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A Computational Model of Lakatosstyle Reasoning
, 2007
"... Lakatos outlined a theory of mathematical discovery and justification, which suggests ways in which concepts, conjectures and proofs gradually evolve via interaction between mathematicians. Different mathematicians may have different interpretations of a conjecture, examples or counterexamples of i ..."
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Cited by 7 (6 self)
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Lakatos outlined a theory of mathematical discovery and justification, which suggests ways in which concepts, conjectures and proofs gradually evolve via interaction between mathematicians. Different mathematicians may have different interpretations of a conjecture, examples or counterexamples of it, and beliefs regarding its value or theoremhood. Through discussion, concepts are refined and conjectures and proofs modified. We hypothesise that (i) it is possible to computationally represent Lakatos’s theory, and (ii) it is useful to do so. In order to test our hypotheses we have developed a computational model of his theory. Our model is a multiagent dialogue system. Each agent has a copy of a preexisting theory formation system, which can form concepts and make conjectures which empirically hold for the objects of interest supplied. Distributing the objects of interest between agents means that they form different theories, which they communicate to each other. Agents then find counterexamples and use methods identified by Lakatos to suggest modifications to conjectures, concept definitions and proofs. Our main aim is to provide a computational reading of Lakatos’s theory, by interpreting it as a
The HOL Light manual (1.1)
, 2000
"... ion is in a precise sense a converse operation to application. Given 49 50 CHAPTER 5. PRIMITIVE BASIS OF HOL LIGHT a variable x and a term t, which may or may not contain x, one can construct the socalled lambdaabstraction x: t, which means `the function of x that yields t'. (In HOL's ASCII concr ..."
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Cited by 6 (0 self)
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ion is in a precise sense a converse operation to application. Given 49 50 CHAPTER 5. PRIMITIVE BASIS OF HOL LIGHT a variable x and a term t, which may or may not contain x, one can construct the socalled lambdaabstraction x: t, which means `the function of x that yields t'. (In HOL's ASCII concrete syntax the backslash is used, e.g. \x. t.) For example, x: x + 1 is the function that adds one to its argument. Abstractions are not often seen in informal mathematics, but they have at least two merits. First, they allow one to write anonymous functionvalued expressions without naming them (occasionally one sees x 7! t[x] used for this purpose), and since our logic is avowedly higher order, it's desirable to place functions on an equal footing with rstorder objects in this way. Secondly, they make variable dependencies and binding explicit; by contrast in informal mathematics one often writes f(x) in situations where one really means x: f(x). We should give some idea of how ordina...
The HOL Light manual (1.0)
, 1998
"... ion is in a precise sense a converse operation to application. Given 49 50 CHAPTER 5. PRIMITIVE BASIS OF HOL LIGHT a variable x and a term t, which may or may not contain x, one can construct the socalled lambdaabstraction x: t, which means `the function of x that yields t'. (In HOL's ASCII concr ..."
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Cited by 1 (0 self)
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ion is in a precise sense a converse operation to application. Given 49 50 CHAPTER 5. PRIMITIVE BASIS OF HOL LIGHT a variable x and a term t, which may or may not contain x, one can construct the socalled lambdaabstraction x: t, which means `the function of x that yields t'. (In HOL's ASCII concrete syntax the backslash is used, e.g. "x. t.) For example, x: x + 1 is the function that adds one to its argument. Abstractions are not often seen in informal mathematics, but they have at least two merits. First, they allow one to write anonymous functionvalued expressions without naming them (occasionally one sees x 7! t[x] used for this purpose), and since our logic is avowedly higher order, it's desirable to place functions on an equal footing with firstorder objects in this way. Secondly, they make variable dependencies and binding explicit; by contrast in informal mathematics one often writes f(x) in situations where one really means x: f(x). We should give some idea of how ordinary...
Bibliography
, 1996
"... interpretation and optimising transformations of applicative programs. Technical report CST1581, Computer Science Department, Edinburgh University, King's Buildings, Mayfield Road, Edinburgh EH9 3JZ, UK. Neumann, P. G. (1995) Computerrelated risks. AddisonWesley. Oppen, D. (1980) Prettyprintin ..."
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interpretation and optimising transformations of applicative programs. Technical report CST1581, Computer Science Department, Edinburgh University, King's Buildings, Mayfield Road, Edinburgh EH9 3JZ, UK. Neumann, P. G. (1995) Computerrelated risks. AddisonWesley. Oppen, D. (1980) Prettyprinting. ACM Transactions on Programming Languages and Systems, 2, 465483. Paulson, L. C. (1983) A higherorder implementation of rewriting. Science of Computer Programming , 3, 119149. Paulson, L. C. (1991) ML for the Working Programmer. Cambridge University Press. Pelletier, F. J. (1986) Seventyfive problems for testing automatic theorem provers. Journal of Automated Reasoning , 2, 191216. Errata, JAR 4 (1988), 235236. Peterson, I. (1996) Fatal Defect : Chasing Killer Computer Bugs. Arrow. Potts, P. (1996) Computable real arithmetic using linear fractional transformations. Unpublished draft for PhD thesis, available on the Web as http://theory.doc.ic.ac.uk/~pjp/pub/phd/draft.ps.gz....
The Irrelevance of the Concept of Relevance to the Concept of Relevant Consequence
"... It is often suggested that truthpreservation is insufficient for logical consequence, and that consequence needs to satisfy a further condition of relevance. Premises and conclusion in a valid consequence must be relevant to one another, and truthpreservation is too coarsegrained a notion to ..."
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It is often suggested that truthpreservation is insufficient for logical consequence, and that consequence needs to satisfy a further condition of relevance. Premises and conclusion in a valid consequence must be relevant to one another, and truthpreservation is too coarsegrained a notion to guarantee that. Thus logical consequence is the intersection of truthpreservation and relevance.
An Emerging Domain Science  A Rôle for Stanisław Leshniewski’s Mereology and Bertrand Russell’s . . .
 HIGHERORDER AND SYMBOLIC COMPUTATION, A SPRINGER JOURNAL
"... Domain engineers describe universes of discourse such as bookkeeping, the financial service industry, container shipping lines, logistics, oil pipelines, railways, etc. In doing so domain engineers have to decide on such issues as identification of that which is to be described; which of the describ ..."
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Domain engineers describe universes of discourse such as bookkeeping, the financial service industry, container shipping lines, logistics, oil pipelines, railways, etc. In doing so domain engineers have to decide on such issues as identification of that which is to be described; which of the describable phenomena and concepts are (to be described as) entities, operations, events, and which as behaviours; which entities are (to be described as) continuous which are (...) discrete, which are (...) atomic, which are (...) composite and what are the attributes of either and the mereology of composite entities, i.e., the way in which they are put together from subentities. For each of these issues and their composite presentation the domain engineer has to decide on levels of abstraction, what to include and what to exclude. In doing so the domain engineer thus has to have a firm grasp on the a robust understanding and practice of the very many issues of description: what can be described, identifying what is to be described, how to describe, description principles, description techniques, description tools and laws of description. This paper will outline the issues in the slanted type font.
Leonard, Goodman, and the Development of the Calculus of Individuals
, 2009
"... This paper investigates the relation of the Calculus of Individuals presented by Henry S. Leonard and Nelson Goodman in their joint paper, and an earlier version of it, the socalled Calculus of Singular Terms, introduced by Leonard in his Ph.D. dissertation thesis Singular Terms. The latter calculu ..."
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This paper investigates the relation of the Calculus of Individuals presented by Henry S. Leonard and Nelson Goodman in their joint paper, and an earlier version of it, the socalled Calculus of Singular Terms, introduced by Leonard in his Ph.D. dissertation thesis Singular Terms. The latter calculus is shown to be a proper subsystem of the former. Further, Leonard’s projected extension of his system is described, and the definition of an nonextensional partrelation in his system is proposed. The final section discusses to what extent Goodman might have contributed to the formulation of the Calculus of Individuals.