Fixed Point vs. First-Order Logic on Finite Ordered Structures with Unary Relations

We prove that first order logic is strictly weaker than fixed point logic over every infinite classes of finite ordered structures with unary relations: Over these classes there is always an inductive unary relation which cannot be defined by a first-order formula, even when every inductive sentence (i.e., closed formula) can be expressed in first-order over this particular class. Our proof first establishes a property valid for every unary relation definable by first-order logic over these classes which is peculiar to classes of ordered structures with unary relations. In a second step we show that this property itself can be expressed in fixed point logic and can be used to construct a non-elementary unary relation.

Efficient Support for Common Relations in Lightweight Formal Reasoning Systems

In work that involves mathematical rigor, there are numerous benefits to adopting a representation of models and arguments that can be supplied to a formal reasoning or verification system: reusability, automatic evaluation of examples, and verification of consistency and correctness. However, accessibility has not been a priority in the design of formal verification tools that can provide these benefits. In earlier work [Lap09a], we attempt to address this broad problem by proposing several specific design criteria organized around the notion of a natural context: the sphere of awareness a working human user maintains of the relevant constructs, arguments, experiences, and background materials necessary to accomplish the task at hand. This work expands one aspect of the earlier work by considering more extensively an essential capability for any formal reasoning system whose design is oriented around simulating the natural context: native support for a collection of mathematical relations that deal with common constructs in arithmetic and set theory. We provide a formal definition for a context of relations that can be used to both validate and assist formal reasoning activities. We provide a proof that any algorithm that implements this formal structure faithfully will necessary converge. Finally, we consider the efficiency of an implementation of this formal structure that leverages modular implementations of well-known data structures: balanced search trees and transitive closures of hypergraphs.

Anglo-Irish relations in the framework of the EEC and Northern Ireland, 1969-1975

This thesis will analyse Anglo-Irish relations between 1969 and 1975, when two topics dominated the relationship: Northern Ireland and the entry of Britain and Ireland into the European Economic Community (hereafter EEC). In 1969 entry to the EEC was still only a possibility and awaited political developments, while the Northern Ireland problem had yet to escalate. 1975 on the other hand confirmed that Ireland would remain in the EEC even if Britain left while Direct Rule for Northern Ireland was confirmed as the British policy for the foreseeable future. These dates are significant because they encompass firstly pre and post entry to the EEC and how this transformed Anglo-Irish relations. Secondly they contain the commencement and then deterioration of the Northern Ireland problem and the attempts to resolve it that finally led to direct rule by Westminster. The study will examine the fluctuating nature of the relationship between Britain and Ireland. Special regard will be devoted to the demands of internal British politics and how such demands affected the relationship. Overall, the study will demonstrate how the bilateral relationship evolved under the pressure of events in Northern Ireland and adapted to the multilateral context of the EEC. It will compare the dynamics of the states’ interactions in two extremely different areas. The thesis will demonstrate how entry to the EEC transformed the unequal Anglo-Irish economic relationship and created one of partners within the EEC. It will also analyse how the developing Northern Ireland problem caused changes to British policy. In particular, it will examine how the British Government came to recognise the beneficial role that the Republic of Ireland might play in resolving the Troubles in Northern Ireland.

Towards enhanced divine-human-earth relations: a Christian-Buddhist contribution

The central claim of the dissertation is that lesser known and somewhat neglected, yet influential thinkers, within classical religious traditions have something worthwhile to contribute to the kind of ethos we should adopt in the face of the world’s various environmental crises. Moreover an exploration of such perspectives is best done in dialogue, particularly between Eastern and Western thought. I examine this claim primarily through a dialogue between the Christian philosopher John Scottus Eriugena and the Japanese Buddhist philosopher Kūkai (Kōbō Daishi). This dialogue, framed by the triad of divine-human-earth relations, primarily emphasises the oneness of all reality, and it finds expression in Eriugena’s concept of natura or phusis and Kūkai’s central teaching that the phenomenal world is the cosmic Buddha Dainichi. By highlighting this focus, I contribute to the existing academic field of ecology and religion on the subject of holism. However, I go beyond the materialist focus that generally marks such ecological holism within that field, offering instead a more metaphysical approach. This is indicated through my use of the concept of ‘immanental transcendence’ to describe Eriugena’s and Kūkai’s dynamic, numinous and mysterious notion of reality, as well as my exploration of Eriugena’s concept of theophany and Kūkai’s notion of kaji. I further explore how both philosophers highlight the human role in the process of reaching enlightenment—understood as attaining union with the whole. In that regard, I note significant differences in their positions: in particular, I note that Kūkai’s emphasis on bodily practices contrasts with Eriugena’s more conceptual approach. Finally to bolster my claim, I examine some ecologically oriented understandings of contemporary phenomenological approaches found particularly in the work of Jean-Luc Marion and to a lesser extent Merleau-Ponty, arguing that these reflect notions of reality and of the human role similar to those of the medieval philosophers.

Relations, residuals, regular interiors, and relative regular equivalence

Given a relation α (a binary sociogram) and an a priori equivalence relation π, both on the same set of individuals, it is interesting to look for the largest equivalence πo that is contained in and is regular with respect to α. The equivalence relation πo is called the regular interior of π with respect to α. The computation of πo involves the left and right residuals, a concept that generalized group inverses to the algebra of relations. A polynomial-time procedure is presented (Theorem 11) and illustrated with examples. In particular, the regular interior gives meet in the lattice of regular equivalences: the regular meet of regular equivalences is the regular interior of their intersection. Finally, the concept of relative regular equivalence is defined and compared with regular equivalence.

Ecological Relations Between the Herring and the Plankton off The North-East Coast of England

I. 430 plankton samples, which were taken by several herring drifters using the Continuous Plankton Recorder in the Shields fishing area during the summer seasons of 1931 to 1933, are analysed to show the main changes in the plankton during those seasons. 2. A comparison is made between the proportions of the different zooplankton organisms found in the plankton and the proportions of these recorded by Savage (1937) in the stomachs of herring obtained from drifters working in the same area and during the same time. The comparisons are made for 29 ten-day periods in the seasons 1931 to 1933, and in addition, for 6 ten-day periods relating to a single drifter which obtained both plankton and stomach samples at the same time in 1932. 3. The comparisons in 2 provide evidence that the herring feeds by selecting certain organisms by individual acts of capture and not by swimming open-mouthed to strain out the plankton indiscriminately: (a) Calanus and Temora in the stomachs either correspond fairly closely to the proportions in the plankton or they may be in very much higher proportions. The latter is always true regarding Anomalocera. (b) Acartia, Oithona, Cladocera and Lamellibranch larvae are always in larger proportions in the plankton than in the stomachs; this applies also to Centropages with two insignificant exceptions. (c) There is a close correspondence between the numbers of Limacina and Sagitta in the plankton and stomachs in the latter half of the 1931 season, but not during 1932 and 1933, when the numbers in the stomachs were insignificant ; during the former period there was a great scarcity of Calanus in the plankton.