Thomas Müller

The article puts forward a branching-style framework for the analysis of determinism and indeterminism of scientific theories, starting from the core idea that an indeterministic system is one whose present allows for more than one alternative possible future. We describe how a definition of determinism stated in terms of branching models supplements and improves current treatments of determinism of theories of physics. In these treatments, we identify three main approaches: one based on the study of equations, one based on mappings between temporal realizations, and one based on branching models. We first give an overview of these approaches and show that current orthodoxy advocates a combination of the mapping- and the equations-based approaches. After giving a detailed formal explication of a branching-based definition of determinism, we consider three concrete applications and end with a formal comparison of the branching- and the mapping-based approach. We conclude that the branching-based definition of determinism most usefully combines formal clarity, connection with an underlying philosophical notion of determinism, and relevance for the practical assessment of theories. 1 Introduction2 Determinism in Philosophy of Science: Three Approaches2.1 Determinism: The core idea and how to spell it out2.2 The three approaches in more detail2.3 Representing indeterminism3 Orthodoxy: DMAP, with Invocations of DEQN4 Branching-Style Determinism 4.1 Models and realizations4.2 Faithfulness4.3 Two types of branching topologies 5 Comparing the Approaches5.1 Case studies5.2 Formal comparison of the DMAP and DBRN frameworks6 Conclusions.

In the consistent histories formalism one specifies a family of histories as an exhaustive set of pairwise exclusive descriptions of the dynamics of a quantum system. We define branching families of histories, which strike a middle ground between the two available mathematically precise definitions of families of histories, viz., product families and Isham’s history projector operator formalism. The former are too narrow for applications, and the latter’s generality comes at a certain cost, barring an intuitive reading of the “histories”. Branching families retain the intuitiveness of product families, they allow for the interpretation of a history’s weight as a probability, and they allow one to distinguish two kinds of coarse-graining, leading to reconsidering the motivation for the consistency condition

Paragraph 50 of Wittgenstein'sPhilosophical Investigationsfamously says that there is one thing of which one can neither state that it is 1 m long nor that it isn't: the standard metre in Paris. Consensus appears to be that (1) exegetically speaking, Wittgenstein affirms this claim, and (2) systematically, whether or not one agrees with it, the practice of using a material artefact as a measurement standard has important philosophical consequences. In this paper, in contrast, we show that (1') Wittgenstein does not affirm the standard metre claim and (2') historical facts about material length standards do not warrant deeper philosophical conclusions. References to material measurement standards in the literature gravely distort the actual metrological practices of developing and using such standards.

Quantum mechanics predicts non-local correlations in spatially extended entangled quantum systems, and these correlations are empirically very well confirmed. This raises philosophical questions of how nature could be that way, prompting the study of purported completions of quantum mechanics by hidden variables. Bell-type theorems connect assumptions about hidden variables with empirical predictions for the outcome of quantum correlation experiments. From among these assumptions, the Setting Independence assumption has received the least formal attention. Its violation is, however, central in the recent discussion about super-deterministic models for quantum correlation experiments. In this paper, we focus on the non-local modal correlations in the GHZ experiment. We model the introduction of hidden variables in the form of instruction sets via structure extensions in the framework of Branching Space-Times. This framework allows us to show in formal detail how the introduction of non-contextual instruction sets results in a specific violation of Setting Independence; a similar result is derived for contextual instruction sets. Our discussion provides additional reasons for foregoing the introduction of local hidden variables, and for accepting non-local quantum correlations as a resource provided by nature.

The metaphor of a branching tree of future possibilities has a number of important philosophical and logical uses. In this paper we trace this metaphor through some of its uses and argue that the metaphor works the same way in physics as in philosophy. We then give an overview of formal systems for branching possibilities, viz., branching time and (briefly) branching space-times. In a next step we describe a number of different notions of possibility, thereby sketching a landscape of possibilities. In the final section of the paper we look at the place of branching-based possibilities in that larger landscape of possibilities. Our main message is that far from being an outlandish metaphysical extravagancy, branching-based possibilities are epistemically as well as metaphysically basic.

The paper gives a physicist's view on the framework of branching space-time, 385--434). Branching models are constructed from physical state assignments. The models are then employed to give a formal semantics for the modal operators ``possibly'' and ``necessarily'' and for the counterfactual conditional. The resulting formal language can be used to analyze quantum correlation experiments. As an application sketch, Stapp's premises LOC1 and LOC2 from his purported proof of non-locality, 300--304) are analyzed

Prior's project of tense logic has both a formal-logical and a philosophical side. Both aspects were important for Prior. The paper suggests viewing Prior's philosophical project as a continuation of the tradition of "logic as language", or "universalism", identified by van Heijenoort and Hintikka, respectively. The label "tense-logical universalism" is chosen in order to stress Prior's emphasis on the foundational role of natural language for the three fields of logic, semantics, and philosophy of science. The paper draws on Prior's works in all three mentioned areas. In particular, we comment on Prior's universalist approach to model theory, which has recently given rise to hybrid logic, and on the interesting perspective on relativity theory offered through his writings.

Towards the end of her famous 1971 paper “Causality and Determination”, Elizabeth Anscombe discusses the controversial idea that “ ‘physical haphazard’ could be the only physical correlate of human freedom of action”. In order to illustrate how the high-level freedom of human action can go together with micro-indeterminism without creating a problem for micro-statistics, she provides the analogy of a glass box filled with minute coloured particles whose micro-dynamics is subject to statistical laws, while its outside reliably displays a recognisable variation of a certain image created by an external cause. Anscombe’s example is somewhat enigmatic, and she provides no details to support its viability. In this paper we discuss the context of Anscombe’s analogy, and we provide basics of the theory and practice of building an Anscombe box.

In this paper we describe a novel approach to defining an ontologically fundamental notion of co-presentness that does not go against the tenets of relativity theory. We survey the possible reactions to the problem of the present in relativity theory, introducing a terminological distinction between a static role of the present, which is served by the relation of simultaneity, and a dynamic role of the present, with the corresponding relation of co-presentness. We argue that both of these relations need to be equivalence relations, but they need not coincide. Simultaneity, the sharing of a temporal coordinate, need not have fundamental ontological import, so that a relativizing strategy with respect to simultaneity seems promising. The notion of co-presentness, on the other hand, does have ontological import, and can therefore not be relativized to an observer or to an arbitrarily chosen frame. We argue that a formal representation of indeterminism can provide the structure needed to anchor the relation of co-presentness, and that this addition is in fact congenial to the notion of dynamic time as requiring real change. The resulting picture is one of an extended dynamic present, implying a formal distinction between static simultaneity and dynamic co-presentness. After working out the basics of our approach in the simpler framework of branching time, we provide our full analysis in the framework of branching space-times, which allows for a formal definition of modal correlations. The spatial extension of the dynamic present can reach as far as the modal correlations do. In the limit, the dynamic present could extend across a maximal space-like hypersurface.

Anscombe’s 1971 inaugural lecture at Cambridge, entitled ‘Causality and Determination’, has had a lasting influence on a remarkably broad range of philosophers and philosophical debates, touching on fundamental topics in philosophy of science, action theory, the free will debate, epistemology, philosophy of mind, and metaphysics. Especially where anti-reductionist or pluralist strands of philosophical thought are being seriously considered, one should not be surprised to find references to Anscombe’s lecture. Moreover, there appears to be a growing interest in Anscombe’s comprehensive philosophical outlook, as attested by the recent publication of a weighty collection of essays spanning that outlook in its full breadth in the prestigious Routledge Philosophical Minds series. Against this background it is apt that now, 50 years after the original lecture, a Topical Collection sees the light, circling around the most central themes from Anscombe’s lecture, with a particular emphasis on the question how these hang together, how they form part of the larger philosophical project that Anscombe obviously intended the lecture to highlight. This Introduction motivates the Topical Collection, and introduces the various contributions against that background.

Branching space-times (BST; Belnap, Synthese 92:385–434, 1992 ) is the most advanced formal framework for representing indeterminism. BST is however based on continuous partial orderings, while our natural way of describing indeterministic scenarios may be called discrete. This paper establishes a theorem providing a discrete data format for BST: it is proved that a discrete representation of indeterministic scenarios leading to BST models is possible in an important subclass of cases. This result enables the representation of limited indeterminism in BST and hopefully paves the way for the representation of substances with capacities in that framework

The present volume contains primarily the invited papers of the 28th Inter-national Wittgenstein Symposium that was held in Kirchberg am Wech-sel (Lower Austria) in August 2005. It was dedicated to the topic Time and History (Zeit und Geschichte) in an interdisciplinary perspective, ranging from the philosophy of time, in the narrower sense, the approaches of the single scientific disciplines, in so far as they are informed by foundational and philosophical issues, to culture and art. As usual, the contributed papers (Beitr age) were already published prior to the symposium. While the latter volume contains, in a special section, papers dedicated to all aspects of Wittgenstein's work, the present volume focuses on his views about time. The editors are well aware that both time and history are prominently discussed within the phenomenological and hermeneutic traditions in philosophy. This was well reflected in the contributed papers, as can be seen in the Beitr age volume, and in some papers dealing with time and history from a cultural perspective. For reasons of thematic coherence and as a consequence of the general orientation of this book series, however, the editors have given priority to philosophers belonging to the analytic tradition in the broad sense. The editors nonetheless hope to have succeeded in presenting an equally focused and comprehensive picture of the contemporary debate.

Since the validity of Bell's inequalities implies the existence of joint probabilities for non-commuting observables, there is no universal consensus as to what the violation of these inequalities signifies. While the majority view is that the violation teaches us an important lesson about the possibility of explanations, if not about metaphysical issues, there is also a minimalist position claiming that the violation is to be expected from simple facts about probability theory. This minimalist position is backed by theorems due to A. Fine and I. Pitowsky.Our paper shows that the minimalist position cannot be sustained. To this end,we give a formally rigorous interpretation of joint probabilities in thecombined modal and spatiotemporal framework of `stochastic outcomes inbranching space-time' (SOBST) (Kowalski and Placek, 1999; Placek, 2000). We show in this framework that the claim that there can be no joint probabilities fornon-commuting observables is incorrect. The lesson from Fine's theorem is notthat Bell's inequalities will be violated anyhow, but that an adequate modelfor the Bell/Aspect experiment must not define global joint probabilities. Thus we investigate the class of stochastic hidden variable models, whichprima facie do not define such joint probabilities. The reasonwhy these models fail supports the majority view: Bell's inequalities are notjust a mathematical artifact.

The theory of branching space-times is designed as a rigorous framework for modelling indeterminism in a relativistically sound way. In that framework there is room for "funny business", i.e., modal correlations such as occur through quantummechanical entanglement. This paper extends previous work by Belnap on notions of "funny business". We provide two generalized definitions of "funny business". Combinatorial funny business can be characterized as "absence of prima facie consistent scenarios", while explanatory funny business characterizes situations in which no localized explanation of inconsistency can be given. These two definitions of funny business are proved to be equivalent, and we provide an example that shows them to be strictly more general than the previously available definitions of "funny business"

Branching theories are popular frameworks for modeling objective indeterminism in the form of a future of open possibilities. In such theories, the notion of a history plays a crucial role: it is both a basic ingredient in the axiomatic definition of the framework, and it is used as a parameter of truth in semantics for languages with a future tense. Furthermore, histories—complete possible courses of events—ground the notion of modal consistency: a set of events is modally consistent iff there is a history containing that set. We will explain these roles of histories and highlight some critical aspects having to do with the fact that histories are global and, in a relevant sense, “big” objects. The notion of modal consistency, on the other hand, has both local and global aspects. We ask in how far a local notion of modal consistency can serve as an alternative to the common uses of histories, and work out two recent approaches to alternatives to histories. Combining these approaches, we develop a novel semantics for branching time

Philosophy of mathematics is moving in a new direction: away from a foundationalism in terms of formal logic and traditional ontology, and towards a broader range of approaches that are united by a focus on mathematical practice. The scientific research network PhiMSAMP (Philosophy of Mathematics: Sociological Aspects and Mathematical Practice) consisted of researchers from a variety of backgrounds and fields, brought together by their common interest in the shift of philosophy of mathematics towards mathematical practice. Hosted by the Rheinische Friedrich- Wilhelms-Universitat Bonn and funded by the Deutsche Forschungsgemeinschaft (DFG) from 2006-2010, the network organized and contributed to a number of workshops and conferences on the topic of mathematical practice. The refereed contributions in this volume represent the research results of the network and consists of contributions of the network members as well as selected paper versions of presentations at the network's mid-term conference, "Is mathematics special?" (PhiMSAMP-3) held in Vienna 2008.

This paper follows Part I of our essay on case-intensional first-order logic (CIFOL; Belnap and Müller (2013)). We introduce a framework of branching histories to take account of indeterminism. Our system BH-CIFOL adds structure to the cases, which in Part I formed just a set: a case in BH-CIFOL is a moment/history pair, specifying both an element of a partial ordering of moments and one of the total courses of events (extending all the way into the future) that that moment is part of. This framework allows us to define the familiar Ockhamist temporal/modal connectives, most notably for past, future, and settledness. The novelty of our framework becomes visible in our discussion of substances in branching histories, i.e., in its first-order part. That discussion shows how the basic idea of tracing an individual thing from case to case via an absolute property is applicable in a branching histories framework. We stress the importance of keeping apart extensionality and moment-definiteness, and give a formal account of how the specification of natural sortals and natural qualities turns out to be a coordination task in BH-CIFOL. We also provide a detailed answer to Lewis’s well-known argument against branching histories, exposing the fallacy in that argument.

This paper follows Part I of our essay on case-intensional first-order logic ). We introduce a framework of branching histories to take account of indeterminism. Our system BH-CIFOL adds structure to the cases, which in Part I formed just a set: a case in BH-CIFOL is a moment/history pair, specifying both an element of a partial ordering of moments and one of the total courses of events that that moment is part of. This framework allows us to define the familiar Ockhamist temporal/modal connectives, most notably for past, future, and settledness. The novelty of our framework becomes visible in our discussion of substances in branching histories, i.e., in its first-order part. That discussion shows how the basic idea of tracing an individual thing from case to case via an absolute property is applicable in a branching histories framework. We stress the importance of keeping apart extensionality and moment-definiteness, and give a formal account of how the specification of natural sortals and natural qualities turns out to be a coordination task in BH-CIFOL. We also provide a detailed answer to Lewis’s well-known argument against branching histories, exposing the fallacy in that argument.

The logical theory of branching space-times, which provides a relativistic framework for studying objective indeterminism, remains mostly disconnected from discussions of space-time theories in philosophy of physics. Earman has criticized the branching approach and suggested “pruning some branches from branching space-time.” This article identifies the different—order-theoretic versus topological—perspective of both discussions as a reason for certain misunderstandings and tries to remove them. Most important, we give a novel, topological criterion of modal consistency that usefully generalizes an earlier criterion, and we introduce a differential-geometrical version of branching space-times as a non-Hausdorff (generalized) manifold

According to a mainstream position in contemporary cognitive science and philosophy, the use of abstract compositional concepts is amongst the most characteristic indicators of meaningful deliberative thought in an organism or agent. In this article, we show how the ability to develop and utilise abstract conceptual structures can be achieved by a particular kind of learning agent. More specifically, we provide and motivate a concrete operational definition of what it means for these agents to be in possession of abstract concepts, before presenting an explicit example of a minimal architecture that supports this capability. We then proceed to demonstrate how the existence of abstract conceptual structures can be operationally useful in the process of employing previously acquired knowledge in the face of new experiences, thereby vindicating the natural conjecture that the cognitive functions of abstraction and generalisation are closely related.