It is a metaphysical orthodoxy that interesting non-symmetric relations cannot be reduced to symmetric ones. This orthodoxy is wrong. I show this by exploring the expressive power of symmetric theories, i.e. theories which use only symmetric predicates. Such theories are powerful enough to raise the possibility of Pythagrapheanism, i.e. the possibility that the world is just a vast, unlabelled, undirected graph.

In this paper, we provide a detailed review of previous classifications of 2 × 2 games and suggest a mathematically simple way to classify the symmetric 2 × 2 games based on a decomposition of the payoff matrix into a cooperative and a zero-sum part. We argue that differences in the interaction between the parts is what makes games interesting in different ways. Our claim is supported by evolutionary computer experiments and findings in previous literature. In addition, we provide (...) a method for using a stereographic projection to create a compact 2-d representation of the game space. (shrink)

Kit Fine (2000) breaks with tradition, arguing that, pace Russell (e.g., 1903: 228), relations have neither directions nor converses. He considers two ways to conceive of these new "neutral" relations, positionalism and anti-positionalism, and argues that the latter should be preferred to the former. Cody Gilmore (2013) argues for a generalization of positionalism, slot theory, the view that a property or relation is n-adic if and only if there are exactly n slots in it, and (very roughly) that each slot (...) may be occupied by at most one entity. Slot theory (and with it, positionalism) bears the full brunt of Fine's (2000) symmetric completions and conflicting adicities problems. I fully develop an alternative, plural slot theory (or pocket theory), which avoids these problems, key elements of which are first considered by Yi (1999: 168 ff.). Like the slot theorist, the pocket theorist posits entities (pockets) in properties and relations that can be occupied. But unlike the slot theorist, the pocket theorist denies that at most one entity can occupy any one of them. As a result, she must also deny that the adicity of a property or relation is equal to the number of occupiable entities in it. By abandoning these theses, however, the pocket theorist is able to avoid Fine's problems, resulting in a stronger theory about the internal structure of properties and relations. Pocket theory also avoids a serious drawback of anti-positionalism. (shrink)

Physicists routinely claim that the fundamental laws of physics are 'time symmetric' or 'time reversal invariant' or 'reversible'. In particular, it is claimed that the theory of quantum mechanics is time symmetric. But it is shown in this paper that the orthodox analysis suffers from a fatal conceptual error, because the logical criterion for judging the time symmetry of probabilistic theories has been incorrectly formulated. The correct criterion requires symmetry between future-directed laws and past-directed laws. This criterion (...) is formulated and proved in detail. The orthodox claim that quantum mechanics is reversible is re-evaluated. The property demonstrated in the orthodox analysis is shown to be quite distinct from time reversal invariance. The view of Satosi Watanabe that quantum mechanics is time asymmetric is verified, as well as his view that this feature does not merely show a de facto or 'contingent' asymmetry, as commonly supposed, but implies a genuine failure of time reversal invariance of the laws of quantum mechanics. The laws of quantum mechanics would be incompatible with a time-reversed version of our universe. (shrink)

An Omega Point Theory says that reality is making progress from some initial state to some final state. It moves from some Alpha Point (the initial state) to some Omega Point (the final state). The progress is an increase in some quality. For example, reality is making progress from the chaotic to the orderly; or it is making progress from the simple to the complex; or from the mindless to the mental; or from evil to good. Here we focus on (...) the Omega Point theory of Peirce. An Omega Point Theory is an evolutionary cosmology – it says that the universe is evolving from the Alpha Point to the Omega Point. Peirce refers to his evolutionary cosmology as a Cosmogonic Philosophy would suppose that in the beginning – infinitely remote – there was a chaos of unpersonalized feeling, which being without connection or regularity would properly be without existence. This feeling, sporting here and there in pure arbitrariness, would have started the germ of a generalizing tendency. Its other sportings would be evanescent, but this would have a growing virtue. Thus the tendency to habit would be started; and from this, with the other principles of evolution, all the regularities of the universe would be evolved. At any time, however, an element of pure chance survives and will remain until the world becomes an absolutely perfect, rational, and symmetrical system, in which mind is at last crystallized in the infinitely distant future. (Peirce, Collected Papers , 6.33). (shrink)

When an entity ontologically depends on another entity, the former ‘presupposes’ or ‘requires’ the latter in some metaphysical sense. This paper defends a novel view, Dependence Deflationism, according to which ontological dependence is what I call an aggregative cluster concept: a concept which can be understood, but not fully analysed, as a ‘weighted total’ of constructive and modal relations. The view has several benefits: it accounts for clear cases of ontological dependence as well as the source of disagreement in controversial (...) ones; it gives a nice story about the evidential relevance of modal, mereological and set-theoretic facts to ontological dependence; and it makes sense of debates over the relation's formal properties. One important upshot of the deflationary account is that questions of ontological dependence are generally less deep and less interesting than usually thought. (shrink)

In this volume Axel Honneth deepens and develops his highly influential theory of recognition, showing how it enables us both to rethink the concept of justice and to offer a compelling account of the relationship between social reproduction and individual identity formation. Drawing on his reassessment of Hegel’s practical philosophy, Honneth argues that our conception of social justice should be redirected from a preoccupation with the principles of distributing goods to a focus on the measures for creating symmetrical relations of (...) recognition. This theoretical reorientation has far-reaching implications for the theory of justice, as it obliges this theory to engage directly with problems concerning the organization of work and with the ideologies that stabilize relations of domination. In the final part of this volume Honneth shows how the theory of recognition provides a fruitful and illuminating way of exploring the relation between social reproduction and identity formation. Rather than seeing groups as regressive social forms that threaten the autonomy of the individual, Honneth argues that the ‘I’ is dependent on forms of social recognition embodied in groups, since neither self-respect nor self-esteem can be maintained without the supportive experience of practising shared values in the group. This important new book by one of the leading social philosophers of our time will be of great interest to students and scholars in philosophy, sociology, politics and the humanities and social sciences generally. (shrink)

We characterize access to empirical objects in biology from a theoretical perspective. Unlike objects in current physical theories, biological objects are the result of a history and their variations continue to generate a history. This property is the starting point of our concept of measurement. We argue that biological measurement is relative to a natural history which is shared by the different objects subjected to the measurement and is more or less constrained by biologists. We call symmetrization the theoretical (...) and often concrete operation which leads to considering biological objects as equivalent in a measurement. Last, we use our notion of measurement to analyze research strategies. Some strategies aim to bring biology closer to the epistemology of physical theories, by studying objects as similar as possible, while others build on biological diversity. (shrink)

TABLE OF CONTENTS: Introduction; de Broglie's paradox.; Quantum theory of distant particles; The EPR paradox; Einstein locality and Bell's inequality; Recent research on Bell's inequality; General consequences of Einstein locality; Nonloeality and relativity; Time-symmetric theories; The Bohm-Aharonov hypothesis; Experiments on Einstein locality; Reduction of the wave packet; Measurements, reality and consciousness; Conclusions.

The mathematical structure of realist quantum theories has given rise to a debate about how our ordinary 3-dimensional space is related to the 3N-dimensional configuration space on which the wave function is defined. Which of the two spaces is our (more) fundamental physical space? I review the debate between 3N-Fundamentalists and 3D-Fundamentalists and evaluate it based on three criteria. I argue that when we consider which view leads to a deeper understanding of the physical world, especially given the deeper (...) topological explanation from the unordered configurations to the Symmetrization Postulate, we have strong reasons in favor of 3D-Fundamentalism. I conclude that our evidence favors the view that our fundamental physical space in a quantum world is 3-dimensional rather than 3N-dimensional. I outline lines of future research where the evidential balance can be restored or reversed. Finally, I draw lessons from this case study to the debate about theoretical equivalence. (shrink)

In the pure powers ontology (PPO), basic physical properties have wholly dispositional essences. PPO has clear advantages over categoricalist ontologies, which suffer from familiar epistemological and metaphysical problems. However, opponents argue that because it contains no qualitative properties, PPO lacks the resources to individuate powers, and generates a regress. The challenge for those who take such arguments seriously is to introduce qualitative properties without reintroducing the problems that PPO was meant to solve. In this paper, I distinguish the core claim (...) of PPO: (i) basic physical properties have dispositional essences, from a hitherto unnoticed assumption: (ii) the dispositional essences of basic physical properties exclusively involve type-causal relations to other basic physical properties. I reject (ii), making room for a structuralist ontology in which all basic physical properties are pure powers, individuated by their places in a causal structure that includes not only other powers, but also physically realized qualitative properties such as shapes, patterns and structures. Such qualities individuate pure powers in the way that non-mental input and output properties individuate realized mental properties in functionalist theories of mind, except that here it is basic physical powers that are individuated by relations to realized non-powers. I distinguish one Platonic and two Aristotelian version of this theory, and argue that the Aristotelian versions require that grounding is not always a relative fundamentality relation, because the powers ground the qualities that individuate them. By considering ontic structural realism, I argue that symmetric grounding is the best way to make sense of relational individuation in structuralist ontologies, and is therefore no additional commitment of the one proposed here. (shrink)

The trope bundle theories of objects are capable of analyzing monadic inherence (objects having tropes), which is one of their main advantage. However, the best current trope theoretical account of relational tropes, namely, the relata specific view leaves relational inherence (a relational trope relating two or more entities) primitive. This article presents the first trope theoretical analysis of relational inherence by generalizing the trope theoretical analysis of inherence to relational tropes. The analysis reduces the holding of relational inherence to (...) the obtaining of certain other facts about entities of the trope theoretical category system. Moreover, I show that the analysis can deal with asymmetric and non-symmetric relations by assuming that all relation-like tropes are quantities. Finally, I provide an account of the spatial location of tropes in the difficult case in which tropes contribute to determining of the location of other entities. (shrink)

A theory of truth is usually demanded to be consistent, but -consistency is less frequently requested. Recently, Yatabe has argued in favour of -inconsistent first-order theories of truth, minimising their odd consequences. In view of this fact, in this paper, we present five arguments against -inconsistent theories of truth. In order to bring out this point, we will focus on two very well-known -inconsistent theories of truth: the classical theory of symmetric truth FS and the non-classical (...) theory of naïve truth based on ᴌukasiewicz infinitely valued logic: PAᴌT. (shrink)

In this article, we re-examine Pascal’s Mugging, and argue that it is a deeper problem than the St. Petersburg paradox. We offer a way out that is consistent with classical decision theory. Specifically, we propose a “many muggers” response analogous to the “many gods” objection to Pascal’s Wager. When a very tiny probability of a great reward becomes a salient outcome of a choice, such as in the offer of the mugger, it can be discounted on the condition that there (...) are other symmetric, non-salient rewards that one may receive if one chooses otherwise. (shrink)

In this article, we analyse the ontological import of adding classes to set theories. We assume that this increment is well represented by going from ZF system to NBG. We thus consider the standard techniques of reducing one system to the other. Novak proved that from a model of ZF we can build a model of NBG (and vice versa), while Shoenfield have shown that from a proof in NBG of a set-sentence we can generate a proof in ZF (...) of the same formula. We argue that the first makes use of a too strong metatheory. Although meaningful, this symmetrical reduction does not equate the ontological content of the theories. The strong metatheory levels the two theories. Moreover, we will modernize Shoenfields proof, emphasizing its relation to Herbrands theorem and that it can only be seen as a partial type of reduction. In contrast with symmetrical reductions, we believe that asymmetrical relations are powerful tools for comparing ontological content. In virtue of this, we prove that there is no interpretation of NBG in ZF, while NBG trivially interprets ZF. This challenges the standard view that the two systems have the same ontological content. (shrink)

This article argues that pseudoscience lacks an adequate philosophical analysis. Using conspiracy theories as a case study, it is claimed that such an analysis needs to go beyond a mere epistemological approach. In the first part, it is shown that the existing philosophical literature shares the assumption that conspiracy theories are primarily deficient scientific hypotheses. This claim is contested, because such an approach can only understand what conspiracy theories fail to be, but not what they are and (...) why people tend to endorse them. To develop an alternative, the second part starts from recent sociology of science, applied to conspiracy theories, that claims that both true and false theories are in need of a symmetrical social explanation. However, this opens the path to relativism because it does not allow for a significant distinction between scientific and conspiracy theories. In the third section an alternative analysis of conspiracy theories is offered, which does not abandon this distinction. Moreover, it is claimed that the most important element of conspiracy theories is not their truth value but rather their symbolic structures of meaning. The aim of conspiracy theories, and perhaps of pseudoscience in general, is then not to know facts about the world, but rather to offer a meaningful framework within which to live. (shrink)

Axel Honneth reconstructs Hegel’s social and political philosophy on the basis of the concept of recognition. For Honneth, recognition is a constitutive relation between individuals that is in principle symmetrical. By conceiving recognition through symmetry, Honneth effectively bans the inclusion of power within recognitive relation. He thus regards the relations of power as cases of non-recognition or misrecognition. In this paper, I develop an alternative theory of the constitutive relation between individuals for Hegel, one that is based on the asymmetrical (...) relation of power. To this aim, I focus on the chapter of “determinations of reflection” in the Science of Logic. Through a close analysis of Hegel’s Logic, I argue that the most fundamental form of relation between individuals is the relation of opposition, that individuals are solely constituted in and through the relation of opposition, and that the relation of opposition is essentially asymmetrical. Together, these claims establish... (shrink)

While many different mechanisms contribute to the generation of spatial order in biological development, the formation of morphogenetic fields which in turn direct cell responses giving rise to pattern and form are of major importance and essential for embryogenesis and regeneration. Most likely the fields represent concentration patterns of substances produced by molecular kinetics. Short range autocatalytic activation in conjunction with longer range “lateral” inhibition or depletion effects is capable of generating such patterns (Gierer and Meinhardt, 1972). Non-linear reactions are (...) required, and mathematical criteria were derived to design molecular models capable of pattern generation. The classical embryological feature of proportion regulation can be incorporated into the models. The conditions are mathematically necessary for the simplest two-factor case, and are likely to be a fair approximation in multi-component systems in which activation and inhibition are systems parameters subsuming the action of several agents. Gradients, symmetric and periodic patterns, in one or two dimensions, stable or pulsing in time, can be generated on this basis. Our basic concept of autocatalysis in conjunction with lateral inhibition accounts for self-regulatory biological features, including the reproducible formation of structures from near-uniform initial conditions as required by the logic of the generation cycle. Real tissue form, for instance that of budding Hydra, may often be traced back to local curvature arising within an initially relatively flat cell sheet, the position of evagination being determined by morphogenetic fields. Shell theory developed for architecture may also be applied to such biological processes. (shrink)

It is widely held that our beliefs can be epistemically faultless despite being morally flawed. Theories of moral encroachment challenge this, holding that moral considerations bear on the epistemic status of our attitudes. According to attitude-based theories of moral encroachment, morality encroaches upon the epistemic standing of our attitudes on the grounds that we can morally injure others with our epistemic practices. In this paper, I aim to show that current attitude-based theories have asymmetric mechanisms: moral features (...) only make it harder for attitudes to secure epistemic merits. I argue that, if attitudes can incur moral injury, failure to form attitudes can too. To make sense of this, I contend, attitude-based accounts require symmetric mechanisms, allowing that moral considerations make it both harder and easier for attitudes to attain epistemic merits. I maintain that, once we recognize this, attitude-based encroachment views must soon concede that they sometimes demand we believe against the evidence. (shrink)

The paper is an exercise in descriptive ontology, with specific applications to problems in the geographical sphere. It presents a general typology of spatial boundaries, based in particular on an opposition between bona fide or physical boundaries on the one hand, and fiat or human-demarcation-induced boundaries on the other. Cross-cutting this opposition are further oppositions in the realm of boundaries, for example between: crisp and indeterminate, complete and incomplete, enduring and transient, symmetrical and asymmetrical. The resulting typology generates a corresponding (...) categorization of the different sorts of objects which (complete) boundaries determine or demarcate. The theory is applied first of all in the areas of geography and of administrative and property law. Indications are then given as to how the typology may be applied also in other fields where physical and fiat boundaries are at work, including the field of cognitive linguistics and the related field of the ontology of truth. (shrink)

Conventional wisdom dictates that proofs of mathematical propositions should be treated as necessary, and sufficient, for entailing `significant' mathematical truths only if the proofs are expressed in a---minimally, deemed consistent---formal mathematical theory in terms of: * Axioms/Axiom schemas * Rules of Deduction * Definitions * Lemmas * Theorems * Corollaries. Whilst Andrew Wiles' proof of Fermat's Last Theorem FLT, which appeals essentially to geometrical properties of real and complex numbers, can be treated as meeting this criteria, it nevertheless leaves two (...) questions unanswered: (i) Why is x^n +y^n = z^n solvable only for n < 3 if x, y, z, n are natural numbers? (ii) What technique might Fermat have used that led him to, even if only briefly, believe he had `a truly marvellous demonstration' of FLT? Prevailing post-Wiles wisdom---leaving (i) essentially unaddressed---dismisses Fermat's claim as a conjecture without a plausible proof of FLT. -/- However, we posit that providing evidence-based answers to both queries is necessary not only for treating FLT as significant, but also for understanding why FLT can be treated as a true arithmetical proposition. We thus argue that proving a theorem formally from explicit, and implicit, premises/axioms using rules of deduction---as currently accepted---is a meaningless game, of little scientific value, in the absence of evidence that has already established---unambiguously---why the premises/axioms and rules of deduction can be treated, and categorically communicated, as pre-formal truths in Marcus Pantsar's sense. Consequently, only evidence-based, pre-formal, truth can entail formal provability; and the formal proof of any significant mathematical theorem cannot entail its pre-formal truth as evidence-based. It can only identify the explicit/implicit premises that have been used to evidence the, already established, pre-formal truth of a mathematical proposition. Hence visualising and understanding the evidence-based, pre-formal, truth of a mathematical proposition is the only raison d'etre for subsequently seeking a formal proof of the proposition within a formal mathematical language (whether first-order or second order set theory, arithmetic, geometry, etc.) By this yardstick Andrew Wiles' proof of FLT fails to meet the required, evidence-based, criteria for entailing a true arithmetical proposition. -/- Moreover, we offer two scenarios as to why/how Fermat could have laconically concluded in his recorded marginal noting that FLT is a true arithmetical proposition---even though he either did not (or could not to his own satisfaction) succeed in cogently evidencing, and recording, why FLT can be treated as an evidence-based, pre-formal, arithmetical truth (presumably without appeal to properties of real and complex numbers). It is primarily such a putative, unrecorded, evidence-based reasoning underlying Fermat's laconic assertion which this investigation seeks to reconstruct; and to justify by appeal to a plausible resolution of some philosophical ambiguities concerning the relation between evidence-based, pre-formal, truth and formal provability. (shrink)

The use of evolutionary game theory to explain the evolution of human norms and the behavior of humans who act according to those norms is widespread. Both the aims and motivation for its use are clearly articulated by Harms and Skyrms (2008) in the following passage: "A good theory of evolution of norms might start by explaining the evolution of altruism in Prisoner’s Dilemma, of Stag Hunting, and of the equal split in the symmetric bargaining game. These are not (...) well-explained by classical game theory based on rational choice. From a technical point of view, they present different theoretical challenges. In the bargaining game, there are an infinite number of equilibria with no principled (rational choice) way to select the cooperative one. In Stag Hunt there are only two, but the non-cooperative one is selected by risk-dominance. In Prisoner’s Dilemma the state of mutual cooperation is not a Nash equilibrium at all, and cooperation flies in the face of the rational-choice principle that one does not choose less rather than more. In contrast to rational choice theory, the most common tool of evolutionary game theory is the replicator dynamics, in which the propagation rate of each strategy is determined by its current payoffs. These dynamics have a rationale in both biological and cultural evolutionary modeling, and sometimes tell us things that rational choice theory does not." We agree with the first sentence in this quotation: a good theory about the behavior under norms ought to explain altruism in the Prisoner’s Dilemma (PD), playing Stag in Stag Hunt (SH), and offering equal splits in the symmetric Nash bargaining game (NB). We also agree with Harms and Skyrms about the difference in technical challenges each of these games poses. Finding a single mechanism, even one as broadly understood as evolution, that could solve these challenges en masse is no doubt a tall order. Nonetheless, in this paper, we present a single, simple, modification to SH, NB, and a general n-player PD that does just that: we introduce deontological autonomy into the models. (shrink)

Cooperation has been analyzed primarily in the context of theories of collective intentionality. These discussions have primarily focused on interactions between pairs or small groups of agents who know one another personally. Cooperative game theory has also been used to argue for a form of cooperation in large unorganized groups. Here I consider a form of minimal cooperation that can arise among members of potentially large organized groups (e.g., corporate teams, committees, governmental bodies). I argue that members of organized (...) groups can be minimally cooperative in virtue of playing roles in an organizational structure and having a common goal. The minimal form of cooperation I argue for is not grounded in collective intentions involving symmetric mental states, special collective intentional modes, or joint commitments. More generally, I show how considering minimal cooperation in the context of organized groups provides an opportunity to reevaluate the extent to which the social world and social phenomena depend on internalist mental factors (e.g., intentions, beliefs) and externalist non-mental factors (e.g., documents, laws, job descriptions). The view of minimal cooperation among members of organized groups I offer provides support for an externalist rather than internalist theory of at least one social phenomenon. (shrink)

Mirror neuron research has come a long way since the early 1990s, and many theorists are now stressing the heterogeneity and complexity of the sensorimotor properties of fronto-parietal circuits. However, core aspects of the initial ‘ mirror mechanism ’ theory, i.e. the idea of a symmetric encapsulated mirroring function translating sensory action perceptions into motor formats, still appears to be shaping much of the debate. This article challenges the empirical plausibility of the sensorimotor segregation implicit in the original mirror (...) metaphor. It is proposed instead that the teleological organization found in the broader fronto-parietal circuits might be inherently sensorimotor. Thus the idea of an independent ‘purely perceptual’ goal understanding process is questioned. Further, it is hypothesized that the often asymmetric, heterogeneous and contextually modulated mirror and canonical neurons support a function of multisensory mapping and tracking of the perceiving agents affordance space. Such a shift in the interpretative framework offers a different theoretical handle on how sensorimotor processes might ground various aspects of intentional action choice and social cognition. Mirror neurons would under the proposed “social affordance model” be seen as dynamic parts of larger circuits, which support tracking of currently shared and competing action possibilities. These circuits support action selection processes—but also our understanding of the options and action potentials that we and perhaps others have in the affordance space. In terms of social cognition ‘ mirror ’ circuits might thus help us understand not only the intentional actions others are actually performing—but also what they could have done, did not do and might do shortly. (shrink)

It is commonly assumed that grounding relations are asymmetric. Here I develop and argue for a theory of metaphysical structure that takes grounding to be nonsymmetric rather than asymmetric. Even without infinite descending chains of dependence, it might be that every entity is grounded in some other entity. Having first addressed an immediate objection to the position under discussion, I introduce two examples of symmetric grounding. I give three arguments for the view that grounding is nonsymmetric (I call this (...) view ‘metaphysical interdependence’). These arguments are: (i) that metaphysical interdependence is the only theory able to reconcile competing intuitions about grounding; (ii) that it is the only theory consistent with both ‘gunk’ and ‘junk’; and (iii) that offers a satisfactory solution to the problem concerning whether or not grounding is itself grounded. (shrink)

The cosmological constant problem arises because the magnitude of vacuum energy density predicted by the Quantum Field Theory is about 120 orders of magnitude larger then the value implied by cosmological observations of accelerating cosmic expansion. We pointed out that the fractal nature of the quantum space-time with negative Hausdorff-Colombeau dimensions can resolve this tension. The canonical Quantum Field Theory is widely believed to break down at some fundamental high-energy cutoff ∗ Λ and therefore the quantum fluctuations in the vacuum (...) can be treated classically seriously only up to this high-energy cutoff. In this paper we argue that the Quantum Field Theory in fractal space-time with negative Hausdorff-Colombeau dimensions gives high-energy cutoff on natural way. We argue that there exists hidden physical mechanism which cancels divergences in canonical QED4 ,QCD4 , Higher-Derivative-Quantum gravity, etc. In fact we argue that corresponding supermassive Pauli-Villars ghost fields really exist. It means that there exists the ghost-driven acceleration of the universe hidden in cosmological constant. In order to obtain the desired physical result we apply the canonical Pauli-Villars regularization up to ∗ Λ . This would fit in the observed value of the dark energy needed to explain the accelerated expansion of the universe if we choose highly symmetric masses distribution between standard matter and ghost matter below the scale ∗ Λ , i.e. , ( ) ( ) . . eff eff s m g m , , , f μ f μ μ mc μ μ μ c ∗  − = ≤ < Λ The small value of the cosmological constant is explained by tiny violation of the symmetry between standard matter and ghost matter. Dark matter nature is also explained using a common origin of the dark energy and dark matter phenomena. (shrink)

Here, the author tries to build the structure of the Theory of computation based on considering time as a fuzzy concept. In fact, there are reasons to consider time as a fuzzy concept. In this article, the author doesn’t go to this side but note that Brower and Husserl views on the concept of time were similar [8]. Some reasons have been given for it in [3]. Throughout this article, the author presents the Theory of Computation with Fuzzy Time. Given (...) the classic definition of Turing Machine, the concept of Time is modified to Fuzzy time. This new term calls as Theory TC* [2] and this type of computation “Fuzzy time Computation”. We have relatively large number of fundamental unsolved problems in Complexity Theory. In the new theory, some of the major obstacles and unsolved problems have been solved [2]. It should be noted that in this article, the writer considers fuzzy number associated to instants of time as a symmetric one. The point about the symmetry is in the proof of Lemma 3, although it is generalizable. In particular, the new classes of complexity Theory, P*, NP*, BPP* in the TC* analogues to the definitions of P, NP, BPP defines as their natural alternative definition. Here, we will see P*≠ NP*, P*= BPP*. Finally, we have Theorem 4. (shrink)

Here, we try to build the structure of a Theory of computation based on considering time as a fuzzy concept. Actually, there are some reasons to consider time as a fuzzy concept. In this article, we don’t go to this side but we remind that Brower and Husserl ideas about the concept of time were similar [14]. Throughout this article, we present the Theory of Computation with Fuzzy Time. Considering the classical definition of Turing Machine we change and modify the (...) concept of Time to Fuzzy time. We call this new Theory TC* and this type of computation “Fuzzy time Computation”. We have relatively large number of fundamental unsolved problems in Complexity Theory. In the new Theory some of the major obstacles and unsolved problems are solved. It should be mentioned that in this article, we consider fuzzy number a symmetric one. The point about the symmetry is in the proof of Lemma 3, although we are able to generalize it. More specifically, we define the new classes of complexity Theory, P*, NP*, BPP* in TC* analogues to the definitions P, NP, BPP as their natural substituted definition. We show P*≠ NP*, P*= BPP*. Finally, we have Theorem 4, P≠NP . (shrink)

In this Part II, I investigate different approaches to the question of what makes imagining different from belief. I find that the sentiment-based approach of David Hume falls short, as does the teleological approach, once advocated by David Velleman. I then consider whether the inferential properties of beliefs and imaginings may differ. Beliefs, I claim, exhibit an anti-symmetric inferential governance over imaginings: they are the background that makes inference from one imagining to the other possible; the reverse is not (...) true, and this allows us to distinguish the two attitudes. I then go on to consider the action theory of imagining and the role that imaginings play in generating emotion. (shrink)

In this paper, we use antecedent-final conditionals to formulate two problems for parsing-based theories of presupposition projection and triviality of the kind given in Schlenker 2009. We show that, when it comes to antecedent-final conditionals, parsing-based theories predict filtering of presuppositions where there is in fact projection, and triviality judgments for sentences which are in fact felicitous. More concretely, these theories predict that presuppositions triggered in the antecedent of antecedent-final conditionals will be filtered (i.e. will not project) (...) if the negation of the consequent entails the presupposition. But this is wrong: John isn’t in Paris, if he regrets being in France intuitively presupposes that John is in France, contrary to this prediction. Likewise, parsing-based approaches to triviality predict that material entailed by the negation of the consequent will be redundant in the antecedent of the conditional; but John isn’t in Paris, if he’s in France and Mary is with him is intuitively felicitous, contrary to these predictions. Importantly, given that the trigger appears in sentence-final position, both incremental (left-to-right) and symmetric versions of such theories make the same predictions. These data constitute a challenge to the idea that presupposition projection and triviality should be computed on the basis of parsing. This issue is important because it relates to the more general question as to whether presupposition and triviality calculation should be thought of as a pragmatic post-compositional phenomenon or as part of compositional semantics (as in the more traditional dynamic approaches). We discuss a solution which allows us to maintain the parsing-based pragmatic approach; it is based on an analysis of conditionals which incorporates a presupposition that their antecedent is compatible with the context, together with a modification to Schlenker’s (2009) algorithm for calculating local contexts so that it takes into account presupposed material. As we will discuss, this solution works within a framework broadly similar to that of Schlenker’s (2009), but it doesn’t extend in an obvious way to other parsing-based accounts (e.g. parsing-based trivalent approaches). We conclude that a parsing-based theory can be maintained, but only if we adopt a substantial change of perspective on the framework. (shrink)

Our ordinary causal concept seems to fit poorly with how our best physics describes the world. We think of causation as a time-asymmetric dependence relation between relatively local events. Yet fundamental physics describes the world in terms of dynamical laws that are, possible small exceptions aside, time symmetric and that relate global time slices. My goal in this paper is to show why we are successful at using local, time-asymmetric models in causal explanations despite this apparent mismatch with fundamental (...) physics. In particular, I will argue that there is an important connection between time asymmetry and locality, namely: understanding the locality of our causal models is the key to understanding why the physical time asymmetries in our universe give rise to time asymmetry in causal explanation. My theory thus provides a unified account of why causation is local and time asymmetric and thereby enables a reply to Russell’s famous attack on causation. (shrink)

This paper develops a novel theory of abstraction—what we call collective abstraction. The theory solves a notorious problem for noneliminative structuralism. The noneliminative structuralist holds that in addition to various isomorphic systems there is a pure structure that can be abstracted from each of these systems; but existing accounts of abstraction fail for nonrigid systems like the complex numbers. The problem with the existing accounts is that they attempt to define a unique abstraction operation. The theory of collective abstraction instead (...) simultaneously defines a collection of distinct abstraction operations, each of which maps a system to its corresponding pure structure. The theory is precisely formulated in an essentialist language. This allows us to throw new light on the question to what extent structuralists are committed to symmetric dependence. Finally, we apply the theory of collective abstraction to solve a problem about converse relations. (shrink)

Unconscious logical inference seems to rely on the syntactic structures of mental representations (Quilty-Dunn & Mandelbaum 2018). Other transitions, such as transitions using iconic representations and associative transitions, are harder to assimilate to syntax-based theories. Here we tackle these difficulties head on in the interest of a fuller taxonomy of mental transitions. Along the way we discuss how icons can be compositional without having constituent structure, and expand and defend the “symmetry condition” on Associationism (the idea that associative links (...) and transitions are perfectly symmetric). In the end, we show how a BIT (“bare inferential transition”) theory can cohabitate with these other non-inferential mental transitions. (shrink)

Consciousness may not only be a problem how to know the brain but also a problem how to understand what known. Understanding is always an ontological system created as the explanation of what known by us. And, if all brains, including human brains, may be defined as the mind, consciousness must be part of our understanding of the mind. The author argues that no mind may exist if not be a life or lives, no life may exist if not be (...) a mind or a part of it, and, if it is the mind that needs to be explained, it must finally and fundamentally be explained as a life or a living system. An ontological definition of life is proposed first in this article, which is also applicable to fields such as physics, biology, psychology, and sociology. Based on an analysis of two kinds of lives and their relationship with matter and energy, a living system is modeled as the organization of two system relations, which explains not only consciousness but also wakefulness, emotion, intelligence, language and mind-body relationship. And finally, a semantic theory is reached, which takes lives as the only meaning of all symmetrical and conservative changes, such as the location changes or form changes of consciousness, or the state changes of memories. The concept of life, the model of living systems and the semantic theory together provide a framework for us to understand what and how consciousness is, why there is consciousness, where and when consciousness occurs. (shrink)

Every quantum state can be represented as a probability distribution over the outcomes of an informationally complete measurement. But not all probability distributions correspond to quantum states. Quantum state space may thus be thought of as a restricted subset of all potentially available probabilities. A recent publication (Fuchs and Schack, arXiv:0906.2187v1, 2009) advocates such a representation using symmetric informationally complete (SIC) measurements. Building upon this work we study how this subset—quantum-state space—might be characterized. Our leading characteristic is that the (...) inner products of the probabilities are bounded, a simple condition with nontrivial consequences. To get quantum-state space something more detailed about the extreme points is needed. No definitive characterization is reached, but we see several new interesting features over those in Fuchs and Schack (arXiv:0906.2187v1, 2009), and all in conformity with quantum theory. (shrink)

SYMMETRY IN PHYSICS: FROM PROPORTION AND HARMONY TO THE TERM OF METALENGUAJE -/- Ruth Castillo Universidad Central de Venezuela -/- The revolutionary changes in physics require a careful exploration of the way in which concepts depend on the theoretical structure in which they are immerse. A historical reconstruction allows us to show how the notion of symmetry evolves from the definition as proportion and harmony to its consideration within the language of contemporary physics, as a linguistic meta-theoretical requirement in physical (...) theories. In contemporary terms, symmetry is a fundamental category of research to which the usual categories of the natural sciences can be reduce in: space, time, causality, interaction, matter, strength, etc ... Thus, symmetry is a concept with different meanings: heuristically symmetric models inspire scientists in the search for solutions to different problems. Methodologically, symmetric structures are use to make theories, laws with invariant properties. A description of nature in terms of symmetric structures and symmetry ruptures seems to be the proper way to describe the complexity of reality. (shrink)

Nguyen argues that only his radically pragmatic account and Sterken’s indexical account can capture what we call the positive data. We present some new data, which we call the negative data, and argue that no theory of generics on the market is compatible with both the positive data and the negative data. We develop a novel version of the indexical account and show that it captures both the positive data and the negative data. In particular, we argue that there is (...) a semantic constraint that, in any context, the semantic value of GEN is upward monotone and non-symmetric. On the other hand, the pragmatic account has difficulty accommodating the negative data. This is because no pragmatic principles have been developed that can explain the negative data. In the paper, we focus on only the pragmatic account and the indexical account, but our discussion has broad implications for the debate on generics: any empirically adequate accounts of generics must be flexible enough to accommodate the positive data and yet constrained enough to accommodate the negative data. (shrink)

The appearance of negative terms in quasiprobability representations of quantum theory is known to be inevitable, and, due to its equivalence with the onset of contextuality, of central interest in quantum computation and information. Until recently, however, nothing has been known about how much negativity is necessary in a quasiprobability representation. Zhu :120404, 2016) proved that the upper and lower bounds with respect to one type of negativity measure are saturated by quasiprobability representations which are in one-to-one correspondence with the (...) elusive symmetric informationally complete quantum measurements. We define a family of negativity measures which includes Zhu’s as a special case and consider another member of the family which we call “sum negativity.” We prove a sufficient condition for local maxima in sum negativity and find exact global maxima in dimensions 3 and 4. Notably, we find that Zhu’s result on the SICs does not generally extend to sum negativity, although the analogous result does hold in dimension 4. Finally, the Hoggar lines in dimension 8 make an appearance in a conjecture on sum negativity. (shrink)

Is the basic mechanism behind presupposition projection fundamentally asymmetric or symmetric? This is a basic question for the theory of presupposition, which also bears on broader issues concerning the source of asymmetries observed in natural language: are these simply rooted in superficial asymmetries of language use— language use unfolds in time, which we experience as fundamentally asymmetric— or can they be, at least in part, directly referenced in linguistic knowledge and representations? In this paper we aim to make progress (...) on these questions by exploring presupposition projection across conjunction, which has typically been taken as a central piece of evidence that presupposition projection is asymmetric. As a number of authors have recently pointed out, however, whether or not this conclusion is warranted is not clear once we take into account independent issues of redundancy. Building on previous work by Chemla & Schlenker (2012) and Schwarz (2015), we approach this question experimentally by using an inference task which controls for redundancy and presupposition suspension. We find strong evidence for left-to-right filtering across conjunctions, but no evidence for right-to-left filtering, suggesting that, at least as a default, presupposition projection across conjunction is indeed asymmetric. (shrink)

In this paper, I present what I call the symmetry conception of God within 1st order, extensional, non-well-founded set theory. The symmetry conception comes in two versions. According to the first, God is that unique being that is universally symmetrical with respect to set membership. According to the second, God is the universally symmetrical set of all sets that are universally symmetrical with respect to set membership. I present a number of theorems, most importantly that any universally symmetrical set is (...) identical to its essence, that show that the two symmetry conceptions intersect with some dominant theological conceptions of God. The theorems also show that both of the symmetry conceptions of God entail that God has a fractal like structure. (shrink)

Suppose rationality requires you to A if you believe you ought to A. Suppose you believe that you ought to A. How can you satisfy this requirement? One way seems obvious. You can satisfy this requirement by A-ing. But can you also satisfy it by stopping to believe that you ought to A? Recently, it has been argued that this second option is not a genuine way of satisfying the above requirement. Conditional requirements of rationality do not have two ‘ (...) class='Hi'>symmetric’, but only one ‘asymmetric’ satisfaction condition. This paper explores the consequences of this argument for a theory of the requirements of rationality. I seek to show that thisview conflicts with another powerful intuition about the requirements of rationality, i.e. ‘rational consistency’: if rationality requires you to X, then it is not the case that rationality requires you to not-X. I shall conclude that ‘asymmetric’ satisfying is based on a misleading intuition, for which we should not sacrifice ‘rational consistency’. (shrink)

The essay discusses the religious and ethical message of Shusaku Endo’s Silence. Briefly focusing first on the plot of the novel, the article proceeds to discuss the moral dilemma that is the core of the novel and asks whether the dilemma is symmetrical or incommensurable. Next, the essay analyzes the dilemma from the point of view of Max Scheler’s theory of the tragic. Finally, to highlight Rodrigues’s tragic situation, it discusses the notion of the hiddenness of God.

Empirical findings may be relevant for aesthetic evaluation in at least two ways. First — within criticism — they may help us to identify the aesthetic value of objects. Second— whithin philosophy — they may help us to decide which theory of aesthetic value and evaluation to prefer. In this paper, I address both kinds of relevance. My focus is thereby on empirical evidence gathered, not by means of first-personal experiences, but by means of third-personal scientific investigations of individual artworks (...) or, more generally, our interaction with art. The main thesis to be defended is that third-personal empirical findings are of limited significance for both critical and philosophical aesthetics. Indeed, they matter only to the extent to which they draw our attention to features or facts that we then iden- tify, from our first-personal perspective, as aesthetically relevant — for instance, as reasons counting for or against certain ascriptions of aesthetic value, or as factors that causally influence our actual as- sessments and thus render them partly inadequate or irrational. This limited significance of empirical findings is in line with the rationalist approach to the formation and justification of aesthetic judgements, that I have already started to defend elsewhere. -/- With respect to critical aesthetics, one problem is that empirical investigations cannot capture the normativity of the aesthetic relevance of lower-level features or facts: the respective studies can tell us what we take to be reason-giving or valuable, but not what is reason-giving or valuable. Furthermore, we cannot use empirically knowable principles to infer the presence of aesthetic values or reasons on the basis of recognising measurable lower-level features because the only available principles are too specific to allow for their actual application to more than one existing object, or even for their actual formulation. For their antecedents make reference to a large number of very determinate properties, as well as sometimes to particular events of creation. The only exception are conceptual principles (such as ‘something symmetrical is balanced’) or default principles (such as ‘something elegant is, by default, beautiful’). But knowledge of these principles is of little inferential use due to the holistic character of the justificatory power of aesthetic reasons and, moreover, does not allow for empirical acquisition. -/- With respect to philosophical aesthetics, I start with considering the limits of evolutionary accounts of aesthetic value. Even if it is true that humans originally came to value artworks because they recognised that artworks reveal certain skills or features of artists (such as imaginativeness or resourcefulness) that are desirable in sexual partners, it does not value that our reasons for valuing art have not changed, or at least not become much more complex. In addition, the answer to the question of why we value art from an evolutionary perspective has no obvious bearing on the issue of how we should appreciate art from an aesthetic perspective, or what such aesthetic appreciation would consist in. On the other hand, experimental studies showing that our actual aesthetic evaluations are influenced by factors (such as exposure e ects, or knowledge of the prices of objects), that undermine the good standing of our responses, just reveal that it is more difficult than perhaps expected to form sound aesthetic judgements; and that we need to improve our future aesthetic judgements by diminishing as much as possible the impact of those factors. (shrink)

The principle of Identity of Indiscernibles has been challenged with various thought experiments involving symmetric universes. In this paper, I describe a fractal universe and argue that, while it is not a symmetric universe in the classical sense, under the assumption of a relational theory of space it nonetheless contains a set of objects indiscernible by pure properties alone. I then argue that the argument against the principle from this new thought experiment resists better than those from classical (...) symmetric universes three main objections put forth against this kind of arguments. (shrink)

The principle of Information Conservation or Determinism is a governing assumption of physical theory. Determinism has counterfactual consequences. It entails that if the present were different, then the future would be different. But determinism is temporally symmetric: it entails that if the present were different, the past would also have to be different. This runs contrary to our commonsense intuition that what has happened in the future depends on the past in a way the past does not depend on (...) the future. To understand how this can be so we observe that while the truth of some counterfactuals is guaranteed by the laws of logic or the laws of nature, some are not. It is among the latter contingent, counterfactuals that we find temporal asymmetry. It is this asymmetry that gives causation a temporal direction. The temporal asymmetry of these counterfactuals is explained by the fact that the dynamical laws of nature are logically irreversible functions from partial states of the world onto other partial states. (Logical reversibility is not to be confused, though it too often is, with time-reversal invariance). Though these irreversible laws are locally indeterministic, they can sum to give a globally deterministic description of the world. This combination of global determinism and local indeterminism gives rise to contingent counterfactual dependence and gives that dependence a direction. That direction is independent of the direction of entropy. The direction of contingent counterfactual dependence is time's arrow. (shrink)

When beliefs are quantified as credences, they are related to each other in terms of closeness and accuracy. The “accuracy first” approach in formal epistemology wants to establish a normative account for credences based entirely on the alethic properties of the credence: how close it is to the truth. To pull off this project, there is a need for a scoring rule. There is widespread agreement about some constraints on this scoring rule, but not whether a unique scoring rule stands (...) above the rest. The Brier score equips credences with a structure similar to metric space and induces a “geometry of reason.” It enjoys great popularity in the current debate. I point out many of its weaknesses in this article. An alternative approach is to reject the geometry of reason and accept information theory in its stead. Information theory comes fully equipped with an axiomatic approach which covers probabilism, standard conditioning, and Jeffrey conditioning. It is not based on an underlying topology of a metric space, but uses a non-commutative divergence instead of a symmetric distance measure. I show that information theory, despite initial promise, also fails to accommodate basic epistemic intuitions; and speculate on its remediation. (shrink)

The present paper explores diffident and dissident practices reflected by the political talk at the end of the 19th-century in Romania. Relying on Jacques Rancière’s theories on the ‘aesthetic regime of politics,’ the introduction sketches a historical frame and proposes a focus change: the relation between ‘politics’ and ‘aesthetics’ does not stand on a set of literary cases, but on political scripts as such. Thus, the hypotheses investigated by the next three parts can be formulated as follows: 1. though (...) determined by an ideological direction (Conservative or Liberal), the political speech still preserves his tendency towards aesthetic autonomy. 2. oratorical merits (hinting at aesthetic autonomy) can turn into practices of political autonomy, diffidence and, then, dissidence. Methodologically, two types of aesthetic practices organize the chosen materials; both the diffident script and the theatre of dissidence help us to perceive how the philosophical and moral meaning of these practices could change into an ideology of dissidence. The formalization of diffident practices, their conversion to outspoken dissidence, also corresponds to the symmetrical symptom of unlimited authority; when old-time politicians warned on ‘Caesarism,’ ‘Vizierate,’ ‘Despotism,’ ‘Omnipotence’ or ‘Tyranny,’ the Romanian society had already been training for a long experience of ‘dictatorship.’. (shrink)

The idea that a being can be identical to its own essence has a long and venerable history in theological speculation. As with many ideas in theology, however, such an idea has never been given an adequate mathematical formulation. The key to such a formulation, I argue, is introducing an essence axiom into non-well-founded set theory. According to such an axiom, for every set, x, there is a set that contains all and only those sets that contain x. With such (...) an axiom, it becomes possible to introduce an essence function that takes each set to its essence. An object that is identical to its essence, then, is simply a fixed point of the essence function. In this paper, I first discuss a theorem according to which any fixed point of the essence function is universally symmetrical with respect to the set membership relation. After proving the theorem, I discuss its theological implications. I go on to argue that the process of counting the members of a set that is identical to its essence is a Sisyphusian process. Such a fact I suggest shows that the absurdity of human existence is not alleviated by the existence of God but rather is entailed by it. (shrink)

The spin-statistics connection is derived in a simple manner under the postulates that the original and the exchange wave functions are simply added, and that the azimuthal phase angle, which defines the orientation of the spin part of each single-particle spin-component eigenfunction in the plane normal to the spin-quantization axis, is exchanged along with the other parameters. The spin factor (−1)2s belongs to the exchange wave function when this function is constructed so as to get the spinor ambiguity under control. (...) This is achieved by effecting the exchange of the azimuthal angle by means of rotations and admitting only rotations in one sense. The procedure works in Galilean as well as in Lorentz-invariant quantum mechanics. Relativistic quantum field theory is not required. (shrink)

In this work we study dimensional theoretical properties of some a±ne dynamical systems. By dimensional theoretical properties we mean Hausdor® dimension and box- counting dimension of invariant sets and ergodic measures on theses sets. Especially we are interested in two problems. First we ask whether the Hausdor® and box- counting dimension of invariant sets coincide. Second we ask whether there exists an ergodic measure of full Hausdor® dimension on these invariant sets. If this is not the case we ask the (...) question, whether at least the variational principle for Haus- dor® dimension holds, which means that there is a sequence of ergodic measures such that their Hausdor® dimension approximates the Hausdor® dimension of the invariant set. It seems to be well accepted by experts that these questions are of great importance in developing a dimension theory of dynamical systems (see the book of Pesin about dimension theory of dynamical systems [PE2]). Dimensional theoretical properties of conformal dynamical systems are fairly well understood today. For example there are general theorems about conformal repellers and hyperbolic sets for conformal di®eomorphisms (see chapter 7 of [PE2]). On the other hand the existence of two di®erent rates of expansion or contraction forces problems that are not captured by a general theory this days. At this stage of de- velopment of the dimension theory of dynamical systems it seems natural to study non conformal examples. This is the ¯rst step to understand the mechanisms that determine dimensional theoretical properties of non conformal dynamical systems. A±ne dynamical systems represent simple examples of non conformal systems. They are easy to de¯ne, but studying their dimensional theoretical properties does never- theless provide challenging mathematical problems and exemplify interesting phe- nomena. We consider here a special class of self-a±ne repellers in dimension two, depending on four parameters (see 2.1.). Furthermore we study a class of attractors of piecewise a±ne maps in dimension three depending on four parameters as well. The last object of our work are projections of these maps that are known as gener- alized Baker's transformations (see 2.2.). The contents of our work is the following: In chapter two we give an overview about some main results in the area of di- mension theory of a±ne dynamical systems and de¯ne the systems we study in this work. We will explain, what is known about the dimensional theoretical properties of these systems and describe what our new results are. In chapter three we then apply symbolic dynamics to our systems. We will introduce explicit shift codings 4 and ¯nd representations of all ergodic measures for our systems using these codings. From chapter four to chapter eight we study dimensional theoretical properties, which our systems generally or generically have. In chapter four we will prove a formula for the box-counting dimension of the repellers and the attractors (see the- orem 4.1.). Then in chapter ¯ve we apply general dimensional theoretical results for ergodic measures found by Ledrappier and Young [LY] and Barreira, Schmeling and Pesin [BPS] to our systems. These results relate the dimension of ergodic measures to metric entropy and Lyapunov exponents. Using this approach we will be able to reduce questions about the dimension of ergodic measures in our context to ques- tions about certain overlapping and especially overlapping self-similar measures on the line. These overlapping self-similar measures are studied in chapter six. Our main theorem extends a result of Peres and Solomyak [PS2] concerning the absolute continuity resp. singularity of symmetric self-similar measures to asymmetric ones (see theorem 6.1.3.). In chapter seven we bring our results together. We prove that we generically (in the sense of Lebesgue measure on a part of the parameter space) have the iden- tity of box-counting and Hausdor® dimension for the repellers and the attractors. (see theorem 7.1.1. and corollary 7.1.2.). This result suggest that one can expect that the identity of box-counting dimension and Hausdor® dimension holds at least generically in some natural classes of non conformal dynamical systems. Furthermore we will see in chapter seven that there generically exists an ergodic measure of full Hausdor® dimension for the repellers. On the other hand the vari- ational principle for Hausdor® dimension is not generic for the attractors. It holds only if we assume a certain symmetry (see theorem 7.1.1.). For generalized Baker's transformations we will ¯nd a part of the parameter space where there generically is an ergodic measure of full dimension and a part where the variational principle for Hausdor® dimension does not hold (see theorem 7.1.3.). Roughly speaking the reason why the variational principle does not hold here is, that if there exists both a stable and an unstable direction one can not generically maximize the dimension in the stable and in the unstable direction at the same time. In an other setting this phenomenon was observed before by Manning and McCluskey [MM]. In chapter eight we extend some results of the last section to invariant sets that correspond to special Markov chains instead of full shifts (see theorem 8.1.1.). In the last two chapters of our work we are interested in number theoretical excep- tions to our generic results. The starting point of our considerations in section nine are results of ErdÄos [ER1] and Alexander and Yorke [AY] that establish singularity and a decrease of dimension for in¯nite convolved Bernoulli measures under special conditions. Using a generalized notion of the Garsia entropy ([GA1/2]) we are able 5 to understand the consequences of number theoretical peculiarities in broader class of overlapping measures (see theorem 9.1.1.). In chapter ten we then analyze number theoretical peculiarities in the context of our dynamical systems. We restrict our attention to a symmetric situation where we generically have the existence of a Bernoulli measure of full dimension and the identity of Hausdor® and box-counting dimension for all of our systems. In the ¯rst section of chapter ten we ¯nd parameter values such that the variational principle for Hausdor® dimension does not hold for the attractors and for the Fat Baker's transformations (see theorem 10.1.1.). These are the ¯rst known examples of dynamical systems for which the variational principle for Hausdor® dimension does not hold because of number theoretical peculiarities of parameter values. For the repellers we have been able to show that under certain number theoretical conditions there is at least no Bernoulli measure of full Hausdor® dimension; the question if the variational principle for Hausdor® dimension holds remains open in this situation. In the second section of chapter ten we will show that the identity for Hausdor® and box-counting dimension can drops because there are number theoretical pecu- liarities. In the context of Weierstrass-like functions this phenomenon was observed by Przytycki and Urbanski [PU]. Our theorem extends this result to a larger class of sets, invariant under dynamical systems (see theorem 10.2.1). At the end of this work the reader will ¯nd two appendices, a list of notations and the list of references. In appendix A we introduce the notions of dimension we use in this work and collect some general facts in dimension theory. In appendix B we state the facts about Pisot-Vijayarghavan number, we need in our analysis of number theoretical peculiarities. The list of notations contains general notations and a table with a summary of notations we use to describe the dynamical systems that we study. Acknowledgments I wish to thank my supervisor JÄorg Schmeling for a lot of valuable discussion and all his help. Also thanks to Luis Barreira for his great hospitality in Lisboa and many interesting comments. This work was done while I was supported by "Promotionstipendium gem. NaFÄoG der Freien UniversitÄat Berlin". (shrink)