Ioannis G. Ntagkas

In the Copenhagen interpretation the Heisenberg inequality ΔQΔP≥ℏ/2 is interpreted as the mathematical expression of the concept of complementarity, quantifying the mutual disturbance necessarily taking place in a simultaneous or joint measurement of incompatible observables. This interpretation was criticized a long time ago and has recently been challenged in an experimental way. These criticisms can be substantiated by using the generalized formalism of positive operator-valued measures, from which an inequality, different from the Heisenberg inequality, can be derived, precisely illustrating the Copenhagen concept of complementarity. The different roles of preparation and measurement in creating uncertainty in quantum mechanics are discussed.

To honor Henry Margenau on the occasion of his 90th birthday, we attempt in this essay to integrate certain aspects of the physics, philosophy, and pedagogy of quantum mechanics in a manner very much inspired by Margenau's idealist scientific epistemology. Over half a century ago, Margenau was perhaps the first philosopher of science to recognize and elaborate upon the essential distinction between thepreparation of a quantum state and themeasurement of an observable associated with a system in that state; yet in contemporary quantum texts that distinction rarely receives adequate emphasis even though, as we demonstrate, it may be explicated through a series of simple illustrations

The main claim of this talk is that mathematical physics and philosophy of physics are not different. This claim, so formulated, is obviously false because it is overstated; however, since no non-tautological statement is likely to be completely true, it is a meaningful question whether the overstated claim expresses some truth. I hope it does, or so I’ll argue. The argument consists of two parts: First I’ll recall some characteristic features of von Neumann’s work on mathematical foundations of quantum mechanics and will claim that von Neumann’s motivation and results are essentially philosophical in their nature; hence, to the extent von Neumann’s work exemplifies what is considered to be mathematical physics, mathematical physics appears as formally explicit philosophy of physics. The second argument is based on a rather trivial interpretation of what mathematical physics is. That interpretation implies that mathematical physics shares some key characteristic features with philosophy of physics which make the two almost indistinguishable.

With the example of a Stern–Gerlach measurement on a spin-1/2 atom, we show that a superposition of both paths may be observed compatibly with properties attributed to state collapse—for example, the singleness (or mutual exclusivity) of outcomes. This is done by inserting a quantum two-state system (an ancilla) in each path, capable of responding to the passage of the atom, and thus acting as a virtual detector. We then consider real measurements on the compound system of atomic spin and two ancillae. Nondestructive measurements of a set of compatible joint observables can be performed, one for a superposition and others for collapse properties. A novel perspective is given as to why, within unitary quantum theory, ordinary measurements are blind to such superpositions. Implications for the theory of measurement are discussed.

This edited volume examines aspects of the mind/consciousness that are relevant to the interpretations of quantum mechanics. In it, an international group of contributors focus on the possible connections between quantum mechanics and consciousness. They look at how consciousness can help us with quantum mechanics as well as how quantum mechanics can contribute to our understanding of consciousness. For example, what do different interpretations aimed at solving the measurement problem in quantum mechanics tell us about the nature of consciousness, such as von Neumann's interpretation? Each interpretation has, associated to it, a corresponding metaphysical framework that helps us think about possible “models” of consciousness. Alternatively, what does the nature of consciousness tell us about the role of the observer and time reversibility in the measurement process? The book features 20 papers on contemporary approaches to quanta and mind. It brings together the work of scholars from different disciplines with diverse views on the connections between quanta and mind, ranging from those who are supportive of a link between consciousness and quantum physics to those who are very skeptical of such link. Coverage includes such topics as free will in a quantum world, contextuality and causality, mind and matter interaction, quantum panpsychism, the quantum and quantum-like brain, and the role of time in brain-mind dynamics.

Consciousness and quantum mechanics are two great mysteries of our time--and recently scholars have postulated a deeper connection between them. Exploring this possible connection can be fruitful: an analysis of the conscious mind and psychophysical connection can be indispensable in understanding quantum mechanics and solving the notorious measurement problem, and there is also likely some kind of intimate connection between quantum mechanics--the most fundamental theory of the physical world--and our efforts to explain, naturalistically, the phenomenon of consciousness. The sixteen newly written chapters in this volume are divided into three sections: Consciousness and the Wave Function Collapse, Consciousness in Quantum Theories, and Quantum Approaches to Consciousness. This is the first volume to provide a comprehensive review and thorough analysis of intriguing conjectures about the connection between consciousness and quantum mechanics. Written by leading experts in physics, philosophy, and cognitive science, Consciousness and Quantum Mechanics will be of value to students and researchers working on the foundations of quantum mechanics and the philosophy of mind.

Does consciousness collapse the quantum wave function? This idea was taken seriously by John von Neumann and Eugene Wigner but is now widely dismissed. We develop the idea by combining a mathematical theory of consciousness (integrated information theory) with an account of quantum collapse dynamics (continuous spontaneous localization). Simple versions of the theory are falsified by the quantum Zeno effect, but more complex versions remain compatible with empirical evidence. In principle, versions of the theory can be tested by experiments with quantum computers. The upshot is not that consciousness-collapse interpretations are clearly correct, but that there is a research program here worth exploring.

The paradigmatic mereological relation is the relation of spatial part. Already much less paradigmatic is the relation of temporal part. The realm of abstract entities seems to be the ontological region where the notion of part and whole has no application at all. In what follows, I will contend that this is not true. There are part-whole relationships between abstract entities, and indeed relationships that are systematic to the point of constituting mereologically structured universes of abstract entities, “intelligible worlds”, as I will call them (in translation of the Latin “mundi intelligibiles”). The part-whole relations between abstract entities differ significantly from those between spatial, or temporal, or spatio-temporal entities. However, there are also significant analogies between abstract and concrete part-whole relations.

How might one tackle the subject of causation with the least amount of preformed conceptions – and arrive by a series of well-motivated conceptual decisions at a concept of causation that captures the “heart of the matter”? This essay is a sustained attempt to answer this question. On the way, causal predicates of various degrees of importance are defined and causal principles of various degrees of plausibility discussed, all of this in the service of approaching, step by step, “the heart of the matter.”

The aim of this paper is to give a systematic account of the so-called “measurement problem” in the frame of the standard interpretation of quantum mechanics. It is argued that there is not one but five distinct formulations of this problem. Each of them depends on what is assumed to be a “satisfactory” description of the measurement process in the frame of the standard interpretation. Moreover, the paper points out that each of these formulations refers not to a unique problem, but to a set of sub-problems.

It is quite extraordinary, philosophically speaking, that according to the orthodox interpretation: (a) quantum mechanics is a complete and comprehensive theory of microphysics, and yet (b) the role of measurement, in quantum mechanics, cannot be analyzed in terms of the collective effects of the microphysical particles making up the apparatus. It follows that, if the orthodox interpretation is correct, the measurement apparatus and its quantum physical effects cannot be accounted for microreductively. This is significant because it is widely believed that the relation between physical wholes and parts is microreductive. Indeed, many philosophers are persuaded of the inevitability of universal microreduction to the basic elements of microphysics. This is the viewpoint embodied in the program for the unity of science, espoused in recent years, most notably by Robert Causey (1977)

Metaphysicians should pay attention to quantum mechanics. Why? Not because it provides definitive answers to many metaphysical questions-the theory itself is remarkably silent on the nature of the physical world, and the various interpretations of the theory on offer present conflicting ontological pictures. Rather, quantum mechanics is essential to the metaphysician because it reshapes standard metaphysical debates and opens up unforeseen new metaphysical possibilities. Even if quantum mechanics provides few clear answers, there are good reasons to think that any adequate understanding of the quantum world will result in a radical reshaping of our classical world-view in some way or other. Whatever the world is like at the atomic scale, it is almost certainly not the swarm of particles pushed around by forces that is often presupposed. This book guides readers through the theory of quantum mechanics and its implications for metaphysics in a clear and accessible way. The theory and its various interpretations are presented with a minimum of technicality. The consequences of these interpretations for metaphysical debates concerning realism, indeterminacy, causation, determinism, holism, and individuality are explored in detail, stressing the novel form that the debates take given the empirical facts in the quantum domain. While quantum mechanics may not deliver unconditional pronouncements on these issues, the range of possibilities consistent with our knowledge of the empirical world is relatively small-and each possibility is metaphysically revisionary in some way. This book will appeal to researchers, students, and anybody else interested in how science informs our world-view.

Formal verification involves the use of logical and computational methods to establish claims that are expressed in precise mathematical terms. These can include ordinary mathematical theorems, as well as claims that pieces of hardware or software, network protocols, and mechanical and hybrid systems meet their specifications. In practice, there is not a sharp distinction between verifying a piece of mathematics and verifying the correctness of a system: formal verification requires describing hardware and software systems in mathematical terms, at which point establishing claims as to their correctness becomes a form of theorem proving. Conversely, the proof of a mathematical theorem may require a lengthy computation, in which case verifying the truth of the theorem requires verifying that the computation does what it is supposed to do. The gold standard for supporting a mathematical claim is to provide a proof, and twentieth-century developments in logic show most if not all conventional proof methods can be reduced to a small set of axioms and rules in any of a number of foundational systems. With this reduction, there are two ways that a computer can help establish a claim: it can help find a proof in the first place, and it can help verify that a purported proof is correct. Automated theorem proving focuses on the “finding” aspect. Resolution theorem provers, tableau theorem provers, fast satisfiability solvers, and so on provide means of establishing the validity of formulas in propositional and first-order logic. Other systems provide search procedures and decision procedures for specific languages and domains, such as linear or nonlinear expressions over the integers or the real numbers. Architectures like SMT combine domain-general search methods with domain-specific procedures. Computer algebra systems and specialized mathematical software packages provide means of carrying out mathematical computations, establishing mathematical bounds, or finding mathematical objects. A calculation can be viewed as a proof as well, and these systems, too, help establish mathematical claims. Automated reasoning systems strive for power and efficiency, often at the expense of guaranteed soundness. Such systems can have bugs, and it can be difficult to ensure that the results they deliver are correct. In contrast, interactive theorem proving focuses on the “verification” aspect of theorem proving, requiring that every claim is supported by a proof in a suitable axiomatic foundation. This sets a very high standard: every rule of inference and every step of a calculation has to be justified by appealing to prior definitions and theorems, all the way down to basic axioms and rules. In fact, most such systems provide fully elaborated “proof objects” that can be communicated to other systems and checked independently. Constructing such proofs typically requires much more input and interaction from users, but it allows us to obtain deeper and more complex proofs. The Lean Theorem Prover aims to bridge the gap between interactive and automated theorem proving, by situating automated tools and methods in a framework that supports user interaction and the construction of fully specified axiomatic proofs. The goal is to support both mathematical reasoning and reasoning about complex systems, and to verify claims in both domains.

Lean is a new open source theorem prover being developed at Microsoft Research and Carnegie Mellon University, with a small trusted kernel based on dependent type theory. It aims to bridge the gap between interactive and automated theorem proving, by situating automated tools and methods in a framework that supports user interaction and the construction of fully specified axiomatic proofs. Lean is an ongoing and long-term effort, but it already provides many useful components, integrated development environments, and a rich API which can be used to embed it into other systems. It is currently being used to formalize category theory, homotopy type theory, and abstract algebra. We describe the project goals, system architecture, and main features, and we discuss applications and continuing work.

Interactive theorem provers might seem particularly impractical in the history of philosophy. Journal articles in this discipline are generally not formalized. Interactive theorem provers involve a learning curve for which the payoffs might seem minimal. In this article I argue that interactive theorem provers have already demonstrated their potential as a useful tool for historians of philosophy; I do this by highlighting examples of work where this has already been done. Further, I argue that interactive theorem provers can continue to be useful tools for historians of philosophy in the future; this claim is defended through a more conceptual analysis of what historians of philosophy do that identifies argument reconstruction as a core activity of such practitioners. It is then shown that interactive theorem provers can assist in this core practice by a description of what interactive theorem provers are and can do. If this is right, then computer verification for historians of philosophy is in the offing.

The sheer multitude of criteria of empirical significance has been taken as evidence that the pre-analytic notion being explicated is too vague to be useful. I show instead that a significant number of these criteria—by Ayer, Popper, Przełęcki, Suppes, and David Lewis, among others—not only form a coherent whole, but also connect directly to the theory of definition, the notion of empirical content as explicated by Ramsey sentences, and the theory of measurement; two criteria by Carnap and Sober are trivial, but can be saved and connected to the other criteria by slight modifications. A corollary is that the ordinary language defense of Lewis, the conceptual arguments by Ayer and Popper, the theoretical considerations by Przełęcki, and the practical considerations by Suppes all apply to the same criterion or closely related criteria. Furthermore, the equivalence of some criteria allows for their individual justifications to be taken cumulatively and, together with the entailment relations between nonequivalent criteria, suggest criteria for general auxiliary assumptions, comparative criteria, and more liberal conceptions of observation.

In this paper I explore informationalism, a pragmatic theory of modality that seems to solve some serious problems in the familiar possible worlds accounts of modality. I view the theory as an elaboration of Stalnaker's moderate modal realism, though it also derives from Dretske's semantic theory of information. Informationalism is presented in Section 2 after the prerequisite stage setting in Section 1. Some applications are sketched in Section 3. Finally, a mathematical model of the theory is developed in Section 4.How many times have I said to you that when you have eliminated the impossible, whatever remains, however improbable, must be the truth? (Arthur Conan Doyle)You've put me in an impossible situation. (Anonymous)[N]othing we imagine is absolutely impossible. (David Hume)

Neurodevelopmental disorders (NDDs), such as autism and ADHD, are behaviorally defined adaptive functioning difficulties arising from variations, alterations and atypical maturation of the brain. While it is widely agreed that NDDs are complex conditions with their presentation and functional impact underpinned by diverse genetic and environmental factors, contemporary and polarizing debate has focused on the appropriateness of the biomedical as opposed to the neurodiverse paradigm in framing conceptions of these conditions. Despite being largely overlooked by both research and practice, the International Classification of Functioning Disability and Health (ICF) endorsed by the World Health Organization in 2001 views functioning dynamically, offering a framework for investigating, assessing and treating NDDs holistically. Exemplified by autism and ADHD, we argue that the ICF provides not only a multitude of opportunities in accounting for the environmental determinants in researching and clinically managing NDDs, but opportunities for harmonizing the seemingly irreconcilable biomedical and neurodiverse paradigms. Also see the video abstract here:https://youtu.be/YwuWPDUOs5k.

We propose a solution to the problem of logical omniscience in what we take to be its fundamental version: as concerning arbitrary agents and the knowledge attitude per se. Our logic of knowledge is a spin-off from a general theory of thick content, whereby the content of a sentence has two components: an intension, taking care of truth conditions; and a topic, taking care of subject matter. We present a list of plausible logical validities and invalidities for the logic of knowledge per se for arbitrary agents, and isolate three explanatory factors for them: the topic-sensitivity of content; the fragmentation of knowledge states; the defeasibility of knowledge acquisition. We then present a novel dynamic epistemic logic that yields precisely the desired validities and invalidities, for which we provide expressivity and completeness results. We contrast this with related systems and address possible objections.

We reply to various arguments by Otavio Bueno and Edward Zalta (‘Object Theory and Modal Meinongianism’) against Modal Meinongianism, including that it presupposes, but cannot maintain, a unique denotation for names of fictional characters, and that it is not generalizable to higher-order objects. We individuate the crucial difference between Modal Meinongianism and Object Theory in the former’s resorting to an apparatus of worlds, possible and impossible, for the representational purposes for which the latter resorts to a distinction between two kinds of predication, exemplification and encoding. We show that encoding has fewer forerunners in the history of philosophy than Bueno and Zalta want, and that there’s a reason why the notion has been found baffling by some.

We study imagination as reality-oriented mental simulation (ROMS): the activity of simulating nonactual scenarios in one’s mind, to investigate what would happen if they were realized. Three connected questions concerning ROMS are: What is the logic, if there is one, of such an activity? How can we gain new knowledge via it? What is voluntary in it and what is not? We address them by building a list of core features of imagination as ROMS, drawing on research in cognitive psychology and the philosophy of mind. We then provide a logic of imagination as ROMS which models such features, combining techniques from epistemic logic, action logic, and subject matter semantics. Our logic comprises a modal propositional language with non-monotonic imagination operators, a formal semantics, and an axiomatization.

We propose a dynamic hyperintensional logic of belief revision for non-omniscient agents, reducing the logical omniscience phenomena affecting standard doxastic/epistemic logic as well as AGM belief revision theory. Our agents don’t know all a priori truths; their belief states are not closed under classical logical consequence; and their belief update policies are such that logically or necessarily equivalent contents can lead to different revisions. We model both plain and conditional belief, then focus on dynamic belief revision. The key idea we exploit to achieve non-omniscience focuses on topic- or subject matter-sensitivity: a feature of belief states which is gaining growing attention in the recent literature.

We propose a new account of indicative conditionals, giving acceptability and logical closure conditions for them. We start from Adams’ Thesis: the claim that the acceptability of a simple indicative equals the corresponding conditional probability. The Thesis is widely endorsed, but arguably false and refuted by empirical research. To fix it, we submit, we need a relevance constraint: we accept a simple conditional 'If φ, then ψ' to the extent that (i) the conditional probability p(ψ|φ) is high, provided that (ii) φ is relevant for ψ. How (i) should work is well-understood. It is (ii) that holds the key to improve our understanding of conditionals. Our account has (i) a probabilistic component, using Popper functions; (ii) a relevance component, given via an algebraic structure of topics or subject matters. We present a probabilistic logic for simple indicatives, and argue that its (in)validities are both theoretically desirable and in line with empirical results on how people reason with conditionals.

Impossible Worlds focuses on an exciting new theory in philosophy, with applications in metaphysics, logic, and the theory of meaning. Its central topic is: how do we meaningfully talk and reason about situations which, unbeknownst to us, are impossible? This issue emerges as a central problem in contemporary philosophical accounts of meaning, information, knowledge, belief, fiction, conditionality, and counterfactual supposition. The book is written bytwo of the leading philosophers in the area and contains original research of relevance to professional philosophers and logicians working in metaphysics, philosophy of language, formal logic, and adjacentareas.

Computers may help us to better understand (not just verify) arguments. In this article we defend this claim by showcasing the application of a new, computer-assisted interpretive method to an exemplary natural-language ar- gument with strong ties to metaphysics and religion: E. J. Lowe’s modern variant of St. Anselm’s ontological argument for the existence of God. Our new method, which we call computational hermeneutics, has been particularly conceived for use in interactive-automated proof assistants. It aims at shedding light on the meanings of words and sentences by framing their inferential role in a given argument. By employing automated theorem reasoning technology within interactive proof assistants, we are able to drastically reduce (by several orders of magnitude) the time needed to test the logical validity of an argu- ment’s formalization. As a result, a new approach to logical analysis, inspired by Donald Davidson’s account of radical interpretation, has been enabled. In computational hermeneutics, the utilization of automated reasoning tools ef- fectively boosts our capacity to expose the assumptions we indirectly commit ourselves to every time we engage in rational argumentation and it fosters the explicitation and revision of our concepts and commitments.