Viewing the language of modal logic as a language for describing directed graphs, a natural type of directed graph to study modally is one where the nodes are sets and the edge relation is the subset or superset relation. A well-known example from the literature on intuitionistic logic is the class of Medvedev frames $\langle W,R\rangle$ where $W$ is the set of nonempty subsets of some nonempty finite set $S$, and $xRy$ iff $x\supseteq y$, or more liberally, where (...) $\langle W,R\rangle$ is isomorphic as a directed graph to $\langle \wp(S)\setminus\{\emptyset\},\supseteq\rangle$. Prucnal [32] proved that the modal logic of Medvedev frames is not finitely axiomatizable. Here we continue the study of Medvedev frames with extended modal languages. Our results concern definability. We show that the class of Medvedev frames is definable by a formula in the language of tense logic, i.e., with a converse modality for quantifying over supersets in Medvedev frames, extended with any one of the following standard devices: nominals (for naming nodes), a difference modality (for quantifying over those $y$ such that $x\not= y$), or a complement modality (for quantifying over those $y$ such that $x\not\supseteq y$). It follows that either the logic of Medvedev frames in one of these tense languages is finitely axiomatizable---which would answer the open question of whether Medvedev's [31] "logic of finite problems'' is decidable---or else the minimal logics in these languages extended with our defining formulas are the beginnings of infinite sequences of frame-incomplete logics. (shrink)

Intermediate prepositional logics we consider here describe the setI() of regular informational types introduced by Yu. T. Medvedev [7]. He showed thatI() is a Heyting algebra. This algebra gives rise to the logic of infinite problems from [13] denoted here asLM 1. Some other definitions of negation inI() lead to logicsLM n (n ). We study inclusions between these and other systems, proveLM n to be non-finitely axiomatizable (n ) and recursively axiomatizable (n ). We also show that formulas (...) in one variable do not separateLM from Heyting's logicH, andLM n (n ) from Scott's logic (H+S). (shrink)

Medvedev's intermediate logic (MV) can be defined by means of Kripke semantics as the family of Kripke frames given by finite Boolean algebras without units as partially ordered sets. The aim of this paper is to present a proof of the theorem: For every set of connectives such that the-fragment ofMV equals the fragment of intuitionistic logic. The final part of the paper brings the negative solution to the problem set forth by T. Hosoi and H. Ono, namely: (...) is an intermediate logic based on the axiom (abc) (ab)(a c) separable? (shrink)

We carry out some of Galois' work in the setting of an arbitrary first-order theory T. We replace the ambient algebraically closed field by a large model M of T, replace fields by definably closed subsets of M, assume that T codes finite sets, and obtain the fundamental duality of Galois theory matching subgroups of the Galois group of L over F with intermediate extensions F ≤ K ≤ L. This exposition of a special case of [10] has the advantage (...) of requiring almost no background beyond familiarity with fields, polynomials, first-order formulae, and automorphisms. (shrink)

Name der Zeitschrift: Byzantinische Zeitschrift Jahrgang: 109 Heft: 2 Seiten: 821-836.

This paper began as a generalization of a part of the author's PhD thesis about ACFA and ended up with a characterization of groups definable in T A . The thesis concerns minimal formulae of the form x ∈ A ∧ σ(x) = f(x) for an algebraic curve A and a dominant rational function f: A → σ(A). These are shown to be uniform in the Zilber trichotomy, and the pairs (A, f) that fall into each of the three cases (...) are characterized. These characterizations are definable in families. This paper covers approximately half of the thesis, namely those parts of it which can be made purely model-theoretic by moving from ACFA, the model companion of the class of algebraically closed fields with an endomorphism, to T A , the model companion of the class of models of an arbitrary totally-transcendental theory T with an injective endomorphism, if this model-companion exists. A T A analog of the characterization of groups definable in ACFA is obtained in the process. The full characterization of the cases of the Zilber trichotomy in the thesis is obtained from these intermediate results with heavy use of algebraic geometry. (shrink)

What is everything made of? -- Do I have free will? -- Can I know the world? -- Does God exist? -- How shall I live? -- What are my rights? -- How can I be saved? -- Appendix: Letters on philosophical topics.

The 1968 My Lai Massacre in South Vietnam and shocking events at Abu Graibh prison in Iraq in 2003 show that the behaviour of some in the American military has descended into barbarism. How strong is the military's commitment to avoiding misconduct and atrocity? This timely and compelling book asks critical questions and raises sobering issues the Australian Army can't ignore. Leading military personnel, aid workers, commentators and scholars discuss the Australian Army's commitment to behaving ethically and the challenges associated (...) with contemporary warfare. With chapters on social media and inciting violence, cyberweapons, and special operations and humanitarian deployments, 'Ethics Under Fire' offers a rare insight into the key issues facing the modern army, from technology and tactics to terrorism"--Back cover. (shrink)

John Frame gives us an accessible introduction to "triperspectival" study-where theological issues are fruitfully viewed from multiple perspectives without compromise to their unity and truth. Book jacket.

A History of Western Philosophy and Theology is the fruit of John Frame's forty-five years of teaching philosophical subjects. No other survey of the history of Western thought offers the same invigorating blend of expositional clarity, critical insight, and biblical wisdom. The supplemental study questions, bibliographies, links to audio lectures, quotes from influential thinkers, twenty appendices, and indexed glossary make this an excellent main textbook choice for seminary- and college-level courses and for personal study. Book jacket.

Charles Darwin found much in Australia to challenge and inform his thinking. This book explores the impact that Darwin’s short visit to Australia in 1836 had on the man himself and on the emerging nation. Now, more than 170 years later, Darwin continues to influence Australian attitudes to life and living.

In this paper we seek to make the case for a teaching and learning strategy that integrates business ethics in the curriculum, whilst not precluding a disciplines based approach to this subject. We do this in the context of specific work experience modules at undergraduate level which are offered by Middlesex University Business School, part of a modern university based in North West London. We firstly outline our educative values and then the modules that form the basis of our research. (...) We then identify and elaborate what we believe are the five dimensions which distinguish an integrated approach based on work experience from a disciplines-based approach, namely: process and content, internal and external, facilitation and teaching, covert and overt, and living wisdom and established wisdom. The last dimension draws on the practical relevance of the Aristotelian notion of phronesis inherent in our approach. We go on to provide two case examples of our practice to illustrate our perspective and in support of our conclusions. These are that reflection integrated into the Business Studies curriculum, using the ASKE typology of learning [Frame, 2001, Proceedings of the 9th Annual Teaching and Learning Conference, p. 80], in respect of personal and group process in a work experience context, provides a useful heuristic for the development of moral sensibility and ethical practice. (shrink)