Online research in philosophy

This is part I of a two-part essay introducing case-intensional first order logic (CIFOL), 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 (Yale University Press 1972 ). 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 (whether the property applies, depends only on an extension in one case) 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 (substance) properties, essential properties, de re vs. de dicto , and the results of possible tests. (shrink)

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. (shrink)

This is Part I of a two-part essay introducing case-intensional first-order logic (CIFOL), 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 (Yale University Press 1972). 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 (whether the property applies, depends only on an extension in one case) 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 (substance) properties, essential properties, de re vs. de dicto, and the results of possible tests. (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)

The distinction between data and phenomena introduced by Bogen and Woodward (Philosophical Review 97(3):303–352, 1988) was meant to help accounting for scientific practice, especially in relation with scientific theory testing. Their article and the subsequent discussion is primarily viewed as internal to philosophy of science. We shall argue that the data/phenomena distinction can be used much more broadly in modelling processes in philosophy.

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)

In this introduction we discuss the motivation behind the workshop “Towards a New Epistemology of Mathematics” of which this special issue constitutes the proceedings. We elaborate on historical and empirical aspects of the desired new epistemology, connect it to the public image of mathematics, and give a summary and an introduction to the contributions to this issue.

We argue that mathematical knowledge is context dependent. Our main argument is that on pain of distorting mathematical practice, one must analyse the notion of having available a proof, which supplies justification in mathematics, in a context dependent way.

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 quantum-mechanical 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)

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''.

We provide a formally rigorous framework for integrating singular causation, as understood by Nuel Belnap's theory of causae causantes, and objective single case probabilities. The central notion is that of a causal probability space whose sample space consists of causal alternatives. Such a probability space is generally not isomorphic to a product space. We give a causally motivated statement of the Markov condition and an analysis of the concept of screening-off. Causal dependencies and probabilities 1.1 Background: causation in branching space-times (...) 1.2 What are probabilities defined for? Basic transitions 2.1 Basics of basic transitions 2.2 Sets of basic transitions Causal probability theory 3.1 Some simple cases 3.2 General causal probabilities 3.3 Application: probability of suprema of a chain. (shrink)

The paper gives a physicist's view on the framework of branching space-time (Belnap, Synthese 92 (1992), 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 (Am. J. Phys. 65 (1997), 300--304) are (...) analyzed. (shrink)

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)