume as Cleanthes and against the amalgamation view while correcting several standard (...) mistakes. I ultimately isolate the interpretation of Philo's Reversal at the end of the Dialogues as of paramount importance, and show how my naturalistic interpretation makes this, and other notoriously difficult passages, unproblematic. (shrink)

Cognitive Science, Volume 45, Issue 8, August 2021.