Let E be a set of n propositions E1, ..., En. We seek a probabilistic measure C(E) of the ‘degree of coherence’ of E. Intuitively, we want C to be a quantitative, probabilistic generalization of the (deductive) logical coherence of E. So, in particular, we require C to satisfy the following..

This chapter provides an overview of a range of probabilistic theories of causality, including those of Reichenbach, Good and Suppes, and the contemporary causal net approach. It discusses two key problems for probabilistic accounts: counterexamples to these theories and their failure to account for the relationship between causality and mechanisms. It is argued that to overcome the problems, an epistemic theory of causality is required.

According to probabilistic theories of reasoning in psychology, people's degree of belief in an indicative conditional `if A, then B' is given by the conditional probability, P(B|A). The role of language pragmatics is relatively unexplored in the new probabilistic paradigm. We investigated how consequent relevance a ects participants' degrees of belief in conditionals about a randomly chosen card. The set of events referred to by the consequent was either a strict superset or a strict subset of the set (...) of events referred to by the antecedent. We manipulated whether the superset was expressed using a disjunction or a hypernym. We also manipulated the source of the dependency, whether in long-term memory or in the stimulus. For subset-consequent conditionals, patterns of responses were mostly conditional probability followed by conjunction. For superset-consequent conditionals, conditional probability responses were most common for hypernym dependencies and least common for disjunction dependencies, which were replaced with responses indicating inferred consequent irrelevance. Conditional probability responses were also more common for knowledge-based than stimulus-based dependencies. We suggest. (shrink)

This paper examines Patrick Suppes’ probabilistic theory of causality understood as a theory of causal inference, and draws some lessons for empirical economics and contemporary debates in the foundations of econometrics. It argues that a standard method of empirical economics, multiple regression, is inadequate for most but the simplest applications, that the Bayes’ nets approach, which can be understood as a generalisation of Suppes’ theory, constitutes a considerable improvement but is still subject to important limitations, and (...) that the currently fashionable ‘design-based approach’ suffers from the same flaws Suppes anticipated a long time ago. It then sketches an alternative in response, one that differs drastically from the formalisms Suppes endorsed but is consistent with his pragmatic general take on science. (shrink)

Bonjour (1985: 101 and 1999: 124) and other coherence theorists of justification before him (e.g. Ewing, 1934: 246) have complained that we do not have a satisfactory analysis of the notion of coherence. The problem with existing accounts of coherence is that they try to bring precision to our intuitive notion of coherence independently of the particular role that it is meant to play within the coherence theory of justification (e.g Lewis, 1946: 338). This is a mistake: it does (...) not make any sense to ask what precisely makes for a more coherent information set independently of the particular role that coherence is supposed to play within the context in question. What is this context and what is this role? The coherence theory of justification rides on a particular common sense intuition: when we gather information from less than fully reliable sources, then the more coherent the story that materializes is, the more confident we may be, ceteris paribus. Within the context of information gathering from certain types of sources, coherence is a property of stories which plays a confidence boosting role. But what features should the information sources have, so that the coherence of the information set is indeed a determinant of our degree of confidence in question? And what goes into the ceteris paribus clause? In other words, what other factors affect our confidence in the information set in question? (shrink)

Larry Wright and others have advanced causal accounts of functional explanation, designed to alleviate fears about the legitimacy of such explanations. These analyses take functional explanations to describe second order causal relations. These second order relations are conceptually puzzling. I present an account of second order causation from within the framework of Eells' probabilistic theory of causation; the account makes use of the population-relativity of causation that is built into this theory.

In this paper I provide a probabilistic account of factual knowledge,[1] based on the notion of chance.[2] This account has some affinity with my chance account of token causation,[3] but it neither relies on it nor presupposes it. Here I concentrate on the core cases of perceptual knowledge and of knowledge by memory (based on perception). The analysis of knowledge presented below is externalist; but pursuing such an analysis need not detract from the significance of attempts to flesh out (...) justificational analyses knowledge; indeed, I have pursued such an analysis in great detail myself. (shrink)

This paper attempts to set up a probabilistic framework for understanding the notion of causal necessity. What results is a relaxed and relativized probabilistic theory of epsilon-Causal necessity and an explicit attempt to avoid deterministic assumptions. The theory developed emphasizes the notions of partial cause, Causal contribution, And the degree of contribution. Implications for causal overdetermination, Causal preemption, And causal discourse are discussed.

Algebraic theories for extensive measurement are traditionally framed in terms of a binary relation $\lesssim $ and a concatenation (x,y)→ xy. For situations in which the data is "noisy," it is proposed here to consider each expression $y\lesssim x$ as symbolizing an event in a probability space. Denoting P(x,y) the probability of such an event, two theories are discussed corresponding to the two representing relations: p(x,y)=F[m(x)-m(y)], p(x,y)=F[m(x)/m(y)] with m(xy)=m(x)+m(y). Axiomatic analyses are given, and representation theorems are proven in detail.

In any probabilistic theory, we say that a bipartite state ω on a composite system AB steers its marginal state ω B if, for any decomposition of ω B as a mixture ω B =∑ i p i β i of states β i on B, there exists an observable {a i } on A such that the conditional states $\omega_{B|a_{i}}$ are exactly the states β i . This is always so for pure bipartite states in quantum mechanics, (...) a fact first observed by Schrödinger in 1935. Here, we show that, for weakly self-dual state spaces (those isomorphic, but perhaps not canonically isomorphic, to their dual spaces), the assumption that every state of a system A is steered by some bipartite state of a composite AA consisting of two copies of A, amounts to the homogeneity of the state cone. If the state space is actually self-dual, and not just weakly so, this implies (via the Koecher-Vinberg Theorem) that it is the self-adjoint part of a formally real Jordan algebra, and hence, quite close to being quantum mechanical. (shrink)

I advance a new theory of causal relevance, according to which causal claims convey information about conditional probability functions. This theory is motivated by the problem of disjunctive factors, which haunts existing probabilistic theories of causation. After some introductory remarks, I present in Section 3 a sketch of Eells's (1991) probabilistic theory of causation, which provides the framework for much of the discussion. Section 4 explains how the problem of disjunctive factors arises within this framework. (...) After rejecting three proposed solutions, I offer in Section 6 a new approach to causation that avoids the problem. Decision-theoretic considerations also support the new approach. Section 8 develops the consequences of the new theory for causal explanation. The resulting theory of causal explanation incorporates the new insights while respecting important work on scientific explanation by Salmon (1971), Railton (1981), and Humphreys (1989). My conclusions are enumerated in Section 9. (shrink)

Germund Hesslow has argued recently [2] that a probabilistic theory of causality as advocated by Patrick Suppes [4] has two problems that a deterministic theory avoids. In this paper, I argue that Suppes' probabilistic causal calculus is free of each of these problems and, moreover, that several broader issues raised by Hesslow's discussion tend to support a probabilistic conception of causes.

This paper develops axiomatic foundations for a probabilistic-interventionist theory of causal strength. Transferring methods from Bayesian confirmation theory, I proceed in three steps: I develop a framework for defining and comparing measures of causal strength; I argue that no single measure can satisfy all natural constraints; I prove two representation theorems for popular measures of causal strength: Pearl's causal effect measure and Eells' difference measure. In other words, I demonstrate these two measures can be derived from a (...) set of plausible adequacy conditions. The paper concludes by sketching future research avenues. (shrink)

Probability theory, epistemically interpreted, provides an excellent, if not the best available account of inductive reasoning. This is so because there are general and definite rules for the change of subjective probabilities through information or experience; induction and belief change are one and same topic, after all. The most basic of these rules is simply to conditionalize with respect to the information received; and there are similar and more general rules. 1 Hence, a fundamental reason for the epistemological success (...) of probability theory is that there at all exists a well-behaved concept of conditional probability. Still, people have, and have reasons for, various concerns over probability theory. One of these is my starting point: Intuitively, we have the notion of plain belief; we believe propositions2 to be true (or to be false or neither). Probability theory, however, offers no formal counterpart to this notion. Believing A is not the same as having probability 1 for A, because probability 1 is incorrigible3; but plain belief is clearly corrigible. And believing A is not the same as giving A a probability larger than some 1 - c, because believing A and believing B is usually taken to be equivalent to believing A & B.4 Thus, it seems that the formal representation of plain belief has to take a non-probabilistic route. Indeed, representing plain belief seems easy enough: simply represent an epistemic state by the set of all propositions believed true in it or, since I make the common assumption that plain belief is deductively closed, by the conjunction of all propositions believed true in it. But this does not yet provide a theory of induction, i.e. an answer to the question how epistemic states so represented are changed tbrough information or experience. There is a convincing partial answer: if the new information is compatible with the old epistemic state, then the new epistemic state is simply represented by the conjunction of the new information and the old beliefs. This answer is partial because it does not cover the quite common case where the new information is incompatible with the old beliefs. It is, however, important to complete the answer and to cover this case, too; otherwise, we would not represent plain belief as conigible. The crucial problem is that there is no good completion. When epistemic states are represented simply by the conjunction of all propositions believed true in it, the answer cannot be completed; and though there is a lot of fruitful work, no other representation of epistemic states has been proposed, as far as I know, which provides a complete solution to this problem. In this paper, I want to suggest such a solution. In [4], I have more fully argued that this is the only solution, if certain plausible desiderata are to be satisfied. Here, in section 2, I will be content with formally defining and intuitively explaining my proposal. I will compare my proposal with probability theory in section 3. It will turn out that the theory I am proposing is structurally homomorphic to probability theory in important respects and that it is thus equally easily implementable, but moreover computationally simpler. Section 4 contains a very brief comparison with various kinds of logics, in particular conditional logic, with Shackle's functions of potential surprise and related theories, and with the Dempster - Shafer theory of belief functions. (shrink)

Acknowledgements Andrea Scarantino, Nicholas Shea, Mark Sprevak, and three anonymous referees provided incisive and constructive comments, for which I am very grateful. In 2012, earlier versions of this paper were delivered in Edinburgh, at the Joint Session in Stirling, and at a workshop on natural information in Aberdeen. I thank participants for their feedback.

The formalism of general probabilistic theories provides a universal paradigm that is suitable for describing various physical systems including classical and quantum ones as particular cases. Contrary to the usual no-restriction hypothesis, the set of accessible meters within a given theory can be limited for different reasons, and this raises a question of what restrictions on meters are operationally relevant. We argue that all operational restrictions must be closed under simulation, where the simulation scheme involves mixing and classical (...) post-processing of meters. We distinguish three classes of such operational restrictions: restrictions on meters originating from restrictions on effects; restrictions on meters that do not restrict the set of effects in any way; and all other restrictions. We fully characterize the first class of restrictions and discuss its connection to convex effect subalgebras. We show that the restrictions belonging to the second class can impose severe physical limitations despite the fact that all effects are accessible, which takes place, e.g., in the unambiguous discrimination of pure quantum states via effectively dichotomic meters. We further demonstrate that there are physically meaningful restrictions that fall into the third class. The presented study of operational restrictions provides a better understanding on how accessible measurements modify general probabilistic theories and quantum theory in particular. (shrink)

This paper derives a mathematical expression giving the development of the probability of a scientific hypothesis with the number of confirming tests, as determined by Bayes's theorem, in a special case in which all the tests are "independent" of one another. The simple expression obtained shows clearly how the various factors influence the growth of the probability. The result is used to set a numerical lower bound on the probabilities representing the a priori beliefs of humans in generalizations that become (...) accepted. By making a comparison with the predictions of a "logical atomic" model in the case of physical laws, it is argued that humans have significant a priori "knowledge" in a weak sense. (shrink)

We investigate the connection between interference and computational power within the operationally defined framework of generalised probabilistic theories. To compare the computational abilities of different theories within this framework we show that any theory satisfying four natural physical principles possess a well-defined oracle model. Indeed, we prove a subroutine theorem for oracles in such theories which is a necessary condition for the oracle model to be well-defined. The four principles are: causality, purification, strong symmetry, and informationally consistent composition. (...) Sorkin has defined a hierarchy of conceivable interference behaviours, where the order in the hierarchy corresponds to the number of paths that have an irreducible interaction in a multi-slit experiment. Given our oracle model, we show that if a classical computer requires at least n queries to solve a learning problem, because fewer queries provide no information about the solution, then the corresponding “no-information” lower bound in theories lying at the kth level of Sorkin’s hierarchy is \. This lower bound leaves open the possibility that quantum oracles are less powerful than general probabilistic oracles, although it is not known whether the lower bound is achievable in general. Hence searches for higher-order interference are not only foundationally motivated, but constitute a search for a computational resource that might have power beyond that offered by quantum computation. (shrink)

Physicists routinely claim that the fundamental laws of physics are 'time symmetric' or 'time reversal invariant' or 'reversible'. In particular, it is claimed that the theory of quantum mechanics is time symmetric. But it is shown in this paper that the orthodox analysis suffers from a fatal conceptual error, because the logical criterion for judging the time symmetry of probabilistic theories has been incorrectly formulated. The correct criterion requires symmetry between future-directed laws and past-directed laws. This criterion is (...) formulated and proved in detail. The orthodox claim that quantum mechanics is reversible is re-evaluated. The property demonstrated in the orthodox analysis is shown to be quite distinct from time reversal invariance. The view of Satosi Watanabe that quantum mechanics is time asymmetric is verified, as well as his view that this feature does not merely show a de facto or 'contingent' asymmetry, as commonly supposed, but implies a genuine failure of time reversal invariance of the laws of quantum mechanics. The laws of quantum mechanics would be incompatible with a time-reversed version of our universe. (shrink)

In this paper, we take a fresh look at three Popperian concepts: riskiness, falsifiability, and truthlikeness of scientific hypotheses or theories. First, we make explicit the dimensions that underlie the notion of riskiness. Secondly, we examine if and how degrees of falsifiability can be defined, and how they are related to various dimensions of the concept of riskiness as well as the experimental context. Thirdly, we consider the relation of riskiness to truthlikeness. Throughout, we pay special attention to probabilistic (...) theories and we offer a tentative, quantitative account of verisimilitude for probabilistic theories. (shrink)

David Danks. Psychological Theories of Categorizations as Probabilistic Models.

Two radically different views about time are possible. According to the first, the universe is three dimensional. It has a past and a future, but that does not mean it is spread out in time as it is spread out in the three dimensions of space. This view requires that there is an unambiguous, absolute, cosmic-wide "now" at each instant. According to the second view about time, the universe is four dimensional. It is spread out in both space and time (...) - in space-time in short. Special and general relativity rule out the first view. There is, according to relativity theory, no such thing as an unambiguous, absolute cosmic-wide "now" at each instant. However, we have every reason to hold that both special and general relativity are false. Not only does the historical record tell us that physics advances from one false theory to another. Furthermore, elsewhere I have shown that we must interpret physics as having established physicalism - in so far as physics can ever establish anything theoretical. Physicalism, here, is to be interpreted as the thesis that the universe is such that some unified "theory of everything" is true. Granted physicalism, it follows immediately that any physical theory that is about a restricted range of phenomena only, cannot be true, whatever its empirical success may be. It follows that both special and general relativity are false. This does not mean of course that the implication of these two theories that there is no unambiguous cosmic-wide "now" at each instant is false. It still may be the case that the first view of time, indicated at the outset, is false. Are there grounds for holding that an unambiguous cosmic-wide "now" does exist, despite special and general relativity, both of which imply that it does not exist? There are such grounds. Elsewhere I have argued that, in order to solve the quantum wave/particle problem and make sense of the quantum domain we need to interpret quantum theory as a fundamentally probabilistic theory, a theory which specifies how quantum entities - electrons, photons, atoms - interact with one another probabilistically. It is conceivable that this is correct, and the ultimate laws of the universe are probabilistic in character. If so, probabilistic transitions could define unambiguous, absolute cosmic-wide "nows" at each instant. It is entirely unsurprising that special and general relativity have nothing to say about the matter. Both theories are pre-quantum mechanical, classical theories, and general relativity in particular is deterministic. The universe may indeed be three dimensional, with a past and a future, but not spread out in four dimensional space-time, despite the fact that relativity theories appear to rule this out. These considerations, finally, have implications for views about the arrow of time and free will. (shrink)

A probabilistic theory of causation is a theory which holds that the central feature of causation is that causes raise the probability of their effects. In this dissertation, I defend Hans Reichenbach's original version of the probabilistic theory of causation, which analyses causal relations in terms of a three place statistical betweenness relation. Unlike most discussions of this theory, I hold that the statistical relation should be taken as a sufficient, but not as a (...) necessary , condition for causal betweenness. With this difference in interpretation, Reichenbach's theory is shown to be immune to all of the criticisms which have been raised against it in the last fifty years. ;Reichenbach's main purpose in defending a probabilistic theory of causation was to use it in giving an analysis of the direction of time. I defend this view, arguing that the arrow of causation is more basic than any of the other arrows of time---metaphysical, psychological or physical. In particular, I argue against the popular view that the direction of time is given by the physical fact that entropy only increases in one direction. I further show that the probabilistic theory of causation is particularly suited to the task of analysing the various arrows of time. In my final chapter, I also suggest that the probabilistic theory of causation may be fruitfully applied to the analysis of some important puzzles in the interpretation of quantum mechanics. (shrink)

A recent development in the philosophy of religion has been the attempt to justify belief in God using Bayesian confirmation theory. My dissertation critically discusses two prominent spokesmen for this approach--Richard Swinburne and J. L. Mackie. Using probabilistic confirmation theory, these philosophers come to wildly divergent conclusions with respect to the hypothesis of theism; Swinburne contends that the evidence raises the overall probability of the hypothesis of theism, whereas Mackie argues that the evidence disconfirms the existence of (...) God. After a careful analysis of the individual authors, this dissertation critically examines the assumptions of the project. The authors consider arguments for the existence of God separately--the cosmological argument, the argument from design, apparent miracles, the problem of evil, and arguments from religious experience--and then conjoin them to assess the cumulative effect of the arguments. The assumptions are that the probability calculus may be interpreted for a successful inductive argument for or against the existence of God, that inference to best explanation is applicable to the justification of religious belief, that the justification of belief in God is analogous to the justification of a scientific theory, and that a classical foundationalist conception of rationality is correct. It is the contention of this dissertation that all of these assumptions are incorrect. All of the major interpretations of the probability calculus are examined and it is argued that there are either no interpretations for the probability assignments or no interpretations which will advance an argument for or against the existence of God. It is also argued that the usual methods for interpreting the explanatory power of theism will not determine any non-zero probabilities; however, density measuring will provide a non-zero interpretation. In the final chapter it is argued that a subjective Bayesian conception of rationality is neither a necessary nor a sufficient condition of rationality and that classical foundationalism is self-referentially inconsistent. An alternative approach to the rationality of religious belief is offered that proceeds along Reidian and Plantingan lines, belief in God is properly basic. (shrink)

We study probabilistic characterisation of a random model of a finite set of first order axioms. Given a set of first order axioms.

I defend an analog of probabilism that characterizes rationally coherent estimates for chances. Specifically, I demonstrate the following accuracy-dominance result for stochastic theories in the C*-algebraic framework: supposing an assignment of chance values is possible if and only if it is given by a pure state on a given algebra, your estimates for chances avoid accuracy-dominance if and only if they are given by a state on that algebra. When your estimates avoid accuracy-dominance (roughly: when you cannot guarantee that other (...) estimates would be more accurate), I say that they are sufficiently coherent. In formal epistemology and quantum foundations, the notion of rational coherence that gets more attention requires that you never allow for a sure loss (or “Dutch book”) in a given sort of betting game; I call this notion full coherence. I characterize when these two notions of rational coherence align, and I show that there is a quantum state giving estimates that are sufficiently coherent, but not fully coherent. (shrink)