We characterize the modal logics of elementary classes of Kripke frames as precisely those modal logics that are axiomatized by modal axioms synthesized in a certain effective way from "quasi-positive" sentences of hybrid logic. These are pure positive hybrid sentences with arbitrary existential and relativized universal quantification over nominals. The proof has three steps. The first step is to use the known result that the modal logic of any elementary class of Kripke frames is also the modal logic of the (...) closure of this class under disjoint unions, generated subframes, bounded morphic images, and ultraroots. This latter class can be defined by the first-order sentences of a special syntactic form (called pseudo-equations by Goldblatt) that are valid in the former class. The second step is to translate these pseudo-equations into equivalent quasi-positive hybrid sentences. In the third and main step, we show that any quasi-positive sentence S generates an infinite set of modal formulas called "approximants," which together axiomatize a canonical modal logic that is sound and complete for the class of frames validating S. The proof is analogous to standard proofs of Sahlqvist's theorem. It generalizes to sets of quasi-positive sentences. The main result now follows. (shrink)

We prove that everyn-modal logic betweenKnandS5nis undecidable, whenever n ≥ 3. We also show that each of these logics is non-finitely axiomatizable, lacks the product finite model property, and there is no algorithm deciding whether a finite frame validates the logic. These results answer several questions of Gabbay and Shehtman. The proofs combine the modal logic technique of Yankov–Fine frame formulas with algebraic logic results of Halmos, Johnson and Monk, and give a reduction of the representation problem of finite relation (...) algebras. (shrink)

We show that there exist 2 ℵ 0 equational classes of Boolean algebras with operators that are not generated by the complex algebras of any first-order definable class of relational structures. Using a variant of this construction, we resolve a long-standing question of Fine, by exhibiting a bimodal logic that is valid in its canonical frames, but is not sound and complete for any first-order definable class of Kripke frames (a monomodal example can then be obtained using simulation results of (...) Thomason). The constructions use the result of Erd $\H{o}$ s that there are finite graphs with arbitrarily large chromatic number and girth. (shrink)

We study a canonical modal logic introduced by Lemmon, and axiomatised by an infinite sequence of axioms generalising McKinsey’s formula. We prove that the class of all frames for this logic is not closed under elementary equivalence, and so is non-elementary. We also show that any axiomatisation of the logic involves infinitely many non-canonical formulas.

‘The problem of Spartan land tenure is one of the most vexed in the obscure field of Spartan institutions.’ Walbank's remark is as true today as when it was written nearly thirty years ago. Controversy surrounding this subject has a long tradition going back to the nineteenth century and the last thirty years have witnessed no diminution in the level of disagreement, as is demonstrated by a comparison of the differing approaches in the recent works by Cartledge, Cozzoli, David and (...) Marasco. Although another study runs the risk of merely adding one more hypothesis to the general state of uncertainty, a fundamental reassessment of the question is required, not least because of its significance for the historian's interpretation of the overall character of Spartiate society. Through the introduction of a new perspective it may be possible to advance our understanding of the subject. In Section I of this essay I shall attempt to review several influential scholarly theories and to examine their feasibility and the reliability of the evidence upon which they are based. Section II will begin to construct a more plausible alternative account which is based upon more trustworthy evidence. Finally, Section III will discuss a comparatively underemphasised aspect of the topic, the property rights of Spartiate women, which suggests a rather different interpretation of the character of land tenure and inheritance from those more usually adopted. (shrink)

This article examines the impact on Spartan historiography of Chapter IV of de Ste. Croix’s Origins of the Peloponnesian War, focusing on his discussions of Spartan politics and society in Sections v–vi. These sections fit oddly within the overall chapter, but they blew a breath of fresh air into Spartan studies through their revisionist approach, intimations of the socio-economic bases of policy-making, and extended accounts of ‘real-life’ political episodes across the classical period. Along with Moses Finley’s near-contemporary article on Sparta, (...) OPW significantly influenced the following generation of British historians (including the author), although they often adopted different interpretations or developed new perspectives on Spartan society only hinted at by de Ste. Croix. OPW also had an important impact on Western European historiography on Spartan politics. Its combination of constitutional and societal approaches gives it an enduring currency in the context of developing Historical Institutionalist approaches to political studies. (shrink)

We prove that everyn-modal logic betweenKnandS5nis undecidable, whenever n ≥ 3. We also show that each of these logics is non-finitely axiomatizable, lacks the product finite model property, and there is no algorithm deciding whether a finite frame validates the logic. These results answer several questions of Gabbay and Shehtman. The proofs combine the modal logic technique of Yankov–Fine frame formulas with algebraic logic results of Halmos, Johnson and Monk, and give a reduction of the representation problem of finite relation (...) algebras. (shrink)

We study relation algebras with n-dimensional relational bases in the sense of Maddux. Fix n with 3nω. Write Bn for the class of non-associative algebras with an n-dimensional relational basis, and RAn for the variety generated by Bn. We define a notion of relativised representation for algebras in RAn, and use it to give an explicit equational axiomatisation of RAn, and to reprove Maddux's result that RAn is canonical. We show that the algebras in Bn are precisely those that have (...) a complete relativised representation of this type. Then we prove that whenever 4nshrink)

We prove, for each 4⩽ n ω , that S Ra CA n+1 cannot be defined, using only finitely many first-order axioms, relative to S Ra CA n . The construction also shows that for 5⩽n S Ra CA n is not finitely axiomatisable over RA n , and that for 3⩽m S Nr m CA n+1 is not finitely axiomatisable over S Nr m CA n . In consequence, for a certain standard n -variable first-order proof system ⊢ m (...) , n of m -variable formulas, there is no finite set of m -variable schemata whose m -variable instances, when added to ⊢ m , n as axioms, yield ⊢ m , n +1. (shrink)

We characterise the class S Ra CA n of subalgebras of relation algebra reducts of n -dimensional cylindric algebras by the notion of a ‘hyperbasis’, analogous to the cylindric basis of Maddux, and by representations. We outline a game–theoretic approximation to the existence of a representation, and how to use it to obtain a recursive axiomatisation of S Ra CA n.

Let S be a signature of operations and relations definable in relation algebra, let R be the class of all S-structures isomorphic to concrete algebras of binary relations with concrete interpretations for symbols in S, and let F be the class of S-structures isomorphic to concrete algebras of binary relations over a finite base. To prove that membership of R or F for finite S-structures is undecidable, we reduce from a known undecidable problem—here we use the tiling problem, the partial (...) group embedding problem and the partial group finite embedding problem to prove undecidability of finite membership of R or F for various signatures S. It follows that the equational theory of R is undecidable whenever S includes the boolean operators and composition. We give an exposition of the reduction from the tiling problem and the reduction from the group embedding problem, and summarize what we know about the undecidability of finite membership of R and of F for different signatures S. (shrink)

This much-needed book provides a thorough account of temporal logic, one of the most important areas of logic in computer science today. The book begins with a solid introduction to semantical and axiomatic approaches to temporal logic. It goes on to cover predicate temporal logic, meta-languages, general theories of axiomatization, many dimensional systems, propositional quantifiers, expressive power, Henkin dimension, temporalization of other logics, and decidability results. With its inclusion of cutting-edge results and unifying methodologies, this book is an indispensable reference (...) for both the pure logician and the theoretical computer scientist. (shrink)

We show that for finite n⩾3n⩾3, every first-order axiomatisation of the varieties of representable n-dimensional cylindric algebras, diagonal-free cylindric algebras, polyadic algebras, and polyadic equality algebras contains an infinite number of non-canonical formulas. We also show that the class of structures for each of these varieties is non-elementary. The proofs employ algebras derived from random graphs.

Using a combinatorial theorem of Herwig on extending partial isomorphisms of relational structures, we give a simple proof that certain classes of algebras, including Crs, polyadic Crs, and WA, have the `finite base property' and have decidable universal theories, and that any finite algebra in each class is representable on a finite set.

Using a combinatorial theorem of Herwig on extending partial isomorphisms of relational structures, we give a simple proof that certain classes of algebras, including Crs, polyadic Crs, and WA, have the `finite base property' and have decidable universal theories, and that any finite algebra in each class is representable on a finite set.

We define analogues of modal Sahlqvist formulas for the modal mu-calculus, and prove a correspondence theorem for them.

For any finite n 3 there are two atomic n-dimensional cylindric algebras with the same atom structure, with one representable, the other, not.Hence, the complex algebra of the atom structure of a representable atomic cylindric algebra is not always representable, so that the class RCAn of representable n-dimensional cylindric algebras is not closed under completions. Further, it follows by an argument of Venema that RCAn is not axiomatisable by Sahlqvist equations, and hence nor by equations where negation can only occur (...) in constant terms.Similar results hold for relation algebras. (shrink)

In this paper, we introduce a new fragment of the first-order temporal language, called the monodic fragment, in which all formulas beginning with a temporal operator have at most one free variable. We show that the satisfiability problem for monodic formulas in various linear time structures can be reduced to the satisfiability problem for a certain fragment of classical first-order logic. This reduction is then used to single out a number of decidable fragments of first-order temporal logics and of two-sorted (...) first-order logics in which one sort is intended for temporal reasoning. Besides standard first-order time structures, we consider also those that have only finite first-order domains, and extend the results mentioned above to temporal logics of finite domains. We prove decidability in three different ways: using decidability of monadic second-order logic over the intended flows of time, by an explicit analysis of structures with natural numbers time, and by a composition method that builds a model from pieces in finitely many steps. (shrink)

We prove that every normal extension of the bi-modal system S52 is finitely axiomatizable and that every proper normal extension has NP-complete satisfiability problem.

Abstract Truth-telling is a key issue within the nurse–patient relationship. Nurses make decisions on a daily basis regarding what information to tell patients. This paper analyses truth-telling within an end of life scenario. Virtue ethics provides a useful philosophical approach for exploring decisions on information disclosure in more detail. Virtue ethics allows appropriate examination of the moral character of the nurse involved, their intention, ability to use wisdom and judgement when making decisions and the virtue of truth-telling. It is appropriate (...) to discuss nursing as a 'practice' in relation to virtue ethics. This is achieved through consideration of the implications of arguments made by Alasdair MacIntyre who believes that qualities such as honesty, courage and justice are virtues because they enable us to achieve the internal goods of practices. (shrink)

We provide a canonical construction of conformal covers for finite hypergraphs and present two immediate applications to the finite model theory of relational structures. In the setting of relational structures, conformal covers serve to construct guarded bisimilar companion structures that avoid all incidental Gaifman cliques-thus serving as a partial analogue in finite model theory for the usually infinite guarded unravellings. In hypergraph theoretic terms, we show that every finite hypergraph admits a bisimilar cover by a finite conformal hypergraph. In terms (...) of relational structures, we show that every finite relational structure admits a guarded bisimilar cover by a finite structure whose Gaifman cliques are guarded. One of our applications answers an open question about a clique constrained strengthening of the extension property for partial automorphisms (EPPA) of Hrushovski, Herwig and Lascar. A second application provides an alternative proof of the finite model property (FMP) for the clique guarded fragment of first-order logic CGF, by reducing (finite) satisfiability in CGF to (finite) satisfiability in the guarded fragment, GF. (shrink)

We show that the loosely guarded and packed fragments of first-order logic have the finite model property. We use a construction of Herwig and Hrushovski. We point out some consequences in temporal predicate logic and algebraic logic.

The tangle modality is a propositional connective that extends basic modal logic to a language that is expressively equivalent over certain classes of finite frames to the bisimulation-invariant fragments of both first-order and monadic second-order logic. This paper axiomatises several logics with tangle, including some that have the universal modality, and shows that they have the finite model property for Kripke frame semantics. The logics are specified by a variety of conditions on their validating frames, including local and global connectedness (...) properties. Some of the results have been used to obtain completeness theorems for interpretations of tangled modal logics in topological spaces. (shrink)

We show that the loosely guarded and packed fragments of first-order logic have the finite model property. We use a construction of Herwig and Hrushovski. We point out some consequences in temporal predicate logic and algebraic logic.

We prove decidability of satisfiability of sentences of the monodic packed fragment of first-order temporal logic with equality and connectives Until and Since, in models with various flows of time and domains of arbitrary cardinality. We also prove decidability over models with finite domains, over flows of time including the real order.

This volume contains eighteen essays by established and younger historians that examine non-democratic alternative political systems and ideologies--oligarchies, monarchies, mixed constitutions--along with diverse forms of communal and regional associations such as ethnoi, amphiktyonies, and confederacies. The papers, which span the length and breadth of the Hellenic world highlight the immense political flexibility and diversity of ancient Greek civilization.

A conjecture of Gabbay (1981) states that any class of flows of time having the property known as finite H-dimension admits a finite set of expressively complete one-dimensional temporal connectives. Here we show that the class of 'circular' structures refutes the generalisation of this conjecture to Kripke frames. We then construct from this class, by a general method, a new class of irreflexive transitive flows of time that refutes the original conjecture. Our paper includes full descriptions of a method for (...) establishing finite H-dimension for a class of structures and of the technique for extending finite H-dimension to other classes, and an introduction surveying the area of expressive completeness. (shrink)

We show that the satisfiability problem for the monodic guarded, loosely guarded, and packed fragments of first-order temporal logic with equality is 2Exptime-complete for structures with arbitrary first-order domains, over linear time, dense linear time, rational number time, and some other classes of linear flows of time. We then show that for structures with finite first-order domains, these fragments are also 2Exptime-complete over real number time and hence over most of the commonly used linear flows of time, including the natural (...) numbers, integers, rationals, and any first-order definable class of linear flows of time. (shrink)

Many religious people are alarmed about features of the current age - violence in the media, a pervasive hedonism, a marginalization of religion, and widespread abortion. These concerns influence politics, but just as there should be a separation between church and state, so should there be a balance between religious commitments and secular arguments calling for social reforms. Robert Audi offers a principle of secular rationale, which does not exclude religious grounds for action but which rules out restricting freedom except (...) on grounds that any rational citizen would accept. The book describes the essential commitments of free democracy, explains how religious and secular moral considerations can be integrated to facilitate co-operation in a world of religious pluralism, and proposes ideals of civic virtue that express the mutual respect on which democracy depends. Audi offers a balanced and sophisticated treatment of the relations between religion and politics in a modern, secular society. (shrink)

We define analogues of modal Sahlqvist formulas for the modal mu-calculus, and prove a correspondence theorem for them.

We prove that every first-order formula that is invariant under quasi-injective bisimulations is equivalent to a formula of the hybrid logic . Our proof uses a variation of the usual unravelling technique. We also briefly survey related results, and show in a standard way that it is undecidable whether a first-order formula is invariant under quasi-injective bisimulations.