Reasoning under uncertainty, that is, making judgements with only partial knowledge, is a major theme in artificial intelligence. Professor Paris provides here an introduction to the mathematical foundations of the subject. It is suited for readers with some knowledge of undergraduate mathematics but is otherwise self-contained, collecting together the key results on the subject, and formalising within a unified framework the main contemporary approaches and assumptions. The author has concentrated on giving clear mathematical formulations, analyses, justifications and consequences of the (...) main theories about uncertain reasoning, so the book can serve as a textbook for beginners or as a starting point for further basic research into the subject. It will be welcomed by graduate students and research workers in logic, philosophy, and computer science as a textbook for beginners, a starting point for further basic research into the subject, and not least, an account of how mathematics and artificial intelligence can complement and enrich each other. (shrink)

This article shows a clear sense in which general relativity allows for a type of ‘machine’ that can bring about a spacetime structure suitable for the implementation of ‘supertasks’. 1Introduction2Preliminaries3Malament–Hogarth Spacetimes4Machines5Malament–Hogarth Machines6Conclusion.

In the Foundations of the Metaphysics of Morals Kant presented a method for discovering what morality requires us to do in any situation and claimed that it is a method everyone can use. The method consists in testing one's maxim against the requirement stated in the formulations of the categorical imperative. There has been endless discussion of the adequacy of Kant's method in giving moral guidance, but there has been little effort to situate Kant's view of ethical method in its (...) historical context. In this paper I try to do so. I take the label "method of ethics" from the work of Henry Sidgwick. A method of ethics, he says, is any rational procedure by which we determine what it is right for an individual to do or what an individual ought to do. Since the moralists I want to consider do not all think of morality as rational, I shall broaden the notion by saying that a method is any systematic or regular procedure, rational or not, by which we determine what morality requires... (shrink)

I analyze a number of the quantum no-signalling proofs (Ghirardi et al. 1980, Bussey 1982, Jordan 1983, Shimony 1985, Redhead 1987, Eberhard and Ross 1989, Sherer and Busch 1993). These purport to show that the EPR correlations cannot be exploited for transmitting signals, i.e., are not causal. First, I show that these proofs can be mathematically unified; they are disguised versions of a single theorem. Second, I argue that these proofs are circular. The essential theorem relies upon the tensor product (...) representation for combined systems, which has no physical basis in the von Neumann axioms. Historically, the construction of this representation scheme by von Neumann and Weyl built no-signalling assumptions into the quantum theory. Signalling between the wings of the EPR-Bell experiments is unlikely but is not ruled out empirically by the class of proofs considered. (shrink)

We give an account of some relationships between the principles of Constant and Atom Exchangeability and various generalizations of the Principle of Instantial Relevance within the framework of Inductive Logic. In particular we demonstrate some surprising and somewhat counterintuitive dependencies of these relationships on ostensibly unimportant parameters, such as the number of predicates in the overlying language.

We show a sense in which the spacetime property of effective completeness—a type of “local hole-freeness” or “local inextendibility”—is not stable.

This introduction to one of the liveliest and most popular fields in philosophy is written specifically for a beginning readership with no background in philosophy or science. Step-by-step analyses of the key arguments are provided and the philosophical heart of the issues is revealed without recourse to jargon, maths, or logical formulas. The book introduces Einstein's revolutionary ideas in a clear and simple way, along with the concepts and arguments of philosophers, both ancient and modern that have proved of lasting (...) value. Specifically, the theories of the ancient Greek philosophers, Zeno, Euclid and Parmenides are considered alongside the ideas of Newton, Leibniz and Kant as well as the giants of twentieth-century physics, Einstein and Lorentz. The problems at the heart of the philosophy of space and time, such as change, motion, infinity, shape, and inflation, are examined and the seismic impact made by relativity theory and quantum theory is assessed in the light of the latest research. The writing is lucid and entertaining, allowing a beginning readership to grasp some difficult concepts while offering the more experienced reader a succinct and illuminating presentation of the state of the debate. "Space, Time and Einstein" shows the reader the excitement of scientific discovery and the beauty of theory in the search for answers to these fundamental questions. (shrink)

We answer some problems set by Priest in [11] and [12], in particular refuting Priest's Conjecture that all LP-models of Th(N) essentially arise via congruence relations on classical models of Th(N). We also show that the analogue of Priest's Conjecture for I δ₀ + Exp implies the existence of truth definitions for intervals [0,a] ⊂ₑ M ⊨ I δ₀ + Exp in any cut [0,a] ⊂e K ⊆ M closed under successor and multiplication.

We present six significant open problems in Pure Inductive Logic, together with their background and current status, with the intention of raising awareness and leading ultimately to their resolution.

Shortly before his death in 1934, the British historian of chemistry, A. N. Meldrum, published two lengthy articles on Lavoisier's early career in science. After a careful investigation of the collection of manuscripts at the Académie des Sciences in Paris and in light of a detailed and penetrating analysis of Lavoisier's published work, Meldrum concluded that as a youth, Lavoisier was concerned with chemistry only to the extent that he found it useful for his mineralogical and geological researches. Lavoisier began (...) his career as a mineralogist; he became a chemist only in 1772, the “crucial year” when he turned his attention to chemical theory for its own sake and started his famous course of experiments on the nature of combustion and fixed air. Although some details—notably concerning Lavoisier's early education and geological work—have been added to this account since Meldrum's time, the broad conclusions of Meldrum's study are still generally accepted by historians of the chemical revolution. (shrink)

Biologists have as yet taken but little cognizance of the revolution in human thought which has been inaugurated by physicists in the last five years, and philosophers have stressed its negative rather than its positive side.

We investigate the relative probabilistic support afforded by the combination of two analogies based on possibly different, structural similarity (as opposed to e.g. shared predicates) within the context of Pure Inductive Logic and under the assumption of Language Invariance. We show that whilst repeated analogies grounded on the same structural similarity only strengthen the probabilistic support this need not be the case when combining analogies based on different structural similarities. That is, two analogies may provide less support than each would (...) individually. (shrink)

What is the function of moral principles within the body of moral knowledge? And what must be the nature of moral principles in order for them to carry out this function? A specific set of answers to these questions is widely accepted among moral philosophers – so widely accepted as almost to constitute a sort of orthodoxy. The answers embody a view of the place of principles within the body of morality which crosses the lines between cognitivism and non-cognitivism. Though (...) I have put the question in cognitivist terms and shall discuss it in those terms, I think a similar question and a more or less parallel discussion could be given in non-cognitivist terms. Perhaps the time-honoured debate between the two positions can be suspended, at least temporarily, while we examine, not the nature of morality, but its structure. (shrink)

Here we discuss the challenge posed by self-organization to the Darwinian conception of evolution. As we point out, natural selection can only be the major creative agency in evolution if all or most of the adaptive complexity manifest in living organisms is built up over many generations by the cumulative selection of naturally occurring small, random mutations or variants, i.e., additive, incremental steps over an extended period of time. Biological self-organization—witnessed classically in the folding of a protein, or in the (...) formation of the cell membrane—is a fundamentally different means of generating complexity. We agree that self-organizing systems may be fine-tuned by selection and that self-organization may be therefore considered a complementary mechanism to natural selection as a causal agency in the evolution of life. But we argue that if self-organization proves to be a common mechanism for the generation of adaptive order from the molecular to the organismic level, then this will greatly undermine the Darwinian claim that natural selection is the major creative agency in evolution. We also point out that although complex self-organizing systems are easy to create in the electronic realm of cellular automata, to date translating in silico simulations into real material structures that self-organize into complex forms from local interactions between their constituents has not proved easy. This suggests that self-organizing systems analogous to those utilized by biological systems are at least rare and may indeed represent, as pre-Darwinists believed, a unique ascending hierarchy of natural forms. Such a unique adaptive hierarchy would pose another major challenge to the current Darwinian view of evolution, as it would mean the basic forms of life are necessary features of the order of nature and that the major pathways of evolution are determined by physical law, or more specifically by the self-organizing properties of biomatter, rather than natural selection. (shrink)

A cutting-edge survey of formal methods directed specifically at dealing with the deep mathematical problems engendered by the study of developing systems, in particular dealing with developing phase spaces, changing components, structures and functionalities, and the problem of emergence. Several papers deal with the modelling of particular experimental situations in population biology, economics and plant and muscle developments in addition to purely theoretical approaches. Novel approaches include differential inclusions and viability theory, growth tensors, archetypal dynamics, ensembles with variable structures, and (...) complex system models. The papers represent the work of theoreticians and experimental biologists, psychologists and economists. The areas covered embrace complex systems, the development of artificial life, mathematics, computer science, biology and psychology. (shrink)

If $\mathbf{S}$ is a semilattice with operators, then there is an implicational theory $\mathscr{Q}$ such that the congruence lattice $\operatorname{Con}$ is isomorphic to the lattice of all implicational theories containing $\mathscr{Q}$.

We characterize those identities and independencies which hold for all probability functions on a unary language satisfying the Principle of Atom Exchangeability. We then show that if this is strengthen to the requirement that Johnson's Sufficientness Principle holds, thus giving Carnap's Continuum of inductive methods for languages with at least two predicates, then new and somewhat inexplicable identities and independencies emerge, the latter even in the case of Carnap's Continuum for the language with just a single predicate.