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. (shrink)
Against the background of the theory of branching space-times (BST), the paper sketches a concept of individuals. It discusses Kripkean modal intuitions concerning individuation, and, finally it addresses Lewis’s objections to branching individuals.
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.
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.
The objective of the paper is to present a comprehensive picture of Bell-type theorems, by giving both the theorems and the proofs of them.Special care is given to specifying the assumptions of the arguments and their physical or metaphysical significance. Taking the EPR argument as a point of departure, the paper discusses four probabilitic Bell-type theorems,which are then followed by two versions on non-probailitic (GHZ) arguments.The final section provides the reader with a classification of the assumptions, which specifies which assumption (...) is used in which proof. (shrink)
The aim of this paper is to reconstruct Brouwer’s justification for the intuitionistic revision of logic and mathematics. It is attempted to show that pivotal premisses of his argument are supplied by his philosophy. To this end, the basic tenets of his philosophical doctrine are discussed: the concepts of mind, causal attention, intuition of two-ity and his repudiation of realism.The restriction of intuitionistically allowable objects to spreads and species is traced back to Brouwer’s concept of intuition that is a defining (...) feature of his notion of mind. On the other hand, it is argued that his objections to some laws of classical logic result from the rejection of the rule of double negation elimination, which in turn follows from both, the claim that rules of logic should preserve evidence for assertions rather than truth, and too restrictive a concept of evidence. (shrink)
We assess Cartwright's models for probabilistic causality and, in particular, her models for EPR-like experiments of quantum mechanics. Our first objection is that, contrary to econometric linear models, her quasi-linear models do not allow for the unique estimation of parameters. We next argue that although, as Cartwright proves, Reichenbach's screening-off condition has only limited validity, her generalized condition is not empirically applicable. Finally, we show that her models for the EPR are mathematically incorrect and physically implausible.
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. (shrink)
The paper extends the framework of outcomes in branching space-time (Kowalski and Placek ) by assigning probabilities to outcomes of events, where these probabilities are interpreted either epistemically or as weighted possibilities. In resulting models I define the notion of common cause of correlated outcomes of a single event, and investigate which setups allow for the introduction of common causes. It turns out that a deterministic common cause can always be introduced, but (surprisingly) only special setups permit the introduction of (...) truly stochastic common causes. I analyse next the Bell-Aspect experiment and derive the Bell-CH inequalities. I observe that we postulate there not a common cause for outcomes of a single event but rather a common common cause that accounts for outcomes of many events, where 'events' mean 'measurements with (different) directions of polarization'. Since the inequalities are violated, I claim that no causal story can be told about the Bell correlations, where causality is subliminal and restricted by screening-off condition. Similarly, given certain intuitive principles, no deterministic story can be told about these correlations. (shrink)
The paper intends to provide an algebraic framework in which subluminal causation can be analysed. The framework merges Belnap's 'outcomes in branching time' with his 'branching space-time' (BST). it is shown that an important structure in BST, called 'family of outcomes of an event', is a boolean algebra. We define next non-stochastic common cause and analyse GHZ-Bell theorems. We prove that there is no common cause that accounts for results of GHZ-Bell experiment but construct common causes for two other quantum (...) mechanical setups. Finally, we investigate why some setups allow for common causes whereas other setups do not. (shrink)
In the article three Zeno's paradoxes are reconstructed. They are: „Achilles and the turtle”, „Arrow” and „Stadium”. Together with the paradox of „Dichotomy” (which was analysed by the author elsewhere) they form the question about the nature of continuum. In the paper the following hypothesis is accepted: „Dichotomy” is principally connected with the mathematical theory of continuum, whereas other paradoxes concern the application of this theory to the description of physical motion.
My aim in this paper is to investigate the notions of comparative similarity definable in the framework of branching space-times. A notion of this kind is required to give a rigorous Lewis-style semantics of space-time counterfactuals, which is the task undertaken by Thomas Muller (PITT-PHIL-SCI00000509, this archive). In turn, the semantical analysis is needed to decide whether the recently proposed proofs of the non-locality of quantum mechanics are correct. From among the three notions of comparative similarity I select two which (...) appear equally good as far as their intuitiveness and algebraic properties are concerned. However, the relations are not transitive, and thus cannot be used in the semantics proposed by (Lewis 1973), which requires transitivity. Yet they are adequate for the account of (Lewis 1981). (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)