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Supporting the Use of External Representations in Problem Solving: the Need for Flexible Learning Environments
, 1995
"... External representations (ERs) are effective in reasoning due to their cognitive and semantic properties. We investigated subjects' use of ERs in their solutions to analytical reasoning problems. Two sources of data were analysed. The first consisted of a large corpus of ERs (`workscratchings&a ..."
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External representations (ERs) are effective in reasoning due to their cognitive and semantic properties. We investigated subjects' use of ERs in their solutions to analytical reasoning problems. Two sources of data were analysed. The first consisted of a large corpus of ERs (`workscratchings') used by students in their solutions to problems administered via paper and pencil tests. The second source of data was collected using switchER, a computerbased system that administered the problems, provided a range of ER construction environments for the subject to choose between and which dynamically logged usersystem interactions. SwitchER was developed in order to study the process and timecourse of ER use and to investigate the mechanisms (such as ER switching) by which subjects resolve impasses in reasoning. The results showed great diversity of ER use across subjects, allowing the utility of various ERs under differing task conditions to be studied. The range of ERs used by subjects ...
A History of Satisfiability
, 2009
"... 1.1. Preface: the concept of satisfiability Interest in Satisfiability is expanding for a variety of reasons, not in the least because nowadays more problems are being solved faster by SAT solvers than other means. This is probably because Satisfiability stands at the crossroads of logic, graph theo ..."
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1.1. Preface: the concept of satisfiability Interest in Satisfiability is expanding for a variety of reasons, not in the least because nowadays more problems are being solved faster by SAT solvers than other means. This is probably because Satisfiability stands at the crossroads of logic, graph theory, computer science, computer engineering, and operations research. Thus, many problems originating in one of these fields typically have multiple translations to Satisfiability and there exist many mathematical tools available to the SAT solver to assist in solving them with improved performance. Because of the strong links to so many fields, especially logic, the history of Satisfiability can best be understood as it unfolds with respect to its logic roots. Thus, in addition to timelining events specific to Satisfiability, the chapter follows the presence of Satisfiability in logic as it was developed to model human thought and scientific reasoning through its use in computer design and now as modeling tool for solving a variety of practical problems. In order to succeed in this, we must introduce many ideas that have arisen during numerous attempts to reason
Faceted Information Representation
 PROCEEDINGS OF THE 8TH INTERNATIONAL CONFERENCE ON CONCEPTUAL STRUCTURES, SHAKER
, 2000
"... This paper presents an abstract formalization of the notion of "facets". Facets are ..."
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This paper presents an abstract formalization of the notion of "facets". Facets are
A Diagrammatic Formal System for Euclidean Geometry
, 2001
"... It has long been commonly assumed that geometric diagrams can only be used as aids to human intuition and cannot be used in rigorous proofs of theorems of Euclidean geometry. This work gives a formal system FG whose basic syntactic objects are geometric diagrams and which is strong enough to formali ..."
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It has long been commonly assumed that geometric diagrams can only be used as aids to human intuition and cannot be used in rigorous proofs of theorems of Euclidean geometry. This work gives a formal system FG whose basic syntactic objects are geometric diagrams and which is strong enough to formalize most if not all of what is contained in the first several books of Euclid’s Elements. Thisformal system is much more natural than other formalizations of geometry have been. Most correct informal geometric proofs using diagrams can be translated fairly easily into this system, and formal proofs in this system are not significantly harder to understand than the corresponding informal proofs. It has also been adapted into a computer system called CDEG (Computerized Diagrammatic Euclidean Geometry) for giving formal geometric proofs using diagrams. The formal system FG is used here to prove metamathematical and complexity theoretic results about the logical structure of Euclidean geometry and the uses of diagrams in geometry.
Cognitive processes in prepositional reasoning
 Psychological Review
, 1983
"... Propositional reasoning is the ability to draw conclusions on the basis of sentence connectives such as and, if, or, and not. A psychological theory of prepositional reasoning explains the mental operations that underlie this ability. The ANDS (A Natural Deduction System) model, described in the fol ..."
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Propositional reasoning is the ability to draw conclusions on the basis of sentence connectives such as and, if, or, and not. A psychological theory of prepositional reasoning explains the mental operations that underlie this ability. The ANDS (A Natural Deduction System) model, described in the following pages, is one such theory that makes explicit assumptions about memory and control in deduction. ANDS uses natural deduction rules that manipulate propositions in a hierarchically structured working memory and that apply in either a forward or a backward direction (from the premises of an argument to its conclusion or from the conclusion to the premises). The rules also allow suppositions to be introduced during the deduction process. A computer simulation incorporating these ideas yields proofs that are similar to those of untrained subjects, as assessed by their decisions and explanations concerning the validity of arguments. The model also provides an account of memory for proofs in text and can be extended to a theory of causal connectives. The importance of deductive reasoning lo
Voting in the Medieval Papacy and Religious Orders
"... We take institutions seriously as both a rational response to dilemmas in which agents found themselves and a frame to which later rational agents adapted their behaviour in turn. Medieval corporate bodies knew that they needed choice procedures. Although the social choice advances of ancient Greece ..."
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We take institutions seriously as both a rational response to dilemmas in which agents found themselves and a frame to which later rational agents adapted their behaviour in turn. Medieval corporate bodies knew that they needed choice procedures. Although the social choice advances of ancient Greece and Rome were not rediscovered until the high middle ages, the rational design of choice institutions predated their rediscovery and took some new paths. Both Ramon Lull (ca 12321316) and Nicolaus of Cusa (a.k.a Cusanus; 140164) made contributions which had been believed to be centuries more recent. Lull promotes the method of pairwise comparison, and proposes the Copeland rule to select a winner. Cusanus proposes the Borda rule, which should properly be renamed the Cusanus rule. Voting might be needed in any institution ruled by more than one person, where decisions could not simply be handed down from above. Medieval theologians no doubt believed that God’s word was handed down from above; but they well knew that they often had to decide among rival human interpretations of it. The Church faced its own decision problem every time a new Pope needed to be elected. Bodies not directly in the hierarchy of the Church had to evolve their own decision procedures. The chief such bodies were commercial and urban corporations; religious orders; and universities. Voting in the Medieval Papacy and Religious Orders 1.
Seeing and Visualizing: It’s Not What You Think, An Essay On Vision And Visual Imagination
"... ..."
On the poverty of Venn diagrams for teaching probability: their history and replacement by Eikosograms
, 2002
"... Diagrams convey information, some intended some not. A history of the information content of ringed diagrams and their use by Euler and Venn is given. It is argued that for the purposes of teaching introductory probability, Venn diagrams are either inappropriate or inferior to other diagrams. A diag ..."
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Diagrams convey information, some intended some not. A history of the information content of ringed diagrams and their use by Euler and Venn is given. It is argued that for the purposes of teaching introductory probability, Venn diagrams are either inappropriate or inferior to other diagrams. A diagram we call an eikosogram is shown to be coincident with what is meant by probability and so visually introduces all the rules of probability including Bayes’ theorem and the product rule for independent events. Eikosograms clearly demonstrate unconditional and conditional independence – both of events and of random variables. An approach to teaching probability via the eikosogram and other more familiar diagrams is described. It is recommended that Venn diagrams no longer be used to teach probability.
“Drawing Illusions ” – a case study in the incorrectness of diagrammatic reasoning
, 1999
"... In “Something to Reckon With ” [6], a system for diagramming syllogistic inferences using straight line segments is presented (see also Englebretsen [5]). In the light of recent research on the representational power of diagrammatic representation systems (Lemon and Pratt [12, 13]) we point out some ..."
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In “Something to Reckon With ” [6], a system for diagramming syllogistic inferences using straight line segments is presented (see also Englebretsen [5]). In the light of recent research on the representational power of diagrammatic representation systems (Lemon and Pratt [12, 13]) we point out some problems with the proposal, and indeed, with any proposal for representing logically possible situations diagrammatically. We shall first outline the proposed linear diagrammatic system of Englebretsen [5], and then show by means of counterexamples that it is inadequate as a representation scheme for general logical inferences (the task for which the system is intended). We also show that modifications to the system fail to remedy the problems. The considerations we present are not limited to the particular proposals of Englebretsen [5, 6]; we thus draw a more general moral about the use of spatial relations in representation systems. 1 1 Diagrammatic representation systems
Critical notice Logical machines:
"... Abstract This essay discusses Peirce’s appeal to logical machines as an argument against psychologism. It also contends that some of Peirce’s antipsychologistic remarks on logic contain interesting premonitions arising from his perception of the asymmetry of proof complexity in monadic and relation ..."
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Abstract This essay discusses Peirce’s appeal to logical machines as an argument against psychologism. It also contends that some of Peirce’s antipsychologistic remarks on logic contain interesting premonitions arising from his perception of the asymmetry of proof complexity in monadic and relational logical calculi that were only given full formulation and explication in the early twentieth century through Church’s Theorem and Hilbert’s broadranging Entscheidungsproblem. In Gulliver’s Travels, Jonathan Swift relates that in his voyage to Balnibarbi Gulliver comes across a professor of the grand academy of Lagado who shows him a machine capable of improving and extending knowledge by ‘mechanical operations. ’ (Swift 1735: 195) The academician explains that ‘by his contrivance, the most ignorant person at a reasonable charge, and with a little bodily labour, may write books in philosophy, poetry, politicks, law, mathematicks and theology, without the least assistance from genius or study. ’ (Swift 1735: