Nuel Belnap

(? - 2024)

The chapter constructs a notion of the present that is both relativity-friendly and serves the metaphysical role required by presentism. It draws a distinction between a static present based on simultaneity and a dynamic present based on co-presentness. Co-presentness points to a dynamic role of the present in separating a fixed past from an open future. That dynamical role is linked to the idea that dynamic change must be based on the indeterministic realization of possibilities for the future. In working out the formal details of this idea, the chapter makes use of the rich notion of modal correlations that BST offers. Depending on what exactly the modal correlations are like, the analysis delivers different regions of the past, present, and future of a given event.

The chapter shows how local indeterminism underlying BST combines with relativistic space-times. First it defines particular BST structures in which histories are isomorphic to Minkowski space-times. It further argues that many general relativistic space-times are one-history structures of BST. It introduces the notion of non-Hausdorff differential manifolds and investigates if they can be interpreted modally, as structures of BST with multiple histories. It investigates bifurcating curves in non-Hausdorff manifolds, which are natural representations of alternative evolutions of point-like objects. As the required bifurcating curves are unlikely in General Relativity, whereas there are cases of indeterministic general relativistic space-times, the chapter concludes that General Relativity is globally indeterministic, but locally deterministic.

This chapter analyzes the phenomenon of quantum mechanical correlations using the BST notions of transitions, propensities, and funny business. It considers two ways of understanding such correlations: First, as modal correlations (exhibited, e.g., in the GHZ setup) and, second, as probabilistic correlations (exhibited, e.g., in the Bell-Aspect setup). Having introduced the notion of structure extensions, it asks if it is possible to extend an initial BST structure harboring correlations (modal or probabilistic) in such a way that the resulting extended structure harbors no such correlations. An important distinction, rigorously stated in BST, is that between agents-induced indeterminism and Nature-induced indeterminism. A main result concerning structure extensions is that in the mentioned cases, the procedure is not possible unless the extended structure violates the initially given division between cases of agents-induced indeterminism and cases of Nature-induced indeterminism.

This chapter offers a BST theory of propensities (i.e., of objective single-case probabilities), which builds on the account of indeterministic causation developed in Chapter 6. Propensities are shown to deliver classical (Kolmogorovian) probability spaces. The chapter draws a distinction between propensities and probability measures. The former are assigned to sets of BST transitions, in particular to sets of _causae causantes_ of transitions, and are interpreted as degrees of possibility of these transitions. The latter are defined in terms of propensities and are measures of Komogorovian probability spaces. Features of propensities are derived from a logico-causal analysis. Finally, the chapter discusses how the theory developed here handles well-known objections to propensities due to Humphreys and to Salmon, especially Humphreys’s paradox.

The focus of this chapter is modal correlations, rigorously analysed in terms of BST transitions. In the simplest case, two transitions are modally correlated if each is possible, but their joint occurrence is not possible. The chapter singles out a class of “interesting” modal correlations, called modal funny business, and offers two approaches to them. On the first analysis, a set of basic transition exhibits modal funny business if it is combinatorially consistent but not consistent as a whole. The other approach draws on explanation: in a set of transitions harboring modal funny business there is no pair of blatantly inconsistent transitions that could explain why the whole set is inconsistent. These two analyses yield precise definitions of combinatorial and explanatory funny business. The main result consists of theorems showing that the two concepts are co-extensive, which stresses the stability of the notion of modal funny business.

The chapter analyses singular causation within an indeterministic context. It assumes that effects are transitions and causes are basic indeterministic transitions, called _causae causantes_. It considers a variety of transitions as effects, depending on what their outcomes are (outcome chains, scattered outcomes, or disjunctive outcomes). By this analysis, a _causa causans_ for a given transition occurs at a risky junction, where alternative basic transitions could prohibit the occurrence of the given transition. The _causa causans_ keeps the occurrence of this transition possible. As an argument for the adequacy of this analysis, the chapter offers a few theorems showing that _causae causantes_ satisfy inus-like conditions as proposed by Mackie.

This chapter introduces a variety of events that are definable in BST and discusses in which histories these events occur. This gives rise to the concept of the occurrence proposition for events of various kinds. Of particular interest are transitions, defined as pairs of events, one of which is appropriately below the other. Transitions play a crucial role in later chapters. The chapter then discusses the topological aspects of BST, which are picked up again in Chapter 9. It defines a natural topology for BST: the diamond topology, and describes some important facts about it, focusing on the Hausdorff property and local Euclidicity. The chapter also gives an overview of how BST structures may be used to build semantic models for languages with temporal and modal operators.

The chapter discusses how the histories in a common BST structure are related. By the axioms of the core theory of BST, any two histories share some past, but there are different ways to implement this. These are distinguished by the so-called prior choice principles, which make specific demands on the way in which histories branch. On one option (which yields structures of BST 92 ), histories branch, or remain undivided, at points, which means that there is a maximal element in the overlap of any two histories. The other option (which yields BST NF structures) prohibits the existence of such maximal elements and works with so-called choice sets. The chapter discusses the pattern of branching in the two theories, BST 92 and BST NF, also with respect to topology. As it turns out, the two theories are are intertranslatable. The chapter ends with a sketch of these translatability results.

In this chapter the reader is guided through the construction of the core theory of Branching Space-Times. This discursive approach culminates in proposing a set of postulates that a structure of the core theory of Branching Space-Times (common BST) has to satisfy. The theory’s basic notion is that of a set of events, partially ordered by a pre-causal relation. Histories are then defined as maximal directed subsets of the base set. The chapter proves essential facts about histories and the postulates that the core of BST is assumed to satisfy. Among other things, it proves the so-called M-property that determines how any two point event in a common BST structure are related.

This introductory chapter explains the aim of the book: the analysis of real possibilities as anchored in a spatio-temporal world that is rudimentarily relativistic. It contrasts real possibilities to other possibilities discussed in the philosophical literature. It explains how branching is related to the possible worlds framework made popular, e.g., by David Lewis’s works. It offers philosophical comments on crucial notions and assumptions of BST, such as events, histories, and temporal directedness. It ends up with some hints about how the BST project is situated in modal metaphysics, touching themes such as the concept of actuality or the distinction between possibilities as “alternatives to” vs. “alternatives for”.

tic sequenzen-kalkul of Gentzen, into rules for PCc, the classical sequenzenkalkul. We shall limit ourselves here to sequenzen or turnstile statements of the form A„A„..., A„ I- B, where A„A„..., A„(n ~ 0), and B are wffs consisting of propositional variables, zero or more of the connectives '5', "v', ' ', ')', and '=', and zero or more parentheses. One can pass from PCi to PCc by amending the intelim rules for ' a result of long standing, or by amending the intelim rules for either one of..

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.

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 book develops a rigorous theory of indeterminism as a local and modal concept. Its crucial insight is that our world contains events or processes with alternative, really possible outcomes. The theory aims at clarifying what this assumption involves, and it does it in two ways. First, it provides a mathematically rigorous framework for local and modal indeterminism. Second, we support that theory by spelling out the philosophically relevant consequences of this formulation and by showing its fruitful applications in metaphysics. To this end, we offer a formal analysis of modal correlations and of causation, which is applicable in indeterministic and non-local contexts as well. We also propose a rigorous theory of objective single-case probabilities, intended to represent degrees of possibility. In a third step, we link our theory to current physics, investigating how local and modal indeterminism relates to issues in the foundations of physics, in particular, quantum non-locality and spatio-temporal relativity. The book also ventures into the philosophy of time, showing how the theory's resources can be used to explicate the dynamic concept of the past, present, and future based on local indeterminism"--

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"

This is part I of a two-part essay introducing case-intensional first order logic, an easy-to-use, uniform, powerful, and useful combination of first-order logic with modal logic resulting from philosophical and technical modifications of Bressan’s General interpreted modal calculus. CIFOL starts with a set of cases; each expression has an extension in each case and an intension, which is the function from the cases to the respective case-relative extensions. Predication is intensional; identity is extensional. Definite descriptions are context-independent terms, and lambda-predicates and -operators can be introduced without constraints. These logical resources allow one to define, within CIFOL, important properties of properties, viz., extensionality and absoluteness, Bressan’s chief innovation that allows tracing an individual across cases without recourse to any notion of “rigid designation” or “trans-world identity.” Thereby CIFOL abstains from incorporating any metaphysical principles into the quantificational machinery, unlike extant frameworks of quantified modal logic. We claim that this neutrality makes CIFOL a useful tool for discussing both metaphysical and scientific arguments involving modality and quantification, and we illustrate by discussing in diagrammatic detail a number of such arguments involving the extensional identification of individuals via absolute properties, essential properties, de re vs. de dicto, and the results of possible tests.

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.

Indeterminism, understood as a notion that an event may be continued in a few alternative ways, invokes the question what a region of chanciness looks like. We concern ourselves with its topological and spatiotemporal aspects, abstracting from the nature or mechanism of chancy processes. We first argue that the question arises in Montague-Lewis-Earman conceptualization of indeterminism as well as in the branching tradition of Prior, Thomason and Belnap. As the resources of the former school are not rich enough to study topological issues, we investigate the question in the framework of branching space-times of Belnap (Synthese 92:385–434, 1992). We introduce a topology on a branching model as well as a topology on a history in a branching model. We define light-cones and assume four conditions that guarantee the light-cones so defined behave like light-cones of physical space-times. From among various topological separation properties that are relevant to our question, we investigate the Hausdorff property. We prove that each history in a branching model satisfies the Hausdorff property. As for the satisfaction of the Hausdorff property in the entire branching model, we prove that it is related to the phenomenon of passive indeterminism, which we describe in detail