A Clifford-algebraic interpretation is proposed of the charge, mass, spin relationship found recently by Cooperstock and Faraoini, which was based on the Kerr–Newman metric solutions of the Einstein–Maxwell equations. The components of the polymomentum associated with a Clifford polyparticle in four dimensions provide for such a charge, mass, spin relationship without the problems encountered in Kaluza–Klein compactifications which furnish an unphysically large value for the electron charge. A physical reasoning behind such charge, mass, spin relationship is provided, followed by a (...) discussion on the geometrical derivation of the fine structure constant by Wyler, Smith, Gonzalez-Martin and Smilga. To finalize, the renormalization of electric charge is discussed and some remarks are made pertaining the modifications of the charge–scale relationship, when the spin of the polyparticle changes with scale, that may cast some light into the alleged Astrophysical variations of the fine structure constant. (shrink)

The fine structure constant α ≡ e2/ c ≈ 1/137 is one of the fundamental parameters of the standard model of particle physics. There is a long history of attempts to derive the measured value of α from an underlying theory, or exhibit it in the form of a compact mathematical expression [2–4, 6, 8, 14–16]. The most significant advance in this endeavour was made by Dirac, who showed that if magnetic monopoles exist, with magnetic charge μ, (...) then.. (shrink)

A modified form of Mach's principle is proposed, and its consequences are discussed.

Two particles, described by an irreducible two-particle representation of the Poincaré group, are correlated by the constraints that the constancy of the Casimir operators imposes on the state space. This correlation can be understood as a geometrically caused interaction between the particles, the strength of which is related to the normalisation constant \ of the two-particle states by \. The numerical value of \ is found to match the experimental value of the electromagnetic fine structure constant (...) \. This strongly suggests that the correlation of two particles in an irreducible two-particle representation of the Poincaré group manifests itself in the electromagnetic interaction. (shrink)

of a logarithmic time dependence of the fine structure constant is apparently within the limits discussed if there is a corresponding logarithmic time dependence of the strong coupling constant also. Moreover the recent discover> of naturally occurring ' Pu places the Gamow hypothesis of e' r much nearer the allov'able limits than had previously been supposed.

Precision testing of the quantum electrodynamics (QED) and the standard model provides some of the most secure knowledge in the history of physics. These tests can also be used to constrain and search for new physics going beyond the standard model. We examine the evidential structure of relationships between theoretical predictions from QED, precision measurements of these phenomena, and the indirect determination of the fine structure constant. We argue that "pure QED" is no longer sufficient to (...) predict the electron's anomalous magnetic moment, and that standard model effects are needed. The details relevant to low-energy QED are robust against future theory change. (shrink)

The reason for physics’ failure to find a final theory of the universe is examined. Problems identified are: the lack of unequivocal definitions for its fundamental elements (time, length, mass, electric charge, energy, work, matter-waves); the danger of relying too much on mathematics for solutions; especially as philosophical arguments conclude the universe cannot have a mathematical basis. It does not even need the concept of number to exist. Numbers and mathematics are human inventions arising from the human predilection for measurement. (...) Following Aristotle, a single fundamental cause is proposed to explore the efficacy of using pure non-mathematical philosophy to explain the universe, contrary to current quantum physical views. The cause is taken as Time, which is then defined, surprisingly leading automatically to a definition of a three-dimensional space answering the question into what can a universe be placed (space before space seems non-sensical). This enables the philosophical arguments to derive a universal rule giving clear-cut descriptions (definitions) of force (both gravitational and electromagnetic), motion, energy, and particles. The formation of atomic nuclei and atomic properties together with an unexpected role for neutrinos follow. In particular, a Popper-test can be prepared by introducing the concept of measurement to allow the philosophy arguments to be tested in another discipline - mathematics. The solution agrees with human physical observations to a remarkable degree of accuracy, automatically explaining and predicting values for Planck’s and the fine-structure constants. Although mathematical, it is included as confirming the efficacy of philosophy in attaining a final theory. (shrink)

Is the mathematical description of the Universe quantum, classical, both or neither? The mandated assumption of rationalism is that if an argument is inconsistent, it is flawed for a conclusion. However, suppose the structural basis of the Universe is fundamentally inconsistent. In that case, paradoxes in the frameworks of logic and mathematics would not be anomalies. A geometric model with a counter-rational framework of inconsistent relationships is applied to analyze Hardy’s paradox, the fine structure constant, and the (...) general relationship between the correlated quantum and classical EPR-type structures. The model conjectures that the well-studied paradoxes found in theoretical arguments and empirically in EPR phenomena are not anomalies and instead point to a new framework for modelling universal structures that incorporates inconsistency. (shrink)

The electric field of an orbiting charge or electron observed in the rotating frame takes on a circular trajectory with a maximum radius of \. The resultant extended electromagnetic structure is used to derive the spin–orbit energy of the orbiting electron. A surprising result of the derived expression is that the orbital velocity has a specific value ) in close agreement ) with the experimentally determined value for the fine structure constant ). Furthermore, the derived spin–orbit (...) expression does not include a g-factor ) which means that the Larmor and Thomas precessions are equal and opposite, resulting in zero net precession in the lab frame. These results suggest that the quantised fine structure constant and by extension Planck’s constant, are a natural extension of classical electromagnetism and conservation of energy in a rotating frame. (shrink)

Matter is pictured as a primitive fluid substratum having the fundamental property of fluctuating at a constant frequency. From this are derived the discrete properties of space and time, and it follows that, at the microlevel, talk of pure space and pure time involves us in ambiguities. A new interpretation of Planck's constant emerges according to which it is a quantum of matter-time combination. Thus, a quantum of matter-space combination should exist. On pursuing further the hydrodynamic model, such (...) a constant is in fact discovered as the drag-quantum of the quantum fluid. A fourth-degree differential equation is considered which, with the help of this new constant, generates spectra of frequency, mass, and fine structure constants. The theory seems to answer some important fundamental questions. (shrink)

Modern physics describes the mechanics of the Universe. We have discovered a new foundation for physics, which explains the components of the Universe with precision and depth. We quantify the existence of Aether, subatomic particles, and the force laws. Some aspects of the theory derive from the Standard Model, but much is unique. A key discovery from this new foundation is a mathematically correct Unified Force Theory. Other fundamental discoveries follow, including the origin of the fine structure (...) class='Hi'>constant and subatomic particle g-factors, a slight correction of neutron magnetic moment, a geometrical structure for charge, the quantification of electromagnetic charge as separate from electrostatic charge, a more precise meaning of spin, the quantification of space-resonance in five dimensions, and a new system of quantum units. The Aether quantifies as a fabric of quantum rotating magnetic fields with electromagnetic, electrostatic, and gravitational dipole structures. Subatomic particles quantify as angular momentum encapsulated in a quantum, rotating magnetic field. All quantum, atomic, and molecular processes can be precisely modeled, leading to discrete physics with new understandings and insights. (shrink)

A nontechnical review is given of a topological bootstrap theory, with emphasis on theraison d'être for an electromagnetism whose fine-structure constant is of order10 −2.

A model of the extended classical charged particle is developed further to prove that the electron potential can be expressed as a superposition of null waves. The null waves are solutions of the homogeneous wave equation and are related to some recently discovered types of solutions which are localized and propagate without dispersion. Connections with quantum electrodynamics and the fine structure constant are indicated.

Unitary, irreducible representations of the proper, orthochronous Lorentz group comprise the main series and the supplementary series. The main series is spanned by the complete set of eigenstates of the self-adjoint Casimir operator C 1=−(1/2)M μν M μν , where M μν are generators of Lorentz transformations. The supplementary series has no such interpretation; moreover it is spurious from the point of view of functional analysis as it does not enter into the integral representation of an arbitrary test function. The (...) author describes the physical context within which the supplementary series arises, nevertheless, in a natural way: it arises if we exponentiate a massless scalar quantum field living in three-dimensional de Sitter space-time and if the appropriate coupling constant is small enough. (shrink)

Fundamental Theory has been called an "unfinished symphony" and "a challenge to the musicians among natural philosophers of the future". This book, written in 1944 but left unfinished because Eddington died too soon, proved to be his final effort at a vision for harmonization of quantum physics and relativity. The work is less connected and internally integrated than 'Protons and Electrons' while representing a later point in the author's thought arc. The really interested student should read both books together.The physical (...) and mathematical reasoning are very deep, but it is possible to enjoy much of the flavor of the book even without following everything. The tidbits given below illustrate Eddington's beautiful and mellifluous English style. This edition contains a clickable table of contents and some interesting illustrations.WARNING: Due to the complexity of the mathematical typesetting, the only way to reproduce this book has been as a series of scanned page images. We believe this work is scientifically and culturally important, and despite a few imperfections and the inevitably small print, have brought it back into print as part of our continuing commitment to the preservation of Eddington's work. On a portable electronic reader the pages come out pretty small, but they are legible. On a desktop computer running the Amazon app, they are perfectly fine. We appreciate your understanding, and hope you enjoy this valuable book.TIDBITS:The number of protons and electrons in the universe would, in itself, be merely a matter of idle curiosity. But N has a more general significance as a fundamental constant which enters into many physical formulae. Its special interpretation as the number of particles in the universe arises in the following way. If we consider a distribution of hydrogen in equilibrium at zero temperature, the presence of the matter produces a curvature of space, and the curvature causes the space to close when the number of particles contained in it reaches a certain total; that number is N. The present investigation seeks to determine N directly from the principles of measurement. We have to show, not that there are N particles in the universe, but that anyone who accepts certain elementary principles of measurement must, if he is consistent, think there are. We have to express in mathematical symbolism what we think we are doing when we measure things; for if we had no conception of what we were doing, the results of the measurements would not persuade us to believe anything in particular. All our results are derived from the condition that the conceptual interpretation which we place on the results of measurement must be consistent with our conceptual interpretation of the process of measurement.If we play about with a pin and a meter standard, we do not become aware of any number in particular unless the standard has been graduated, or unless we use a process of displacement which we interpret as adding pin-extensions. The fact is that, although measurement is primarily a process involving four entities, the conceptual interpretation of measurement postulates in addition the existence (in the structure contemplated) as something referred to as Z. The existence of Z is the condition that makes graduation an exact concept. In the actual universe there exists a basis which is uniform to a very high approximation, and this serves for almost all purposes; but to the much higher approximation required in the calculation of N the ideal exact basis of uniformity does not exist. The finitude of N is, in fact, the cause of the failure of the approximation.It must be remembered that we are now working to what would ordinarily be considered a fantastically high approximation. For ordinary purposes a measurable has 16 eigenvalues; but under the super-magnification here employed (which is not content to overlook 1 part in 10E39) there is a fine structure which splits each of them into 16 components. (shrink)

Fundamental Theory has been called an "unfinished symphony" and "a challenge to the musicians among natural philosophers of the future". This book, written in 1944 but left unfinished because Eddington died too soon, proved to be his final effort at a vision for harmonization of quantum physics and relativity. The work is less connected and internally integrated than 'Protons and Electrons' while representing a later point in the author's thought arc. The really interested student should read both books together.The physical (...) and mathematical reasoning are very deep, but it is possible to enjoy much of the flavor of the book even without following everything. The tidbits given below illustrate Eddington's beautiful and mellifluous English style. This edition contains a clickable table of contents and some interesting illustrations.WARNING: Due to the complexity of the mathematical typesetting, the only way to reproduce this book has been as a series of scanned page images. We believe this work is scientifically and culturally important, and despite a few imperfections and the inevitably small print, have brought it back into print as part of our continuing commitment to the preservation of Eddington's work. On a portable electronic reader the pages come out pretty small, but they are legible. On a desktop computer running the Amazon app, they are perfectly fine. We appreciate your understanding, and hope you enjoy this valuable book.TIDBITS:The number of protons and electrons in the universe would, in itself, be merely a matter of idle curiosity. But N has a more general significance as a fundamental constant which enters into many physical formulae. Its special interpretation as the number of particles in the universe arises in the following way. If we consider a distribution of hydrogen in equilibrium at zero temperature, the presence of the matter produces a curvature of space, and the curvature causes the space to close when the number of particles contained in it reaches a certain total; that number is N. The present investigation seeks to determine N directly from the principles of measurement. We have to show, not that there are N particles in the universe, but that anyone who accepts certain elementary principles of measurement must, if he is consistent, think there are. We have to express in mathematical symbolism what we think we are doing when we measure things; for if we had no conception of what we were doing, the results of the measurements would not persuade us to believe anything in particular. All our results are derived from the condition that the conceptual interpretation which we place on the results of measurement must be consistent with our conceptual interpretation of the process of measurement.If we play about with a pin and a meter standard, we do not become aware of any number in particular unless the standard has been graduated, or unless we use a process of displacement which we interpret as adding pin-extensions. The fact is that, although measurement is primarily a process involving four entities, the conceptual interpretation of measurement postulates in addition the existence (in the structure contemplated) as something referred to as Z. The existence of Z is the condition that makes graduation an exact concept. In the actual universe there exists a basis which is uniform to a very high approximation, and this serves for almost all purposes; but to the much higher approximation required in the calculation of N the ideal exact basis of uniformity does not exist. The finitude of N is, in fact, the cause of the failure of the approximation.It must be remembered that we are now working to what would ordinarily be considered a fantastically high approximation. For ordinary purposes a measurable has 16 eigenvalues; but under the super-magnification here employed (which is not content to overlook 1 part in 10E39) there is a fine structure which splits each of them into 16 components. (shrink)

How is mind related to matter? This ancient question in philosophy is rapidly becoming a core problem in science, perhaps the most important of all because it probes the essential nature of man himself. The origin of the problem is a conflict between the mechanical conception of human beings that arises from the precepts of classical physical theory and the very different idea that arises from our intuition: the former reduces each of us to an automaton, while the latter allows (...) our thoughts to guide our actions. The dominant contemporary approaches to the problem attempt to resolve this conflict by clinging to the classical concepts, and trying to explain away our misleading intuition. But a detailed argument given here shows why, in a scientific approach to this problem, it is necessary to use the more basic principles of quantum physics, which bring the observer into the dynamics, rather than to accept classical precepts that are profoundly incorrect precisely at the crucial point of the role of human consciousness in the dynamics of human brains. Adherence to the quantum principles yields a dynamical theory of the mind/brain/body system that is in close accord with our intuitive idea of what we are. In particular, the need for a self-observing quantum system to pose certain questions creates a causal opening that allows mind/brain dynamics to have three distinguishable but interlocked causal processes, one micro-local, one stochastic, and the third experiential. Passing to the classical limit in which the critical difference between zero and the finite actual value of Planck's constant is ignored not only eliminates the chemical processes that are absolutely crucial to the functioning of actual brains, it simultaneously blinds the resulting theoretical construct to the physical fine structure wherein the effect of mind on matter lies: the use of this limit in this context is totally unjustified from a physics perspective. (shrink)

There is long-standing conflict between genealogical and developmental accounts of homology. This paper provides a general framework that shows that these accounts are compatible and clarifies precisely how they are related. According to this framework, understanding homology requires both an abstract genealogical account that unifies the application of the term to all types of characters used in phylogenetic systematics and locally enriched accounts that apply only to specific types of characters. The genealogical account serves this unifying role by relying on (...) abstract notions of ‘descent’ and ‘character’. As a result, it takes for granted the existence of such characters. This requires theoretical justification that is provided by enriched accounts, which incorporate the details by which characters are inherited. These enriched accounts apply to limited domains, providing the needed theoretical justification for recognizing characters within that domain. Though connected to the genealogical account of homology in this way, enriched accounts include phenomena that fall outside the scope of the genealogical account. They therefore overlap, but are not nested within, the genealogical account. Developmental accounts of homology are to be understood as enriched accounts of body part homology. Once they are seen in this light, the conflict with the genealogical account vanishes. It is only by understanding the fine conceptual structure undergirding the many uses of the term ‘homology’ that we can understand how these uses hang together. (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)

We extend the construction of a global square sequence in extender models from Zeman [8] to a construction of coherent non-threadable sequences and give a characterization of stationary reflection at inaccessibles similar to Jensen’s characterization in L.

This chapter contains section titled: The Structure of the Focus of Appreciation Performance and Appreciation Ontology After Empiricism.

Before one can construct scales of minimal complexity in the Real Core Model, K(R), one needs to develop the fine-structure theory of K(R). In this paper, the fine structure theory of mice, first introduced by Dodd and Jensen, is generalized to that of real mice. A relative criterion for mouse iterability is presented together with two theorems concerning the definability of this criterion. The proof of the first theorem requires only fine structure; whereas, the (...) second theorem applies to real mice satisfying AD and follows from a general definability result obtained by abstracting work of John Steel on L(R). In conclusion, we discuss several consequences of the work presented in this paper relevant to two issues: the complexity of scales in K(R) and the strength of the theory ZF + AD + ¬ DC R. (shrink)

Traditionally, philosophers have focused mostly on the logical template of inference. The paradigm-case has been deductive inference, which is topic-neutral and context-insensitive. The study of deductive rules has engendered the search for the Holy Grail: syntactic and topic-neutral accounts of all prima facie reasonable inferential rules. The search has hoped to find rules that are transparent and algorithmic, and whose following will just be a matter of grasping their logical form. Part of the search for the Holy Grail has been (...) to show that the so-called scientific method can be formalised in a topic-neutral way. We are all familiar with Carnap’s inductive logic, or Popper’s deductivism or the Bayesian account of scientific method. (shrink)

It is shown how certain generic extensions of a fine structural model in the sense of Mitchell and Steel [MiSt] can be reorganized as relativizations of the model to the generic object. This is then applied to the construction of Steel's core model for one Woodin cardinal [St] and its generalizations.

Lines of identity in Peirce's existential graphs are logically complex structures that comprise both identity and existential quantification. Yet geometrically they are simple: linear continua that cannot have “furcations” or cross “cuts.” By contrast Peirce's “ligatures” are geometrically complex: they can both have furcations and cross cuts. Logically they involve not only identity and existential quantification but also negation. Moreover, Peirce makes clear that ligatures are composed of lines of identity by virtue of the fact that such lines can be (...) “connected” with one another and can “abut upon” one another at a cut. This paper shows in logical detail how ligatures are composed and how they relate to identity, existential quantification, and negation. In so doing, it makes use of Peirce's non-standard account of the linear continuum, according to which, when a linear continuum is separated into two parts, the parts are symmetric rather than asymmetric, and the one point at which separation occurs actually becomes two points, each of which is a Doppelgänger of the other. (shrink)

We model Correia's hybrid algebraic/structural semantics within a form of the truthmaker semantics in which the propositions of the semantics are taken to to be closed under a well-behaved closure operation.

Let M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{M}}$$\end{document} be a fine structural mouse. Let D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{D}}$$\end{document} be a fully backgrounded L[E]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L[\mathbb{E}]}$$\end{document}-construction computed inside an iterable coarse premouse S. We describe a process comparing M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{M}}$$\end{document} with D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{D}}$$\end{document}, through forming iteration trees on M\documentclass[12pt]{minimal} (...) \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{M}}$$\end{document} and on S. We then prove that this process succeeds. (shrink)

A number of philosophers have recently become receptive to the idea that, in addition to scientific or causal explanation, there may be a distinctive kind of metaphysical explanation, in which explanans and explanandum are connected, not through some sort of causal mechanism, but through some constitutive form of determination. I myself have long been sympathetic to this idea of constitutive determination or ‘ontological ground’; and it is the aim of the present paper to help put the idea on a firmer (...) footing - to explain how it is to be understood, how it relates to other ideas, and how it might be of use in philosophy. (shrink)

I lay down a system of structural rules for various notions of ground and establish soundness and completeness.

In this paper we introduce a collection of isols having some interesting properties. Imagine a collection W of regressive isols with the following features: u, v ϵ W implies that u ⩽ v or v ⩽ u, u ⩽ v and v ϵ W imply u ϵ W, W contains ℕ = {0,1,2,…} and some infinite isols, and u eϵ W, u infinite, and u + v regressive imply u + v ϵ W. That such a collection W exists is (...) proved in our paper. It has many nice features. It also satisfies u, v ϵ W, u ⩽ v and u infinite imply v ⩽ g for some recursive combinatorial function g, and each u ϵ W is hereditarily odd-even and is hereditarily recursively strongly torre. The collection W that we obtain may be characterized in terms of a semiring of isols D introduced by J. C. E. Dekker in [5]. We will show that W = D, where c is an infinite regressive isol that is called completely torre. (shrink)

We pose a puzzle for forms and show how it might be solved by appeal to the theory of arbitrary objects. We also discuss how the resulting account of forms relates to issues concerning structural universals and the nature of abstraction.

The interest in "social sustainability" has recently increased in the field of urban development. We want societies, cities, and neighborhoods to be economically and environmentally sustainable, but we also want urban areas that are safe, diverse, walkable, and relaxing, just to mention a few examples. Strikingly, however, there is no consensus regarding what definition of "social sustainability" should be employed. Additionally, some people are skeptical about the prospect of finding a useful definition at all and claim it is impossible to (...) satisfactorily define the concept for various reasons, such as its complexity. A potential first step towards navigating this conceptual maze is to provide desiderata for a definition of social sustainability. We defend a list of nine desiderata and thereby create a theoretical framework for analyzing and constructing a definition of "social sustainability". We also examine the skeptical arguments and find that it is premature to conclude that the goal of finding a useful definition is hopeless. With the criteria in place, the future debate can proceed by assessing definitions of "social sustainability" in a more structured and transparent manner. This activity is of upmost importance if we want to create just cities. Keywords: Social Sustainability, Definition, Purposes and Aims, Conditions of Adequacy. (shrink)

In this thesis, I deal with the notions of a condition holding for some proposition and a proposition being true in a certain number of possible worlds. These notions are called propositional quantifiers and numerical modalizers respectively. In each chapter, I attempt to dispose of a system. A system consists of: a language; axioms and rules of inference; and an interpretation. To dispose of a system is to prove its decidability and its consistency and completeness for the given interpretation. I (...) shall, in passing, make applications to de definability, translatability and other topics. In Chapter 1, I consider the system 5SQ. Its language is that of 55 with Q as a fresh unary operator. Its axioms and rules of inference are those for 85 plus the following special axiom-schemes for Q: 1) QA::>MA 2) Q A ::> L V L 3) L ::> 4) Q A ::> L Q A. 'QA' is interpreted as 'A is true in exactly one possible world.' I dispose of the system by showing that every formula in it is equivalent to one in normal form. In Chapter 2 I consider the system S5n. Its language is that of S5 but with the unary operators Qk for each non-negative integer k. Its axioms and rules are those of S5 plus the following special axiom-schemes for Qk: 1) Qk A>~Q1 A, 1˂K 2) Qk A ≡ Vki≡oQi ˄ Qk-I 3) L ˂ 4) Qk A ˂ L Qk A 5) Qo A ≡ L ~ A, 1 ≥ 0, k ≥ 1 ‘Qk A’ is interpreted as 'A is true in exactly k possible worlds.' I dispose of this system by generalising the normal forms of S5Q. In the chapters 3-5, I consider three systems which result from adding propositional quantifiers to S5. The first two systems, S5╥+ and S5╥, contain the usual axioms and rules for quantifiers. The first contains, in addition, the axiom-scheme G = )). The last, S5╥-, results from S5╥ by restricting the Scheme of Specification, viz., A > A, B free for P in A, to formulas B of the propositional calculus. To interpret these systems we must specify which propositions the variable P ranges over. For St╥-, we merely require that if p and q are propositions, then and are also propositions. For S5╥+, we also require that each possible world be describable i.e. that there be a proposition which is true in that world alone. And for S5╥, we require not that each possible world be describable but that there be a proposition which is true in just those possible worlds which are describable. Again, we dispose of the systems by normal forms. This requires that we eliminate quantifiers and nested occurrences of L by adding new symbols to the language. For S5╥+, the operators Qk suffice. For S5╥, the operators Qk suffice. For S5╥, we also require the constant g and a fresh unary operator N. For S5╥-, even greater additions are required. In the last two chapters, 6 and 7, I turn to systems which have essentially the same language as S5╥. However, ‘Qk A' is now interpreted as 'A is true in exactly k possible worlds accessible from the given world.' Different conditions on R, the relation of accessibility, lead to different axioms. In chapter 6 I consider the conditions of reflexivity, symmetry and transitivity, and in Chapter 7 the conditions of being a partial, convergent, total or dense order. I prove consistency and completeness by the method of maximally consistent systems. The method can yield decidability results, but I do not go into the matter. I have, as a rule, not given acknowledgements for well-established results or terminology. The main references are at the end of each chapter. Fuller references are in the bibliography. (shrink)

Micro-sociological theory has traditionally stressed interactional pressures towards alignment: actors’ attempts to co-construct a shared definition of the situation. We argue that this model provides an insufficient account of the coordination of action and of the emergence of intersubjectivity among actors. To complement the focus on alignment, we develop a theory of disruption—a perceived misalignment of the dramaturgical structure of interaction in coordinating expected lines of action. We develop a theory of the interaction order that takes the interplay between (...) interactional alignment and disruption as a foundational challenge both for sociology and for actors in their everyday lives. We focus on the practical ways in which actors negotiate both interactional breaches and wider relational ruptures, and how they differentiate between disruptions-of relations and disruptions-for them. By doing so, we connect the interaction order to a wider relational order, providing a bridge between micro-level interactionism and the sociology of culture. (shrink)