We introduce the notion of a Minkowskian Branching Structure ("MBS" for short). Then we prove some results concerning the phenomenon of funny business in its finitary and infinitary variants.

I discuss various formulations of stochastic Einstein locality (SEL), which is a version of the idea of relativistic causality, that is, the idea that influences propagate at most as fast as light. SEL is similar to Reichenbach's Principle of the Common Cause (PCC), and Bell's Local Causality. My main aim is to discuss formulations of SEL for a fixed background spacetime. I previously argued that SEL is violated by the outcome dependence shown by Bell correlations, both in quantum mechanics and (...) in quantum field theory. Here I reassess those verdicts in the light of some recent literature which argues that outcome dependence does not violate the PCC. I argue that the verdicts about SEL still stand. Finally, I briefly discuss how to formulate relativistic causality if there is no fixed background spacetime. (shrink)

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". (shrink)

The paper develops a theory of branching spatiotemporal histories that accommodates indeterminism and the insights of general relativity. A model of this theory can be viewed as a collection of overlapping histories, where histories are defined as maximal consistent subsets of the model's base set. Subsequently, generalized manifolds are constructed on the theory's models, and the manifold topology is introduced. The set of histories in a model turns out to be identical with the set of maximal subsets of the model's (...) base set with respect to being Hausdorff and downward closed. Further postulates ensure that the topology is connected, locally Euclidean, and satisfies the countable sub-cover condition. (shrink)

We investigate the concepts of past, present, and future that build upon a modal distinction between the settled past and the open future. The concepts are defined in terms of a pre-causal ordering and of qualitative differences between alternative histories. Finally, we look what an event's past, present, and future look like in the so-called Minkowskian Branching Structures, in which histories are isomorphic to Minkowski spacetime.

“Branching space-times” (BST) is intended as a representation of objective, event-based indeterminism. As such, BST exhibits both a spatio-temporal aspect and an indeterministic “modal” aspect of alternative possible historical courses of events. An essential feature of BST is that it can also represent spatial or space-like relationships as part of its (more or less) relativistic theory of spatio-temporal relations; this ability is essential for the representation of local (in contrast with “global”) indeterminism. This essay indicates how BST might be seen (...) to grow out of Newton’s deterministic and non-relativistic theory by two independent moves: (1) Taking account of indeterminism, and (2) attending to spatio-temporal relationships in a spirit derived from Einstein’s theory of special relativity. Since (1) and (2) are independent, one can see that there is room for four theories: Newtonian determinism, branching time indeterminism, relativistic determinism, and (finally) branching space-times indeterminism. (shrink)

“Branching space-times” is intended as a representation of objective, event-based indeterminism. As such, BST exhibits both a spatio-temporal aspect and an indeterministic “modal” aspect of alternative possible historical courses of events. An essential feature of BST is that it can also represent spatial or space-like relationships as part of its relativistic theory of spatio-temporal relations; this ability is essential for the representation of local indeterminism. This essay indicates how BST might be seen to grow out of Newton ’s deterministic and (...) non-relativistic theory by two independent moves: Taking account of indeterminism, and attending to spatio-temporal relationships in a spirit derived from Einstein’s theory of special relativity. Since and are independent, one can see that there is room for four theories: Newtonian determinism, branching time indeterminism, relativistic determinism, and branching space-times indeterminism. (shrink)

The paper defends an Aristotelian notion of indeterminism, as rigorously formulated in the framework of branching space-times (BST) of Belnap (1992), against the model-theoretic characterization of indeterminism that Montague (1962) introduced into the philosophy of science. It delineates BST branching against the background provided by Earman's (2008) distinction between individual vs. ensemble branching. It describes a construction of physically-motivated BST models, in which histories are isomorphic to Minkowski spacetime. Finally it responds to criticism leveled against BST by addressing some semantical (...) questions, a topological issue, and the past/future asymmetry. (shrink)

The paper puts forward a theory of historical modalities that is framed in terms of possible continuations rather than possible worlds or histories. The proposal is tested as a semantic theory for a language with historical modalities, tenses, and indexicals.

This paper investigates the prospects of developing a branching modal framework while keeping with the spirit of Humean Supervenience. It is argued that such an approach is bound to face hard problems regarding haecceitism and the notion of recombination. Possible directions for future philosophical developments of branching frameworks are suggested.

The paper investigates, in the framework of branching space–times, whether an infinite EPR-like correlation which does not involve finite EPR-like correlations is possible.

We suggest a rigorous theory of how objective single-case transition probabilities fit into our world. The theory combines indeterminism and relativity in the “branching space–times” pattern, and relies on the existing theory of causae causantes (originating causes). Its fundamental suggestion is that (at least in simple cases) the probabilities of all transitions can be computed from the basic probabilities attributed individually to their originating causes. The theory explains when and how one can reasonably infer from the probabilities of one “chance (...) set-up” to the probabilities of another such set-up that is located far away. (shrink)

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. (shrink)

(No abstract is available for this citation).

This paper suggests an epistemic interpretation of Belnap’s branching space-times theory based on Everett’s relative state formulation of the measurement operation in quantum mechanics. The informational branching models of the universe are evolving structures defined from a partial ordering relation on the set of memory states of the impersonal observer. The totally ordered set of their information contents defines a linear “time” scale to which the decoherent alternative histories of the informational universe can be referred—which is quite necessary for assigning (...) them a probability distribution. The “historical” state of a physical system is represented in an appropriate extended Hilbert space and an algebra of multi-branch operators is developed. An age operator computes the informational depth of historical states and its standard deviation can be used to provide a universal information/energy uncertainty relation. An information operator computes the encoding complexity of historical states, the rate of change of its average value accounting for the process of correlation destruction inherent to the branching dynamics. In the informational branching models of the universe, the asymmetry of phenomena in nature appears as a mere consequence of the subject’s activity of measuring, which defines the flow of time-information. (shrink)

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. (shrink)

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. (shrink)

Popper’s introduction of ‘‘propensity’’ was intended to provide a solid conceptual foundation for objective single-case probabilities. By considering the partly opposed contributions of Humphreys and Miller and Salmon, it is argued that when properly understood, propensities can in fact be understood as objective single-case causal probabilities of transitions between concrete events. The chief claim is that propensities are well-explicated by describing how they fit into the existing formal theory of branching space-times, which is simultaneously indeterministic and causal. Several problematic examples, (...) some commonsense and some quantum-mechanical, are used to make clear the advantages of invoking branching space-times theory in coming to understand propensities. (shrink)

We show that truth conditions for counterfactuals need not always be given in terms of a vague notion of similarity. To this end, we single out the important class of historical counterfactuals and give formally rigorous truth conditions for these counterfactuals, employing a partial ordering relation called "comparative closeness" that is defined in the framework of branching space-times. Among other applications, we provide a detailed analysis of counterfactuals uttered in the context of lost bets. In an appendix we compare our (...) theory with the branching space-times based reading of counterfactuals recently proposed by Belnap. (shrink)

In this contribution, I comment on Raffaella Campaner’s defense of explanatory pluralism in psychiatry (in this volume). In her paper, Campaner focuses primarily on explanatory pluralism in contrast to explanatory reductionism. Furthermore, she distinguishes between pluralists who consider pluralism to be a temporary state on the one hand and pluralists who consider it to be a persisting state on the other hand. I suggest that it would be helpful to distinguish more than those two versions of pluralism – different understandings (...) of explanatory pluralism both within philosophy of science and psychiatry – namely moderate/temporary pluralism, anything goes pluralism, isolationist pluralism, integrative pluralism and interactive pluralism. Next, I discuss the pros and cons of these different understandings of explanatory pluralism. Finally, I raise the question of how to implement or operationalize explanatory pluralism in scientific practice; how to structure the “genuine dialogue” or shape “the pluralistic attitude” Campaner is referring to. As tentative answers, I explore a question-based framework for explanatory pluralism as well as social-epistemological procedures for interaction among competing approaches and explanations. (shrink)

The paper develops models of statistical experiments that combine propensities with frequencies, the underlying theory being the branching space-times (BST) of Belnap (1992). The models are then applied to analyze Bell's theorem. We prove the so-called Bell-CH inequality via the assumptions of a BST version of Outcome Independence and of (non-probabilistic) No Conspiracy. Notably, neither the condition of probabilistic No Conspiracy nor the condition of Parameter Independence is needed in the proof. As the Bell-CH inequality is most likely experimentally falsified, (...) the choice is this: contrary to the appearances, experimenters cannot choose some measurement settings, or some transitions, with spacelike related initial events, are correlated; or both. (shrink)

This volume seeks to further the use of formal methods in clarifying one of the central problems of philosophy: that of our free human agency and its place in our indeterministic world. It celebrates the important contributions made in this area by Nuel Belnap, American logician and philosopher. Philosophically, indeterminism and free action can seem far apart, but in Belnap’s work, they are intimately linked. This book explores their philosophical interconnectedness through a selection of original research papers that build forth (...) on Belnap’s logical and philosophical work. Some contributions take the form of critical discussions of Belnap's published work, some develop points made in his publications in new directions, and others provide additional insights on the topics of indeterminism and free action. In Nuel Belnap’s work on indeterminism and free action, three formal frameworks figure prominently: the simple branching histories framework known as "branching time;" its relativistic spatio-temporal extension, branching space-times; and the “seeing to it that” logic of agency. As those frameworks provide the formal background for the contributed papers, the volume introduction gives an overview of the current state of their development. It also introduces case-intensional first order logic, a general intensional logic offering resources for a first-order extension of the mentioned frameworks and a recent research focus of Belnap’s. The volume also contains an extended biographical interview with Nuel Belnap. (shrink)

We investigate the concepts of past, present, and future that build upon a modal distinction between a settled past and an open future. The concepts are defined in terms of a pre-causal ordering that is determined by the qualitative differences between alternative possible histories. We look what an event’s past, present, and future look like in the so-called Minkowskian Branching Structures, one in which histories are isomorphic to Minkowski space-time.

The logical theory of branching space-times, which is intended to provide a framework for studying objective indeterminism, remains at a certain distance from the discussion of space-time theories in the philosophy of physics. In a welcome attempt to clarify the connection, Earman has recently found fault with the branching approach and suggested ``pruning some branches from branching space-time''. The present note identifies the different---order theoretic vs. topological---points of view of both discussion as a reason for certain misunderstandings, and tries to (...) remove them. Most importantly, we give a novel, topological criterion of modal consistency that usefully generalizes the order-theoretic criterion of directedness, and we introduce a differential-geometrical version of BST based on the theory of non-Hausdorff manifolds. (shrink)

In this paper I assess the prospects for combining contemporary Everettian quantum mechanics (EQM) with branching-time semantics in the tradition of Kripke, Prior, Thomason and Belnap. I begin by outlining the salient features of ‘decoherence-based’ EQM, and of the ‘consistent histories’ formalism that is particularly apt for conceptual discussions in EQM. This formalism permits of both ‘branching worlds’ and ‘parallel worlds’ interpretations; the metaphysics of EQM is in this sense underdetermined by the physics. A prominent argument due to Lewis (On (...) the Plurality of Worlds, 1986 ) supports the non-branching interpretation. Belnap et al. (Facing the Future: Agents and Choices in Our Indeterministic World, 2001 ) refer to Lewis’ argument as the ‘Assertion problem’, and propose a pragmatic response to it. I argue that their response is unattractively ad hoc and complex, and that it prevents an Everettian who adopts branching-time semantics from making clear sense of objective probability. The upshot is that Everettians are better off without branching-time semantics. I conclude by discussing and rejecting an alternative possible motivation for branching time. (shrink)

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. (shrink)

The theory of branching space-times, put forward by Belnap, considers indeterminism as local in space and time. In the axiomatic foundations of that theory, so-called choice points mark the points at which the possible future can turn out in different ways. Working under the assumption of choice points is suitable for many applications, but has an unwelcome topological consequence that makes it difficult to employ branching space-times to represent a range of possible physical space-times. Therefore it is interesting to develop (...) a branching space-times theory without choice points. This is what we set out to do in this paper, providing new foundations for branching space-times in terms of choice sets rather than choice points. After motivating and developing the resulting theory in formal detail, we show that it is possible to translate structures of one style into structures of the other style and vice versa. This result shows that the underlying idea of indeterminism as the branching of spatio-temporal histories is robust with respect to different implementations, making a choice between them a matter of expediency rather than of principle. (shrink)

In this paper we describe some first steps for bringing the framework of branching space-times to bear on quantum information theory. Our main application is quantum error correction. It is shown that branching space-times offers a new perspective on quantum error correction: as a supplement to the orthodox slogan, ``fight entanglement with entanglement'', we offer the new slogan, ``fight indeterminism with indeterminism''.

I shall argue that there is no such property of an event as its “probability.” This is why standard interpretations cannot give a sound definition in empirical terms of what “probability” is, and this is why empirical sciences like physics can manage without such a definition. “Probability” is a collective term, the meaning of which varies from context to context: it means different — dimensionless [0, 1]-valued — physical quantities characterising the different particular situations. In other words, probability is a (...) reducible concept, supervening on physical quantities characterising the state of affairs corresponding to the event in question. On the other hand, however, these “probability-like” physical quantities correspond to objective features of the physical world, and are objectively related to measurable quantities like relative frequencies of physical events based on finite samples — no matter whether the world is objectively deterministic or indeterministic. (shrink)

An overview of the German philosophy of science community is given for the years 1992–2012, based on a survey in which 159 philosophers of science in Germany participated. To this end, the institutional background of the German philosophy of science community is examined in terms of journals, centers, and associations. Furthermore, a qualitative description and a quantitative analysis of our survey results are presented. Quantitative estimates are given for: (a) academic positions, (b) research foci, (c) philosophers’ of science most important (...) publications, and (d) externally funded projects, where for (c) all survey participants had indicated their five most important publications in philosophy of science. In addition, the survey results for (a)–(c) are also qualitatively described, as they are interesting in their own right. With respect to (a), we estimated the gender distribution among academic positions. Concerning (c), we quantified philosophers’ of science preference for (i) journals and publishers, (ii) publication format, (iii) language, and (iv) coauthorship for their most important publications. With regard to research projects, we determined their (i) prevalence, (ii) length, and (iii) trend (an increase in number?) as well as their most frequent (iv) research foci and (v) funding organizations. We also distinguished between German-based and non-German-based journals, publishers, and funding institutions, making it thereby possible to evaluate the involvement of the German philosophy of science community in the international research landscape. Finally, we discuss some implications of our findings. (shrink)

In this paper we describe some first steps for bringing the framework of branching space-times to bear on quantum information theory. Our main application is quantum error correction. It is shown that branching space-times offers a new perspective on quantum error correction: as a supplement to the orthodox slogan, ``fight entanglement with entanglement'', we offer the new slogan, ``fight indeterminism with indeterminism''.

There is a remarkable similarity between some mathematical objects used in the Branching Space-Times framework and those appearing in computer science in the fields of event structures for concurrent processing and Chu spaces. This paper introduces the similarities and formulates a few open questions for further research, hoping that both BST theorists and computer scientists can benefit from the project.

The paper re-evaluates Prior's tenets about indeterminism and relativity from the point of view of the current state of the debate. We first discuss Prior's claims about indeterministic tense logic and about relativity separately and confront them with new technical developments. Then we combine the two topics in a discussion of indeterministic approaches to space-time logics. Finally we show why Prior would not have to "dig his heels in" when it comes to relativity: We point out a way of combining (...) the existential import of the distinction between past, present, and future with a frame-relative notion of the present. (shrink)