The work described in this dissertation is mainly a study of some ordinal-theoretic properties of logic programs that are related to the downward powers of their immediate-consequence functions. The downward powers for any program give rise to an interesting non-increasing sequence of interpretations, whose point of convergence is called the downward closure ordinal of that program. The last appearance of ground atoms that get eliminated somewhere in this sequence is called their downward order. While it is well-known that there is no general procedure that can determine downward orders of atoms in any program, we present some rules for constructing such a procedure for a restricted class of programs. Another existing result is that for every ordinal up to and including the least non-recursive ordinal [special characters omitted] there is a logic program having that ordinal as its downward closure ordinal. However, the literature contains only a few examples of programs, constructed in an ad hoc manner, with downward closure ordinal greater than the least transfinite ordinal (ω). We contribute to bridging this wide gap between the abstract and concrete knowledge by showing the connection between some of the existing examples and the well-known concept of the order of a vertex in a graph. Using this connection and a convenient notation system for ordinals involving ground terms as bases, we construct a family [special characters omitted] of logic programs where [special characters omitted] is the least fixpoint of the function λβ[ωβ] and any member Pα of the family has downward closure ordinal ω + α. We also present an organization of a general transformation system, in which the objective is to search for transformations on syntax objects that satisfy pre-established semantic constraints. As desired transformations are not always guaranteed to exist, we present necessary and sufficient conditions for their existence. In this framework, we proceed to give transformations on logic programs for the successor and addition operations on their downward closure ordinals. / Graduate

It is commonly agreed that subjective evaluation tends to be less reliable than objective evaluation. This study represents an attempt to discover whether or not there are predictable characteristics of subjective evaluation which account for part of this unreliability. After logical analysis, three possible effects were proposed, (1) A contrast effect, in which the difference between a sample and the preceding sample or samples is minimized or exaggerated. (2) An experience effect, in which there is a long term shift in values. (3) An end effect, in which the last few samples in a series are judged according to different standards. Several possible test materials were considered. Finally, handwriting samples were chosen as satisfying the criteria of explicitness and necessity for subjective rather than disguised objective evaluation. Two experiments were devised; the first to test for a contrast effect, and the second to test for the experience and end effects. Evidence significant at the one percent level was presented for rejecting the hypothesis that there is no contrast effect. It was shown that there is a long term increase in the marks given, in which the increase was spread over the whole range of marks. There was evidence significant at the five percent level for rejecting the hypothesis that there is no end effect. It was shown that the last three samples of a long series tend to be downgraded. / Education, Faculty of / Graduate

The title of my dissertation is "confirmation theory & confirmation logic", and it consists of five Parts. The motivation of the dissertation was to construct an adequate confirmation theory that could solve "the paradoxes of confirmation" discovered by Carl G. Hempel. In Part One I try mainly to do the three things, (i) introduce the fundamentals of Hempel's theory of qualitative confirmation as the common background for subsequent discussions, (ii) review the major views of the paradoxes of confirmation, (iii) present a new view, which is more radical than other known views, and argue that a solution to the paradoxes of confirmation may require a change of logic. In Part Two I construct a number of promising three-valued logics. I employ these "quasi confirmation logics" as the underlying logics of some new confirmation theories which, I had hoped, would solve the paradoxes of confirmation. I consider three-valued logics instead of any other many-valued logics as the underlying logic for any promising confirmation theory, because I believe that there is some intimate relationship or, even, a one-to-one correspondence between the (controversial) three truth-values of "truth", "falsity" and "neither truth nor falsity" and, respectively, the (non-controversial) three confirmation-statuses of "confirmation", "disconfirmation" and "neutrality". Unfortunately, these theories were found to be semantically inadequate. This became clear after a complete semantics for them had been developed. Thus, one negative result of Part Two is that our syntactical approach to confirmation theory is wrong from the very beginning. However, from this negative result we learn a positive lesson: a semantical approach is more fundamental and decisive than a syntactical one, at least this is so for constructing an adequate theory of confirmation. It is rewarding to note that the three-valued semantics worked out in Part Two is simple, complete and the first of its kind. In fact, the new three-valued semantics is in the spirit of Frege, although the line of thought is much neglected (even by Frege himself). In Part Three I shift the search for a confirmation logic and an adequate theory of confirmation from a syntactical to a semantical approach because of the lesson learned in Part Two. After a systematic search through several promising three-valued logics I come, at last, to a plausible confirmation logic and to a confirmation theory that could solve all known paradoxes of confirmation. The promising three-valued confirmation theory is called "the internal confirmation theory". In Part Four I review and appraise the adequacy conditions laid down by Hempel as the necessary conditions for any adequate confirmation theory. Under the criticisms of Carnap, Goodman and, especially, with the help of Hanen's thorough studies, I come to almost an identical conclusion to Hanen's we should not impose a priori in a theory of qualitative confirmation any adequacy conditions laid down by Hempel except perhaps the Entailment Condition, although the internal confirmation theory also adopts the Equivalence Condition for some intrinsic reasons. In the last Part Five I try to appraise the three most important confirmation theories discussed and/or constructed in this dissertation. They are Hempel's theory of confirmation, Goodman's and Scheffler's theory of selective confirmation and the internal confirmation theory. After some more vigorous criticisms are made and some new paradoxes of confirmation are unexpectedly derived in both the theory of selective confirmation and the internal confirmation theory, I arrive at, perhaps reluctantly, this more reasonable conclusion under the present situation when there is no obvious way to overcome the new difficulties the best thing that we can do is to dissolve (i.e. to live with) all new and old paradoxes of confirmation, for Hempel may be after all right to say that the paradoxes of confirmation are not genuine and to think otherwise is to have psychological illusions as Hempel says. / Arts, Faculty of / Philosophy, Department of / Graduate

M.A. (Philosophy) / The purpose of my project is to provide necessary and sufficient conditions for a prediction to be considered good. Alex Broadbent (2013) claims that a good prediction is a stable prediction, thereby providing an internalist account of judging predictions. In contrast, this project demonstrates that an externalist approach to identifying good predictions is not only possible but, on the proposed view, more reliable too. Robert Nozick’s notion of sensitivity provides a means of understanding what makes a good prediction. It is argued that a good prediction is a sensitive prediction; one where a prediction activity tracks the truth of the claims and assumptions used to produce prediction claims. To gauge whether a prediction activity tracks the truth it is suggested that we ask the following question of the prediction: if the claims and assumptions appealed to in the prediction activity were false, would the same prediction claims have been made if at all? If the prediction claims would have been made in spite of this, then the prediction is not sensitive. Otherwise, the prediction satisfies the following tracking condition for good prediction: in the closest possible world to our own where one or more of the claims and assumptions appealed to in the prediction activity are false the prediction claims would be different or not made at all.

Schmidt's game, and other similar intersection games have played an important role in recent years in applications to number theory, dynamics, and Diophantine approximation theory. These games are real games, that is, games in which the players make moves from a complete separable metric space. The determinacy of these games trivially follows from the axiom of determinacy for real games,ADR, which is a much stronger axiom than that asserting all integer games are determined, AD. One of our main results is a general theorem which under the hypothesis AD implies the determinacy of intersection games which have a property allowing strategies to be simplified. In particular, we show that Schmidt's (α,β,ρ) game on R is determined from AD alone, but on Rn for n≥3 we show that AD does not imply the determinacy of this game. We then give an application of simple strategies and prove that the winning player in Schmidt's (α,β,ρ) game on R has a winning positional strategy, without appealing to the axiom of choice. We also prove several other results specifically related to the determinacy of Schmidt's game. These results highlight the obstacles in obtaining the determinacy of Schmidt's game from AD

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The theme of logic runs through all of Husserl's writings, from his earliest Philosophy of Arithmetic, to his final Experience and Judgment. Husserl has even characterized phenomenology as a transcendental logic. I examine Husserl's notion of logic, and it turns out to be an interesting vehicle for bringing together two diverse aspects of Husserl's phenomenology. Which by many critics are thought to be incompatible with each other, namely the purely formal aspect of phenomenology, and the aspect of phenomenology when deals with life. I show how Husserl develops a transcendental logic by going through the tradition of formal logic. He argues that traditional formal logic is not a pure logic, but is one which presupposes the world. Husserl goes beyond this and develops a logic which is pure, one based on the pure transcendental ego in which no world is presupposed. This introduces a "subjective" factor into logic. I show that no logical psychologism is implied. At the same time I show that this pure ego is the centre of life, and that it can do justice to the speculative demands contained in the concept of life. This is done in three ways: by demonstrating that i) the notion of pure transcendental ego is compatible with and can do justice to the life related concepts contained in Freudian psychology, ii) the notion of pure transcendental ego can account of other selves, Also a notion found in the concept of life, and iii) Husserl's essentialist approach is compatible with a teleological-historical approach, the latter introducing a factual element. Finally I show that Husserl's transcendental logic, with its notion of the constituting subject, is compatible with the logic of Frege. / Thesis / Doctor of Philosophy (PhD)

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