Traditionally, logic has been the dominant formal method within philosophy. Are logical methods still dominant today, or have the types of formal methods used in philosophy changed in recent times? To address this question, we coded a sample of philosophy papers from the late 2000s and from the late 2010s for the formal methods they used. The results indicate that the proportion of papers using logical methods remained more or less constant over that time period but the (...) proportion of papers using probabilistic methods was approximately three times higher in the late 2010s than it was in the late 2000s. Further analyses explored this change by looking more closely at specific methods, specific levels of technical engagement, and specific subdisciplines within philosophy. These analyses indicate that the increasing proportion of papers using probabilistic methods was pervasive, not confined to particular probabilistic methods, levels of sophistication, or subdisciplines. (shrink)

The purpose of this paper is twofold. First, it aims at introducing the ontological argument through the analysis of five historical developments: Anselm’s argument found in the second chapter of his Proslogion, Gaunilo’s criticism of it, Descartes’ version of the ontological argument found in his Meditations on First Philosophy, Leibniz’s contribution to the debate on the ontological argument and his demonstration of the possibility of God, and Kant’s famous criticisms against the (cartesian) ontological argument. Second, it intends to critically (...) examine the enterprise of formally analyzing philosophical arguments and, as such, contribute in a small degree to the debate on the role of formalization in philosophy. My focus will be mainly on the drawbacks and limitations of such enterprise; as a guideline, I shall refer to a Carnapian, or Carnapian-like theory of argument analysis. (shrink)

In his paper “Flaws of Formal Relationism”, Mahrad Almotahari argues against the sort of response to Frege's Puzzle I have defended elsewhere, which he dubs ‘Formal Relationism’. Almotahari argues that, because of its specifically formal character, this view is vulnerable to objections that cannot be raised against the otherwise similar Semantic Relationism due to Kit Fine. I argue in response that Formal Relationism has neither of the flaws Almotahari claims to identify.

A speculative exploration of the distinction between a relational formal ontology and a classical formal ontology for modelling phenomena in nature that exhibit relationally-mediated wholism, such as phenomena from quantum physics and biosemiotics. Whereas a classical formal ontology is based on mathematical objects and classes, a relational formal ontology is based on mathematical signs and categories. A relational formal ontology involves nodal networks (systems of constrained iterative processes) that are dynamically sustained through signalling. The nodal networks are hierarchically ordered and (...) exhibit characteristics of deep learning. (shrink)

Formal epistemology is just what it sounds like: epistemology done with formal tools. Coinciding with the general rise in popularity of experimental philosophy, formal epistemologists have begun to apply experimental methods in their own work. In this entry, I survey some of the work at the intersection of formal and experimental epistemology. I show that experimental methods have unique roles to play when epistemology is done formally, and I highlight some ways in which results from formal epistemology have been (...) used fruitfully to advance epistemically-relevant experimental work. The upshot of this brief, incomplete survey is that formal and experimental methods often constitute mutually informative means to epistemological ends. (shrink)

he general theme of this paper is the issue of formalization in philosophy; in a more specific way, it deals with the issue of formalization of arguments in analytic philosophy of religion. One argument in particular – Anselm’s Proslogion II ontological argument – and one specific attempt to formalize it – Robert Adams’ formalization found in his paper “The Logical Structure of Anselm’s Arguments”, published in The Philosophical Review in 1971 – are taken as study cases. The (...) purpose of the paper is to critically analyze Adams’ formalization with the intent to shed some light on the following questions: What are the virtues of formally analyzing arguments and the contributions, if any, of such an enterprise to the debate on Anselm’s argument? Which lessons can Adam’s work teach us about the dangers and limitations of formalization? Do these virtues and dangers teach us something about analysis of arguments in general? (shrink)

A brief review of Carnap's formal program in philosophy of science.

Since the beginning of the 20th Century to the present day, it has rarely been doubted that whenever formal aesthetic methods meet their iconological counterparts, the two approaches appear to be mutually exclusive. In reality, though, an ahistorical concept is challenging a historical analysis of art. It is especially Susanne K. Langer´s long-overlooked system of analogies between perceptions of the world and of artistic creations that are dependent on feelings which today allows a rapprochement of these positions. Krois’s insistence on (...) a similar point supports this analysis. - I - Unbestritten bis heute gilt, formwissenschaftliche und ikonologische Methoden scheinen sich grundsätzlich auszuschließen, da die ersteren auf ahistorischen und die letzteren auf historischen Grundlagen aufbauen. Dem entgegen soll mit diesem Beitrag gezeigt werden, wie insbesondere die Forschungen Susanne K. Langers und ergänzend diejenigen von John M. Krois eine Annäherung beider Positionen ermöglichen. (shrink)

Philosophy of Education in South Africa during the latter half of the 20th century was characterised by three ideological strands. The first was known as ‘Fundamental Pedagogics’, the second ‘Liberalism’, and the third ‘Liberation Socialism’ (i.e., Marxism/Freire). When apartheid formally ended in 1994 these strands lost their impetus and faded from educational debates, arguably because of the disappearance of apartheid itself, as the locus relative to which these ideological strands positioned themselves. This paper characterises these three positions and some (...) of their major proponents and ideas, and then present an overview of where philosophy of education in South Africa is now, nearly 30 years after apartheid. I suggest that since 1994 the strong, competing philosophical and ideological commitments that informed philosophy of education during apartheid have significantly dissipated, and have not been meaningfully replaced. Instead, education policy and curriculum are now arguably underpinned by neo-liberal concerns, and the results have been, and continue to be, an unmitigated disaster for schooling in South Africa. Part of the reason for this is because philosophy of education is much diminished, and no longer directly meaningfully contributes to, and informs, educational discourse, policy, teacher training and curriculum in South African schooling. (shrink)

This paper introduces the special issue on Formal Approaches to the Ontological Argument of the Journal of Applied Logics (College Publications). The issue contains the following articles: Formal Approaches to the Ontological Argument, by Ricardo Sousa Silvestre and Jean-Yves Béziau; A Brief Critical Introduction to the Ontological Argument and its Formalization: Anselm, Gaunilo, Descartes, Leibniz and Kant, by Ricardo Sousa Silvestre; A Mechanically Assisted Examination of Begging the Question in Anselm’s Ontological Argument, by John Rushby; A Tractarian Resolution to (...) the Ontological Argument, by Erik Thomsen; On Kant’s Hidden Ambivalence Toward Existential Generalization in his Critique of the Ontological Argument, by Giovanni Mion; The Totality of Predicates and the Possibility of the Most Real Being, by Srećko Kovač; An Even More Leibnizian Version of Gödel’s Ontological Argument, by Kordula Świętorzecka and Marcin Łyczak; A Case Study On Computational Hermeneutics: E. J. Lowe’s Modal Ontological Argument, by David Fuenmayor. (shrink)

(This is for the Cambridge Handbook of Analytic Philosophy, edited by Marcus Rossberg) In this handbook entry, I survey the different ways in which formal mathematical methods have been applied to philosophical questions throughout the history of analytic philosophy. I consider: formalization in symbolic logic, with examples such as Aquinas’ third way and Anselm’s ontological argument; Bayesian confirmation theory, with examples such as the fine-tuning argument for God and the paradox of the ravens; foundations of mathematics, with (...) examples such as Hilbert’s programme and Gödel’s incompleteness theorems; social choice theory, with examples such as Condorcet’s paradox and Arrow’s theorem; ‘how possibly’ results, with examples such as Condorcet’s jury theorem and recent work on intersectionality theory; and the application of advanced mathematics in philosophy, with examples such as accuracy-first epistemology. (shrink)

I submit that epistemic progress in key areas of contemporary academic philosophy has been compromised by politically correct ideology. First, guided by an evolutionary account of ideology, results from social and cognitive psychology and formal philosophical methods, I expose evidence for political bias in contemporary Western academia and sketch a formalization for the contents of beliefs from the PC worldview taken to be of core importance, the theory of social oppression and the thesis of anthropological mental egalitarianism. Then, (...) aided by discussions from contemporary epistemology on epistemic values, I model the problem of epistemic appraisal using the frameworks of multi-objective optimization theory and multi-criteria decision analysis and apply it to politically correct philosophy. I conclude that philosophy guided by politically correct values is bound to produce constructs that are less truth-conducive and that spurious values which are ideologically motivated should be abandoned. Objections to my framework stemming from contextual empiricism, the feminine voice in ethics and political philosophy are considered. I conclude by prescribing the epistemic value of empirical adequacy, the contextual value of political diversity and the moral virtue of moral courage to reverse unwarranted trends in academic philosophy due to PC ideology. (shrink)

Gödel's Philosophical Legacy Kurt Gödel's contributions to philosophy extend beyond his incompleteness theorems. He engaged deeply with the work of other philosophers, including Immanuel Kant and Edmund Husserl, and explored topics such as the nature of time, the structure of the universe, and the relationship between mathematics and reality. Gödel's philosophical writings, though less well-known than his mathematical work, offer rich insights into his views on the nature of existence, the limits of human knowledge, and the interplay between the (...) finite and the infinite. His work continues to inspire and challenge philosophers, mathematicians, and scientists, inviting them to explore the profound and often enigmatic questions at the heart of human understanding. -/- Kurt Gödel's Broader Contributions to Philosophy Kurt Gödel, while primarily known for his monumental incompleteness theorems, made significant contributions that extended beyond the realm of mathematical logic. His philosophical pursuits deeply engaged with the works of eminent philosophers like Immanuel Kant and Edmund Husserl. Gödel's explorations into the nature of time, the structure of the universe, and the relationship between mathematics and reality reveal a profound and multifaceted intellectual legacy. -/- Engagement with Immanuel Kant Gödel held a deep interest in the philosophy of Immanuel Kant. He admired Kant's critical philosophy, particularly the distinction between the noumenal and phenomenal worlds. Kant posited that human experience is shaped by the mind’s inherent structures, leading to the conclusion that certain aspects of reality (the noumenal world) are fundamentally unknowable. Gödel’s incompleteness theorems echoed this Kantian theme, illustrating the limits of formal systems in capturing the totality of mathematical truth. Gödel believed that mathematical truths exis t independently of human thought, akin to Kant's noumenal realm. This philosophical alignment provided a robust foundation for Gödel's Platonism, which asserted the existence of mathematical objects as real, albeit abstract, entities. -/- Influence of Edmund Husserl Gödel was also profoundly influenced by Edmund Husserl, the founder of phenomenology. Husserl's phenomenology emphasizes the direct investigation and description of phenomena as consciously experienced, without preconceived theories about their causal explanation. Gödel saw Husserl's work as a pathway to bridge the gap between the abstract world of mathematics and concrete human experience. Husserl's ideas about the structures of consciousness and the intentionality of thought resonated with Gödel's views on mathematical intuition. Gödel believed that human minds could access mathematical truths through intuition, a concept that draws on Husserlian phenomenological methods. -/- The Nature of Time and the Universe Gödel's philosophical inquiries extended to the nature of time and the structure of the universe. His collaboration with Albert Einstein at the Institute for Advanced Study led to the development of the "Gödel metric" in 1949. This solution to Einstein's field equations of general relativity described a rotating universe where time travel to the past was theoretically possible. Gödel's model challenged conventional notions of time and causality, suggesting that the universe might have a more intricate structure than previously thought. Gödel's exploration of time was not just a mathematical curiosity but a profound philosophical statement about the nature of reality. He questioned whether time was an objective feature of the universe or a construct of human consciousness. His work hinted at a timeless realm of mathematical truths, aligning with his Platonist view. -/- Mathematics and Reality Gödel's philosophical outlook extended to the broader relationship between mathematics and reality. He believed that mathematics provided a more profound insight into the nature of reality than empirical science. For Gödel, mathematical truths were timeless and unchangeable, existing independently of human cognition. This perspective led Gödel to critique the materialist and mechanistic views that dominated 20th-century science and philosophy. He argued that a purely physicalist interpretation of the universe failed to account for the existence of abstract mathematical objects and the human capacity to understand them. Gödel's philosophy suggested a more integrated view of reality, where both physical and abstract realms coexist and inform each other. -/- Gödel's Exploration of Time Kurt Gödel, one of the most profound logicians of the 20th century, ventured beyond the confines of mathematical logic to explore the nature of time. His inquiries into the concept of time were not merely theoretical musings but were grounded in rigorous mathematical formulations. Gödel's exploration of time challenged conventional views and opened new avenues of thought in both physics and philosophy. -/- Gödel and Einstein Gödel’s interest in the nature of time was significantly influenced by his friendship with Albert Einstein. Both were faculty members at the Institute for Advanced Study in Princeton, where they engaged in deep discussions about the nature of reality, time, and space. Gödel's exploration of time culminated in his solution to Einstein's field equations of general relativity, known as the Gödel metric. -/- The Gödel Metric In 1949, Gödel presented a model of a rotating universe, which became known as the Gödel metric. This solution to the equations of general relativity depicted a universe where time travel to the past was theoretically possible. Gödel’s rotating universe contained closed timelike curves (CTCs), paths in spacetime that loop back on themselves, allowing for the possibility of traveling back in time. The Gödel metric posed a significant philosophical challenge to the conventional understanding of time. If time travel were possible, it would imply that time is not linear and absolute, as commonly perceived, but rather malleable and subject to the geometry of spacetime. This raised profound questions about causality, the nature of temporal succession, and the very structure of reality. -/- Philosophical Implications Gödel’s exploration of time extended beyond the mathematical implications to broader philosophical inquiries: Nature of Time: Gödel questioned whether time was an objective feature of the universe or a construct of human consciousness. His work suggested that our understanding of time as a linear progression from past to present to future might be an illusion, shaped by the limitations of human perception. -/- Causality and Free Will: The existence of closed timelike curves in Gödel’s model raised questions about causality and free will. If one could travel back in time, it would imply that future events could influence the past, potentially leading to paradoxes and challenging the notion of a deterministic universe. -/- Temporal Ontology: Gödel's work contributed to debates in temporal ontology, particularly the debate between presentism (the view that only the present exists) and eternalism (the view that past, present, and future all equally exist). Gödel’s rotating universe model seemed to support eternalism, suggesting a block universe where all points in time are equally real. Philosophy of Science: Gödel’s exploration of time had implications for the philosophy of science, particularly in the context of understanding the limits of scientific theories. His work underscored the importance of considering philosophical questions when developing scientific theories, as they shape our fundamental understanding of concepts like time and space. -/- Legacy Gödel’s exploration of time remains a significant and controversial contribution to both physics and philosophy. His work challenged established notions and encouraged deeper inquiries into the nature of reality. Gödel’s rotating universe model continues to be a topic of interest in theoretical physics and cosmology, inspiring new research into the nature of time and the possibility of time travel. In philosophy, Gödel’s inquiries into time have prompted ongoing debates about the nature of temporal reality, the relationship between mathematics and physical phenomena, and the limits of human understanding. His work exemplifies the intersection of mathematical rigor and philosophical inquiry, demonstrating the profound insights that can emerge from such an interdisciplinary approach. The Temporal Ontology of Kurt Gödel Kurt Gödel's profound contributions to mathematics and logic extend into the realm of temporal ontology—the philosophical study of the nature of time and its properties. Gödel's insights challenge conventional perceptions of time and suggest a more intricate, layered understanding of temporal reality. This essay explores Gödel's contributions to temporal ontology, particularly through his engagement with relativity and his philosophical reflections. -/- Gödel's Rotating Universe One of Gödel’s most notable contributions to temporal ontology comes from his work in cosmology, specifically his solution to Einstein’s field equations of general relativity, known as the Gödel metric. Introduced in 1949, the Gödel metric describes a rotating universe with closed timelike curves (CTCs). These curves imply that, in such a universe, time travel to the past is theoretically possible, presenting a significant challenge to conventional views of linear, unidirectional time. -/- Implica tions for Temporal Ontology Gödel's rotating universe model has profound implications for our understanding of time: Eternalism vs. Presentism: Gödel’s model supports the philosophical stance known as eternalism, which posits that past, present, and future events are equally real. In contrast to presentism, which holds that only the present moment exists, eternalism suggests a "block universe" where time is another dimension like space. Gödel’s rotating universe, with its CTCs, reinforces this view by demonstrating that all points in time could, in principle, be interconnected in a consistent manner. Non-linearity of Time: The possibility of closed timelike curves challenges the idea of time as a linear sequence of events. In Gödel’s universe, time is not merely a straight path from past to future but can loop back on itself, allowing for complex interactions between different temporal moments. This non-linearity has implications for our understanding of causality and the nature of temporal succession. Objective vs. Subjective Time: Gödel’s work invites reflection on the distinction between objective time (the time that exists independently of human perception) and subjective time (the time as experienced by individuals). His model suggests that our subjective experience of a linear flow of time may not correspond to the objective structure of the universe. This raises questions about the relationship between human consciousness and the underlying temporal reality. -/- Gödel and Philosophical Reflections on Time Gödel’s engagement with temporal ontology was not limited to his cosmological work. He also reflected deeply on philosophical questions about the nature of time and reality, drawing on the ideas of other philosophers and integrating them into his own thinking. Kantian Influences: Gödel was influenced by Immanuel Kant’s distinction between the noumenal world (things as they are in themselves) and the phenomenal world (things as they appear to human observers). Gödel’s views on time echoed this distinction, suggesting that our perception of time might be a phenomenon shaped by the limitations of human cognition, while the true nature of time (the noumenal aspect) might be far more complex and non-linear. Husserlian Phenomenology: Gödel’s interest in Edmund Husserl’s phenomenology also informed his views on time. Husserl’s emphasis on the structures of consciousness and the intentionality of thought resonated with Gödel’s belief in the importance of intuition in accessing mathematical truths. Gödel’s reflections on time incorporated a phenomenological perspective, considering how temporal experience is structured by human consciousness. Mathematical Platonism: Gödel’s Platonist views extended to his understanding of time. Just as he believed in the independent existence of mathematical objects, Gödel saw time as an objective entity with a structure that transcends human perception. His work on the Gödel metric can be seen as an attempt to uncover this objective structure, revealing the deeper realities that underlie our experience of time. (shrink)

The task of this essay is to put biological individuals in formal terms. This approach is not directly interested in matters of time (for example, in evolution), but rather in the formal shape of biological objects. So it is different from, but not opposed to, natural science. In his later years, Wittgenstein made similar investigations in psychology and mathematics. Unfortunately, he found no time to make extensive remarks on philosophy of biology. This is what we are going to advance (...) here. (shrink)

A collection of material on Husserl's Logical Investigations, and specifically on Husserl's formal theory of parts, wholes and dependence and its influence in ontology, logic and psychology. Includes translations of classic works by Adolf Reinach and Eugenie Ginsberg, as well as original contributions by Wolfgang Künne, Kevin Mulligan, Gilbert Null, Barry Smith, Peter M. Simons, Roger A. Simons and Dallas Willard. Documents work on Husserl's ontology arising out of early meetings of the Seminar for Austro-German Philosophy.

As large language models (LLMs) continue to demonstrate remarkable abilities across various domains, computer scientists are developing methods to understand their cognitive processes, particularly concerning how (and if) LLMs internally represent their beliefs about the world. However, this field currently lacks a unified theoretical foundation to underpin the study of belief in LLMs. This article begins filling this gap by proposing adequacy conditions for a representation in an LLM to count as belief-like. We argue that, while the project of belief (...) measurement in LLMs shares striking features with belief measurement as carried out in decision theory and formal epistemology, it also differs in ways that should change how we measure belief. Thus, drawing from insights in philosophy and contemporary practices of machine learning, we establish four criteria that balance theoretical considerations with practical constraints. Our proposed criteria include accuracy, coherence, uniformity, and use, which together help lay the groundwork for a comprehensive understanding of belief representation in LLMs. We draw on empirical work showing the limitations of using various criteria in isolation to identify belief representations. (shrink)

Edward Nieznański developed two logical systems to deal with the problem of evil and to refute religious determinism. However, when formalized in first-order modal logic, two axioms of each system contradict one another, revealing that there is an underlying minimal set of axioms enough to settle the questions. In this article, we develop this minimal system, called N3, which is based on Nieznański’s contribution. The purpose of N3 is to solve the logical problem of evil through the defeat of a (...) version of religious determinism. On the one hand, these questions are also addressed by Nieznański’s systems, but, on the other hand, they are obtained in N3 with fewer assumptions. Our approach can be considered a case of logic of religion, that is, of logic applied to religious discourse, as proposed by Józef Maria Bocheński; in this particular case, it is a discourse in theodicy, which is situated in the context of the philosophy of religion. (shrink)

Over the years many mathematicians have voiced a preference for proofs that stay “close” to the statements being proved, avoiding “foreign”, “extraneous”, or “remote” considerations. Such proofs have come to be known as “pure”. Purity issues have arisen repeatedly in the practice of arithmetic; a famous instance is the question of complex-analytic considerations in the proof of the prime number theorem. This article surveys several such issues, and discusses ways in which logical considerations shed light on these issues.

A central aim for philosophers of science has been to understand scientific theory change, or more specifically the rationality of theory change. Philosophers and historians of science have suggested that not only theories but also scientific methods and standards of rational inquiry have changed through the history of science. The topic here is methodological change, and what kind of theory of rational methodological change is appropriate. The modest ambition of this paper is to discuss in what ways results in formal (...) theories of belief revision can throw light on the question of what an appropriate theory of methodological change would look like. (shrink)

In this paper I try to answer four basic questions: (1) How the concept of God is to be represented? (2) Are there any logical principles governing it? (3) If so, what kind of logic lies behind them? (4) Can there be a logic of the concept of God? I address them by presenting a formal-logical account to the concept of God. I take it as a methodological desideratum that this should be done within the simplest existing logical formalism. I (...) start with first-order logic (FOL) with identity, and then show that its simplest modal extension (SQML, or the simplest quantified modal logic) is enough for us to formalize a minimally satisfactory theory of the concept of God. I focus exclusively on the monotheistic concept of God. (shrink)

In the era of “big data,” science is increasingly information driven, and the potential for computers to store, manage, and integrate massive amounts of data has given rise to such new disciplinary fields as biomedical informatics. Applied ontology offers a strategy for the organization of scientific information in computer-tractable form, drawing on concepts not only from computer and information science but also from linguistics, logic, and philosophy. This book provides an introduction to the field of applied ontology that is (...) of particular relevance to biomedicine, covering theoretical components of ontologies, best practices for ontology design, and examples of biomedical ontologies in use. After defining an ontology as a representation of the types of entities in a given domain, the book distinguishes between different kinds of ontologies and taxonomies, and shows how applied ontology draws on more traditional ideas from metaphysics. It presents the core features of the Basic Formal Ontology (BFO), now used by over one hundred ontology projects around the world, and offers examples of domain ontologies that utilize BFO. The book also describes Web Ontology Language (OWL), a common framework for Semantic Web technologies. Throughout, the book provides concrete recommendations for the design and construction of domain ontologies. (shrink)

I argue that normative formal epistemology (NFE) is best understood as modelling, in the sense that this is the reconstruction of its methodology on which NFE is doing best. I focus on Bayesianism and show that it has the characteristics of modelling. But modelling is a scientific enterprise, while NFE is normative. I thus develop an account of normative models on which they are idealised representations put to normative purposes. Normative assumptions, such as the transitivity of comparative credence, are characterised (...) as modelling idealisations motivated normatively. I then survey the landscape of methodological options: what might formal epistemologists be up to? I argue the choice is essentially binary: modelling or theorising. If NFE is theorising it is doing very poorly: generating false claims with no clear methodology for separating out what is to be taken seriously. Modelling, by contrast, is a successful methodology precisely suited to the management of useful falsehoods. Regarding NFE as modelling is not costless, however. First, our normative inferences are less direct and are muddied by the presence of descriptive idealisations. Second, our models are purpose-specific and limited in their scope. I close with suggestions for how to adapt our practice. (shrink)

The scientific study of consciousness, also referred to as consciousness science, is a young scientific field devoted to understanding how conscious experiences and the brain relate. It comprises a host of theories, experiments, and analyses that aim to investigate the problem of consciousness empirically, theoretically, and conceptually. This thesis addresses some of the questions that arise in these investigations from a formal and mathematical perspective. These questions concern theories of consciousness, experimental paradigms, methodology, and artificial consciousness. -/- Regarding theories of (...) consciousness, the thesis contributes to the understanding of the mathematical structure that some of the formal theories in the field propose. The work presented here targets the theory of consciousness known as Integrated Information Theory (IIT) and the neuroscientific theory known as Predictive Processing Theory or Free Energy Principle in its Active Inference form (AI-PP). The thesis provides axiomatic definitions of the mathematical structures that constitute these theories, and uses these definitions to address some of the open questions surrounding the theories. For AI-PP, this includes a rigorous derivation of the formula for Active Inference via Free Energy minimisation and a proof of compositionality of Free Energy. For IIT, this includes resolutions of some of the criticisms of IIT's formal scope and applications, but also the identification of new issues that concern the formalism and its derivation. When possible, the definitions are provided in the mathematical framework of category theory. -/- Regarding experiments, the thesis addresses the main paradigm for testing and falsifying theories of consciousness currently applied in the field. This paradigm consists of comparing the conscious experience that a theory predicts with the conscious experience that is inferred from behavioural data or report by use of measures of consciousness. The thesis provides a formal model of this paradigm and shows that under a certain condition—if inference and prediction are independent—, any minimally informative theory of consciousness can always be falsified. This is deeply problematic since the field’s reliance on report or behaviour to infer conscious experiences, in conjunction with the general structure of most contemporary theories of consciousness, implies such independence. This observation provides the exact formal underpinning of the well-known unfolding argument. The thesis analyses the origin of the problem and identifies precisely which changes are required to avoid this problem in future research. The thesis furthermore shows that the problem of falsifying theories of consciousness, and of empirical comparisons of theories of consciousness more generally, follows from a pervasive closure paradigm in consciousness science, that consists of taking a neuroscientific account of the brain as input to a theory of consciousness, so as to explain what consciousness is, without allowing for modifications or adaptations of the neuroscientific account that would accommodate consciousness as part of the brain's functioning. As is shown in the thesis, this paradigm has implications that point to a fundamental need of revision. -/- Regarding methodological and conceptual questions, the thesis contributes to the foundations of structural research in consciousness science. Structural research aims to use mathematical structures or mathematical spaces, instead of verbal descriptions or simple categorisations, to represent conscious experiences scientifically, for example when building theories of consciousness, or when exploring new empirical avenues to measure consciousness. Despite considerable advances in this realm, there was, prior to this thesis, no explicit definition of what a mathematical structure of conscious experience should be; that is, how the attribution of mathematical structure to conscious experiences should be systematically understood. Perhaps the most important contribution of this thesis to the field is to propose such a definition. The definition, a structural concept, extends existing approaches wherever available, and provides a basis for developing a common formal language to study consciousness, bridging developments as far apart as psychophysics and phenomenology. In addition, and independently of this proposal, the thesis offers a critical analysis of which metaphysical premises need to be presumed in structural research, whether the use of particular formal tools (such as structure-preserving mappings or homomorphisms) is justified, and how structural theories of consciousness could otherwise be built in the first place. An attempt to expand the results from consciousness to more general problems in philosophy of science is made in the context of the well-known Newman problem. -/- Regarding the question of artificial consciousness—can AI feel?—, the thesis contributes two results that take the form of no-go theorems, as known from physics. The first no-go theorem shows that if consciousness is relevant for the temporal evolution of a system's states—if it is dynamically relevant—, then contemporary AI systems cannot be conscious. That is because AI systems run on CPUs, GPUs, TPUs or other processors which have been designed and verified to adhere to computational dynamics that systematically preclude or suppress deviations. The second no-go theorem is situated in the context of computational functionalism, a view which posits that consciousness is a computation. The theorem shows that if computational functionalism holds true, consciousness cannot be a Turing computation. Rather, it must be a novel type of computation that has recently been proposed by Geoffrey Hinton, called mortal computation. -/- This thesis is part of a global effort to pioneer a mathematical perspective in consciousness science, now called Mathematical Consciousness Science. The hope behind the research carried out in this PhD is to illustrate the power and usefulness of mathematical approaches in different areas of consciousness science, and in doing so, to lay the foundations for future mathematical work that complements and supports empirical and theoretical work in the further development of this exciting field. (shrink)

Carnap and Twentieth-Century Thought: Explication as En lighten ment is the first book in the English language that seeks to place Carnap's philosophy in a broad cultural, political and intellectual context. According to the author, Carnap synthesized many different cur rents of thought and thereby arrived at a novel philosophical perspective that remains strik ing ly relevant today. Whether the reader agrees with Carus's bold theses on Carnap's place in the landscape of twentieth-century philosophy, and his even bolder (...) claims concerning the role that philosophy in Carnap's style should play in the thought of our century, does not matter so much as the excellent opportunity Carus's book offers to thoroughly rethink one's ideas about Carnap's philosophy. One reason why Carnap and Twentieth-Century Thought might change one's ideas is that Carus has unearthed much hitherto unknown material from the archives that sheds new light on Carnap's early life and thought. Indeed, the many archival findings presented in CTT for the first time suffice to make the book re warding reading for philosophers and historians of philosophy alike. CTT exhibits a high standard of historical scholarship, and the book itself is a beautiful example of high-quality academic publishing. Up to now, Carnap has remained a controversial figure on the philo sophical scene. On the one hand, he has a solid reputation as a leading figure of logical positivism . According to conventional wisdom, this was a school of thought characterized by its formal and technical philosophy, as well as being rather dismissive of other ways of doing philosophy, dogmatically sticking to its own theses. As a typical example of this arrogant logical empiricist attitude, one usually refers to Carnap's notorious Overcoming Metaphysics by Logical Analysis of Language , written when the Vienna Circle's Logical Empiricism had entered its most radical phase. Self-proclaimed postpositivist philosophers of science dismissed logical positivism, in particular Carnap's, as the dogmatic and orthodox “received view.” The tendency to portray logical empiricism as an obsolete doctrine centering around certain “dogmas” started with Quine's Two Dogmas of Empiricism and reached its somewhat ridiculous culmination in the early 1980s when allegedly “six or seven dogmas” were discovered . Thereby an allegedly un brid geable gap between classical “dogmatic” logical em pi ricism and its modern “enlightened” suc ces sors was construed. (shrink)

How does one formalize the structure of structures necessary for the foundations of physics? This work is an attempt at conceptualizing the metaphysics of pregeometric structures, upon which new and existing notions of quantum geometry may find a foundation. We discuss the philosophy of pregeometric structures due to Wheeler, Leibniz as well as modern manifestations in topos theory. We draw attention to evidence suggesting that the framework of formal language, in particular, homotopy type theory, provides the conceptual building blocks (...) for a theory of pregeometry. This work is largely a synthesis of ideas that serve as a precursor for conceptualizing the notion of space in physical theories. In particular, the approach we espouse is based on a constructivist philosophy, wherein ``structureless structures'' are syntactic types realizing formal proofs and programs. Spaces and algebras relevant to physical theories are modeled as type-theoretic routines constructed from compositional rules of a formal language. This offers the remarkable possibility of taxonomizing distinct notions of geometry using a common theoretical framework. In particular, this perspective addresses the crucial issue of how spatiality may be realized in models that link formal computation to physics, such as the Wolfram model. (shrink)

This essay discusses Wittgenstein's conception of logic, early and late, and some of the types of logical system that he constructed. The essay shows that the common view according to which Wittgenstein had stopped engaging in logic as a philosophical discipline by the time of writing Philosophical Investigations is mistaken. It is argued that, on the contrary, logic continued to figure at the very heart of later Wittgenstein's philosophy; and that Wittgenstein's mature philosophy of logic contains many interesting (...) thoughts that have gone widely unnoticed. (shrink)

In a recent paper, “The Concept of Logical Consequence,” W. H. Hanson criticizes a formal-structural characterization of logical consequence in Tarski and Sher. Hanson accepts many principles of the formal-structural view. Relating to Sher 1991 and 1996a, he says.

Formalizations in first-order logic are standardly used to represent logical forms of sentences and to show the validity of ordinary-language arguments. Since every sentence admits of a variety of formalizations, a challenge arises: why should one valid formalization suffice to show validity even if there are other, invalid, formalizations? This paper suggests an explanation with reference to criteria of adequacy which ensure that formalizations are related in a hierarchy of more or less specific formalizations. This proposal is then compared (...) with stronger criteria and assumptions, especially the idea that sentences essentially have just one logical form. (shrink)

This paper formalizes part of the cognitive architecture that Kant develops in the Critique of Pure Reason. The central Kantian notion that we formalize is the rule. As we interpret Kant, a rule is not a declarative conditional stating what would be true if such and such conditions hold. Rather, a Kantian rule is a general procedure, represented by a conditional imperative or permissive, indicating which acts must or may be performed, given certain acts that are already being performed. These (...) acts are not propositions; they do not have truth-values. Our formalization is related to the input/ output logics, a family of logics designed to capture relations between elements that need not have truth-values. In this paper, we introduce KL3 as a formalization of Kant’s conception of rules as conditional imperatives and permissives. We explain how it differs from standard input/output logics, geometric logic, and first-order logic, as well as how it translates natural language sentences not well captured by first-order logic. Finally, we show how the various distinctions in Kant’s much-maligned Table of Judgements emerge as the most natural way of dividing up the various types and sub-types of rule in KL3. Our analysis sheds new light on the way in which normative notions play a fundamental role in the conception of logic at the heart of Kant’s theoretical philosophy. (shrink)

In the past few years, social epistemologists have developed several formal models of the social organisation of science. While their robustness and representational adequacy has been analysed at length, the function of these models has begun to be discussed in more general terms only recently. In this article, I will interpret many of the current formal models of the scientific community as representing the latest development of what I will call the ‘Kuhnian project’. These models share with Kuhn a number (...) of questions about the relation between individuals and communities. At the same time, they also inherit some of Kuhn’s problematic characterisations of the scientific community. In particular, current models of the social organisation of science represent the scientific community as essentially value-free. This may put into question both their representational adequacy and their normative ambitions. In the end, it will be shown that the discussion on the formal models of the scientific community may contribute in fruitful ways to the ongoing debates on value judgements in science. (shrink)

In this text I propose a formal framework for the study of phenomenal objectness - the distinction in an a priori undifferenciated experience of the phenomenal field of certain 'objects'. The purpose of this framework is to represent (even partially) the reality of phenomenal experience in its structure (which participates conceptually to consciousness as such) and at the same time to allow the production of a tractable formalism in order to search for a mathematical explanation for the fundamental phenomenon of (...) objectness. The constitution of this framework is based on intuitions from consciousness literature as well as the recent integrated information theory. Before coming to the framework's definition, I analyse this theory from the point of view of the discourse, through concepts that I introduce here (statical and dynamical phases of the discourse), and explain some severe epistemic difficulties it encounters. The definition of a general combination criterion for the inter-discourse between philosophy and mathematics (which is necessary for the project of explaining consciousness scientifically) in these terms leads in this text to disentangle this theory into the proposed framework. (shrink)

There are two fundamentally distinct kinds of biological theorizing. "Formal biology" focuses on the relations, captured in formal laws, among mathematically abstracted properties of abstract objects. Population genetics and theoretical mathematical ecology, which are cases of formal biology, thus share methods and goals with theoretical physics. "Compositional biology," on the other hand, is concerned with articulating the concrete structure, mechanisms, and function, through developmental and evolutionary time, of material parts and wholes. Molecular genetics, biochemistry, developmental biology, and physiology, which are (...) examples of compositional biology, are in serious need of philosophical attention. For example, the very concept of a "part" is understudied in both philosophy of biology and philosophy of science. ;My dissertation is an attempt to clarify the distinction between formal biology and compositional biology and, in so doing, provide a clear philosophical analysis, with case studies, of compositional biology. Given the social, economic, and medical importance of compositional biology, understanding it is urgent. For my investigation, I draw on the philosophical fields of metaphysics and epistemology, as well as philosophy of biology and philosophy of science. I suggest new ways of thinking about some classic philosophy of science issues, such as modeling, laws of nature, abstraction, explanation, and confirmation. I hint at the relevance of my study of two kinds of biological theorizing to debates concerning the disunity of science. (shrink)

Hilbert arithmetic in a wide sense, including Hilbert arithmetic in a narrow sense consisting by two dual and anti-isometric Peano arithmetics, on the one hand, and the qubit Hilbert space (originating for the standard separable complex Hilbert space of quantum mechanics), on the other hand, allows for an arithmetic version of Gentzen’s cut elimination and quantum measurement to be described uniformy as two processes occurring accordingly in those two branches. A philosophical reflection also justifying that unity by quantum neo-Pythagoreanism links (...) it to the opposition of propositional logic, to which Gentzen’s cut rule refers immediately, on the one hand, and the linguistic and mathematical theory of metaphor therefore sharing the same structure borrowed from Hilbert arithmetic in a wide sense. An example by hermeneutical circle modeled as a dual pair of a syllogism (accomplishable also by a Turing machine) and a relevant metaphor (being a formal and logical mistake and thus fundamentally inaccessible to any Turing machine) visualizes human understanding corresponding also to Gentzen’s cut elimination and the Gödel dichotomy about the relation of arithmetic to set theory: either incompleteness or contradiction. The metaphor as the complementing “half” of any understanding of hermeneutical circle is what allows for that Gödel-like incompleteness to be overcome in human thought. (shrink)

The formal sciences - mathematical as opposed to natural sciences, such as operations research, statistics, theoretical computer science, systems engineering - appear to have achieved mathematically provable knowledge directly about the real world. It is argued that this appearance is correct.

In the paper, various notions of the logical semiotic sense of linguistic expressions – namely, syntactic and semantic, intensional and extensional – are considered and formalised on the basis of a formal-logical conception of any language L characterised categorially in the spirit of certain Husserl's ideas of pure grammar, Leśniewski-Ajdukiewicz's theory of syntactic/semantic categories and, in accordance with Frege's ontological canons, Bocheński's and some of Suszko's ideas of language adequacy of expressions of L. The adequacy ensures their unambiguous syntactic and (...) semantic senses and mutual, syntactic and semantic correspondence guaranteed by the acceptance of a postulate of categorial compatibility of syntactic and semantic (extensional and intensional) categories of expressions of L. This postulate defines the unification of these three logical senses. There are three principles of compositionality which follow from this postulate: one syntactic and two semantic ones already known to Frege. They are treated as conditions of homomorphism of partial algebra of L into algebraic models of L: syntactic, intensional and extensional. In the paper, they are applied to some expressions with quantifiers. Language adequacy connected with the logical senses described in the logical conception of language L is, obviously, an idealisation. The syntactic and semantic unambiguity of its expressions is not, of course, a feature of natural languages, but every syntactically and semantically ambiguous expression of such languages may be treated as a schema representing all of its interpretations that are unambiguous expressions. (shrink)

This paper reviews the central points and presents some recent developments of the epistemic approach to paraconsistency in terms of the preservation of evidence. Two formal systems are surveyed, the basic logic of evidence (BLE) and the logic of evidence and truth (LET J ), designed to deal, respectively, with evidence and with evidence and truth. While BLE is equivalent to Nelson’s logic N4, it has been conceived for a different purpose. Adequate valuation semantics that provide decidability are given for (...) both BLE and LET J . The meanings of the connectives of BLE and LET J , from the point of view of preservation of evidence, is explained with the aid of an inferential semantics. A formalization of the notion of evidence for BLE as proposed by M. Fitting is also reviewed here. As a novel result, the paper shows that LET J is semantically characterized through the so-called Fidel structures. Some opportunities for further research are also discussed. (shrink)

Focusing upon the five books of his early Hermes series, this article argues that Michel Serres furnishes an accomplished, unconventional philosophical account of communication and mediation-a structuralist epistemology designed to comprehend the sciences in their complexity and plurality-that, even decades after its first publication, has significant value for media theory. Two key themes within this pentalogy are highlighted: firstly, its emphasis upon motifs of communication, transport, and circulation, attempting to grasp the scientific field in topological terms, as a kind of (...) networked encyclopaedia; and secondly, its attempt to account for the intricate relationship between the formal and the empirical in all theorization. (shrink)

Attempts to unify Marxist and feminist social critique have been vexed by the fact that ‘patriarchy’ predates the advent of capitalism (its transhistorical status). Feminists within the Marxist, socialist, and materialist traditions have responded to this point by either granting patriarchy a certain autonomy relative to capitalism (the ‘dual/triple systems’ approach), or by suggesting that patriarchal relations have a foundational and necessary status in the history of capitalist development (which we term the ‘origins-subsistence’ approach). This paper offers an alternative account (...) of the relationship between capitalism and the transhistorical status of ‘patriarchy’. In aid of a ‘unitary theory’ of Marxist Feminism, we argue that the transhistorical status of patriarchy is better understood through an application of Marx’s concepts of formal and real subsumption. A modified version of these concepts can illuminate not only capitalist appropriation of antecedent social and economic forms, but also its capacity to produce new forms of gendered exploitation and oppression. (shrink)

I defend the thesis that Kantian analytic judgments are about objects (as opposed to concepts) against two challenges raised by recent scholars. First, can it accommodate cases like “A two-sided polygon is two-sided”, where no object really falls under the subject-concept as Kant sees it? Second, is it compatible with Kant’s view that analytic judgments make no claims about objects in the world and that we can know them to be true without going beyond the given concepts? I address these (...) challenges in two steps. First, given Kant’s distinction between an object in general = x from an object of sensible intuition, I argue that analytic judgments are about objects in the former sense, no matter whether the purported objects can be given in our intuition. Second, using Kant’s method of representing certain logical relations of concepts with such figures as circles, I construct a model to show that analytic truths are truths about objects in general = x and yet can be determined solely by the intensional relation between the given concepts plus certain Kantian-logical laws. Analytic truths are thus shown as formal in the Kantian sense that they do not presuppose the purported objects as givable in our intuition. This account of the formality of analytic truths captures Kant’s diagnosis of the Leibnizian illusion that we can make material claims about the world by analytically true judgments. (shrink)

In his book “Minds and Computers: An Introduction to the Philosophy of Artificial Intelligence”, Matt Carter presents a comprehensive exploration of the philosophical questions surrounding artificial intelligence (AI). Carter argues that the development of AI is not merely a technological challenge but fundamentally a philosophical one. He delves into key issues like the nature of mental states, the limits of introspection, the implications of memory decay, and the functionalist framework that allows for the possibility of AI. Carter contrasts functionalism (...) with reductive materialism, highlighting how the former accommodates the concept of artificial minds. He also emphasizes the significance of computationalism, which combines functionalism with computational theory to provide a robust explanation for mental processes. The book further discusses formal systems, the role of register machines in computation, and the inherent challenges in AI research, particularly in developing natural language processing systems. Carter’s work underscores the limitations of current computational models of cognition while remaining hopeful that advances in neuroscience may lead to stronger AI systems in the future. (shrink)

Truthmaker semantics is a recent development in formal and philosophical semantics, with similar motivation and scope to possible worlds semantics. The technical background is rather different, however, and results in a more fine-grained hyperintensional notion of content, allowing us to distinguish between classically equivalent propositions. After briefly introducing the main ideas, this entry will describe the technical apparatus of state spaces and the central notions of content and partial content. It will then outline applications of truthmaker semantics in language, logic, (...) and other areas of philosophy. (shrink)

This submission constitutes six (core) pages in literary prose expounding the philosophy of Parmenides of Elea outside the reductionist framework based in formal logic that has become de rigeur in philosophy departments of the English language world — at least prior to Dr. Papa-Grimaldi’s research. To this end, my paper, in chronological order, by way of introduction simulates Plato’s Parmenides; in the main body reconstructs Taran’s argument in “Much Ado About ‘Nothing,’” in Apeiron, Vol. XXXV, No. 2, June (...) 2002, pp. 87-104; and in the final section which will be presented at the conference, breaks away into an original hermeneutics in dialogical format featuring Socrates as a guest dramatis personae charged with the task of reconstructing in a new key, on behalf of the author, Parmenides’s philosophy. (shrink)

This paper introduces new logical systems which axiomatize a formal representation of inconsistency (here taken to be equivalent to contradictoriness) in classical logic. We start from an intuitive semantical account of inconsistent data, fixing some basic requirements, and provide two distinct sound and complete axiomatics for such semantics, LFI1 and LFI2, as well as their first-order extensions, LFI1* and LFI2*, depending on which additional requirements are considered. These formal systems are examples of what we dub Logics of Formal Inconsistency (LFI) (...) and form part of a much larger family of similar logics. We also show that there are translations from classical and paraconsistent first-order logics into LFI1* and LFI2*, and back. Hence, despite their status as subsystems of classical logic, LFI1* and LFI2* can codify any classical or paraconsistent reasoning. (shrink)

It is commonly said that some standards, such as morality, are ‘normatively authoritative’ in a way that other standards, such as etiquette, are not; standards like etiquette are said to be ‘not really normative’. Skeptics deny the very possibility of normative authority, and take claims like ‘etiquette is not really normative’ to be either empty or confused. I offer a different route to defeat skeptics about authority: instead of focusing on what makes standards like morality special, we should focus on (...) what makes standards like etiquette ‘not really normative’. I defend a fictionalist theory on which etiquette is ‘not really normative’ in roughly the same way that Sherlock is ‘not really a detective’, and show that fictionalism about some normative standards helps us explain the possibility of normative authority. (shrink)

I review Michael T. Ferejohn's "Formal Causes: Definition, Explanation, and Primacy in Socratic and Aristotelian Thought".

This paper surveys Bolzano's Beyträge zu einer begründeteren Darstellung der Mathematik (Contributions to a better-grounded presentation of mathematics) on the 200th anniversary of its publication. The first and only published issue presents a definition of mathematics, a classification of its subdisciplines, and an essay on mathematical method, or logic. Though underdeveloped in some areas (including,somewhat surprisingly, in logic), it is nonetheless a radically innovative work, where Bolzano presents a remarkably modern account of axiomatics and the epistemology of the formal sciences. (...) We also discuss the second, unfinished and unpublished issue, where Bolzano develops his views on universal mathematics. Here we find the beginnings of his theory of collections, for him the most fundamental of the mathematical disciplines. Though not exactly the same as the later Cantorian set theory, Bolzano's theory of collections was used in very similar ways in mathematics, notably in analysis. In retrospect, Bolzano's debut in philosophy was a remarkably successful one, though its fruits would only become generally known much later. (shrink)

Formal relationism in the philosophy of mind is the thesis that folk psychological states should be individuated, at least partially, in terms of the purely formal inference-licensing relations between underlying mental representations. It's supposed to provide a Russellian alternative to a Fregean theory of propositional attitudes. I argue that there's an inconsistency between the motivation for formal relationism and the use to which it's put in defense of Russellian propositions. Furthermore, I argue that formal relationism is committed to epiphenomenalism (...) about singular mental content. (shrink)

This contribution unfolds an existential-ontological response to the question of sexual difference in the context of Heidegger’s formally indicative concept of “Dasein.” The question of Dasein’s “neutrality” concerns how formal indication formalizes, empties, and neutralizes the givenness of factical human existence. Ostensibly “given” biological and anthropological facts, such as sexual difference, are interpreted from an emptied and neutralized perspective that appears abstract and fictional to Heidegger’s critics. How, then, is the “neutrality” of formalizing emptying related to the “facticity” of in (...) each case factically existing as a sexual and gendered being? The answer lies in Heidegger’s elucidation of the existential-ontological structure of Dasein: Dasein is always relationally instituted in its factical entangled life through “being-with.” The sexual difference of human existence is not merely biologically or anthropologically given, but disclosed and enacted according to one’s relations with others, the environment, and oneself. The formally indicating concept “Dasein” indicates, accordingly, the concrete diversity and plurality of individuated sexual and gendered ways of being. (shrink)

This short article pairs the realms of “Mathematics”, “Philosophy”, and “Poetry”, presenting some corners of intersection of this type of scientocreativity. Poetry have long been following mathematical patterns expressed by stern formal restrictions, as the strong metrical structure of ancient Greek heroic epic, or the consistent meter with standardized rhyme scheme and a “volta” of Italian sonnets. Poetry was always connected to Philosophy, and further on, notable mathematicians, like the inventor of quaternions, William Rowan Hamilton, or Ion Barbu, (...) the creator of the Barbilian spaces, have written appreciated poems. We will focus here on an avant-garde movement in literature, art, philosophy, and science, called Paradoxism, founded in Romania in 1980 by a mathematician, philosopher and poet, and on the laboured writing exercises of the Oulipo group, founded in Paris in 1960 by mathematicians and poets, both of them still in act. (shrink)

Should we believe our controversial philosophical views? Recently, several authors have argued from broadly conciliationist premises that we should not. If they are right, we philosophers face a dilemma: If we believe our views, we are irrational. If we do not, we are not sincere in holding them. This paper offers a way out, proposing an attitude we can rationally take toward our views that can support sincerity of the appropriate sort. We should arrive at our views via a certain (...) sort of ‘insulated’ reasoning – that is, reasoning that involves setting aside certain higher-order worries, such as those provided by disagreement – when we investigate philosophical questions. (shrink)