The “four-color” theorem seems to be generalizable as follows. The four-letter alphabet is sufficient to encode unambiguously any set of well-orderings including a geographical map or the “map” of any logic and thus that of all logics or the DNA plan of any alive being. Then the corresponding maximally generalizing conjecture would state: anything in the universe or mind can be encoded unambiguously by four letters. That admits to be formulated as a “four-letter theorem”, (...) and thus one can search for a properly mathematical proof of the statement. It would imply the “four colour theorem”, the proof of which many philosophers and mathematicians believe not to be entirely satisfactory for it is not a “human proof”, but intermediated by computers unavoidably since the necessary calculations exceed the human capabilities fundamentally. It is furthermore rather unsatisfactory because it consists in enumerating and proving all cases one by one. Sometimes, a more general theorem turns out to be much easier for proving including a general “human” method, and the particular and too difficult for proving theorem to be implied as a corollary in certain simple conditions. The same approach will be followed as to the four colour theorem, i.e. to be deduced more or less trivially from the “four-letter theorem” if the latter is proved. References are only classical and thus very well-known papers: their complete bibliographic description is omitted. (shrink)

The Four-Colour Theorem (4CT) proof, presented to the mathematical community in a pair of papers by Appel and Haken in the late 1970's, provoked a series of philosophical debates. Many conceptual points of these disputes still require some elucidation. After a brief presentation of the main ideas of Appel and Haken’s procedure for the proof and a reconstruction of Thomas Tymoczko’s argument for the novelty of 4CT’s proof, we shall formulate some questions regarding the connections between the (...) points raised by Tymoczko and some Wittgensteinian topics in the philosophy of mathematics such as the importance of the surveyability as a criterion for distinguishing mathematical proofs from empirical experiments. Our aim is to show that the “characteristic Wittgensteinian invention” (Mühlhölzer 2006) – the strong distinction between proofs and experiments – can shed some light in the conceptual confusions surrounding the Four-Colour Theorem. (shrink)

All accepted proofs of the Four Colour Theorem (4CT) are computer-dependent; and appeal to the existence, and manual identification, of an ‘unavoidable’ set containing a sufficient number of explicitly defined configurations—each evidenced only by a computer as ‘reducible’—such that at least one of the configurations must occur in any chromatically distinguished, minimal, planar map. For instance, Appel and Haken ‘identified’ 1,482 such configurations in their 1977, computer-dependent, proof of 4CT; whilst Neil Robertson et al ‘identified’ 633 configurations (...) as sufficient in their 1997, also computer-dependent, proof of 4CT. However, treating any specific number of ‘reducible’ configurations in an ‘unavoidable’ set as sufficient entails a minimum number as necessary and sufficient. We now show that the minimum number of such configurations can only be the one corresponding to the ‘unavoidable’ set of a ‘reducible’, 4-sided configuration identified by Alfred Kempe in his, seemingly fatally flawed, 1879 ‘proof’ of 4CT. Although Kempe appealed to putative properties of ‘Kempe’ chains in a graphical representation to fallaciously argue that a 5-sided configuration was also in the ‘unavoidable’ set, and ‘reducible’, we shall show why the flaw in his ‘proof’ is not fatal when the argument is expressed geometrically; and that, essentially, Kempe correctly argued that any planar map which admits a chromatic differentiation with a five-sided area C that shares non-zero boundaries with four, all differently coloured, neighbours can be 4-coloured. (shrink)

All acknowledged proofs of the Four Colour Theorem (4CT) are computerdependent. They appeal to the existence, and manual identification, of an ‘unavoidable’ set containing a sufficient number of explicitly defined configurations—each evidenced only by a computer as ‘reducible’—such that at least one of the configurations must occur in any chromatically distinguished, putatively minimal, planar map. For instance, Appel and Haken ‘identified’ 1,482 such configurations in their 1977, computer-dependent, proof of 4CT; whilst Neil Robertson et al ‘identified’ 633 (...) configurations as sufficient in their 1997, also computer-dependent, proof of 4CT. However, treating any specific number of ‘reducible’ configurations in an ‘unavoidable’ set as sufficient entails a minimum number as necessary and sufficient. We now show that the minimum number of such configurations can only be the one corresponding to the ‘unavoidable’ set of the single, ‘reducible’, 4-sided configuration identified by Alfred Kempe in his, seemingly fatally flawed, 1879 ‘proof’ of 4CT. We shall further show that although Kempe fallaciously concluded that a 5-sided configuration was also in the ‘unavoidable’ set, and appealed to unproven properties of ‘Kempe’ chains in a graphical representation to then argue for its ‘reducibility’, neither flaw in his ‘proof’ is fatal when the argument is expressed geometrically; and that, essentially, Kempe correctly argued that any planar map which admits a chromatic differentiation with a five-sided area C that shares non-zero boundaries with four, all differently coloured, neighbours can be 4-coloured. (shrink)

Although the Four Colour Theorem is passe, we give an elementary pre-formal proof that transparently illustrates why four colours suffice to chromatically differentiate any set of contiguous, simply connected and bounded, planar spaces; by showing that there is no minimal 4-coloured planar map M. We note that such a pre-formal proof of the Four Colour Theorem highlights the significance of differentiating between: (a) Plato's knowledge as justified true belief, which seeks a formal proof (...) in a first-order mathematical language in order to justify a belief as true; and (b) Piccinini's knowledge as factually grounded belief, which seeks a pre-formal proof, in Pantsar's sense, in order to justify the axioms and rules of inference of a first-order mathematical language which can, then, formally prove the belief as justifiably true under a well-defined interpretation of the language. (shrink)

The paper continues the consideration of Hilbert mathematics to mathematics itself as an additional “dimension” allowing for the most difficult and fundamental problems to be attacked in a new general and universal way shareable between all of them. That dimension consists in the parameter of the “distance between finiteness and infinity”, particularly able to interpret standard mathematics as a particular case, the basis of which are arithmetic, set theory and propositional logic: that is as a special “flat” case of Hilbert (...) mathematics. The following four essential problems are considered for the idea to be elucidated: Fermat’s last theorem proved by Andrew Wiles; Poincaré’s conjecture proved by Grigori Perelman and the only resolved from the seven Millennium problems offered by the Clay Mathematics Institute (CMI); the four-color theorem proved “machine-likely” by enumerating all cases and the crucial software assistance; the Yang-Mills existence and mass gap problem also suggested by CMI and yet unresolved. They are intentionally chosen to belong to quite different mathematical areas (number theory, topology, mathematical physics) just to demonstrate the power of the approach able to unite and even unify them from the viewpoint of Hilbert mathematics. Also, specific ideas relevant to each of them are considered. Fermat’s last theorem is shown as a Gödel insoluble statement by means of Yablo’s paradox. Thus, Wiles’s proof as a corollary from the modularity theorem and thus needing both arithmetic and set theory involves necessarily an inverse Grothendieck universe. On the contrary, its proof in “Fermat arithmetic” introduced by “epoché to infinity” (following the pattern of Husserl’s original “epoché to reality”) can be suggested by Hilbert arithmetic relevant to Hilbert mathematics, the mediation of which can be removed in the final analysis as a “Wittgenstein ladder”. Poincaré’s conjecture can be reinterpreted physically by Minkowski space and thus reduced to the “nonstandard homeomorphism” of a bit of information mathematically. Perelman’s proof can be accordingly reinterpreted. However, it is valid in Gödel (or Gödelian) mathematics, but not in Hilbert mathematics in general, where the question of whether it holds remains open. The four-color theorem can be also deduced from the nonstandard homeomorphism at issue, but the available proof by enumerating a finite set of all possible cases is more general and relevant to Hilbert mathematics as well, therefore being an indirect argument in favor of the validity of Poincaré’s conjecture in Hilbert mathematics. The Yang-Mills existence and mass gap problem furthermore suggests the most general viewpoint to the relation of Hilbert and Gödel mathematics justifying the qubit Hilbert space as the dual counterpart of Hilbert arithmetic in a narrow sense, in turn being inferable from Hilbert arithmetic in a wide sense. The conjecture that many if not almost all great problems in contemporary mathematics rely on (or at least relate to) the Gödel incompleteness is suggested. It implies that Hilbert mathematics is the natural medium for their discussion or eventual solutions. (shrink)

Computation is central to contemporary mathematics. Many accept that we can acquire genuine mathematical knowledge of the Four Color Theorem from Appel and Haken's program insofar as it is simply a repetitive application of human forms of mathematical reasoning. Modern LLMs / DNNs are, by contrast, opaque to us in significant ways, and this creates obstacles in obtaining mathematical knowledge from them. We argue, however, that if a proof-checker automating human forms of proof-checking is attached to such machines, (...) then we can obtain apriori mathematical knowledge from them, even though the original machines are entirely opaque to us and the proofs they output are not human-surveyable. (shrink)

The paper introduces and utilizes a few new concepts: “nonstandard Peano arithmetic”, “complementary Peano arithmetic”, “Hilbert arithmetic”. They identify the foundations of both mathematics and physics demonstrating the equivalence of the newly introduced Hilbert arithmetic and the separable complex Hilbert space of quantum mechanics in turn underlying physics and all the world. That new both mathematical and physical ground can be recognized as information complemented and generalized by quantum information. A few fundamental mathematical problems of the present such as Fermat’s (...) last theorem, four-color theorem as well as its new-formulated generalization as “four-letter theorem”, Poincaré’s conjecture, “P vs NP” are considered over again, from and within the new-founding conceptual reference frame of information, as illustrations. Simple or crucially simplifying solutions and proofs are demonstrated. The link between the consistent completeness of the system mathematics-physics on the ground of information and all the great mathematical problems of the present (rather than the enumerated ones) is suggested. (shrink)

The orthodox monadic determination thesis holds that we represent colour relations by virtue of representing colours. Against this orthodoxy, I argue that it is possible to represent colour relations without representing any colours. I present a model of iconic perceptual content that allows for such primitive relational colour representation, and provide four empirical arguments in its support. I close by surveying alternative views of the relationship between monadic and relational colour representation.

Now with extra footnotes, by editorial demand! Final version accepted by Oxford Studies in Ancient Philosophy. -/- This essay consists in a trick and a potential insight. The trick consists in a minimalist interpretation of color mixture. The account of color mixture is minimalist in the sense that, given certain background assumptions, there is no more to Timaeus’ account of color mixture than the list of the chromatic pathēmata and the list of how these combine to elicit perceptions of all (...) the colors. The only potential controversial elements of the minimalist interpretation are the relevant background assumptions and the interpretation of the chromatic pathēmata. The potential insight concerns a motive that Plato, in the guise of Timaeus, may have for presenting an account of color mixture. Specifically, I shall argue that on the minimalist interpretation, Plato may be read as reconciling the Democritrean four color scheme with an older tradi- tion where white and black are the fundamental chromatic opposition. As we shall see, this bears on the interpretation of the chromatic pathēmata. (shrink)

We study the first-order theories of some natural and important classes of coloured trees, including the four classes of trees whose paths have the order type respectively of the natural numbers, the integers, the rationals, and the reals. We develop a technique for approximating a tree as a suitably coloured linear order. We then present the first-order theories of certain classes of coloured linear orders and use them, along with the approximating technique, to establish complete axiomatisations of the (...) class='Hi'>four classes of trees mentioned above. (shrink)

This work describes a seminal framework of law by one of the founders of the field of law and economics, Judge Guido Calabresi. It broadens what is known as the framework of law among legal scholars, and posits a calabresi theorem which is developed and explained, in part, in comparison to the coase theorem. The framework provides policymakers a tool for creating balanced policies.

REVIEW OF: Automated Development of Fundamental Mathematical Theories by Art Quaife. (1992: Kluwer Academic Publishers) 271pp. Using the theorem prover OTTER Art Quaife has proved four hundred theorems of von Neumann-Bernays-Gödel set theory; twelve hundred theorems and definitions of elementary number theory; dozens of Euclidean geometry theorems; and Gödel's incompleteness theorems. It is an impressive achievement. To gauge its significance and to see what prospects it offers this review looks closely at the book and the proofs it presents.

Democratic decision-making is often defended on grounds of the ‘wisdom of crowds’: decisions are more likely to be correct if they are based on many independent opinions, so a typical argument in social epistemology. But what does it mean to have independent opinions? Opinions can be probabilistically dependent even if individuals form their opinion in causal isolation from each other. We distinguish four probabilistic notions of opinion independence. Which of them holds depends on how individuals are causally affected by (...) environmental factors such as commonly perceived evidence. In a general theorem, we identify causal conditions guaranteeing each kind of opinion independence. These results have implications for whether and how ‘wisdom of crowds’ arguments are possible, and how truth-conducive institutions can be designed. (shrink)

Many mathematicians have cited depth as an important value in their research. However, there is no single widely accepted account of mathematical depth. This article is an attempt to bridge this gap. The strategy is to begin with a discussion of Szemerédi's theorem, which says that each subset of the natural numbers that is sufficiently dense contains an arithmetical progression of arbitrary length. This theorem has been judged deep by many mathematicians, and so makes for a good case (...) on which to focus in analyzing mathematical depth. After introducing the theorem, four accounts of mathematical depth will be considered. (shrink)

We report on an exploratory study of the way eight mid-level undergraduate mathematics majors read and reflected on four student-generated arguments purported to be proofs of a single theorem. The results suggest that mid-level undergraduates tend to focus on surface features of such arguments and that their ability to determine whether arguments are proofs is very limited -- perhaps more so than either they or their instructors recognize. We begin by discussing arguments (purported proofs) regarded as texts and (...) validations of those arguments, i.e., reflections of individuals checking whether such arguments really are proofs of theorems. We relate the way the mathematics research community views proofs and their validations to ideas from reading comprehension and literary theory. We then give a detailed analysis of the four student-generated arguments and finally analyze the eight students' validations of them. (shrink)

In spite of the many efforts made to clarify von Neumann’s methodology of science, one crucial point seems to have been disregarded in recent literature: his closeness to Hilbert’s spirit. In this paper I shall claim that the scientific methodology adopted by von Neumann in his later foundational reflections originates in the attempt to revaluate Hilbert’s axiomatics in the light of Gödel’s incompleteness theorems. Indeed, axiomatics continues to be pursued by the Hungarian mathematician in the spirit of Hilbert’s school. I (...) shall argue this point by examining four basic ideas embraced by von Neumann in his foundational considerations: a) the conservative attitude to assume in mathematics; b) the role that mathematics and the axiomatic approach have to play in all that is science; c) the notion of success as an alternative methodological criterion to follow in scientific research; d) the empirical and, at the same time, abstract nature of mathematical thought. Once these four basic ideas have been accepted, Hilbert’s spirit in von Neumann’s methodology of science will become clear. (shrink)

I propose a relevance-based independence axiom on how to aggregate individual yes/no judgments on given propositions into collective judgments: the collective judgment on a proposition depends only on people’s judgments on propositions which are relevant to that proposition. This axiom contrasts with the classical independence axiom: the collective judgment on a proposition depends only on people’s judgments on the same proposition. I generalize the premise-based rule and the sequential-priority rule to an arbitrary priority order of the propositions, instead of a (...) dichotomous premise/conclusion order resp. a linear priority order. I prove four impossibility theorems on relevance-based aggregation. One theorem simultaneously generalizes Arrow’s Theorem (in its general and indifference-free versions) and the well-known Arrow-like theorem in judgment aggregation. (shrink)

Robert Aumann presents his Agreement Theorem as the key conditional: “if two people have the same priors and their posteriors for an event A are common knowledge, then these posteriors are equal” (Aumann, 1976, p. 1236). This paper focuses on four assumptions which are used in Aumann’s proof but are not explicit in the key conditional: (1) that agents commonly know, of some prior μ, that it is the common prior; (2) that agents commonly know that each of (...) them updates on the prior by conditionalization; (3) that agents commonly know that if an agent knows a proposition, she knows that she knows that proposition (the “K K” principle); (4) that agents commonly know that they each update only on true propositions. It is shown that natural weakenings of any one of these strong assumptions can lead to countermodels to Aumann’s key conditional. Examples are given in which agents who have a common prior and commonly know what probability they each assign to a proposition nevertheless assign that proposition unequal probabilities. To alter Aumann’s famous slogan: people can “agree to disagree”, even if they share a common prior. The epistemological significance of these examples is presented in terms of their role in a defense of the Uniqueness Thesis: If an agent whose total evidence is E is fully rational in taking doxastic attitude D to P, then necessarily, any subject with total evidence E who takes a different attitude to P is less than fully rational. (shrink)

There exist two widely used notions concerning the structure of phenomenal color space. The first is the notion of unique/binary hue structure, which maintains that there are four unique hues from which all other hues are composed. The second notion is the similarity structure of hues, which describes the interrelations between the hues and hence does not divide hues into two types as the first notion does. Philosophers have considered the existence of the unique/binary hue structure to be empirically (...) and phenomenally well-grounded, and the structure has been considered to be primary because this can account for the similarity structure. Consequently, the unique/binary hue structure has played a central role in color philosophy. This calls for the assessment of the justification for its existence carried out in this paper. It is concluded that, despite the prevalent view among philosophers, none of their reasons for endorsing the existence of the unique/binary hue structure are justified. Since the notion of the unique/binary hue structure appears intuitively plausible for many, however, a sketch explaining this intuition is outlined at the end. (shrink)

Does pre-voting group deliberation improve majority outcomes? To address this question, we develop a probabilistic model of opinion formation and deliberation. Two new jury theorems, one pre-deliberation and one post-deliberation, suggest that deliberation is beneficial. Successful deliberation mitigates three voting failures: (1) overcounting widespread evidence, (2) neglecting evidential inequality, and (3) neglecting evidential complementarity. Formal results and simulations confirm this. But we identify four systematic exceptions where deliberation reduces majority competence, always by increasing Failure 1. Our analysis recommends deliberation (...) that is 'participatory', 'neutral', but not necessarily 'equal', i.e., that involves substantive sharing, privileges no evidences, but might privilege some persons. (shrink)

Suppose that the members of a group each hold a rational set of judgments on some interconnected questions, and imagine that the group itself has to form a collective, rational set of judgments on those questions. How should it go about dealing with this task? We argue that the question raised is subject to a difficulty that has recently been noticed in discussion of the doctrinal paradox in jurisprudence. And we show that there is a general impossibility theorem that (...) that difficulty illustrates. Our paper describes this impossibility result and provides an exploration of its significance. The result naturally invites comparison with Kenneth Arrow's famous theorem (Arrow, 1963 and 1984; Sen, 1970) and we elaborate that comparison in a companion paper (List and Pettit, 2002). The paper is in four sections. The first section documents the need for various groups to aggregate its members' judgments; the second presents the discursive paradox; the third gives an informal statement of the more general impossibility result; the formal proof is presented in an appendix. The fourth section, finally, discusses some escape routes from that impossibility. (shrink)

Four experiments investigated the extent to which abstract quantitative information can be conveyed by basic visual features. This was done by asking observers to estimate and discriminate Pearson correlation in graphical representations where the first data dimension of each element was encoded by its horizontal position, and the second by the value of one of its visual features; perceiving correlation then requires combining the information in the two encodings via a common abstract representation. Four visual features were examined: (...) luminance, color, orientation, and size. All were able to support the perception of correlation. Indeed, despite the strikingly different appearances of the associated stimuli, all gave rise to performance that was much the same: just noticeable difference was a linear function of distance from complete correlation, and estimated correlation a logarithmic function of this distance. Performance differed only in regards to the level of noise in the feature, with these values compatible with estimates of channel capacity encountered in classic experiments on absolute perceptual magnitudes. These results suggest that quantitative information can be conveyed by visual features that are abstracted at relatively low levels of visual processing, with little representation of the original sensory property. It is proposed that this is achieved via an abstract parameter space in which the values in each perceptual dimension are normalized to have the same means and variances, with perceived correlation based on the shape of the joint probability density function of the resultant elements. (shrink)

The notion that natural selection is a process of fitness maximization gets a bad press in population genetics, yet in other areas of biology the view that organisms behave as if attempting to maximize their fitness remains widespread. Here I critically appraise the prospects for reconciliation. I first distinguish four varieties of fitness maximization. I then examine two recent developments that may appear to vindicate at least one of these varieties. The first is the ‘new’ interpretation of Fisher's fundamental (...) theorem of natural selection, on which the theorem is exactly true for any evolving population that satisfies some minimal assumptions. The second is the Formal Darwinism project, which forges links between gene frequency change and optimal strategy choice. In both cases, I argue that the results fail to establish a biologically significant maximization principle. I conclude that it may be a mistake to look for universal maximization principles justified by theory alone. A more promising approach may be to find maximization principles that apply conditionally and to show that the conditions were satisfied in the evolution of particular traits. (shrink)

We show that true colors as defined by Chevreul (1839) produce unsuspected simultaneous brightness induction effects on their immediate grey backgrounds when these are placed on a darker (black) general background surrounding two spatially separated configurations. Assimilation and apparent contrast may occur in one and the same stimulus display. We examined the possible link between these effects and the perceived depth of the color patterns which induce them as a function of their luminance contrast. Patterns of square-shaped inducers of a (...) single color (red, green, blue, yellow, or grey) were placed on background fields of a lighter and a darker grey, presented on a darker screen. Inducers were always darker on one side of the display and brighter on the other in a given trial. The intensity of the grey backgrounds varied between trials only. This permitted generating four inducer luminance contrasts, presented in random order, for each color. Background fields were either spatially separated or consisted of a single grey field on the black screen. Experiments were run under three environmental conditions: dark-adaptation, daylight, and rod-saturation after exposure to bright light. In a first task, we measured probabilities of contrast, assimilation, and no effect in a three-alternative forced-choice procedure (background appears brighter on the ‘left’, on the ‘right’ or the ‘same’). Visual adaptation and inducer contrast had no significant influence on the induction effects produced by colored inducers. Achromatic inducers produced significantly stronger contrast effects after dark-adaptation, and significantly stronger assimilation in daylight conditions. Grouping two backgrounds into a single one was found to significantly decrease probabilities of apparent contrast. Under the same conditions, we measured probabilities of the inducers to be perceived as nearer to the observer inducers. These, as predicted by Chevreul’s law of contrast, were determined by the luminance contrast of the inducers only, with significantly higher probabilities of brighter inducers to be seen as nearer, and a marked asymmetry between effects produced by inducers of opposite sign. Implications of these findings for theories which attempt to link simultaneous induction effects to the relative depth of object surfaces in the visual field are discussed. (shrink)

Here are four examples of “seeing.” You see that something green is wriggling. You see that an iguana is in distress. You see that someone is wrongfully harming an iguana. You see that torturing animals is wrong. The first is an example of low-level perception. You visually represent color and motion. The second is an example of high-level perception. You visually represent kind properties and mental properties. The third is an example of moral perception. You have an impression of (...) moral properties. The fourth is an example of intuition. You intellectually grasp a general moral truth. Should moral perceptions be thought of as high-level perceptions or as intuitions? Most proponents of moral perception have thought of them as high-level perceptions. I give epistemological and methodological reasons for thinking that at least some are examples of what I call low-level intuitions—experiences in which we both apprehend abstract generalities and apply them to concrete particulars. (shrink)

The present paper intends to report two results. It is shown that the formula P(x)=∀y∀z[¬G(x, y)→¬M(z)] provides the logic underlying gauge symmetry, where M denotes the predicate of being massive. For the logic of spontaneous symmetry breaking, by Higgs mechanism, we have P(x)=∀y∀z[G(x, y)→M(z)]. Notice that the above two formulas are not logically equivalent. The results are obtained by integrating four components, namely, gauge symmetry and Higgs mechanism in quantum field theory, and Gödel's incompleteness theorem and Tarski's indefinability (...) theorem in mathematical logic. Gödel numbering is the key for arithmetic modeling applied in this paper. (shrink)

I argue that perception is necessarily situation-dependent. The way an object is must not just be distinguished from the way it appears and the way it is represented, but also from the way it is presented given the situational features. First, I argue that the way an object is presented is best understood in terms of external, mind-independent, but situation-dependent properties of objects. Situation-dependent properties are exclusively sensitive to and ontologically dependent on the intrinsic properties of objects, such as their (...) shape, size, and color, and the situational features, such as the lighting conditions and the perceiver’s location in relation to the perceived object. Second, I argue that perceiving intrinsic properties is epistemically dependent on representing situation-dependent properties. Recognizing situation-dependent properties yields four advantages. It makes it possible to embrace the motivations that lead to phenomenalism and indirect realism by recognizing that objects are presented a certain way, while holding on to the intuition that subjects directly perceive objects. Second, it acknowledges that perceptions are not just individuated by the objects they are of, but by the ways those objects are presented given the situational features. Third, it allows for a way to accommodate the fact that there is a wide range of viewing conditions or situational features that can count as normal. Finally, it makes it possible to distinguish perception and thought about the same object with regard to what is represented. (shrink)

Benchmarking automated theorem proving (ATP) systems using standardized problem sets is a well-established method for measuring their performance. However, the availability of such libraries for non-classical logics is very limited. In this work we propose a library for benchmarking Girard's (propositional) intuitionistic linear logic. For a quick bootstrapping of the collection of problems, and for discussing the selection of relevant problems and understanding their meaning as linear logic theorems, we use translations of the collection of Kleene's intuitionistic theorems in (...) the traditional monograph "Introduction to Metamathematics". We analyze four different translations of intuitionistic logic into linear logic and compare their proofs using a linear logic based prover with focusing. In order to enhance the set of problems in our library, we apply the three provability-preserving translations to the propositional benchmarks in the ILTP Library. Finally, we generate a comprehensive set of reachability problems for Petri nets and encode such problems as linear logic sequents, thus enlarging our collection of problems. (shrink)

This paper focuses on the fact that AI is predominantly portrayed as white—in colour, ethnicity, or both. We first illustrate the prevalent Whiteness of real and imagined intelligent machines in four categories: humanoid robots, chatbots and virtual assistants, stock images of AI, and portrayals of AI in film and television. We then offer three interpretations of the Whiteness of AI, drawing on critical race theory, particularly the idea of the White racial frame. First, we examine the extent to (...) which this Whiteness might simply reflect the predominantly White milieus from which these artefacts arise. Second, we argue that to imagine machines that are intelligent, professional, or powerful is to imagine White machines because the White racial frame ascribes these attributes predominantly to White people. Third, we argue that AI racialised as White allows for a full erasure of people of colour from the White utopian imaginary. Finally, we examine potential consequences of the racialisation of AI, arguing it could exacerbate bias and misdirect concern. (shrink)

This paper indicates how continental philosophy may contribute to a diagnostics of contemporary life sciences research, as part of a “diagnostics of the present”. First, I describe various options for an oblique reading of emerging scientific discourse, bent on uncovering the basic “philosophemes” of science. Subsequently, I outline a number of radical transformations occurring both at the object-pole and at the subject-pole of the current knowledge relationship, namely the technification of the object and the anonymisation or collectivisation of the subject, (...) under the sway of automation, ICT and big machines. Finally, I further elaborate the specificity of the oblique perspective with the help of Lacan’s theorem of the four discourses. Philosophical reflections on contemporary life sciences concur neither with a Master’s discourse, nor with university discourse, nor with what Lacan refers to as hysterical discourse, but rather with the discourse of the analyst, listening with evenly-poised attention to the scientific files in order to bring to the fore the cupido sciendi which both inspires and disrupts contemporary life sciences discourse. (shrink)

A. Accident 1. We call an accident (συμβεβηκὸς) that which attaches to something and can be truly asserted, but neither of necessity nor usually.’ (Met. , Δ, 1025a14-16) 2. Whenever an accident attaches to a subject, it attaches to it not because it is that subject (μὴ διότι τοδὶ ἧν). (Met., Δ, 1025a21-24) 3. ‘There is no definite cause for an accident, but a chance cause, i.e. an indefinite one.’ (Met., Δ, 1025a24-25) 4. ‘The accident has happened or exists, -not (...) in virtue of itself, however, but of something else. (Met., Δ, 1025a28-29) 5. ‘What attaches to each thing in virtue of itself but is not in its substance is an accident.’ (Met., Δ, 1025a30-32) 6. ‘An accident is something which, though it is … neither a definition nor a property nor a genus, yet belongs to the thing.’ (To., I, 5, 102b4-) 7. An accident is ‘something which may either belong or not belong to any one and the self-same thing, as (e.g.) being seated may belong or not belong to some self-same thing.’ (To., I, 5, 102b4-) 8. ‘There is nothing to prevent an accident from becoming a temporary or relative property … [and] there is nothing to prevent an accident from becoming both a relative and a temporary property; but a property absolutely it will never be.’ Aristotle’s example is ‘being seated.’ That it is a temporary property is evident. It is also a relative property to those who are not seating when he is the only person sitting. Therefore, it is a temporary and also a relative property. 9. ‘In the case of accidents and in no other it is possible for something to be true in a certain respect and not universally’ and ‘there is nothing to prevent an attribute belonging in part’ so that it is open to dispute accidents e.g. being white or just about a man because he might be white or just in part only. Thus, ‘conversion is not a necessary process in the case of accidents.’ (To., B, 1, 109a9-) B. Essential versus accidental Aristotle distinguishes between two senses of being (τὸ ὄν): accidental sense (κατὰ συμβεβηκός) and essential sense (καθ᾿ αὑτό). The accidental sense happens when something is said to be another thing while there is in fact a third thing which is both of them. Thus, being the first thing is not by itself the second thing. a) Aristotle speaks of three ways in which one thing is said in an accidental sense to be another. (Met., Δ, 1017a8-22) i) When both the subject and the predicate are accidents of a third thing and it is in fact the third thing which ‘is’ by its own. Thus, when it is said e.g. that ‘The just is musical’ it is said accidentally because it is in fact a third thing, Socrates for example, which is just and musical. (Met., Δ, 1017a13-19) ii) When that to which the attribute belongs is. The example of this must be ‘the man is musical’ because it is in fact the man which is. iii) When the subject which has as an attribute that of which it is itself predicated, itself is. The example of this must be Aristotle’s third example: ‘The musical is man.’ b) Aristotle rejects the essential sense to categories: ‘Those things are said in their own right to be that are indicated by the figures of predication; for the senses of ‘being’ are just as many as these figures.’ (Met., Δ, 1017a22-24) Aristotle does not give us any example of the essential sense. Daniel W. Graham suggests that while essential predication is of the logical form of ‘S is P,’ the logical form of the accidental predication is ‘S has P.’ Although he concedes that the is/has contrast is not Aristotle’s, for which he thinks Aristotle prefers ‘said of’/’in’ terminology, he points to the discussion of ‘have’ in Cat. 15 as an evidence of its potential value for an analysis of predication. Kirwan analyses the distinction of essential and accidental predicate as such: while essential predicates ‘are identical with the subjects of which they are predicated; other predications are true in virtue of the fact that two distinct items, e.g. a substance and a quality, ‘coincide.’ Herman Weidemann believes ‘what divides a predicative statement which predicates essentially from a predicative statement which predicates accidentally is the fact that the former does- whereas the latter does not- answer the question what its subject cannot fail to be without ceasing to exist, as the somewhat its subject goes on existing, no matter what happens to be true of it during its existence, or, what kind of things its subject must be in order to be identifiable as one and the same object as long as it exists.’ C. Essential 1. Either the formula or the name of the subject of an essential attribute is involved in it. For example, animal is involved in female but not in white. (Met., Z, 1030b23-26) 2. Essential attributes, combined together, form a unity. Thus, e.g. animal and biped form a unity because they are both essential of man. (OI., II, 11, 21a14-16) 3. Aristotle distinguishes between two kinds of essential attributes (PsA., A, 4, 73a34-b3; 22, 84a12-17): a) Those attributes that belong to their subject as elements in its essential nature as e.g. line belongs to triangle or point to line. Here the very being of triangle is composed of line or that of line is composed of points: line is contained in the formula of triangle and point in that of line. b) Those attributes whose subjects are contained in their own defining formula as e.g. straight is an essential attribute of line because line is part of the defining formula of straight or odd belongs to number because number is part of the defining formula of odd. Every attribute that is related in neither of these two ways to its subject, it is not an essential but an accidental attribute. (PsA., A, 4, 73b3-5) 4. That attribute which is not predicated of a subject other than itself is an essential attribute. A substance cannot be predicated of anything but itself and is, thus, essential. (PsA., A, 4, 73b5-10) 5. A thing consequently co-stated with anything is essential. For example, if a beast dies when its throat is being cut, its death is also essentially connected with the cutting because the cutting was the cause of death, not death a coincidence of cutting. (PsA., A, 4, 73b10-15) 6. Essential attributes in both of this sense, either in the sense that their subjects are contained in them or in the sense that they are contained in their subjects, are necessary as well as consequently co-stated with their subjects. The reason is that it is impossible for them to inhere in their subjects either simply or in the qualified sense (that one or other of a pair of opposites must inhere in the subject as e.g. one of straight or curve must be necessarily predicable of a line or one of odd or even of number). (PsA., A, 4, 73b16-21; 6, 74b5-12; B, 96b1-5) 7. Only those attributes that are within a genus are essential and possessed by their respective subjects as such and, thus, are necessary. Thereupon, both the conclusion and the premises of demonstrations which produce scientific knowledge are essential. (PsA., A, 6, 75a28-31) Therefore, ‘the extreme and the middle terms must be drawn from the same genus; otherwise, as predicated, they will not be essential and will thus be accidents.’ (PsA., A, 6, 75b10-12) Also, the theorems of a science can be demonstrated by means of another science only when they are related as subordinate to superior. (PsA., A, 6, 75b13-) 8. Our knowledge of the connexion of an attribute with a subject is essential only when ‘we know that connexion through the middle term in virtue of which it inheres (καθ᾿ὃ ὑπἀρχει), and as an inference from basic premises essential and ‘appropriate’ to the subject.’ The essential belonging of the middle to the minor has a necessary connexion with its belonging to the same genus of the major and minor: ‘If that middle term also belongs essentially to the minor, the middle must belong to the same genus as the major and minor terms (ἀνάγκη τὸ μέσον ἐν τῇ αὐτῇ συγγενείᾳ εἶναι). (PsA., A, 9, 76a4-9) 9. In an essential predication, the predicate is signified of a subject identical with itself or with a species of itself. Since only predicates which signify substance signify that the subject is identical with the predicate or with a species of the predicate, predicates signifying substance are essentially predicated and any other predicate is accidental or coincidental. Thus, while animal is essentially predicated of man because man is identical with a species of animal, white is accidentally predicated of man because man is neither identical with white nor a species of it. (PsA., A, 22, 83a24-32) Weidemann discusses the confusion between essential predications and statements of identity. He warns that the expression ‘(to be) just what a man is’ (analogous to the Greek phrase ‘(εἶναι) ὃπερ ἀνθρώπον) ought not to be confused with the expression ‘just what it is for a man to be,’ which renders the phrase τὸ ὃπερ ἀνθρώπῳ εἶναι’ (1007a22, 23, 27-28); a confusion Kirwan discusses. 10. ‘Demonstration proves the inherence of essential attributes in things.’ (PsA., A, 22, 84a11-12) 11. ‘Nothing which does not happen to belong to the genus is essentially the genus; e.g. a white man is not essentially a color.’ Thus, while justice falls within the genus, a just man does not. (To., Γ, 1, ^116a24-) 12. There is no syllogism of essence (ti esti). Nevertheless, we come to know ti esti through a demonstration. (93b15-20; cf. 93b25-28) D. Accidental 1. An accidental attribute cannot be predicated of a subject universally. Thus, we cannot say ‘Every man is musical.’ The reason is that ‘universal attributes belong to things in virtue of their nature.’ Therefore, accidental predicates are predicated only of the individuals. (Met., Δ, 1017b33-1018a2) 2. An accidental is a mere name. (Met., E, 1026b13-) 3. An accidental is obviously akin to non-being. (Met., E, 1026b21) 4. ‘That which is neither always nor for the most part we call accidental.’ (Met., E, 1026b27-33) 5. The matter (ὕλη) is the cause of the accidental because it is matter which is capable of being otherwise than as it for the most part is. (Met., E, 1027a13-16) In fact, there is cause or principle of the incidental of the same kind as there are of the essential because if there were, everything would be of necessity. (Met., K, 1065a6-8) The causes of accidental are unordered and indefinite. (Met., K, 1065a25-26) 6. Since all science is either of that which is always or of that which is for the most part, there is no science of the accidental. (Met., E, 1027a19-24; K, 1064b30-1065a6) Therefore, since an accident may also not inhere and, thus, it is impossible to prove its inherence as a necessary conclusion, there is no demonstrative knowledge of accidents. (PsA., A, 6, 75a18-23) 7. The accidental is not necessary but indeterminate. (Met., K, 1065a24-26) 8. The co-positing of a subject and a predicate which are accidental either to the same subject or to one another, whiteness and being musical for instance, does not form a unity. (OI, II, 11, 21a7-14) 9. Negative predicates, i.e. predicates including a ‘not,’ are accidental predicates. Thus, if good be an essential predicate of a subject, ‘not bad’ cannot be essential but is accidental. (OI., II, 14, 23b15-20) 10. Every attribute which does not belong to its subject in either of two senses of essential attribute (cf. PsA., A, 4, 73a34-b3), namely, belonging to their subjects as an element of its formula or their subjects belonging to them as an element of them, is an accidental attribute. (PsA., A, 4, 73b3-5) 11. Those attributes that are predicated of a subject other than themselves are accidental attributes. Thus, everything except substance is an accidental in this sense. (PsA., A, 4, 73b5-10) 12. A thing not consequentially connected with anything is accidental. For example, in ‘while he was walking it lightened,’ the lightening was not due to his walking but was a coincidence. (PsA., A, 4, 73b10-15) 13. Those attributes that are not within a genus are accidental attributes. (PsA., A, 6, 75a28-31) Therefore, the extreme and middle terms that are not drawn from the same genus are accidents. (PsA., A, 6, 75b10-12) Moreover, to apply an attribute to subjects in different genera afford knowledge of the attribute, only as inhering accidentally, not as belonging to its subject as such.’ (PsA., A, 9, 75b38-76a3) . (shrink)

A three dimensional hypercube representing all of the 4,096 dyadic computations in a standard bivalent system has been created. It has been constructed from the 16 functions arrayed in a table of functional completeness that can compute a dyadic relationship. Each component of the dyad is an operator as well as a function, such as “implication” being a result, as well as an operation. Every function in the hypercube has been color keyed to enhance the display of emerging patterns. At (...) the minimum, the hypercube is a “multiplication table” of dyadic computations and produces values in a way that shortens the time to do operations that normally would take longer using conventional truth table methods. It also can serve as a theorem prover and creator. With the hypercube comes a deductive system without the need for axioms. The main significance of the 3-D hypercube at this point is that it is the most fundamental way of displaying all dyadic computations in binary space, thus serving as a way of normalizing the rendition of uninterpreted, or raw, binary space. The hypercube is a dimensionless entity, a standard by which in binary spaces can be measured and classified, analogous to a meter stick. (shrink)

The visual system adapts to a wide range of visual features, from lower-level features like color and motion to higher-level features like causality and, perhaps, number. According to some, adaptation is a strictly perceptual phenomenon, such that the presence of adaptation licenses the claim that a feature is truly perceptual in nature. Given the theoretical importance of claims about adaptation, then, it is important to understand exactly when the visual system does and does not exhibit adaptation. Here, we take as (...) a case study one specific kind of adaptation: visual adaptation to size. Supported by evidence from four experiments, we argue that, despite robust effects of size adaptation in the lab, (1) size adaptation effects are phenomenologically underwhelming (in some cases, hardly appreciable at all), (2) some effects of size adaptation appear contradictory, and difficult to explain given current theories of size adaptation, and (3) prior studies on size adaptation may have failed to isolate size as the adapted dimension. Ultimately, we argue that while there is evidence to license the claim that size adaptation is genuine, size adaptation is a puzzling and poorly understood phenomenon. (shrink)

A consequence relation is strongly classical if it has all the theorems and entailments of classical logic as well as the usual meta-rules (such as Conditional Proof). A consequence relation is weakly classical if it has all the theorems and entailments of classical logic but lacks the usual meta-rules. The most familiar example of a weakly classical consequence relation comes from a simple supervaluational approach to modelling vague language. This approach is formally equivalent to an account of logical consequence according (...) to which α1, ..., αn entails β just in case □α1, ..., □αn entails □β in the modal logic S5. This raises a natural question: If we start with a different underlying modal logic, can we generate a strongly classical logic? This paper explores this question. In particular, it discusses four related technical issues: (1) Which base modal logics generate strongly classical logics and which generate weakly classical logics? (2) Which base logics generate themselves? (3) How can we directly characterize the logic generated from a given base logic? (4) Given a logic that can be generated, which base logics generate it? The answers to these questions have philosophical interest. They can help us to determine whether there is a plausible supervaluational approach to modelling vague language that yields the usual meta-rules. They can also help us to determine the feasibility of other philosophical projects that rely on an analogous formalism, such as the project of defining logical consequence in terms of the preservation of an epistemic status. (shrink)

Smell 'sensations' are among the most mysterious of conscious experiences, and have been cited in defense of the thesis that the character of perceptual experience is independent of the physical events that seem to give rise to it. Here we review the scientific literature on olfaction, and we argue that olfaction has a distinctive profile in relation to the other modalities, on four counts: in the physical nature of the stimulus, in the sensorimotor interactions that characterize its use, in (...) the structure of its intramodal distinctions and in the functional role that it plays in people's behaviour. We present two thought experiments in which we detail what would be involved in transforming sounds into smells, and also smells into colours. Through these thought-experiments, we argue that the experiential character of smell derives precisely from the structural features of olfaction, and that an embodied account of olfactory phenomenology is called for. (shrink)

This article aims to develop a Lacanian approach to bioethics. Point of departure is the fact that both psychoanalysis and bioethics are practices of language, combining diagnostics with therapy. Subsequently, I will point out how Lacanian linguistics may help us to elucidate the dynamics of both psychoanalytical and bioethical discourse, using the movie One flew over the Cuckoo’s Nest and Sophocles’ tragedy Antigone as key examples. Next, I will explain the ‘topology’ of the bioethical landscape with the help of Lacan’s (...) three dimensions: the imaginary, the symbolical and the real. This will culminate in an assessment of the dynamics of bioethical discourse with the help of Lacan’s theorem of the four discourses. Bioethics, I will argue, is not a homogeneous discourse. Rather, four modalities of bioethical discourse can be distinguished, all of them displaying specific weaknesses and strengths, opportunities and threats. This will be elucidated with the help of two case studies, namely the debates on human reproductive technologies and on the use of animals as biomedical research models. (shrink)

To study conscious experience we must, to some extent, trust introspective reports; yet introspective reports often do not merit our trust. A century ago, E.B. Titchener advocated extensive introspective training as a means of resolving this difficulty. He describes many of his training techniques in his four-volume laboratory manual of 1901- 1905. This paper explores Titchener's laboratory manual with an eye to general questions about the prospects of introspective training for contemporary consciousness studies, with a focus on the following (...) examples: introspective knowledge of the combination tones that arise when a musical interval is played; the 'flight of colours' in the afterimage of a field of bright, broad- spectrum light; and the possibility of non-obvious visual illusions. Introspective training appears to have some merit, but also to involve significant hazards. (shrink)

This paper develops a largely neglected parallel between prudence and population ethics. Prudence is generally understood to be concerned with the balancing of well-being over time. How, precisely, well-being ought to be balanced over time, however, is a fervently debated question. I argue that developing a standard guiding such evaluations is exceedingly challenging. This is due to an often overlooked fact about prudence, namely that it shares a structural similarity with population ethics: In both contexts, we assess the comparative value (...) of populations of personstages/people, which may vary in number and level of well-being. Based on this analogy, I show that the development of an adequate theory of prudence runs into very similar impossibility results as obtain in population ethics. In particular, I prove that Arrhenius’ fifth impossibility theorem can be applied to prudence. I develop and compare four possible answers to this challenge. First, I discuss the possibility of accepting the very repugnant conclusion intrapersonally. Second, I present and further develop Donald Bruckner’s Minimax Regret approach, which gives up Transitivity. Third, I apply Jacob Nebel’s Lexical Threshold View to prudence and critically evaluate it. Lastly, I introduce what I call the Negative Lexicality View, which is based on Lexical Threshold View but overcomes some of its problems. (shrink)

Formal systems are standardly envisaged in terms of a grammar specifying well-formed formulae together with a set of axioms and rules. Derivations are ordered lists of formulae each of which is either an axiom or is generated from earlier items on the list by means of the rules of the system; the theorems of a formal system are simply those formulae for which there are derivations. Here we outline a set of alternative and explicitly visual ways of envisaging and analyzing (...) at least simple formal systems using fractal patterns of infinite depth. Progressively deeper dimensions of such a fractal can be used to map increasingly complex wffs or increasingly complex 'value spaces', with tautologies, contradictions, and various forms of contingency coded in terms of color. This and related approaches, it turns out, offer not only visually immediate and geometrically intriguing representations of formal systems as a whole but also promising formal links (1) between standard systems and classical patterns in fractal geometry, (2) between quite different kinds of value spaces in classical and infinite-valued logics, and (3) between cellular automata and logic. It is hoped that pattern analysis of this kind may open possibilities for a geometrical approach to further questions within logic and metalogic. (shrink)

What is the relationship between degrees of belief and binary beliefs? Can the latter be expressed as a function of the former—a so-called “belief-binarization rule”—without running into difficulties such as the lottery paradox? We show that this problem can be usefully analyzed from the perspective of judgment-aggregation theory. Although some formal similarities between belief binarization and judgment aggregation have been noted before, the connection between the two problems has not yet been studied in full generality. In this paper, we seek (...) to fill this gap. The paper is organized around a baseline impossibility theorem, which we use to map out the space of possible solutions to the belief-binarization problem. Our theorem shows that, except in limiting cases, there exists no belief-binarization rule satisfying four initially plausible desiderata. Surprisingly, this result is a direct corollary of the judgment-aggregation variant of Arrow’s classic impossibility theorem in social choice theory. (shrink)

: Phenomenal knowledge usually comes from experience. But it need not. For example, one could know what it’s like to see red without seeing red—indeed, without having any color experiences. Daniel Dennett (2007) and Pete Mandik (forthcoming) argue that this and related considerations undermine the knowledge argument against physicalism. If they are right, then this is not only a problem for anti‐physicalists. Their argument threatens to undermine any version of phenomenal realism— the view that there are phenomenal properties, or qualia, (...) that are not conceptually reducible to physical or functional properties. I will argue that this threat is illusory. Explaining why will clarify what is and is not at issue in discussions of the knowledge argument and phenomenal realism. This will strengthen the case for physically and functionally irreducible qualia. 2 Phenomenal knowledge usually comes from experience. For example, I know what it’s like to see red because I have done so. Does knowing what it’s like to have an experience with phenomenal character X require having an experience with X? No. A famous counterexample is Hume’s missing shade of blue, in which one can extrapolate from phenomenally similar experiences (A Treatise of Human Nature, Bk. I, Pt. I, Sec. I).1 One might think that a weaker version of the no‐phenomenal‐knowledge‐without‐experience thesis remains tenable, e.g., that knowing what it’s like to see in color requires having or having had color experiences. But this thesis also seems doubtful. Peter Unger (1966) devised plausible counterexamples over four decades ago, and since then others (e.g., Lewis 1988, Alter 1998, Stoljar 2005) have done the same. One could have phenomenal knowledge of color experiences without having such experiences. Indeed, one could have such knowledge without having experiences that are remotely like color experiences. What is the significance of this observation for contemporary debates about consciousness and physicalism? Daniel Dennett (2007) and Pete Mandik (forthcoming) suggest that it undermines the knowledge argument against physicalism.2 That is because they take the claim that someone who has never seen in color could not know what it’s like to see in color to be the basis of the knowledge argument’s main epistemic premise: the premise that (roughly put) 3 no amount of physical knowledge is sufficient for phenomenal knowledge of color experiences. If they are right, then this is not only a problem for anti‐ physicalists. Many physicalists (e.g., Loar 1990/97, Papineau 2002, 2007) accept the knowledge argument’s epistemic premise. Dennett’s and Mandik’s arguments threaten all versions of what David Chalmers (2003) calls phenomenal realism, the view that there are phenomenal properties, or qualia, that are not conceptually reducible to physical or functional properties.3 I will argue that this threat is illusory. Explaining why will clarify what is and is not at issue in discussions of the knowledge argument and phenomenal realism. The net result will be to strengthen the case for physically inexplicable qualia. (shrink)

Bilattices, introduced by Ginsberg as a uniform framework for inference in artificial intelligence, are algebraic structures that proved useful in many fields. In recent years, Arieli and Avron developed a logical system based on a class of bilattice-based matrices, called logical bilattices, and provided a Gentzen-style calculus for it. This logic is essentially an expansion of the well-known Belnap–Dunn four-valued logic to the standard language of bilattices. Our aim is to study Arieli and Avron’s logic from the perspective of (...) abstract algebraic logic . We introduce a Hilbert-style axiomatization in order to investigate the properties of the algebraic models of this logic, proving that every formula can be reduced to an equivalent normal form and that our axiomatization is complete w.r.t. Arieli and Avron’s semantics. In this way, we are able to classify this logic according to the criteria of AAL. We show, for instance, that it is non-protoalgebraic and non-self-extensional. We also characterize its Tarski congruence and the class of algebraic reducts of its reduced generalized models, which in the general theory of AAL is usually taken to be the algebraic counterpart of a sentential logic. This class turns out to be the variety generated by the smallest non-trivial bilattice, which is strictly contained in the class of algebraic reducts of logical bilattices. On the other hand, we prove that the class of algebraic reducts of reduced models of our logic is strictly included in the class of algebraic reducts of its reduced generalized models. Another interesting result obtained is that, as happens with some implicationless fragments of well-known logics, we can associate with our logic a Gentzen calculus which is algebraizable in the sense of Rebagliato and Verdú . We also prove some purely algebraic results concerning bilattices, for instance that the variety of distributive bilattices is generated by the smallest non-trivial bilattice. This result is based on an improvement of a theorem by Avron stating that every bounded interlaced bilattice is isomorphic to a certain product of two bounded lattices. We generalize it to the case of unbounded interlaced bilattices. (shrink)

A uniform construction for sequent calculi for finite-valued first-order logics with distribution quantifiers is exhibited. Completeness, cut-elimination and midsequent theorems are established. As an application, an analog of Herbrand’s theorem for the four-valued knowledge-representation logic of Belnap and Ginsberg is presented. It is indicated how this theorem can be used for reasoning about knowledge bases with incomplete and inconsistent information.

We present an elementary system of axioms for the geometry of Minkowski spacetime. It strikes a balance between a simple and streamlined set of axioms and the attempt to give a direct formalization in first-order logic of the standard account of Minkowski spacetime in Maudlin and Malament. It is intended for future use in the formalization of physical theories in Minkowski spacetime. The choice of primitives is in the spirit of Tarski : a predicate of betwenness and a four (...) place predicate to compare the square of the relativistic intervals. Minkowski spacetime is described as a four dimensional ‘vector space’ that can be decomposed everywhere into a spacelike hyperplane—which obeys the Euclidean axioms in Tarski and Givant, 175–214 1999)—and an orthogonal timelike line. The length of other ‘vectors’ are calculated according to Pythagoras’ theorem. We conclude with a Representation Theorem relating models \ of our system \ that satisfy second order continuity to the mathematical structure \, called ‘Minkowski spacetime’ in physics textbooks. (shrink)

This author’s reply addresses critiques by Daniel Engster, Kelly Gawel, and Andrea Westlund about my 2020 book, Freedom to Care: Liberalism, Dependency Care, and Culture. I begin with a statement of my commitment to liberalism. In section two, I defend the value of a distinction between conceptions of persons in the real world and in contract theory to track inequalities in care when indexed to legitimate needs. I argue, as well, that my variety of contract theory supplies the normative content (...) needed to reject the subordination of women of color. Acknowledging the enduring danger of expressive subordination, I emphasize my theory’s compatibility with the full social inclusion of people with disabilities. Section three then defends liberal dependency care’s compatibility with radical critique and transformative change by emphasizing the abstract nature of its core theoretical module. Finally, in section four, I reaffirm conceptual distinctions between autonomy skills, care skills, and a sense of justice by explicating their theoretical roles. In that section, I also embrace Westlund’s insight that theorists of justice need to have skills enabling responsiveness to other perspectives. To this new requirement for actual theorists of justice, I further add that we must attain capacities to engage critically with our society’s norms. Thus, the final section of this article supplements the justificatory module of liberal dependency care, building from the necessary conditions specified as two-level contract theory toward an account of necessary and sufficient conditions for this liberalism’s justificatory module. (shrink)

Sikhism is the world's fifth-largest religion. It was founded during the late 15th century in the Punjab region of the Indian subcontinent. Its adherents are known as Sikhs. Currently, there are about 30 million Sikhs worldwide. Most of them live in the Indian state of Punjab. As per Sikh tradition, Sikhism was established by Guru Nanak (1469–1539) and subsequently led by a succession of nine other Gurus. Before his death, the tenth Sikh Guru, Guru Gobind Singh (1666–1708), bestowed the status (...) of Guru to the sacred scripture of Sikhs, Adi Granth, which is presently known as Sri Guru Granth Sahib (SGGS) [1]. The Adi Granth was first compiled by Guru Arjan Dev, the fifth Sikh Guru, in 1604. Its second and final version has been the handiwork of Guru Gobind Singh, who added the hymns of his father, Guru Teg Bahadur, the ninth Sikh Guru [2], at Damdama Sahib, Talwandi Sabo, Punjab, in 1705. The holy Sikh scripture, SGGS, contains 1430 pages of text in poetry form. In addition to the hymns of the six Sikh Gurus and four Sikhs, it includes hymns composed by fifteen saints (Bhagats) and eleven poet laureates (Bhats) of the Guru's court. Muslims and Hindus, Brahmins, and "untouchables" all come together in one congregation to create a universal scripture. It is a compendium of mystic, metaphysical and religious poetry written or recited between the 12th and 17th centuries in the Indian sub-continent [3]. Sri Guru Granth Sahib, through its comprehensive worldview, offers a perfect set of values and an applicable code of conduct. Its cardinal message addresses the welfare of all humans irrespective of their caste, colour, creed, culture, and religion. SGGS emphasizes love, respect, empathy, and acceptance of others' existence. It prohibits us from infringing on the freedom and rights of others. The life and works of the Sikh Gurus exemplify the practicability of these ideas. Their inter-faith dialogues highlighted that human unity and oneness could be achieved through tolerance, communication, and respect for others [4]. Besides a matchless elaboration of spirituality, Sri Guru Granth Sahib enshrines a powerful expression of the message of revolutionary ideals of social welfare, human rights, multicultural distinctness, and religious freedom. In the present era, when the threats and fear of interfaith conflicts, military aggression, terrorism, etc., have overpowered human sentiments, the teachings of Sri Guru Granth Sahib are even more relevant to resolving all these problems. (shrink)

Risky prospects represent policies that impose different types of risks on multiple people. I present an example from food safety. A utilitarian following Harsanyi's Aggregation Theorem ranks such prospects according to their mean expected utility or the expectation of the social utility. Such a ranking is not sensitive to any of four types of distributional concerns. I develop a model that lets the policy analyst rank prospects relative to the distributional concerns that she considers fitting in the context (...) at hand. I name this model ‘the Distribution View’ posing an alternative to Parfit's Priority View for risky prospects. (shrink)

June 2022 A Revolutionary New Metaphysics, Based on Consciousness, and a Call to All Philosophers We are in a unique moment of our history unlike any previous moment ever. Virtually all human economies are based on the destruction of the Earth, and we are now at a place in our history where we can foresee if we continue on as we are, our own extinction. As I write, the planet is in deep trouble, heat, fires, great storms, and record flooding, (...) weather patterns drastic everywhere. Capitalism has been so successful in supplying us with an endless array of consumer items, that we have never noticed there is a cost. But there is a cost, the cost is the very life of the Earth herself, and now the day of reckoning is here. Days of reckoning are most often days of rath. But I am hoping that this is a day of major Awakening over the entire planet. We have caused a crisis so great for the Earth we can hardly get our minds around it. But I am proposing that deeply embedded in the crisis are ideas, about who and what the Earth is, Ourselves, and what we are here for. I am proposing that this is a call to all philosophy and philosophers to rethink ourselves from the ground up. The human race has never gotten the Earth right and now we must, for the Earth is the true foundation for civilization. We are in deep trouble with the Earth, and it owes to the fact that the great male thinkers never got the Earth right, they could never make the connection between the life of mind and the life of the Earth. I am proposing that deeply embedded in the crisis we are in are ideas. And that in fact the planet is in the grip of a false metaphysics, and scientific image of nature that really comes most recently from Descartes. His conviction that matter, and with it the Earth, are simply the weighable, the measurable the extended in space and time, utterly devoid of any inner life of mind thoughts or Consciousness. This view of matter and the Earth goes back to the very dawn of thought, Descartes is just its most recent and comprehensive version of things. This view of matter is really a false metaphysics that dominates most of the modern world and underpins the whole scientific image of nature itself. I am proposing this is a false metaphysics and offering a new version of things based on Consciousness. If there is a single idea fueling and driving the destruction of the planet, it is the conviction that the Earth is something without Consciousness. When you intend to ravage and savage the planet, turn it into consumer items to be converted into money in the bank, first pretend that all of the bright beings in the natural world are devoid of Consciousness, just “things” you do not have to have any conscious about destroying them. I am proposing that this term Consciousness, that has appeared as term and a problem for modern science, is the key to getting ourselves right with the Earth. I propose this new theory for science will ramify into every other disciplines as well, and the ways we think just about everything. This old paradigm based on a false theory of matter, has run its course, it has brought us to the brink of our own extinction. We need new ideas, and we need them now. You will find my recent work in the Scientific God Journal under the title: Consciousness as the Organizing Principle, and a very complete version of my arguments for this term Consciousness, along with radical reinterpretation of modern science itself, on the Galileo Commission site. It is a radical moment for all of the thinkers among us, take this as my small contribution to rethinking ourselves in new terms, based on my understanding that the true basis for the whole of Being, the Universe, the Cosmos, and with them Ourselves, is this term Consciousness. And unpack it. And I predict that with this term Consciousness every intractable human problem in both science and the modern world will go down like a line of dominos before it. And so, here are a few new ideas I would offer you. These are the times of the Reappearance of the Feminine, a great Awakening as of Springtime after a long dark and lonely Winter. We are leaving behind us an old civilization based on male dominance, fear lies and control, now in decay, disease, and death disintegrating all around us. We are moving into a whole new civilization based on Love and Truth. A new civilization requires new basic terms values and first principles. And so, I return to Descartes. 1. Just over 300 years ago a young man travelling with the army found himself stranded in a foreign town in a bare room lit by a candle with nothing to do but think. And in a couple of nights Descartes laid the entire thought foundations for modern science and most of the modern world. It isn’t clear that modern philosophers with their funding sources of grants, their graduate students, their reputation and all the perks and privileges of academia can do any better. Descartes claimed there are just two principles in the Universe, Matter and Mind. God was pure mind; human beings were mind and matter. And we are the only ones on the planet with mind. Everything else all of the bright array of beings who share the planet with us are simply matter. And matter? The weighable the measurable the extended in space and time utterly devoid of any inner life of thoughts, feelings, or Consciousness This classification was a boon for developing science. Scientists did not have to worry about any inner life in nature, ideas or Consciousness, they could just set about weighing and measuring, as they have until the present day. But it was a disaster for the Earth. Earth was once worshiped as a Goddess. With Descartes she became almost overnight “things.” Simply commodities and resources, an endless of supply of consumer items, to be turned into money in the bank. And now Descartes concept of matter is up for us. If there is a single idea that is enabling and driving the destruction of the Earth, it is the conviction that Earth is something without Consciousness. This whole scientific image of nature as something devoid of Consciousness is driving the destruction of the Earth, and driving us all insane, and it needs to change. Ever since Descartes formulated Dualism, scientists and philosophers have been trying to get rid of it. They have done this by attempting to make matter the fundamental term of reality, and to explain mind or Consciousness in terms of matter. And now, Consciousness has emerged as term and a problem for modern science. 2. The prevailing Universe picture has been mainly based on the belief that Consciousness appears only at the end of evolution in ourselves and some of the higher animals when matter attains “a certain state of complexity” and it is produced by neurons in the brain. But Consciousness is nothing real in itself, the neurons are doing the work, Consciousness is simply a byproduct of neural activity. As William James summed up this position, not his: Consciousness is to the brain as a shadow to the runner, it runs beside him but never influences his stride. And so, the great problem of neuroscience: How do neurons produce Consciousness? So far, no one has been able to say how. All we have is parallelism. Two processes are running in parallel a train of neural events and a train of Consciousness events. And so, which one is primary? The face of neuroscience and much of the modern world is that the neural events are what is really real, Consciousness is secondary. The conviction that Consciousness is something produced by neurons is so deeply held and has been so for so long, that it seems like an obvious fact. It is in fact an assumption and a theory, and it could all well be the other way around. The scientific image of nature as something devoid of Consciousness is now up for us. It has enabled the destruction of the Earth, and we are now at a unique place in our own evolution where we can foresee if we continue on as we are, our own extinction. And so, there are the best reasons for rethinking Descartes concepts of matter. Scientists and philosophers have been trying to escape dualism by making matter the fundamental term. And now, this has not worked. In a nutshell: The mind body problem, the “hard problem of Consciousness” ---how do neurons in the brain, atoms and molecules give rise to ideas thoughts and feelings and who what and where is the “I” who sees them? And then, the near-death experience. I remind you of the truth of the near-death experience. The body in the bed is brain dead eeg flat, but the patient, pure Consciousness is hovering at the ceiling watching the doctors work over the body below, able to read dials he could not have seen from the bed, sometimes able to events in the corridor beyond the room, all subsequently confirmed by the doctors. Evidence as good as it gets that the patient really was out of his body. And so, the patient is able to see without eyes, hear without ears and to think and remember without a brain. There is not a single scientific principle that can account for this. And I am proposing that the near-death experience is a threshold and a key for a much deeper understanding of the Universe. These three problems are so hard they cannot be solved by present scientific principles and therefore call all present principles and ways of thought into question. When your axioms cannot account for the data, it is time to drop them and find better fundamental terms. We have not been able to solve the problems of Consciousness with the assumption that matter is the most basic and fundamental term. And so, if Cartesian dualism is to be overcome at all, we need to try it the other way around. 3. We are in the position of Copernicus. Remember Copernicus? He was making his calculations on the assumption that the Earth was the center of the Universe until they became so unwieldy that he decided to try it all the other way around. To make the sun the center, the Earth revolving around it. And lo, all of his data fell neatly into place. I am proposing a whole new Universe picture. Evolution makes us continuous with every other being, Consciousness would not be in us if it were not in them also, and then, in the atoms and molecules and cells that make up living beings. I propose to make Consciousness the most basic term and see what follows. I am in fact, proposing a new Copernican Revolution, that Consciousness is the true basis for the Universe and the right fundamental term, for science itself all other disciplines as well, and the whole of the modern world. And I predict that every intractable problem in science or the modern world, will go down like a line of dominos before it. In a nutshell: Consciousness and not matter is “first and fundamental” in the Universe, it is there from the very “Beginning” everything has it and all of the true causalities, the explanatory principles belong to it. Not to matter. What then is matter? All matter is an Appearance of something much more real, Consciousness. All matter is an expression of Consciousness, even the least little bit of matter contains Consciousness, and is an expression of Consciousness. And so, we revise the scientific image of nature. Science at present is two termed, matter/energy. It needs to become three termed: Matter/energy/Consciousness, with the recognition that all of the basic principles, the true causalities of events, and of the Universe belong to Consciousness. And now physics. To accommodate this new Universe picture physics needs to alter just one term, energy. All energy contains Consciousness, that idea alone will change our world forever. The animals the trees the plants are just as conscious as we are, as is the Earth herself, a continuum of many forms of Consciousness of which our own is one. The Earth consists of interacting and intersecting forms of Consciousness, Consciousness within Consciousness within Consciousness within Consciousness. A remarkable work has just appeared entitled “Beyond Words What Animals Feel and Think.” Describing in detail how like ourselves all of the animals are. And the more we can see how like us everything else really is, the harder it will be for us to do them all in, in the interest of turning them into consumer items. And so, here is a new principle of understanding, that entails completely new forms of explanation, and a whole new Universe picture to be spelled out in terms of Consciousness and its various properties. This is a Universe that is aware awake and Enspirited throughout. And so, there is only one principle, not two. And so, how does Consciousness work? There is “no mind/body problem.” My Consciousness is aligning the Consciousness of neurons in the brain, that aligns the Consciousness of all other cells in the body, that aligns the Consciousness of molecules and atoms themselves. Mainly from the top down, but also interactive, and not reduction to the purely physical forces. 4. And so, here is a radical new Universe picture, and with it a new way to think about the Universe itself, the Earth, Ourselves, and Spirit. It requires a complete revision of both of the terms of modern science itself. The problem of the modern world owes in deep measure to Descartes concept of matter as something devoid of interior life. But he is only the most powerful and simple expression of a whole tradition of thought that predates him, that goes back to the very dawn of thought itself. It owes to the conviction of both Plato and Aristotle that the Earth was something devoid of reason, or mind, and therefore something to be held in complete contempt. We are emerging from a civilization based on contempt for the Earth, and down the road, it will become contempt for the body, sex, women, the Native people, and people of color everywhere. And so, this entire tradition is now up for us, it has brought us to the brink of our own extinction, and we need to get beyond it. I am proposing that this term Consciousness is the way out. The missing piece of the puzzle and the idea whose time has come. And it will enable us to move into a completely new civilization based on respect for the Earth, women, sex, people of color, that honors everyone and has a place for all. Revolutions in science occur when new data show up that cannot be accounted for by existing paradigms (Thomas Kuhns: The Structure of Scientific Revolutions). Consciousness, and all the properties and problems of Consciousness are just such data, and they compel a new scientific revolution. But we are a scientific age. A new Copernican revolution in science is such as revolution everywhere. And Consciousness is the key to getting every other reality at all levels and depths right as well. It will enable us finally to get the Earth right, get ourselves right with the Earth, and with one another. Archimedes: Give me a place to stand, and I will move the world. I am proposing a new synthesis among our fundamental terms in which this term Consciousness is the new lynchpin. And so, I return once more to Descartes. Descartes in the Mediations was a dualist, they were two realities interacting in the pineal gland mind and body. But Descartes age was high on “Cartesianisms,” on Descartes concept of matter. Animals were simply cleverly wired machines (Hobbes called them automata). Rumor had it that Descartes followers went about kicking dogs reasoning that their howls were not due to any feelings of pain, but simply the wiring of the machine. We have been doing this to the Earth ever since, and now the Earth is not going to stand for it any longer. It is time we all woke up. There is no matter in Descartes sense of it, or in Plato and Aristotle’s sense of it. This old concept of matter is embedded in all of the crisis and catastrophes---fires, drought, pandemics, over population, the disintegrating economies, our ways of life, capitalism and consumerism themselves. All matter is Enspirited, contains Consciousness, and is an expression of Consciousness. Now I tell you something about Descartes you may not know. In the Mediations was a dualist, there are minds and bodies. But by the end of his life, he knew the truth. Descartes guided his whole life by these vivid dreams, and toward the end of his life he had this dream: He saw the Universe as a giant machine and in the heart of the machine propelling the whole, was Consciousness. And so, out of the shades and shadows and into the light of day. Here is a renewed and a new science, a new cosmology. And with it, a revolutionary new metaphysics. The whole of the modern world is in the grip of a false metaphysics, based on Descartes theory of matter. Here is a new, and finally true metaphysics based on the single term, Consciousness. 5. And in all my works, I spell out this new metaphysics based on Consciousness, and what it means for how we think about virtually everything: The Universe, the Earth, Ourselves, and Spirit. Of who, what and where Spirit is, that we may connect with Spirit everywhere. There is only one term for the Universe, and not two, this single term Consciousness, that manifests itself in two modes: Consciousness in Form, Consciousness Formless. And so, here is a sketch of what this new term Consciousness really means. For four great aspects of the real, four new cornerstones for a whole new civilization on Earth, based on Love and Truth. A. For the Universe: New causalities, all matter is an expression of Consciousness, awake and aware, Enspirited throughout, which has a place for the causalities of love ideas intent plan and purpose, what Aristotle called final causality, that for the sake of which. And it has a place for subjectivity, the greatest mystery in the whole of the Universe, knowable only in ourselves, but there in everything else as well, bedrock in the chain of causalities. The buck stops here. Here are two very different Universe pictures. The Cartesian Universe consists of things, pushed, and pulled around by the physical forces that function blind. The true Universe consists of selves and not things. Selves at least in ourselves, are organized and focused about an I, an I am, or a sense of self in everything, ever other being dear to itself. Just one aspect of subjectivity. And so, it is Consciousness all the way down, and subjectivity all the way down. Science proceeds on the basis of the negligible of what it can leave out as unimportant. Subjectivity is the ultimate animating principle of the Universe. What Descartes really did was to strip from nature what is most essential about it, its animating principle Consciousness. And so, we come out of a long history of “clunk.” The great male thinkers have killed creation. First in concepts and now in the lived life. And so, after centuries of fruitless search, not all of it in vain, finally, and just in time the true basic principle for the Universe has shown up in virtually all disciplines, this term Consciousness. B. And so, the Earth: What does this term Consciousness really mean for the Earth? It means that all of the bright array of beings around us have Consciousness. And the more we can realize how like us they are, the harder it will be to do them all in. Is this what it really means to be “made in the image of Spirit?” To destroy ever other living being? The god of Genesis is a creator god who has created billions and billions of living forms. Man, taking himself to be made in the image of Spirit has destroyed virtually all of it. The Earth is not things, not commodities and resources, not an endless array of consumer goods. The Earth consists of conscious living spiritual beings like ourselves, who need to be honored and respected, nurtured and cared for, to be Loved. And if the human race can finally pull it off and bring Love here and start helping the Earth rather than destroying her, even our vast numbers now at 8 billion, may not matter. Deeply embedded in the modern world is a false cosmology based on the hitherto useful but ultimately false metaphysics of Descartes. In particular it owes to Descartes concepts of matter, and of the Earth, as something devoid of Consciousness. And now this concept is up for us. If there is a single idea enabling and driving the destruction of the planet, it is the conviction that the Earth is something devoid of Consciousness. And we all have a vested interest in keeping this false concept in place. When you intend to ravage and savage the Earth, to turn it into consumer items, and money in the bank, first pretend she has no Consciousness that way we need not have any conscious about it. We are in deep trouble with the planet. Virtually all human economies are based on the destruction of the Earth, and we are now at a place in our history where we can foresee if we continue on as we are, our own extinction. And so, here are a few new guiding principles for the planet, based on Consciousness. Consciousness has field properties. The Earth is Consciousness throughout, connected and interconnected throughout, intersecting and interacting fields of Consciousness of which our own is one. To harm one is to harm all. Only this understanding of the Earth can set our destructive technologies, and capitalism, their goals, and their limits. The real problem with capitalism, and with civilization itself is it has no true understanding of the Earth, all the exquisite networks and webworks that connect all beings with one another. Nor it is about to admit that the Earth is our “life support system,” reason gone insane in the modern world. We are destroying our life support system, capitalism is the full expression of the human death wish in the modern world, and it needs to change. And now, let the barriers between ourselves and all other beings go down. C. And so, Ourselves. This new Universe picture based on Consciousness opens into and has a place for new concepts of human identity. In particular, it has a place for reincarnation, a process whereby one Consciousness dons’ bodies again and again and lays them down. This is a process wholly impossible on materialistic theories. This new view of things puts a higher perspective on Freudian determinism, which is a partial knowing in a greater account. We are something more than the victims of genes drives and childhood. We are immortally eternal evolving spiritual beings who take on a cycle of earthly lives in order to achieve certain goals and abilities, and then, our earthly cycle complete, we depart the Earth plane for other areas of the Universe physical or purely spiritual, to continue our evolution elsewhere. The scientific evidence for reincarnation is increasing on all sides, see for example the work of Michael Newton, a mainstream scientist forced to confront the fact of reincarnation. A completely new reincarnational picture is now emerging everywhere. We choose our parents for abilities we could acquire through them. We all come here with reasons for being here, and we all have free will, an original property of Consciousness, and we work out our issues accordingly. We all come here with the same reason for being here. It is to experience the Earth plane and human society at all levels and depths, in every social role, every climate, both sexes male and female and in all races. We are all here to acquire a deep understanding of the Earth plane with all its challenges and all of its present problems. We are not here to devour and consume the Earth, but to honor her for the learning experience she makes available for us. And so, reincarnation is the great equalizer, the fundamental basis for any bill of rights. Many of the problems of the modern world owe to the fact that reincarnation has not been taught in the West. Reincarnation was the center piece of Jesus teachings. It was banned by a fourth century council that decided the populous could best be controlled through fear. They edited all mention of reincarnation from the scriptures, and inserted fear terms instead, a judgmental god, and so on. If reincarnation had been taught, there would have been no suppression of women, but the gift of both sexes would have been honored and developed. And there would have been no reason for contempt for people of color because who they are, we have been or will be. There is no reason to look down on anyone when we know that our own identity includes membership in theirs. The truth of reincarnation needs to be the foundation for getting ourselves right with the Earth, and with one another. D. And so, Spirit. This new Universe picture, based on Consciousness, has a place for Spirit as simply Infinite Consciousness. All universes all worlds are formed out of this Consciousness, divine and sacred throughout. And Spirit is everywhere in the Earth and can be connected everywhere. And the Native North American people had things right. Everything has its own Spirit, and without contradiction is also an expression of Spirit. And what I am really doing in terms of concepts of modern science is presenting the Native North American version of the Real. And this new Universe picture has a place for the essential insight of all religions almost smothered by church tradition: Love. Not an emotion but a great Oneness, that can be found everywhere, even in a busy city street. But especially in the Earth, that is levels and depths of Oneness everywhere. And so, for the first time in their long and bitter antagonistic history, science and religion now share a common base with this term Spirit, the lynchpin in a new conceptual framework. And I offer an interpretation of where we are in our history, this present moment of historical grace, and these tumultuous times in which we live, the return of the Goddess and the Divine Feminine, the Reappearance of the Feminine in history, the Advent of Women, this great Awakening, as of Springtime after a long dark Winter, the Awakening, the Second Coming. And so, stay tuned! (shrink)

Richard Wright gave a series of lectures in Europe from 1950 to 1956, collected in the following year in the volume, White Man, Listen! One dominant theme in all four essays is that expanding the moral imagination is centrally important in repairing our racism-benighted globe. What makes Wright’s version of this claim unique is his forthright admission that expanding the moral imagination necessarily involves pain and suffering. The best place to hear Wright in regard to the necessary pain of (...) expanding the moral imagination, I would argue, is his poetry collection, This Other World: Projections in the Haiku Manner. To wit, for Wright the necessary pain of expanding one’s moral imagination is the loneliness that results from delegating to others—in the etymological sense of “deputizing or committing”—one’s whiteness qua privilege or social capital. In conclusion, lonely delegation constitutes an imperative template from Wright regarding the painful expansion of our own moral imagination, in the service of social justice for economically oppressed communities of color across the globe. (shrink)