No description available.

No description available.

Gödel proved his Incompleteness theorems for any theory 'strong' enough to represent recursive functions. In the process he showed that the provability predicate can be represented in such theories. Modal logics of provability are modal logics which attempt to express the concept of 'provability' and 'consistency' using the modal operators '[]' and '<>' respectively. This is achieved by forcing '[]' to behave like the provability predicate. GL is a modal logic which has been shown to be complete and sound with respect to arithmetic theories (theories which can represent all recursive functions), hence results about concepts such as 'consistency,' 'provability' and 'decidability' in arithmetic theories can be stated and proved in GL. It has also been proved that GL is complete with respect to the class of finite, transitive, reversely well-founded models. This essentially means that the set of theorems of GL is recursive and hence there exists an effective procedure to determine whether a given wff is a theorem of GL or not. We investigate a weaker version of GL called GH and show that GH is not complete with respect to arithmetic theories. We show this by first showing that GH is a proper subset of GL and then showing that the theorems missing from GH are properties of the provability predicate. We finally, show that GH is not complete with respect to the class of transitive, reversely well-founded models and hence not sound and complete with respect to any frame. / Master of Science

This investigation proposes a powerful formalization of common sense knowledge based on function-free normal deduction graphs (NDGs) which form a powerful tool for deriving Horn and non-Horn clauses without functions. Such formalization allows common sense reasoning since it has the ability to handle not only negative but also incomplete information.

In this dissertation, we investigate how commonsense reasoning can be formalized by using extended logic programs. In this investigation, we first use extended logic programs to formalize inheritance hierarchies with exceptions by adopting McCarthy's simple abnormality formalism to express uncertain knowledge. In our representation, not only credulous reasoning can be performed but also the ambiguity-blocking inheritance and the ambiguity-propagating inheritance in skeptical reasoning are simulated. In response to the anomalous extension problem, we explore and discover that the intuition underlying commonsense reasoning is a kind of forward reasoning. The unidirectional nature of this reasoning is applied by many reformulations of the Yale shooting problem to exclude the undesired conclusion. We then identify defeasible conclusions in our representation based on the syntax of extended logic programs. A similar idea is also applied to other formalizations of commonsense reasoning to achieve such a purpose.

A thesis submitted to the Faculty of Humanities, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy September 2015 / This study explores the mathematical discourse of Grade 11 learners on the topic function through their routines. From a commognitive perspective, it describes routines in terms of exploration and ritual. Data was collected through in-depth interviews with 18 pairs of learners, from six South African secondary schools, capturing a landscape of public schooling, where poor performance in Mathematics predominates. The questions pursued became: why does poor performance persist and what might a commognitive lens bring into view? With the discursive turn in education research, commognition provides an alternate view of learning mathematics. With the emphasis on participation and not on constraints from inherited mental ability, the study explored the nature of learner discourse on the object, function. Function was chosen as it holds significant time and weight in the secondary school curriculum. Examining learners’ mathematical routines with the object was a way to look at their discourse development: what were the signifiers related to the object and what these made possible for learners to realise. Within learners’ routines, I was able to characterise these realisations, which were described and categorised. This enabled a description of learner thinking over three signifiers of function in school Mathematics: the algebraic expression, table and graph. In each school, Grade 11 learners were separated into three groups according to the levels at which they were performing, from summative scores of grade 11 assessments, so as to enable a description of discourse related to performance. Interviews were conducted in pairs, and designed to provoke discussion on aspects of function and its signifiers between learners in each pair. This communication between learners and with the interviewer provided data for description and analysis of rituals and explorations. Zooming in and out again on these routines made a characterisation of the discourse of failure possible, which is seldom done. It became apparent early in the study that learners talked of the object function, without a formal mathematical narrative, a definition in other words, of the object. The object was thus vested in its signifiers. The absence of an individualised formal narrative of the object impacts directly what is made possible for learners to realise, hence to learn. The study makes the following contributions: first, it describes learners’ discursive routines as they work with the object function. Second, it characterises the discourse of learners at different levels of performance. Third, it starts exploration of commognition as an alternate means to look at poor performance. The strengths and limitations of the theory as it pertains to this study, are discussed later in the concluding chapter. Keywords commognition, discourse, communication, participation, routines, exploration, ritual, learners, learning, narratives, endorsed narratives, visual mediators.

"Let P(n) be a statement for every positive integer n. We denote the set of all positive integers by N and consider G = {n [is an element of] N [such that] P(n) is true}. The principle of mathematical induction can now be stated as follows: If [(i) 1 [is an element of] G and, (ii) for all k [is an element of] N if k [is an element of] G, then k + 1 [is an element of] G], then G = N. Now symbolize this statement as follows: P: 1 [is an element of] G. R: k [is an element of] G. S: k + 1 [is an element of] G. Q: G = N. Therefore the statement of the principle of mathematical induction can be seen in the following form. If [P and, [for all] k [is an element of] N (if R, then S)], then Q. One strategy for teaching this principle is to explain that in order to apply the principle of mathematical induction and assert Q, one must appeal to the logical rule of modus ponens (the law of detachment). That is, we must affirm the antecedent [P and, [for all] k [is an element of] N (if R, then S)], and then we can assert Q. Therefore the research hypothesis for this study was that if people have the prerequisite knowledge of logic, and that if they are taught the principle of mathematical induction in terms of logic, then they will perform better on a criterion test over the principle of mathematical induction than people who are not taught in terms of logic"--Introduction. / Typescript. / "June, 1972." / "Submitted to the Department of Mathematics Education in partial fulfillment of the requirements for the degree of Doctor of Education in Mathematics Education." / Advisor: E. D. Nichols, Professor Directing Dissertation. / Includes bibliographical references.

Thesis. 1978. Ph.D.--Massachusetts Institute of Technology. Dept. of Linguistics and Philosophy. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND HUMANITIES. / Bibliography: leaves 131-132. / by David D. Auerbach. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Linguistics and Philosophy, 1982. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND HUMANITIES / Bibliography: leaves 175-176. / by Lon A. Berk. / Ph.D.

No description available.