Abstract not found

This chapter presents an overview of the major interpretations of probability followed by an outline of the objective Bayesian interpretation and a discussion of the key challenges it faces. I discuss the ramifications of interpretations of probability and objective Bayesianism for the philosophy of mathematics in general.

To a majority of the people involved in the study of expertise from a computational perspective, `expertise' tends to refer to domains such as medical diagnosis, aircraft piloting, auditing, etc. The reasoning in domains like these appears to be ready-made for computational packaging. But what if we try to cast a broader, braver net in an attempt to catch varieties of expertise out there in the real world which don't, at least at first glance, look like they can be rendered in computational terms? In particular, what about mathematical expertise? In this chapter I focus on elementary "infinitary " expertise in the domain of mathematical logic. I argue that at least some of this expertise is indeed uncomputable. I end by briefly discussing the implications of this argument for the practice of AI and expert systems.

We prove normalization for a dependently typed lambda-calculus extended with first-order data types and computation schemata for first-order size-change terminating recursive functions. Size-change termination, introduced by C.S. Lee, N.D. Jones and A.M. Ben-Amram, can be seen as a generalized form of structural induction, which allows inductive computations and proofs to be defined in a straight-forward manner. The language can be used as a proof system—an extension of Martin-Löf’s Logical Framework.

This thesis provides a reconstruction of the B-method and sketches an implementation of its tool support.For background, this work investigates the field of formal methods in general and the relevance of formal methods to software engineering in particular. Formal (first-order) logic is also considered: both its development and important points relevant to formal methods. Automated reasoning, particularly its theoretical limits as well as unification and resolution, is discussed. The main part of this thesis is a systematic reconstruction of the B-method, starting from its version of untyped predicate calculus and typed set theory, continuing with the Generalized Substitution Language (GSL) and finishing with the Abstract Machine Notation (AMN). Specification, refinement and implementation of a simple example using the B-method is presented. Both validation and verification of specifications, refinements and implementations using the B-method is discussed. The thesis concludes with a report of the current state of the effort (by the author) to implement the tool support of the B-method, as the Ebba Toolset. The main design decisions are discussed. The use of Unicode as the primary input encoding of AMN and GSL in Ebba is described.

This essay explores the implications of the thesis that implementation is semantic interpretation. Implementation is (at least) a ternary relation: I is an implementation of an ‘Abstraction ’ A in some medium M. Examples are presented from the arts, from language, from computer science and from cognitive science, where both brains and computers can be understood as implementing a ‘mind Abstraction’. Implementations have side effects due to the implementing medium; these can account for several puzzles surrounding qualia. Finally, an argument for benign panpsychism is developed.

Machine for DRT? In our discussion of correctness and full abstractness, we noted that one must specify some notion of behavior and of behavioral equivalence and that one could do this in many different ways and at many different levels of abstraction. In the case mentioned above, of partial correctness behavior, we are interested only in the input-output behavior of programs, with respect only to highly abstract properties of undecomposed states. To get a feel for what might be involved in a more ful-bodied conception of behavior, let us turn briefly to a consideration of Kamp's conception, which, though certainly not straightforwardly computational, is `mentalistic' and oriented toward issues of cognitive processes. One can think of the discourse representation construction algorithm as specifying the reader or parser for an interpreter for an abstract machine whose input is a stream of sentences and whose output (when it is initialized with a starting DRS) is a stream of DRS's or a...

. The logical symbol # (pronounced "for all") denotes universal quantification. For example, the formula "#x#B x # x 2 +1" reads "for all x a member of the real numbers, x is less than or equal to x-squared plus one" (i.e., no real number is greater than one more than its own square). The logical symbol # (pronounced "there 2 exists") denotes existential quantification. For example, the formula "#x#8 | x 2 =5x" states that there exists an integer whose square is equal to 5 times itself (i.e., x is either 5 or 0) . These connectives may be composed in more complicated formulae, as in the following example: "#x#8 #y#8#| y>x" which states that there is no largest integer. The logical connective (pronounced

A set is formally an undefined term, but intuitively it is a (possibly empty) collection of arbitrary objects. A set is usually denoted by curly braces and some (optional) restrictions. Examples of sets are {1,2,3}, {hi, there}, and {k | k is a perfect square}. The symbol ∈ denotes set membership, while the symbol ∉ denotes set non-membership; for example, 7∈{p | p prime} states that 7 is a prime number, while q∉{0,2,4,6,...} states that q is not an even number. Some common sets are denoted by special notation: The natural numbers: = {1,2,3,...}

Abstract: Can an epistemic conception of truth and an endorsement of the excluded middle (together with other principles of classical logic abandoned by the intuitionists) cohabit in a plausible philosophical view? In PART I I describe the general problem concerning the relation between the epistemic conception of truth and the principle of excluded middle. In PART II I give a historical overview of different attitudes regarding the problem. In PART III I sketch a possible holistic solution. Part I The Problem §1. The epistemic conception of truth. The epistemic conception of truth can be formulated in many ways. But the basic idea is that truth is explained in terms of epistemic notions, like experience, argument, proof, knowledge, etc. One way of formulating this idea is by saying that truth and knowability coincide, i.e. for every statement S