A generic logic environment

Imperial Users onl

Aspects of the constructive omega rule within automated deduction

In general, cut elimination holds for arithmetical systems with the w -rule, but not for systems with ordinary induction. Hence in the latter, there is the problem of generalisation, since arbitrary formulae can be cut in. This makes automatic theorem -proving very difficult. An important technique for investigating derivability in formal systems of arithmetic has been to embed such systems into semi- formal systems with the w -rule. This thesis describes the implementation of such a system. Moreover, an important application is presented in the form of a new method of generalisation by means of "guiding proofs" in the stronger system, which sometimes succeeds in producing proofs in the original system when other methods fail

Visualisation and manipulation tools for Modal logic

In this thesis, an investigation into how visualisation and manipulation tools can provide better support for learners of Modal logic is described. Problems associated with learning Modal logic are also researched.Seven areas topics in Modal logic are investigated, as is the influence of domain independent factors (e. g. motivation) on learning. Studies show that students find concepts such as Modal proofs and systems difficult to learn, whilst possible worlds and Modes are fairly straightforward. Areas such as reference, belief and accessibility relations fall between these extremes.Two roles for representations in reasoning are identified: providing a concrete domain for students to reason about, and supporting the process of reasoning. Systems which make use of these complementary representations were found to be more effective for learners than either the syntactic or the diagrammatic representations traditionally used to teach Modal logic.A review of software used to support students learning logic highlights two important features: the use of examples, and automation of routine tasks. A learning environment for Modal logic was designed which incorporated these. The environment was developed using an adapted version of Smalltalk's Model-View-Controller mechanism, and incorporates complementary representations, enhance by direct manipulation.A further study investigates the added benefits of using this tool, as opposed to using the same representation but working with pen and paper. This confirms the importance of using 'concrete' content representations and minimising learners' cognitive load. Performance measures show that software users learnt more, had a deeper style of learning, and found the topics less abstract than their counterparts working with pen & paper.This research shows that complementary representations are an effective way of supporting students studying Modal logic, and that visualisation and manipulation tools which incorporate these systems will provide additional benefits for learners