This paper has been superseded by chapter 3 of my book "God, the Best, and Evil" (OUP 2008). The chapter concerns God's choices in cases in which God has infinitely many better and better options.

William Rowe argues that if an omnipotent, omniscient being were faced with an infinite hierarchy of better and better worlds to create, that being could not also be unsurpassably morally excellent. His argument assumes that, at least in ideal circumstances, degree of moral goodness must be perfectly expressed in the degree of goodness of the outcomes chosen. Reflection upon the application of analogous expression principles for certainty and desire shows that such principles can be expected to fail for anyone (...) capable of facing an infinite range of options. (shrink)

Abstract Though it has been claimed that Frege's commitment to expressions in indirect contexts not having their customary senses commits him to an infinite number of semantic primitives, Terrence Parsons has argued that Frege's explicit commitments are compatible with a two-level theory of senses. In this paper, we argue Frege is committed to some principles Parsons has overlooked, and, from these and other principles to which Frege is committed, give a proof that he is indeed committed to an (...) class='Hi'>infinite number of semantic primitives?an intolerable result. (shrink)

My reply corrects one misstatement in Oppy’s summary of my book, abandons a footnote in the light of one of Oppy’s criticisms, and argues that Oppy’s other criticisms do not succeed in showing either that my claims are mistaken or that the arguments by which I supported them are defective.

“Artworks are not being but a process of becoming” —Theodor W. Adorno, Aesthetic Theory In the everyday use of the concept, saying that something is grotesque rarely implies anything other than saying that something is a bit outside of the normal structure of language or meaning – that something is a peculiarity. But in its historical use the concept has often had more far reaching connotations. In different phases of history the grotesque has manifested its forms as a means of (...) subversive resistance against society’s prevailing notions of form and power. What aids this impact and distinguishes two of its basic stylistic features is the grotesque’s dissolution of form and its hyperbole. Such grotesqueries, however, soon solidify into new forms, new structures of meaning, hierarchy and practice, and in this sense the history of the grotesque is, on one hand, a continual opposition and transgression of the prevailing notions of art as well as of God and humanity, while on the other hand it offers continual resistance to its own solidifying process. As such the grotesque will not and cannot be contained in form without it losing the very thing that makes it grotesque. Even though it is somewhat easy to point out certain stylistic features of the grotesque, the sum of those features tell us very little about what is at play within it. Definitions concerned with the grotesque’s content rather than its form faces similar difficulties. Throughout history, expressions relating to the grotesque have been used in defense of a whole host of different social and cultural discourse including, for instance, Catholicism (See Erhard Schön’s Der Teufel mit der Sackpfeife , 1536), Protestantism (See Lucas Cranach’s Der Papstesel , 1523), a material folk culture (see Mikhail Bakhtin’s Rabelais and his World , 1984) and an idealistic romantic structure of meaning (see Wolfgang Kayser’s The Grotesque in Art and Literature , 1957). Thus though the grotesque cannot be reduced to the expression of certain forms or meanings, in order to examine what the grotesque is about one has to try to see how it’s relationship to flow and process and how, by maintaining this relationship, it attempts to avoid solidifying into form and meaning. The process of the grotesque alluded to here revolves around a play between periphery and center, the marginalized and the dominant. Most often this is expressed as a direct negation of the center of power. In the medieval grotesque tradition of the carnival, for example, it is expressed by its emphasis on the nether regions of the body as the center of creation of meaning. Spirit does not come from above, but from the belly, buttocks and genitals, and there is, expressed in this manner, a mockery of the predominant Christian notion of truth and meaning. Such inversions of the sites of meaning most often explicitly express an increased interest in materiality instead of ideal content which comes to the fore through the play between periphery and center. Even in early grotesque Renaissance art (for example in the works of Raphael) the depiction of the mythological or ideal reveals an exploration of the possibilities of the material. The works of art thus explore their own boundaries rather than act as vessels of the divine and in this way the grotesque explores the limits of form and materiality. This in turn brings to the fore a metaphysical dimension in the workings of the grotesque: It is an immanent exploration of its own boundaries. Materiality and metaphysics are joined together, because it is through the awareness of itself as a structuring (dis)order that it places itself in opposition to the prevailing notions of form and power. This way the grotesque expresses an awareness of the division of sign and reality and a search for this reality. It implicitly expresses the awareness that meaning is not something God-given and static, but fluctuating and man-made. In modern explorations of the history of the grotesque it is commonly seen as a form of realism that manifests a shift from the ideal towards the material, freeing art from representing some hidden higher order and instead making the work of art the very site of creation and meaning. Going even further, the grotesque actively opposes the notions of the ideal by marking a shift from an ahistorical view of the world towards the historical. However, this also makes clear why the grotesque has a tendency to solidify into new static forms of meaning, for by establishing itself as a subversive counterforce there is an idea of transgression and emancipation. That is why the grotesque often historicizes and relativizes only to repeat the mistakes of that which it negates: putting itself in the king’s chair. The exploration of the material seems to create new forms and notions of the ideal. The representation of an otherness outside the prevailing notions of meaning and truth becomes a positive manifestation of this otherness as truth itself. Thus the historical “truths” of the grotesque are superseded by history itself. When the expressions of the grotesque solidify into static form and meaning, they become mere objects like anything else in the world. Instead of being this elusive thing of process and flow, it becomes tangible to the existing hierarchy of power as well as history itself. In this way the grotesque can be expunged, included or simply revealed as yet another false notion of truth, which is precisely what has happened with the grotesque throughout the course of history. If from the beginning the expression of the grotesque has not already been in the service of some structure of meaning and truth, it has quickly been included or has quickly included itself in such a structure. But by doing so it rejects its own basis: the tension between periphery and center is replaced by the attempt to create a new center, a new structure of meaning and truth. The doubleness, ambiguity and flow of the grotesque are then rejected by the grotesque itself. THE NON-FORM OF FORM In the critical thinking of Theodor W. Adorno (1903-69) the problem of art, caught between a reductive and prevalent notion of truth and meaning and a peripheral and alienated otherness, is reexamined and reformulated. For Adorno, the instrumental rationality of Enlightenment itself reverts into a new form of mythology which is every bit as static and exclusive as the hierarchies of religion it supersedes. All is excluded that is non-identical to the instrumental mastery of the objects of the world, "For the Enlightenment, anything which cannot be resolved into numbers, and ultimately into one, is illusion; modern positivism consigns it to poetry." 1 As such the non-identical has been banished from the realm of thought and action, and banished specifically to the realm of art. But the realm of art is not able to provide a decent home for it because even modern art is not something entirely autonomous. While it is excluded from the notions of truth and meaning provided by instrumental rationality, it still originates from society. Just like instrumental rationality art is about mastery, structuring and exclusion. Therefore, the problem for art is that it cannot directly provide a space for the non-identical because its mode of operation is similar to that of instrumental rationality. Even worse, art has very few possibilities to free itself from this. If it becomes l’art pour l’art it is at the same time reduced to the very thing instrumental rationality claims it to be: a subjective judgment of taste.expressionism also stands as an example of this. Conversely, art cannot work within the framework of the identity-thinking of society, because any notion of freedom or emancipation would reproduce the exclusive practices of instrumental rationality. For example: Explicit socialist literature may establish itself in opposition to instrumental rationality while at the same time reproducing it. Its language may seem different, but in reality it merely provides a different set of reductive schematics for human life and experience. The problem for the art of the grotesque was, as I mentioned earlier, that far too often it was merely a symbolic manifestation of a notion of otherness—not otherness itself. Because of this the subversive character of the grotesque ends up shaping and containing otherness instead of providing a space for it. According to Adorno the only way to avoid this is to radicalize the grotesque. In order to avoid solidifying into form the work of art has to be an object which cannot be contained in thought or form—a non-form of form. The work of art has to work in a way that it strives for autonomy but without entirely leaving a discernible reality. In other words, it has to pull in two directions at once. Thus the work of art is a paradox. It is autonomous in the sense that it closes itself off from everything outside of the work of art, trying to shy away from the contaminating influence of the identity thinking of instrumental rationality. At the same time though the work of art is heteronomous in the sense that it originates from a specific historical context and is bound to be what society is not. Autonomy and heteronomy are inseparably intertwined. Unable to display otherness itself, but still trying to be a refuge for it, there is only one option for art: It has to turn against itself and destroy its own logic of form, thereby demonstrating how any representation of otherness is impossible. Art becomes a lamentation of the victims of the identity thinking of instrumental rationality and shows the traces of the otherness that is unable to appear. Adorno’s pessimistic theory of art and history stages art as a negative dialectical process where the work of art closes itself off and rejects everything outside, while the individual elements in the work of art destroys its very own logic of form from within: Artworks synthetize ununifiable, nonidentical elements that grind away at each other; they truly seek the identity of the identical and nonidentical processually because even their unity is only an element and not the magical formula of the whole […]. The resistance to them of otherness, on which they are nevertheless dependent compels them to articulate their own formal language, to leave not the smallest unformed particle as remnant. This reciprocity constitutes art’s dynamic; it is an irresolvable antithesis that is never brought to rest in the state of being. 2 Relative to a traditional understanding of the grotesque, Adorno radicalizes the tension between center and periphery, autonomy and heteronomy. In the traditional understanding of the grotesque, as we find it for example in the writings of Mikhail Bakhtin, the grotesque establishes the display of otherness as a new center of meaning and truth. However Adorno rejects such notions and argues instead for a modernism of the grotesque in which both center and periphery are included in a negative dialectical process and continually synthesizing and destabilizing each other’s positions. A modernism of the grotesque works on the inside of the sign or the artwork, continually outlining and questioning the boundaries between materiality and form. This moves the focus away from the relation between the sign or work of art as a conceptual manifestation on one side and the object it refers to on the other side. Instead it implicitly points to the amorphous mass from which the signs and thereby meaning have been carved, the remnants that have been left behind. Modernism of the grotesque does not imply that it is emancipatory or that it reconciles man with nature, and does not represent a new position or a new ideology. It merely remembers and insists on the double nature of man and art: to write or paint yourself or an artwork whole by inscribing meaning in life or by applying form on a canvas is at the same time both creation and destruction. The very signs we use to center meaning in our lives carry its own alienation in them. Meaning surfaces only when process and flow comes to a halt—solidifying, but at the same time excluding parts that do not fit into that particular structure of meaning. NOTHING DONE AND NOTHING UNDONE It has always been difficult for art historians to find a suitable category for the works of the Danish artist Leif Lage (b. 1933). In his paintings the sensitive figuration seems so fragile that it is on the verge of turning into abstraction. Or conversely: the figurative emerges cautiously and hesitantly out of abstraction. As an art critic once wrote, he can almost paint things away. 3 Working in some undefinable space between abstraction and figuration, the Lage’s works shy away from the realm of words, and as a result few critics have been able to write anything meaningful about his works. In fact even fewer have been able to write more than one or two pages. The Danish author and art critic Leif Hjernøe is one of those few, but even he starts off by saying how impossible it is to write about Lage, confessing that his works exist both as process and in a field of in-between which words cannot reach. He writes: Always you find yourself in a place where nothing is done and nothing left undone, and where the elements of the work of art rest in a continuous motion. Always there is the open wound, where infection rather than surgical intervention is considered as an opportunity. And the conflict between matter and antimatter makes all diagnoses uncertain and allows conditions of uncertainty to spread in a place of raw presence. 4 The art works against the diagnosis and the certainty. Words are being eaten up by what is outside the realm of words: tension, process and doubleness. An approach to an understanding of the work of Lage could be to follow his connection to Samuel Beckett (1906-89). One can note, therefore, that Lage has illustrated several of the Danish translations of Beckett’s works. Though one is an author and the other a painter both seek a place where figuration, where the word, erodes. For both this entails an attempt to question and examine and so move in behind the façade of the surface in an exploration of that something or nothing that may be behind representation. At the same time even in the vanishing point of figuration or words, the meaningless words or the disjointed lines still stand as signs of human activity—signs of movement in emptiness. With very few exceptions Lage’s artworks revolve around one motif: Man—most often “en face.” But unlike traditional portraitures Lage’s people are found within. The artworks do not sense the reality of appearances, but rather the existential dimensions within man. As a seismograph they are sensitive to the utmost subjectivity in order to display an objective, sensuous depiction of the very problem of subjectivity. Instead of a static façade of a man, Lage’s “en face” shows the process-character of man. The people in Lage’s paintings are always in process—in the midst of becoming or dissolving. I will take a closer look in the following at the ways in which Lage establishes this sense of being in constant process, of being in-between modes or categories. THE DOUBLENESS OF THE STROKE Lage’s etchings attract attention in particular for the tension conveyed there between the material and the form. The material physically opposes the violence of the needle. But instead of embracing the violence of the needle and the victory over matter—instead of making the artistic creation and figuration express a heroic mastery of the background it is made out of—Lage uses the stroke to express the very tension between stroke and material, figuration and background. In the drypoint etching Untitled from 1983, which illustrated the Danish translation of Beckett’s Ill Seen, Ill Said , the needle tears up the material in long rough lines: Together, the primarily horizontal and vertical lines form a face. Thus the force needed to etch a line in the metal-plate draws attention to itself: It is there in the absence of curves and nuances. Man emerges on the metal-plate after great exertion and in a very simplistic form. This way Lage enhances the nature of the material, as the material’s resistance is not something to overcome. On the contrary the figuration is smitten by its very resistance and as such expresses the violence of its own becoming. Even more pronounced is the fact that the lines do not just show the violence of creation—it actively erases figuration in the same process that creates it. For example the mouth consists of 8-10 horizontal strokes, of which some seem to be connected to each other as if the mouth has been quickly scratched out. The mouth speaks of the inability to speak. In the same way, the eyes are black holes due to a large amount of criss-cross cuts, which together form a dark middle. Those lines, at the same time, form the eyes of the figuration and scratch out the eyes of the figuration. The extroverted aspects of the face—connected to speech and sight—are portrayed as introverted as well. Just as the etchings down into the metal-plate makes something appear. Leif Lage, Untitled , 1983 (etching) The figuration of the etching is connected to creation. It expresses that something is wrought into existence out of nothing by sheer artistic force. However, this force is at the same time presented as a violence of creation, which makes it impossible for form to truly emerge. Herein lies one of the basic principles of Lage’s art, namely that art cannot be that nothing or something behind the realm of words or signs. Without form, art would lose its ability to express anything at all. Form is a necessity if art is to avoid becoming as devoid of expression as the reality is that we cannot reach behind the appearance of things. On the other hand, art cannot raise itself above its own material in joyous celebration of the beauty of form. This would use the material in the manner of an instrument and thus would suppress the otherness it attempts to rescue. When art becomes mere form, any traces of otherness will have dissolved. Then art would be a question of technique, in no way different from the instrumental rationality of society. Therefore art works in a particular space between form and non-form. As Hjernøe says of the works of Lage, it is in the center of the event, in the in-between where things neither are nor are not, "And everything takes place just before it changes character. As the blood just before it coagulates, as the milk just before it separates, as fried egg just before it turns white […]." 5 The precise emphasis on the in-between of Lage’s art points in particular to the work of art as both outside of and in time. The work of art is a singular point in time, but at the same time it is continually in process. In the borderland between form and non-form, between creation and destruction, it shimmers like a fata morgana . The work is done but at the same time undone. It is only there as a cry in response to the inability to come into being. Akin to the etching is the drawing, but while both, of course, use the stroke or line as their main instrument, the paper does not resist the line in the same sense as the metal-plate of the etching. As a result in Lage’s drawings the stroke frolics in a space with almost no tension at all. There is nothing there to keep the pencil stroke at bay, but at the same time this means that there is nothing there to keep it together. One of the more expressive examples of this is found in an untitled drawing from Tegninger og Text ( Drawings and Text ) from 1999. The drawing consists of one unbroken pencil stroke. It is as if the stroke is confused—it darts around on the paper making doodles on the way or gathers itself in more straight lines in the center of the paper. It is in this manner that the semblance of a body is established. A couple of small circles serve as eyes and are the reason that the rest can be read as both face and body instead of pure abstraction. In fact the unbroken stroke serves as both abstraction and figuration as on the one hand it is the stroke that seemingly outlines the shape of a figure, while on the other hand the unbroken stroke dissolves any truly recognizable figuration by continuously, without end or aim, overflowing the boundaries of the figure. Seen again here is a doubleness to the stroke and the act of creation. In the etching it was all about the tension between the resistance of the material which created the doubleness. Here it is the opposite. It is the lack of tension. The stroke is free to do anything, but has no aim: without some kind of resistance, there is nothing to hold figuration together. It threatens to dissolve into nothingness. Conversely, the drawing can also be seen as a process of becoming. Like a form of automatic writing, the pencil darts around until it finds a point of densification. But even in such an interpretation of the drawing, the figuration appears artificial—it points to itself as the result of the artistic act of creation. The stroke is primarily stroke and only secondarily figuration. With its wild movement it calls attention to itself as stroke. Thus the figuration never becomes something in its own right. It points to the basic principle of creation: Someone has drawn me, I am the result of artistic endeavors, and therefore I am not for myself but always overflowing the boundaries that contain me. It is the stroke that in the very same movement creates figuration and dissolves it. Leif Lage, Untitled , 1999 (drawing) THE LIGHTNESS OF COLOR, THE DENSITY OF PAPER Often the watercolor is used as a precursor for larger paintings, because it has an immediacy in which one can quickly paint the main lines of a composition. However, Lage’s watercolors are not temporary points in a process leading to another final painting. They are not sketches, later to be filled with details in another material. For him watercolor is an end in itself. This also implies that he doesn’t try to work against the paper’s absorption of the water color and its tendency to absorb detail. On the contrary, he examines the possibilities inherent in this blurring of the line and absorption of color. As in the previous examples, there are of course recognizable shapes in Lage’s water colors. But these shapes are rarely separable from color, line or even paper. In other words: the figurative and the abstract more or less become one. One of the reasons for this is that Lage often works on wetted paper which means that the paper hungrily absorbs color into its fabric instead of it being on top of the paper. The paper becomes line because it is the absorption of the color which, together with the brush stroke itself, separates the colors from each other. The line is also color and the color is also line, because there is no separation between the different colors other than the colors themselves. At times this results in almost complete abstraction. In such cases Lage add lines with a pencil as if he wants to hold some sort of figuration together. At other times it is the other way around: the strokes of the pencil are a being dissolved by the water color blurring out the figuration. Untitled 1 from 1999 displays various tones of green. The contour of the colors appears unusually rough. Most of all it is reminiscent of a shoreline which has found its form through the work of thousands of years between water and land. Pockets of resistance surface as islands not yet drowned in color—as paper or color not yet consumed by the dominant color. The shape of the colors does not seem created by the hand of the artist, but rather by the inner workings of the watercolor: color against color over time has resulted in this very image, as if it is all a part of a natural process of change and decay. Despite the fact that there are no strong differences in color there seems to be a struggle taking place in the watercolor. For example, inexplicable holes in the dark color, which cover most of the right side of the watercolor, appear as if the underlying color is eating away the dark color, or that the dark color is in the process of covering up the underlying color. The small pockets of resistance demonstrate a process and development that, almost as a happy accident, evokes a couple of small eyes and a large mouth. Behind the colors a few slanted lines can be discerned, but the hand of the artist has long ago disappeared behind the life of the colors. Thus the watercolor seems to be held at precisely the point in time where figuration randomly appears. The small eyes and wide mouth seemingly express the realization of the figuration’s temporality. It is just a moment in time, soon to be consumed again in the life of the colors. The nature, the background it stems from and is part of, moves on undeterred, continuing its everlasting process of creation and decay. In this sense, the watercolor provides an experience of how neither artist nor man is able to transcend the materiality they consist of. The colors of man, the materials man consists of are “becoming” and “progress” but also “decay” and “disappearance”. The artist’s pencil strokes have been eaten up long ago, and the figuration is living on borrowed time. Human or artistic activity is about creating form, transcending mere matter by shaping and applying meaning to it. But Untitled 1 shows the doubleness of such creation: the transience and fragility of human life. Should the figuration come into being, should it rise and transcend the material, which binds it to its temporal existence, it would become nothing because its very existence is conditioned by the life and work of the colors. In the same sense man cannot escape his own temporality. He too, for better or worse, must accept a life in the volatility of sensation. Leif Lage, Untitled 1 , 1999 (watercolor) In most of Lage’s watercolors there seem to be a stronger artistic control than in Untitled 1 . Most often both the arrangement of colors and the technique in which they are painted express the hand of the artist. An example of this is Untitled 2 from 1999. While figuration in Untitled 1 seemed to appear as a happy accident, the figuration is much more “created” in Untitled 2 . Here the most important parts of the figuration, the eyes, the mouth and part of the skull, are colored black and painted on top of the background’s play of colors. The background primarily consists of blue and yellow tones. Only parts of the paper are covered in color, however, which yet again stresses the act of painting: There is a surface which is being filled by color by an artist. The blue tones are concentrated in the figuration, while the yellow tones surround it. All this means that the figuration stands out more clearly in this watercolor. This is also underlined by the fact that the blue and dark tones cover part of the yellow tones as spots here and there. Thus it is not a background which is in the process of consuming figuration, but rather the emerging figuration which blots out the background. Such an understanding is even suggested by how the mouth and eyes are placed in the larger blue color field. The eyes and mouth are in the center of the watercolor, while the blue color field is placed from center out towards the left side. This gives the impression of a face turned slightly to the right and slightly upwards, which again makes the mouth appear to emerge from the paper. It also helps that the mouth cuts the center axis of the watercolor. In other words, the arrangement of the blue and black colors create some depth and perspective which gives the illusion of a figuration, seen partially in profile, emerging from the background. Seen in profile, the eyes and mouth seem extroverted—they almost seem as if they are put on top of the face. But at the same time they are painted as black holes, sucking in the gaze of the onlooker. The watercolor is even more challenging if one follows what may be the faint outline of the skull, which, on the right side of the watercolor, no longer works together with the blue color, but rather alone postulates the outline of the skull. If not the blue color field, but this line, is the outline of the skull, then all of a sudden the face is not in profile but “en face”—staring directly towards the onlooker. Outline and color seemingly work against each other and establish two different expressions. Without the slightly upward tilted expression of the profile, the eyes and mouth do not seem to emerge from the paper. On the contrary, seen “en face,” the eyes and mouth are really gaping pits of darkness. In this sense Untitled 2 is at the same time surface and depth of perspective. The line gives form but at the same time it dissolves the form created by the colors, and vice versa. Leif Lage, Untitled 2 , 1999 (watercolor) As the last example of Lage’s artistic method, I have chosen another untitled work (as indeed nearly all of Lage’s artworks are)—this time from 2000. This is perhaps one of the most figurative watercolors Lage has created, though that doesn’t tell us too much. The painting depicts a human face in blue and green tones on a yellow toned background. The colors create a clear demarcation between the figure and the background, which compensate for the watercolor’s lack of outline. Beneath the face darker tones imply the beginning of the torso, and nuances of color even hint at something which could be a collar. The hair is in even darker tones and its structure is emphasized using pencil strokes on top of the water color. One eye is marked by a brown spot with a darker spot in the middle, while the other eye is added by pencil. The mouth is a round, red dot. In Untitled from 2000 the work between abstraction and figuration is much less pronounced than in the previous examples cited here. Nonetheless the watercolor has a delicacy to it which almost makes the figuration disappear, even as it appears. This has something to do with the fact that all the individual brush strokes have dissolved. The only outlines are those created by the clash of colors. Even the red mouth seems to be more paint than mouth—unable to be formed truly as anything other than a speck of color with some small semblance to a mouth. In contrast to the other watercolors, there is no movement, no progress or decay, just a vibrating standstill. The figure is not emerging or disappearing but embedded in and as surface, and as such it makes the figure seem stuck in its condition rather than freed by the act of giving form: mouth open as a wound. Pencil strokes add detail and life to the hair as an attempt to help the figure emerge. But all this does is create two levels in the watercolor: The pencil strokes are put on top of the figuration rather than being part of it. Thus they end up highlighting the endless distance between the movement and form-giving of the pencil strokes and the static and stuck figure behind these strokes. Leif Lage, Untitled , 2000 (watercolor) MAN AND THE VIEW OF THE ONLOOKER Lage’s works seem at the same time to be an examination of tensions between form and material in art and a display of an image of man full of the same tensions. This is because the people who are emerging in Lage’s images are not just in his works, They are his works. This means that man is not something emerging from a background but is part of this background and vice versa. The art of Lage inscribes man in its surroundings and background. Man is not the ruler of nature. Man does not rise above matter; there is no truly transcendent position for man to occupy in the art of Lage. Instead man is a part of the very matter, it, at the same time, tries to distance itself from in order to distinguish itself from its surroundings. Man emerges in an attempt not to be line, stroke, color. In an attempt to free itself of matter. Should it succeed in this, however, it would turn into nothing. On the other hand, if man fully accepted itself as mere matter it would disappear in it—in instincts, urges and lack of reflection. The appearance of man is thus subject to a state of being in-between. Only in the tension between material and its negation can the outlines of man be traced. However this also means that in the art of Lage man is never something in itself but always in process—on the way to its freedom and its loss. However, Lage’s art is not just about showing the tensions inherent in man but also about showing such tension on the verge of breaking point. In the moment before everything is done, or when nothing yet has been done, the delicate tension between form and material has been stretched to its utmost in such a way that man’s character of process emerges and is contained on the paper or canvas. Therefore, Lage’s art contain traces of human life. Not only is it an autonomous unity of tensions where man is stretched out between its singularity and the surroundings it is a part of. It is also this pulsation between figuration and material which brings about the traces of human endeavor and activity. Even when man almost seems to have disappeared into the material, strokes and lines are still there as a reminder of human activity and potential creative power—as a reminder that hope is still possible and that man is still there. The view of the onlooker seeks out forms and meaning. It seeks outlines and edges. But first and foremost it is the eyes in Lage’s works of art that allow the onlooker to decode the figuration at play. This creates a distinctive relationship between the onlooker and the artwork. The onlooker tries to wrest the enigma from the artwork, tries to reduce it to solid, meaningful form. But the very same moment that hints at such a possibility sees the view of the onlooker confronted with a gaze of its own. The eyes of the figure in the work of art are most often dark pits, as if they were an illustration of the violence committed against the eyes of the onlookers and their search for meaning. Or as if they were a dark reflection of the onlooker’s failed attempts to delineate the traces of life in the artwork to a solid meaningful form. Therefore, the gaze of the figuration speaks not only of its own powerlessness but also of the powerlessness of the onlooker. When the view or sight has trouble finding solid meaningful forms, it has something to do with the fact that the precondition for sight, light, no longer stems from something transcendent. In Lage’s art there is nothing outside the works of art, nothing outside of human life, and as such nothing to shed light on the works of art or human life and reveal solid meaningful forms. All light comes from within. In a western context, light as a medium for sight is central to the understanding of meaning. People speak of the clarity of thought, while knowledge is tied together with enlightenment, and so forth. But the light no longer originates from a divine point of view. It originates from man, who has created himself in the image of God. The problem with this is that light always covers up its own base. Light illuminates something but at the same time it hides itself. 6 While it was previously God who was unknowable, now it is man. It is also problematic that sight and the medium of sight stem from the same place. Following this thought one can posit two final problems. Firstly, the immediacy and presence established by light by making forms and objects accessible to sight is fake. Light is not some independent entity which present things in a neutral manner for the onlooker. It is inextricably tied together with the gaze’s desire after meaning. In other words: Light perverts the appearance of things because it doesn’t originate from some objective entity which allows them to appear as they are. Instead light is based upon and originates from the subject and its conquering gaze, seeking out meaning and form constantly. Secondly, the joining of sight and light makes introspection within human life impossible. Everything can be illuminated by its use value for the subject but only as long as it is separated from the subject, as long as it is an object. But the subject itself cannot be illuminated. Lage’s art tears down such a dividing line between subject and matter, but it does so without reestablishing a divine position from where the light emanates. Therefore, the light in Lage’s art may not illuminate very well. It may not have the certainty presented by modern science or the religions of yesterday. But it does show what such light tries to hide, namely, the subject as it is in its material reality—inscribed in time and death, which are the basic conditions of materiality, but at the same time also full of life in its joy and will to create. For the onlooker the meeting of the gaze in Lage’s works of art is thus also a destruction of the view of the onlooker. The ability to see is challenged in a radical way. Those outlines will not solidify into static form, as if the light was too dim to see. Just as in Lage’s figurations, the view of the onlooker turns to black: turns towards the onlooker in recognition and becoming—or disappearance. NOTES Max Horkheimer and Theodor W. Adorno. Dialectic of Enlightenment . (Palo Alto: Stanford University Press, 2002): 4-5. Theodor W. Adorno. Aesthetic Theory . (London & New York, 2002): 176. Carsten Bach-Nielsen. “Mere glæde end gru.” Kristeligt Dagblad . September 24, 2002. Leif Hjernøe. “Den yderliggående inderlighed.” LEIF LAGE—Retrospektiv udstilling 1983 . (Copenhagen: Brøndum, 1983): 18. Ibid. Maurice Blanchot. The Infinite Conversation . (Minneapolis: University of Minnesota Press, 1993): 162-163. (shrink)

In this article, we will present a number of technical results concerning Classical Logic, ST and related systems. Our main contribution consists in offering a novel identity criterion for logics in general and, therefore, for Classical Logic. In particular, we will firstly generalize the ST phenomenon, thereby obtaining a recursively defined hierarchy of strict-tolerant systems. Secondly, we will prove that the logics in this hierarchy are progressively more classical, although not entirely classical. We will claim that a logic is to (...) be identified with an infinite sequence of consequence relations holding between increasingly complex relata: formulae, inferences, metainferences, and so on. As a result, the present proposal allows not only to differentiate Classical Logic from ST, but also from other systems sharing with it their valid metainferences. Finally, we show how these results have interesting consequences for some topics in the philosophical logic literature, among them for the debate around Logical Pluralism. The reason being that the discussion concerning this topic is usually carried out employing a rivalry criterion for logics that will need to be modified in light of the present investigation, according to which two logics can be non-identical even if they share the same valid inferences. (shrink)

The question whether Frege’s theory of indirect reference enforces an infinite hierarchy of senses has been hotly debated in the secondary literature. Perhaps the most influential treatment of the issue is that of Burge (1979), who offers an argument for the hierarchy from rather minimal Fregean assumptions. I argue that this argument, endorsed by many, does not itself enforce an infinite hierarchy of senses. I conclude that whether or not the theory of indirect reference can avail itself of (...) only finitely many senses is pending further theoretical development. (shrink)

In this study we use a computational model of language learning called model of syntax acquisition in children (MOSAIC) to investigate the extent to which the optional infinitive (OI) phenomenon in Dutch and English can be explained in terms of a resource-limited distributional analysis of Dutch and English child-directed speech. The results show that the same version of MOSAIC is able to simulate changes in the pattern of finiteness marking in 2 children learning Dutch and 2 children learning English as (...) the average length of their utterances increases. These results suggest that it is possible to explain the key features of the OI phenomenon in both Dutch and English in terms of the interaction between an utterance-final bias in learning and the distributional characteristics of child-directed speech in the 2 languages. They also show how computational modeling techniques can be used to investigate the extent to which cross-linguistic similarities in the developmental data can be explained in terms of common processing constraints as opposed to innate knowledge of universal grammar. (shrink)

This paper argues against the claim that Frege is committed to an infinite hierarchy of senses. Carnap and Kripke, along with many others, argue the contrary; I expose where all such arguments go astray. Invariably these arguments assume (without citation) that Frege holds that sense and reference are always distinct. This is the fulcrum upon which the hierarchy is hoisted. The counter to this assumption is based on two important but neglected passages. The locution ‘indirect sense’ has no ontological (...) significance for Frege. It was just a ‘short expression’ (his term) to indicate that the customary relation between sign, sense, and reference is suspended in what he called the exception case of indirect context. Once the hierarchy is abandoned, the alleged problem of just what indirect sense is, as well as questions as to why Frege said so little about it and why he did not examine iterative indirect contexts are resolved. Of course, with no hierarchy, Davidson’s ‘unlearnability argument’ turns out to be a non-starter. Some objections to my interpretation are considered and answered. Additionally, comparisons to Dummett’s one-level view of sense, which is similar to mine are made, as well as to some two-level views (Parsons and Skiba). (shrink)

This paper argues against the claim that Frege is committed to an infinite hierarchy of senses. Carnap and Kripke, along with many others, argue the contrary; I expose where all such arguments go astray. Invariably these arguments assume (without citation) that Frege holds that sense and reference are always distinct. This is the fulcrum upon which the hierarchy is hoisted. The counter to this assumption is based on two important but neglected passages. The locution ‘indirect sense’ has no ontological (...) significance for Frege. It was just a ‘short expression’ (his term) to indicate that the customary relation between sign, sense, and reference is suspended in what he called the exception case of indirect context. Once the hierarchy is abandoned, the alleged problem of just what indirect sense is, as well as questions as to why Frege said so little about it and why he did not examine iterative indirect contexts are resolved. Of course, with no hierarchy, Davidson’s “unlearnability argument” turns out to be a non-starter. Some objections to my interpretation are considered and answered. Additionally, comparisons to Dummett’s one-level view of sense, which is similar to mine are made, as well as to some two-level views (Parsons and Skiba). (shrink)

This paper discusses the theoretical relationship between the views of Damascius and those of Pseudo-Dionysius the Areopagite. While Damascius’ De principiis is a bold treatise devoted to investigating the hypermetaphysics of apophatism, it anticipates various theoretical positions put forward by Dionysius the Areopagite. The present paper focuses on the following. First, Damascius is the only ancient philoso­pher who systematically demonstrates the first principle to be infinite. Second, Damascius modifies the concept and in several important passages shows the infinite (...) to be superior and prior to the finite. Third, Damascius’ theory of being is the strongest ancient articulation of the nature of the One which is a clear prefiguration of the negative theology developed by Dionysius the Areopagite. (shrink)

Let CH be the class of compacta (i.e., compact Hausdorff spaces), with BS the subclass of Boolean spaces. For each ordinal α and pair $\langle K,L\rangle$ of subclasses of CH, we define Lev ≥α K,L), the class of maps of level at least α from spaces in K to spaces in L, in such a way that, for finite α, Lev ≥α (BS,BS) consists of the Stone duals of Boolean lattice embeddings that preserve all prenex first-order formulas of quantifier rank (...) α. Maps of level ≥ 0 are just the continuous surjections, and the maps of level ≥ 1 are the co-existential maps introduced in [8]. Co-elementary maps are of level ≥α for all ordinals α; of course in the Boolean context, the co-elementary maps coincide with the maps of level ≥ω. The results of this paper include: (i) every map of level ≥ω is co-elementary; (ii) the limit maps of an ω-indexed inverse system of maps of level ≥α are also of level ≥α; and (iii) if K is a co-elementary class, k ≥ k (K,K) = Lev ≥ k+1 (K,K), then Lev ≥ k (K,K) = Lev ≥ω (K,K). A space X ∈ K is co-existentially closed in K if Lev ≥ 0 (K, X) = Lev ≥ 1 (K, X). Adapting the technique of "adding roots," by which one builds algebraically closed extensions of fields (and, more generally, existentially closed extensions of models of universal-existential theories), we showed in [8] that every infinite member of a co-inductive co-elementary class (such as CH itself, BS, or the class CON of continua) is a continuous image of a space of the same weight that is co-existentially closed in that class. We show here that every compactum that is co-existentially closed in CON (a co-existentially closed continuum) is both indecomposable and of covering dimension one. (shrink)

We consider ω n -automatic structures which are relational structures whose domain and relations are accepted by automata reading ordinal words of length ω n for some integer n ≥ 1. We show that all these structures are ω-tree-automatic structures presentable by Muller or Rabin tree automata. We prove that the isomorphism relation for ω 2 -automatic (resp. ω n -automatic for n > 2) boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups) is not (...) determined by the axiomatic system ZFC. We infer from the proof of the above result that the isomorphism problem for ω n -automatic boolean algebras, n ≥ 2, (respectively, rings, commutative rings, non commutative rings, non commutative groups) is neither a ${\mathrm{\Sigma }}_{2}^{1}-\mathrm{s}\mathrm{e}\mathrm{t}$ nor a ${\mathrm{\Pi }}_{2}^{1}-\mathrm{s}\mathrm{e}\mathrm{t}$ . We obtain that there exist infinitely many ω n -automatic, hence also ω-tree-automatic, atomless boolean algebras $\mathcal{B}_{n}$ , n ≥ 1, which are pairwise isomorphic under the continuum hypothesis CH and pairwise non isomorphic under an alternate axiom AT, strengthening a result of [14]. (shrink)

Suppose q is some proposition, and let P(q) = v0 (1) be the proposition that the probability of q is v0.1 How can one know that (1) is true? One cannot know it for sure, for all that may be asserted is a further probabilistic statement like P(P(q) = v0) = v1, (2) which states that the probability that (1) is true is v1. But the claim (2) is also subject to some further statement of an even higher probability: P(P(P(q) (...) = v0) = v1) = v2, (3) and so on. Thus, an infinite regress emerges of probabilities of probabilities, and the question arises as to whether this regress is vicious or harmless. Radical probabilists would like to claim that it is harmless, but Nicholas Rescher (2010), in his scholarly and very stimulating Infinite Regress: The Theory and History of Varieties of Change, argues that it is vicious. He believes that an infinite hierarchy of probabilities makes it impossible to know anything about the probability of the original proposition q: unless some claims are going to be categorically validated and not just adjudged probabilistically, the radically probabilistic epistemology envisioned here is going to be beyond the prospect of implementation. . . . If you can indeed be certain of nothing, then how can you be sure of your probability assessments. If all you ever have is a nonterminatingly regressive claim of the format . . . the probability is .9 that (the probability is .9 that (the probability of q is .9)) then in the face of such a regress, you would know effectively nothing about the condition of q. After all, without a categorically established factual basis of some sort, there is no way of assessing probabilities. But if these requisites themselves are never categorical but only probabilistic, then we are propelled into a vitiating regress of presuppositions. (shrink)

In set theory without the axiom of choice (AC), we study the relative strength of the principle “No decreasing sequence of cardinals,” that is, “There is no function f on ω such that |f(n+1)|<|f(n)| for all n∈ω” (NDS) with regard to its position in the hierarchy of weak choice principles. We establish the following results: (1) The Boolean prime ideal theorem plus countable choice does not imply NDS in ZF; (2) “Every non-well-orderable set has a well-orderable partition into denumerable sets” (...) (Dℵ0) plus the axiom of choice for well-ordered families of nonempty finite sets does not imply NDS ∨ “The axiom of choice for countable families of nonempty countable sets” in ZFA; and (3) The axiom of choice for well-ordered families of nonempty sets, each of cardinality not greater than or equal to 2ℵ0, does not imply NDS ∨ “countable union theorem” in ZFA. The above results answer corresponding questions left open in works by Howard and Rubin (1998) and Howard and Tachtsis (2016). We also show the following: (4) “Every non-well-orderable set has a well-orderable partition into infinite well-orderable sets” (Dw) is strictly weaker than D ℵ0 in ZFA. This resolves an open problem from Keremedis and Tachtsis (2003). (shrink)

Supererogatory acts are those that lie “beyond the call of duty.” There are two standard ways to define this idea more precisely. Although the definitions are often seen as equivalent, I argue that they can diverge when options are infinite, or when there are cycles of better options; moreover, each definition is acceptable in only one case. I consider two ways out of this dilemma.

Some philosophers have claimed that it is meaningless or paradoxical to consider the probability of a probability. Others have however argued that second-order probabilities do not pose any particular problem. We side with the latter group. On condition that the relevant distinctions are taken into account, second-order probabilities can be shown to be perfectly consistent. May the same be said of an infinite hierarchy of higher-order probabilities? Is it consistent to speak of a probability of a probability, and of (...) a probability of a probability of a probability, and so on, {\em ad infinitum}? We argue that it is, for it can be shown that there exists an infinite system of probabilities that has a model. In particular, we define a regress of higher-order probabilities that leads to a convergent series which determines an infinite-order probability value. We demonstrate the consistency of the regress by constructing a model based on coin-making machines. (shrink)

It is known that infinitely many Medvedev degrees exist inside the Muchnik degree of any nontrivial Π10 subset of Cantor space. We shed light on the fine structures inside these Muchnik degrees related to learnability and piecewise computability. As for nonempty Π10 subsets of Cantor space, we show the existence of a finite-Δ20-piecewise degree containing infinitely many finite-2-piecewise degrees, and a finite-2-piecewise degree containing infinitely many finite-Δ20-piecewise degrees 2 denotes the difference of two Πn0 sets), whereas the greatest degrees in (...) these three “finite-Γ-piecewise” degree structures coincide. Moreover, as for nonempty Π10 subsets of Cantor space, we also show that every nonzero finite-2-piecewise degree includes infinitely many Medvedev degrees, every nonzero countable-Δ20-piecewise degree includes infinitely many finite-piecewise degrees, every nonzero finite-2-countable-Δ20-piecewise degree includes infinitely many countable-Δ20-piecewise degrees, and every nonzero Muchnik degree includes infinitely many finite-2-countable-Δ20-piecewise degrees. Indeed, we show that any nonzero Medvedev degree and nonzero countable-Δ20-piecewise degree of a nonempty Π10 subset of Cantor space have the strong anticupping properties. Finally, we obtain an elementary difference between the Medvedev degree structure and the finite-Γ-piecewise degree structure of all subsets of Baire space by showing that none of the finite-Γ-piecewise structures is Brouwerian, where Γ is any of the Wadge classes mentioned above. (shrink)

We extend the hierarchy of finite-dimensional Ellentuck spaces to infinite dimensions. Using uniform barriers [Formula: see text] on [Formula: see text] as the prototype structures, we construct a class of continuum many topological Ramsey spaces [Formula: see text] which are Ellentuck-like in nature, and form a linearly ordered hierarchy under projections. We prove new Ramsey-classification theorems for equivalence relations on fronts, and hence also on barriers, on the spaces [Formula: see text], extending the Pudlák–Rödl theorem for barriers on the (...) Ellentuck space. The inspiration for these spaces comes from continuing the iterative construction of the forcings [Formula: see text] to the countable transfinite. The [Formula: see text]-closed partial order [Formula: see text] is forcing equivalent to [Formula: see text], which forces a non-p-point ultrafilter [Formula: see text]. This work forms the basis for further work classifying the Rudin–Keisler and Tukey structures for the hierarchy of the generic ultrafilters [Formula: see text]. (shrink)

This collection of papers in epistemic logic is oriented towards applications to game theory and individual decision theory. Most of these papers were presented at the inaugural conference of the LOFT (Logic for the Theory and Games and Decisions) conference series, which took place in 1994 in Marseille. Among the notions dealt with are those of common knowledge and common belief, infinite hierarchies of beliefs and belief spaces, logical omniscience, positive and negative introspection, backward induction and rationalizable equilibria (...) in game theory. (shrink)

In the context of [Formula: see text], we analyze a version of Hindman’s finite unions theorem on infinite sets, which normally requires the Axiom of Choice to be proved. We establish the implication relations between this statement and various classical weak choice principles, thus precisely locating the strength of the statement as a weak form of the [Formula: see text].

Le mode infini immédiat de l’étendue (MIIE) est l’un des éléments les plus mystérieux de l’ontologie de Spinoza. Malgré son importance pour le système métaphysique de Spinoza, ce dernier nous dit très peu à propos de ce mode. Dans un effort pour faire progresser l’étude de cette question, j’examine trois hypothèses bien acceptées qui traitent de l’identité de ce mode : l’interprétation de la force, l’interprétation nomique et l’interprétation cinétique. J’affirme premièrement que l’interprétation de la force et l’interprétation nomique doivent (...) faire face à de sérieuses objections. Deuxièmement, je soutiens que malgré les affirmations contraires des spécialistes, l’interprétation cinétique du MIIE est une option valable. (shrink)

Discussions of supererogation usually focus on cases in which the agent can choose among a finite number of options. However, Daniel Muñoz has recently shown that cases in which the agent faces an infinite chain of increasingly less good options make trouble for existing definitions of supererogation. Muñoz proposes a promising new definition as a solution to the problem of infinite cases. I argue that any acceptable account of supererogation must (1) enable us to accurately identify (...) supererogatory acts in both finite and infinite option cases. It must also (2) include a suitably related account of what makes one act more supererogatory than another for finite, infinite, single-choice (one agent choosing among several supererogatory options) and inter-choice (two different agents, each choosing a supererogatory option) cases. I further argue that the best current account of supererogation for finite cases works well for finite cases, but cannot handle infinite cases. However, Muñoz’s proposal cannot handle inter-choice cases in either finite or infinite cases. I conclude we still need an account for infinite cases, and may have to settle for separate definitions of supererogation for finite and infinite cases. (shrink)

We tell a story where an agent who chooses in such a way as to make the greatest possible profit on each of an infinite series of transactions ends up worse off than an agent who chooses in such a way as to make the least possible profit on each transaction. That is, contrary to what one might suppose, it is not necessarily rational always to choose the option that yields the greatest possible profit on each transaction.

The general notions of object- and metalanguage are discussed and as a special case of this relation an arbitrary first order language with an infinite model is expanded by a predicate symbol T0 which is interpreted as truth predicate for . Then the expanded language is again augmented by a new truth predicate T1 for the whole language plus T0. This process is iterated into the transfinite to obtain the Tarskian hierarchy of languages. It is shown that there are (...) natural points for stopping this process. The sets which become definable in suitable hierarchies are investigated, so that the relevance of the Tarskian hierarchy to some subjects of philosophy of mathematics are clarified. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Joseph Margolis THE OPTIONS OF CONTEMPORARY ETHICAL THEORY It may be said, with some prospect ofbeing not altogether idiotic, that the global philosophical question ofour age concerns the possibility of legitimating the conceptual grounds for legitimating claims about anything. The formulation has no interest in the abstract. It merely registers the possibility of an infinite regress; and in that form it has been with us forever. But (...) our own world has inexorably subverted all the standard forms of confidence regarding legitimation—not so much perhaps by intellectual confrontation as, more indirectly, by coming to believe the deep contingencies of human history, by discovering again and again the astounding diversity of actual human societies, by reflecting on the incompletely managed self-manipulation of our immense technology, by the natural fear entailed in the growing chaos of the entire terrestrial world we mean to put into serviceable order. These considerations are certainly local to moral theorizing. But since they have a general currency as well, they lead us to suppose that speculations about legitimizing moral claims cannot be more satisfactory than those of philosophy in general, though they may be weaker. This, in its turn, encourages us to imagine that a reduced assessment of the larger claims of philosophy may conveniendy justify a greater respect for the inherendy modest prospects ofmoral theorizing. Furthermore, the characteristic triple thinking of our age, always incipient in any reflexive appraisal of cognitive powers, has been raised to a particularly fine art — if that is the right term. It is quite impossible nowadays, unless by a confidence or innocence bordering on the Martian, simply to proclaim the principles or grounds or foundations of moral judgment and moral commitment. Nevertheless, what one cannot help noticing is that wher37 38Philosophy and Literature ever theorists are morally serious — and they seem never more serious than when they are theorizing about morality— they are understandably inclined to believe that they absolutely must give some sense of the accessibility of reliable and valid grounds for judgments and commitments of the most important sorts. In yielding to that touching obligation to serve others in the capacity of philosopher, however, they usually yield as well in the direction ofinsinuating— ifnot openly announcing— what their own sense of professional history would otherwise have readily shown to be in desperate need of a smidgen of support. Notably in the last generation, the retreat from older forms of confidence, in moral philosophy, in philosophy in general, and in all those intellectual disciplines either pleased or displeased to be linked to the fate of philosophy, has decidedly accelerated. There is no resisting the current, though that is hardly to say we understand its implications or are obliged to subscribe to its madder exemplars. But when, to take a counter-specimen from the Anglo-American tradition of the last decade, a moral philosopher like Alan Donagan attempts to restore something like the Kantian moral system, we sense its unlikely and alien body even before we prepare our instruments for the required dissection. Thus, Donagan, who can hardly be accused of failing to sustain a most humane document, speaks at the beginning of The Theory ofMorality of his having been persuaded — by George Orwell and others — "that any acceptable form of social order, whether socialist or not, must rest on moral foundations, which are in principle ascertainable at any period, and permanendy valid."1 Donagan had hoped to recover in a compelling way the "common morality" of mankind, the rationally accessible portion of the Biblical tradition, presumably congruent with the classical tradition, ordered in a conceptually convincing way for the first time by Immanuel Kant reflecting, apparently, on his own Pietist upbringing. Donagan is, of course, among other things, a careful student of R. G. Collingwood's work, and here may be thought to exhibit Collingwood's confidence that the historian is capable, under favorable conditions, of "reenacting" at any period the motivating thought of effective historical agents of the past.2 At any rate, Donagan is quite explicit that: (a) social orders must have moral foundations or first principles; (b) social orders do have such moral foundations; (c) those foundations are ascertainable; (d) they are ascertainable in principle at... (shrink)

We introduce a new axiom called inductive dichotomy, a weak variant of the axiom of inductive definition, and analyze the relationships with other variants of inductive definition and with related axioms, in the general second order framework, including second order arithmetic, second order set theory and higher order arithmetic. By applying these results to the investigations on the determinacy axioms, we show the following. (i) Clopen determinacy is consistency-wise strictly weaker than open determinacy in these frameworks, except second order arithmetic; (...) this is an enhancement of Schweber–Hachtman separation of open and clopen determinacy into the consistency-wise separation. (ii) Hausdorff–Kuratowski hierarchy of differences of opens is faithfully reflected by the hierarchy of consistency strengths of corresponding parameter-free determinacies in the aforementioned frameworks; this result is valid also in second order arithmetic only except clopen determinacy. (shrink)

Freivalds defined an acceptable programming system independent criterion for learning programs for functions in which the final programs were required to be both correct and "nearly" minimal size, i.e., within a computable function of being purely minimal size. Kinber showed that this parsimony requirement on final programs limits learning power. However, in scientific inference, parsimony is considered highly desirable. A lim-computablefunction is (by definition) one calculable by a total procedure allowed to change its mind finitely many times about its output. (...) Investigated is the possibility of assuaging somewhat the limitation on learning power resulting from requiring parsimonious final programs by use of criteria which require the final, correct programs to be "not-so-nearly" minimal size, e.g., to be within a lim-computable function of actual minimal size. It is shown that some parsimony in the final program is thereby retained, yet learning power strictly increases. Considered, then, are lim-computable functions as above but for which notations for constructive ordinals are used to bound the number of mind changes allowed regarding the output. This is a variant of an idea introduced by Freivalds and Smith. For this ordinal notation complexity bounded version of lim-computability, the power of the resultant learning criteria form finely graded, infinitely ramifying, infinite hierarchies intermediate between the computable and the lim-computable cases. Some of these hierarchies, for the natural notations determining them, are shown to be optimally tight. (shrink)

This essay examines the problem of evil, and then develops a free will theodicy. Then the paper considers some themes in distinctively Christian theodicy building, in more detail.

In [7], open questions are raised regarding the computational strengths of so-called ∞-α -Turing machines, a family of models of computation resembling the infinite-time Turing machine model of [2], except with α -length tape . Let TαTα denote the machine model of tape length α . Define that TαTα is computationally stronger than TβTβ precisely when TαTα can compute all TβTβ-computable functions ƒ: min2→min2 plus more. The following results are found: Tω1≻TωTω1≻Tω. There are countable ordinals α such that Tα≻TωTα≻Tω, (...) the smallest of which is precisely γ , the supremum of ordinals clockable by TωTω. In fact, there is a hierarchy of countable TαTαs of increasing strength corresponding to the transfinite Turing-jump operator ▼. There is a countable ordinal μ such that neither Tμ⪰Tω1Tμ⪰Tω1 nor Tμ⪯Tω1Tμ⪯Tω1—that is, the models TμTμ and Tω1Tω1 are computation-strength incommensurable . A similar fact holds for any larger uncountable device replacing Tω1Tω1. Further observations are made about countable TαTα. (shrink)

The paper presents an infinite hierarchy of sound and complete axiomatic systems for Two-Dimensional Modal Tense Logic with Historical Necessity, Agents and Acts. A main novelty of these logics is their capacity to represent formally (i) basic action-sentences asserting that such and such an act is performed/omitted by an agent, as well as (ii) causative action-sentences asserting that by performing/omitting a certain act, an agent causes that such and such a state-of-affairs is realized (e.g. comes about/ceases/remains/remains absent). We illustrate (...) how the formal machinery of our systems can be used to reconstruct a number of interesting ideas in the Logic of Agency and Action that have been proposed by authors like von Wright, von Kutschera, Belnap and Segerberg. (shrink)

Where there are infinitely many possible [equiprobable] basic states of the world, a standard probability function must assign zero probability to each state—since any finite probability would sum to over one. This generates problems for any decision theory that appeals to expected utility or related notions. For it leads to the view that a situation in which one wins a million dollars if any of a thousand of the equally probable states is realized has an expected value of zero (since (...) each such state has probability zero). But such a situation dominates the situation in which one wins nothing no matter what (which also has an expected value of zero), and so surely is more desirable. I formulate and defend some principles for evaluating options where standard probability functions cannot strictly represent probability—and in particular for where there is an infinitely spread, uniform distribution of probability. The principles appeal to standard probability functions, but overcome at least some of their limitations in such cases. (shrink)

Let a be a Kleene's ordinal notation of a nonzero computable ordinal. We give a sufficient condition on a, so that for every \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Sigma ^{-1}_a$\end{document}‐computable family of two embedded sets, i.e., two sets A, B, with A properly contained in B, the Rogers semilattice of the family is infinite. This condition is satisfied by every notation of ω; moreover every nonzero computable ordinal that is not sum of any two smaller ordinals has a notation that satisfies this condition. On (...) the other hand, we also give a sufficient condition on a, that yields that there is a \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Sigma ^{-1}_a$\end{document}‐computable family of two embedded sets, whose Rogers semilattice consists of exactly one element; this condition is satisfied by all notations of every successor ordinal bigger than 1, and by all notations of the ordinal ω + ω; moreover every computable ordinal that is sum of two smaller ordinals has a notation that satisfies this condition. We also show that for every nonzero n ∈ ω, or n = ω, and every notation a of a nonzero ordinal there exists a \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Sigma ^{-1}_a$\end{document}‐computable family of cardinality n, whose Rogers semilattice consists of exactly one element. (shrink)

I consider the natural infinitary variations of the games Wordle and Mastermind, as well as their game‐theoretic variations Absurdle and Madstermind, considering these games with infinitely long words and infinite color sequences and allowing transfinite game play. For each game, a secret codeword is hidden, which the codebreaker attempts to discover by making a series of guesses and receiving feedback as to their accuracy. In Wordle with words of any size from a finite alphabet of n letters, including (...) class='Hi'>infinite words or even uncountable words, the codebreaker can nevertheless always win in n steps. Meanwhile, the mastermind number, defined as the smallest winning set of guesses in infinite Mastermind for sequences of length ω over a countable set of colors without duplication, is uncountable, but the exact value turns out to be independent of, for it is provably equal to the eventually different number, which is the same as the covering number of the meager ideal. I thus place all the various mastermind numbers, defined for the natural variations of the game, into the hierarchy of cardinal characteristics of the continuum. (shrink)

In intuitionistic analysis, a subset of a Polish space like or is called positively Borel if and only if it is an open subset of the space or a closed subset of the space or the result of forming either the countable union or the countable intersection of an infinite sequence of (earlier constructed) positively Borel subsets of the space. The operation of taking the complement is absent from this inductive definition, and, in fact, the complement of a positively (...) Borel set is not always positively Borel itself (see Veldman, 2008a). The main result of Veldman (2008a) is that, assuming Brouwer's Continuity Principle and an Axiom of Countable Choice, one may prove that the hierarchy formed by the positively Borel sets is genuinely growing: every level of the hierarchy contains sets that do not occur at any lower level. The purpose of the present paper is a different one: we want to explore the truly remarkable fine structure of the hierarchy. Brouwer's Continuity Principle again is our main tool. A second axiom proposed by Brouwer, his Thesis on Bars is also used, but only incidentally. (shrink)

According to Frege, expressions shift their reference when they occur in indirect contexts: in “Anna believes that Plato is wise” the expression “Plato” no longer refers to Plato but to what is ordinarily its sense. Many philosophers, including Carnap, Davidson, Burge, Parsons, Kripke and Künne, believe that on Frege's view the iteration of indirect context creating operators gives rise to an infinite hierarchy of senses. While the former two take this to be problematic, the latter four welcome the hierarchy (...) with open arms. In this paper I argue that the hierarchy should be avoided and that it can be avoided. (shrink)

The problem of how to rationally aggregate probability measures occurs in particular when a group of agents, each holding probabilistic beliefs, needs to rationalise a collective decision on the basis of a single ‘aggregate belief system’ and when an individual whose belief system is compatible with several probability measures wishes to evaluate her options on the basis of a single aggregate prior via classical expected utility theory. We investigate this problem by first recalling some negative results from preference and (...) judgment aggregation theory which show that the aggregate of several probability measures should not be conceived as the probability measure induced by the aggregate of the corresponding expected utility preferences. We describe how McConway’s :410–414, 1981) theory of probabilistic opinion pooling can be generalised to cover the case of the aggregation of infinite profiles of finitely additive probability measures, too; we prove the existence of aggregation functionals satisfying responsiveness axioms à la McConway plus additional desiderata even for infinite electorates. On the basis of the theory of propositional-attitude aggregation, we argue that this is the most natural aggregation theory for probability measures. Our aggregation functionals for the case of infinite electorates are neither oligarchic nor integral-based and satisfy a weak anonymity condition. The delicate set-theoretic status of integral-based aggregation functionals for infinite electorates is discussed. (shrink)

Scientific progress in the 20th century has shown that the structure of the world is hierarchical. A philosophical analysis of the hierarchy will bear obvious significance for metaphysics and philosophy in general. Jonathan Schaffer’s paper, “Is There a Fundamental Level?”, provides a systematic review of the works in the field, the difficulties for various versions of fundamentalism, and the prospect for the third option, i.e., to treat each level as ontologically equal. The purpose of this paper is to provide an (...) argument for the third option. The author will apply Aristotle’s theory of matter and form to the discussion of the hierarchy and develop a theory of form realism, which will grant every level with “full citizenship in the republic of being.” It constitutes an argument against ontological and epistemological reductionism. A non-reductive theory of causation is also developed against the fundamental theory of causation. (shrink)

Approaches to truth and the Liar paradox seem invariably to face a dilemma: either appeal to some sort of hierarchy, or declare apparently perfectly coherent concepts incoherent. But since both options lead to severe expressive restrictions, neither seems satisfactory. The aim of this paper is a new approach, which avoids the dilemma and the resulting expressive restrictions. Previous approaches tend to appeal to some new sort of semantic value for the truth predicate to take. I argue that such approaches (...) inevitably lead to the dilemma in question. In contrast, the present proposal sticks with classical semantic values, but allows the compositional rules associated with these to admit of exceptions. I show how such an approach can be developed systematically. (shrink)

The case for longtermism depends on the vast potential scale of the future. But that same vastness also threatens to undermine the case for longtermism: If the universe as a whole, or the future in particular, contain infinite quantities of value and/or disvalue, then many of the theories of value that support longtermism (e.g., risk-neutral total utilitarianism) seem to imply that none of our available options are better than any other. If so, then even apparently vast effects on (...) the far future cannot in fact make the world morally better. On top of this, some strategies for avoiding this problem of “infinitarian paralysis” (e.g., exponential pure time discounting) yield views that are much less supportive of longtermism. In this chapter, we explore how the potential infinitude of the future affects the case for longtermism. We argue that (i) there are reasonable prospects for extending risk-neutral totalism and similar views to infinite contexts and (ii) many such extension strategies will still support the case for longtermism, since they imply that when we can only effect (or only predictably affect) a finite, bounded part of an infinite universe, we can ignore the unaffectable rest of the universe and reason as if the finite, affectable part were all there is. (shrink)

Infinite time Turing machines represent a model of computability that extends the operations of Turing machines to transfinite ordinal time by defining the content of each cell at limit steps to be the lim sup of the sequences of previous contents of that cell. In this paper, we study a computational model obtained by replacing the lim sup rule with an ‘eventually constant’ rule: at each limit step, the value of each cell is defined if and only if the (...) content of that cell has stabilized before that limit step and is then equal to this constant value. We call these machines weak infinite time Turing machines. We study different variants of wITTMs adding multiple tapes, heads, or bidimensional tapes. We show that some of these models are equivalent to each other concerning their computational strength. We show that wITTMs decide exactly the arithmetic relations on natural numbers. (shrink)

Limiting identification of r.e. indexes for r.e. languages (from a presentation of elements of the language) and limiting identification of programs for computable functions (from a graph of the function) have served as models for investigating the boundaries of learnability. Recently, a new approach to the study of "intrinsic" complexity of identification in the limit has been proposed. This approach, instead of dealing with the resource requirements of the learning algorithm, uses the notion of reducibility from recursion theory to compare (...) and to capture the intuitive difficulty of learning various classes of concepts. Freivalds, Kinber, and Smith have studied this approach for function identification and Jain and Sharma have studied it for language identification. The present paper explores the structure of these reducibilities in the context of language identification. It is shown that there is an infinite hierarchy of language classes that represent learning problems of increasing difficulty. It is also shown that the language classes in this hierarchy are incomparable, under the reductions introduced, to the collection of pattern languages. Richness of the structure of intrinsic complexity is demonstrated by proving that any finite, acyclic, directed graph can be embedded in the reducibility structure. However, it is also established that this structure is not dense. The question of embedding any infinite, acyclic, directed graph is open. (shrink)

The question of whether the bounded arithmetic theories and are equal is closely connected to the complexity question of whether is equal to. In this paper, we examine the still open question of whether the prenex version of,, is equal to. We give new dependent choice‐based axiomatizations of the ‐consequences of and. Our dependent choice axiomatizations give new normal forms for the ‐consequences of and. We use these axiomatizations to give an alternative proof of the finite axiomatizability of and to (...) show new results such as is finitely axiomatized and that there is a finitely axiomatized theory,, containing and contained in. On the other hand, we show that our theory for splits into a natural infinite hierarchy of theories. We give a diagonalization result that stems from our attempts to separate the hierarchy for. (shrink)

In proving that the language of arithmetic does not contain its own truth predicate, Tarski demonstrated that the claim that a language both satisfies certain minimal conditions and contains its own truth predicate leads to a contradiction – a result that can seem puzzling in light of the fact that it seems obvious that English does satisfy the relevant conditions, while containing its own truth predicate. Chapter 5 explores the well‐known response to this problem, which maintains that English is really (...) an infinite hierarchy of languages defined by a hierarchy of Tarski‐style truth predicates. The construction of the hierarchy is explained, and the ways in which it is used to block different versions of the paradox are illustrated. The discussion then turns to problems with the approach, the most serious being the irresistible urge to violate the hierarchy's restrictions on intelligibility in the very process of setting it up – something we tend to forget because we imagine ourselves taking a position outside the hierarchy from which it can be described. Once we realize that the hierarchy is supposed to apply to the language we are using to describe it, the paradox returns with a vengeance, threatening to destroy the very construction that was introduced to avoid it. (shrink)

We consider a version of so called T x W logic for historical necessity in the sense of R.H. Thomason (1984), which is somewhat special in three respects: (i) it is explicitly based on two-dimensional modal logic in the sense of Segerberg (1973); (ii) for reasons of applicability to interesting fields of philosophical logic, it conceives of time as being discrete and finite in the sense of having a beginning and an end; and (iii) it utilizes the technique of systematic (...) frame constants in order to handle the problem of irreflexivity in tense logics, well known since Gabbay (1981). Axiomatizations are given for two infinite hierarchies of two-dimensional modal tense logics, one without and one with the characteristic operators for historical necessity and possibility. Strong and weak completeness results are obtained for both hierarchies as well as a result to the effect that two approaches to their semantics are equivalent, much in the spirit of Di Maio and Zanardo (1996) and von Kutschera (1997). (shrink)

We prove that if f is a partial Borel function from one Polish space to another, then either f can be decomposed into countably many partial continuous functions, or else f contains the countable infinite power of a bijection that maps a convergent sequence together with its limit onto a discrete space. This is a generalization of a dichotomy discovered by Solecki for Baire class 1 functions. As an application, we provide a characterization of functions which are countable unions (...) of continuous functions with domains of type Πn0, for a fixed n<ω. For Baire class 1 functions, this generalizes analogous characterizations proved by Jayne and Rogers for n=1 and Semmes for n=2. (shrink)

Infinite machines (IMs) can do supertasks. A supertask is an infinite series of operations done in some finite time. Whether or not our universe contains any IMs, they are worthy of study as upper bounds on finite machines. We introduce IMs and describe some of their physical and psychological aspects. An accelerating Turing machine (an ATM) is a Turing machine that performs every next operation twice as fast. It can carry out infinitely many operations in finite time. Many (...) ATMs can be connected together to form networks of infinitely powerful agents. A network of ATMs can also be thought of as the control system for an infinitely complex robot. We describe a robot with a dense network of ATMs for its retinas, its brain, and its motor controllers. Such a robot can perform psychological supertasks - it can perceive infinitely detailed objects in all their detail; it can formulate infinite plans; it can make infinitely precise movements. An endless hierarchy of IMs might realize a deep notion of intelligent computing everywhere. (shrink)