In lieu of an abstract, here is a brief excerpt of the content:G. E STOUT AND THE THEORY OF DESCRIPTIONS ALASDAIR URQUHART Philosophy / Toronto Toronto, ON, Canada M5S rAr R ussell's famous essay "On Denoting" is rightly considered one of the fundamental sources of modern analytic philosophy. The literature of commentary and exegesis devoted to this one short paper is vast. The present note discusses the background ~6 the writing of the paper, and some questions connected with (...) its reception.l I. HOW "ON DENOTING" WAS WRITTEN We know more about the writing of OD than about most of the other short works from the early period of Russell's career. Russell chronicled his progress on the paper very clearly in his manuscripts and correspondence. The first indication ofthe new theory is in the unpublished manuscript "On Fundamentals" (Papers 4: 359-413). This manuscript is dated 1905, to which the note "Begun June 7" has been added; a marginal note on the first leafof the manuscript reads: "Pp. 18ff. contain the reasons for the new theory ofdenoting." This note refers to the series of numbered paragraphs beginning on folio 18, in which the new theory gradually emerges (Papers 4: 369£). On 13 June, Russell wrote Lucy Donnelly about his newideas, and presented her with the famous puzzle about George IV and the author of waverley (Auto. I: 177). The article itselfwas written at the end ofJuly 1905, under peculiarly trying circumstances. On 23 July 1905, Russell wrote in a letter to Louis 1 I would like to thank the Editor ofRussellfor very useful suggestions and for supplying historical information, and Sheila Turcon for correcting my transcription of Stout's letter. russell: the Journal of the Bertrand Russdl Archives McMaster University Library Press n.s. 14 (wimer 1994-95): 163-71 ism 0036-01631 164 ALASDAIR URQUHART Couturat: "As for existence, I shall be able to explain my opinion better when I have written my article (On Denoting Phrases'." On 3August he reported to Lucy Donnelly that she would be able to find the answer to the puzzle in his new paper: "I have written an article on George N for Mind, which will appear in due course; there you will find the (answer' " (Auto. I: 178). We can conclude from this that the article was written within a twelve-day period towards the end oOuly. This is not surprising in itself, given Russell's facility in writing. It becomes surprising if we put it together with the other events in Russell's life at that time. On Tuesday 25 July, one of Russell's closest friends, Theodore Llewelyn Davies, drowned in a pool on the fells near Kirkby Lonsdale, where his father was vicar. He had apparently struck his head on diving in, while on his way to the railway station. He was thirty-four years old. His brother Crompton, who had been particularly close to Theodore, sent the news to Russell in a note dated 27 July 1905 (Auto. 1: 195)· Theodore and Crompton were among the nrst friends Russell made at Cambridge; his reminiscences of the two brothers overflow with admiration and affection (Auto. I: 57-63). Russell had intended to visit Ireland for two weeks with AIys, where they had planned a visit with the Monteagles. 2 On 27 July, Russell wrote in a letter to Louis Couturat that he and Alys would be in Ireland from 29 July to 10 August. The news ofTheodore's death altered these plans. After accompanying AIys to Ireland, Russell returned to London to be·with Crompton. In his note of 27July, Crompton had told Russell: "I shall be back in London on Monday. Let me see you some time soon." On Tuesday, 1 August, Russell wrote again to Couturat from I, Barton Street,3 the quiet out-of-the-way street near Westminster Abbey where Crompton had lived with Theodore: "I just received your letter and postcard (in 1 Thomas Spring-Rice (1849-1926), 2nd Baron Monteagle, had literary and political interests and was the brother-in-law ofthe historian, Sir George Prothero, whom Russdl also knew. The next year The Limerick Chronicle for 28 August noted the Russells' presence at a house party given by the Monteagles and attended by... (shrink)

What I wish to propose in the present paper is a new form of “career induction” for ambitious young logicians. The basic problem is this: if we look at the n-variable fragments of relevant propositional logics, at what point does undecidability begin? Focus, to be definite, on the logic R. John Slaney showed that the 0-variable fragment of R (where we allow the sentential con- stants t and f) contains exactly 3088 non-equivalent propositions, and so is clearly decidable. In the (...) opposite direction, I claimed in my paper of 1984 that the five variable fragment of R is undecidable. The proof given there was sketchy (to put the matter charitably), and a close examination reveals that although the result claimed is true, the proof given is incorrect. In the present paper, I give a detailed and (I hope) correct proof that the four variable fragments of the principal relevant logics are undecidable. This leaves open the question of the decidability of the n-variable fragments for n = 1, 2, 3. At what point does undecidability set in? (shrink)

I discuss a collection of problems in relevance logic. The main problems discussed are: the decidability of the positive semilattice system, decidability of the fragments of R in a restricted number of variables, and the complexity of the decision problem for the implicational fragment of R. Some related problems are discussed along the way.

In lieu of an abstract, here is a brief excerpt of the content:Reviews 91 THE ROOTS OF MODERN LOGIC ALASDAIR URQUHART Philosophy/ U. ofToronto Toronro, ON, Canada M5S IAI URQUHART@CS.TORONTO.EDU I. Grattan-Guinness. The Searchfor Mathematical Roots,r870--r940: logics, Set Theoriesand the Foundations of Mathematicsfrom Cantor through Russellto Godel Princeron: Princeton U. P.,2000. Pp. xiv,690. us$45.oo. Grattan-Guinness's new hisrory of logic is a welcome addition to the literature. The title does not quite do justice ro the book, since it (...) begins with the prehistory of English work in algebraic logic, including the work of the French analycicalschool, and extends to just after Godel's great incompleteness paper of 1931.The core of the book, though, is the philosophical and mathematical developmems leading up to and immediately following from chework ofWhicehead and Russell.Russell is ac chehearcof che book, and as a whole che hisrory forms an imporranr conrribucion ro Russell scudies. Commemarors on Russell have often confined themselves to a rather narrow hisrorical perspeccivein which Russell is seen as the (problemacic) heir of Gottlob Frege, and few ocher hisrorical figures (ocher than Peano) emer imo che picture. Graccan-Guinness corrects chis hisrorica.limbalance by placing Russell in che mucl1 wider context of the development of machemacics on che continent, and in parricu.laremphasizing strongly che key influence of Camor on 92 Reviews Russell's work. Cantor has usually received short shrift from philosophers, as unlike Frege,he appears rarher naive from rhe philosophical point of view. The story proper begins in Chaprer 2 wirh L-igrange'sversion of analysis in which rhe basic concepts were to be defined in rerms of algebraic manipularion of power series. This lead to rhe founding of rhe Analyrical Socierywhere Babbage, Herschel and Peacock were active. It is in chis English tradicion of algebraic analysis char rhe pioneering work of De Morgan and Boole found irs roors. Grarran-Guinness givesa derailed account of the work of both logicians, although his discussion of Boole's merhods does not seem entirely adequate. The question is:what are we to make of Boole'spuzzling insistence chat expressions like x + y are "uninterpretable", while he nevertheless manipulated them freely in his mathematical derivations? It is not correct to say that the addition sign can only link disjoint class symbols, since ir is belied by Boole's formal pracrice. A possible solution has been suggested by Hailperin, in his book on Boole's logic, in which he proposes interpreting the "uninterpretable" expressions as denoting signed mulrisets. Oddly, Grattan-Guinness refers to Hailperin 's work,' bur elsewhere (p. 42) adopts rhe view that Boole's addition sign could only link disjoint classes.The chapter concludes with brief accounts of the work of Cauchy, Weierstrass and Bolz.'lno. The next chapter is a detailed account of rhe work of Cantor and his creation of Mengenlehre. The origins of set theory in the theory of trigonometrical series, and rhe ensuing discoveryof rransfinire ordinal and cardinal numbers, are described clearly and succinctly. In addition, the chapter contains an account of Dedekind's philosophy of arithmetic and Cantor's philosophy of mathematics. Cantor's philosophy is an uneasy blend of formalism, platonism and idealism, and has tmderstandably aroused little enthusiasm among philosophers of matl1ematics, alrhough Michael Hallerr has recently smdied it in detail. GrammGuinness emphasizes, rhough, Cantor's magnificent marhematica.l achievements, in defining and clarifying basic conceprs such as measure, dimension and cardinality of sets. Chaprer 4 is a rather miscellaneous chapter, in which six partly independent, partly interrwined stories are told. It begins with developments in sectheory in Germany and France up to the turn of rhe century, goes on to discuss American logic in rhe work of C. S. Peirce and his students, and continues the rheme of algebraic logic with Schroder and his logic of relatives. The remainder of the chapter is given over to Frege, Husserl and Hilbert. The section on Frege is one of rhe more idiosyncratic parts of the book. Grattan-Guinness distinguishes berween Frege, "a mathematician who wrore in ' T Hailperin, Book's logic and Probability, 2nd ed. (Amsterdam: North-Holland, 1986). Reviews 93 German, in a markedly Platonic spirit", and Frege, "a philosopher of language and founder of... (shrink)

I was born in Auchtermuchty, a small village in Fife, Scotland, the son of a lowland mother, Meta Mowat, daughter of a country headmaster, and a highland father, William Urquhart from Lairg, Sutherland. The family business in Lairg was a butcher’s shop, known as Urquhart the Butcher, even for some years after my father’s family had given it up.

I am deeply grateful to everybody who has contributed to the volume, and wish to express my heartfelt thanks to my old friends and colleagues Ivo Düntsch and Ed Mares who have worked so hard to produce a volume in my honour. I’ve done research in a lot of areas in logic, and the selection of authors provides a good cross-section of my preoccupations.

We show that short bounded-depth Frege proofs of matrix identities, such as PQ=I⊃QP=I (over the field of two elements), imply short bounded-depth Frege proofs of the pigeonhole principle. Since the latter principle is known to require exponential-size bounded-depth Frege proofs, it follows that the propositional version of the matrix principle also requires bounded-depth Frege proofs of exponential size.

Alasdair MacIntyre criticizes the modern morality for having emotivist features and in his cent- ral book After Virtue he points out that David Hume is the main personality who provides these emotivist contents to the modern morality. According to MacIntyre, Hume’s and the modern emotivist moral philosophy include fundamental contrasts generally with the classical moral tradition particularly with Aristotle’s moral philosophy. However, MacIntyre underlines these contrasts in After Virtue, he in his other texts out of After Virtue, distinguishably brings (...) Hume and Aristotle together as they both have similar standpoints about how to understand morality. Therefore, MacIntyre’s interpretation on Hume needs to be examined with a perspective poin- ting these two different and seemingly mutually exclusive aspects. Putting together these two perspectives, this article aims to construct a holistic comprehension about MacIntyre’s placing Hume in the modern emotivism. (shrink)

The reappearance of Professor Alasdair MacIntyre's far-ranging and provocative article, ‘Hume on “is” and “ought”’, is the proximate cause of this short excursion to an old, well-scarred, and still fascinating battleground. Re-reading MacIntyre's brilliant offensive thrust led me to review the counter-attacks and diversionary movements that followed its first appearance. They in turn sent me back, inevitably and ultimately, to look again at the cause of this philosophic skirmishing: Section 1 of Part i of Book III of Hume's Treatise (...) of Human Nature, entitled ‘Moral Distinctions not deriv'd from Reason’. The battles of the past have been waged chiefly round the last paragraph of this Section (pp. 469–470 in Selby-Bigge's edition), but my primary concern here is going to be with those ‘reasonings’ that precede the celebrated ‘is-ought’ paragraph. Closer attention to the bulk of Book III, Part i, Section 1, to which ‘reasonings’ Hume says he ‘cannot forbear adding … an observation’ on ‘is’ and ‘ought’, and some selective attention as well to Books I and II of the Treatise, should help to scale down the exaggerated importance attached to the ‘is-ought’ passage. Unless this is done, one runs the risk of permitting a tailpiece to wag the body of the chapter, if not the whole Book ‘Of Morals’. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:IN PRAISE OF PLURALISM * RONALD F. THIEMANN Harvard Diviinity School Cambridge, Massachusetts K CENTLY A GROUP of scholars at Harvard University met to discuss the question of whether the United States ha;d entered a. period of moral decline. Our conversations ranged over a wide spectrum of topics: the distinction between priva.te and public life, the relation of notions like mol'lality and justice, the issue of how a decline (...) in morality might be dooumented. The discuS'sion was carried on with a sense of intellectual seriousness and passion but was also characterized by a degree of frustration, given the brea;dth of the topic and the diversity of the participants, who were drawn from the faculties of arts and sciences, 1aw, medicine, government, business, and divinity. Our approaches to the topic were diverse and consequentily our disputes were spirited. Whatever the disagreements among the group, however, we were united by a conviction tha.t such conversations a;re essential if we are to under:s.tand the moral complexity of our pluralistic society. Tihis gathering of soho1ars at Harvard exhibits four important ohamcteristics of the current deba.te about moraility and public life. 1) The recognition that the moral issues we are facing cut a;cross traditional disciplinary lines and require a joint effort by scholar:s working in diverse fields and pl'ofessions ; 2) an awaireness that the plurality of moral positions within pwblic and aicademic life threatens the possibility of consensus on issues of public importance; 3) the conviction, nonetheless, that joint scholarly refieob:ion can have an impact *Ethics After Babel. By JEFFREY STOUT. Boston: Beacon Press, 1988. Pp. 338. $27.50. 489 490 RONALD F. THIEMANN on the moral issues currently vexing American public life: 4) the a;eknowledgement that religion has a role to play in the conversation about the future of momlity and public affairs.1 Jeffrey Stout'·s new book Ethics After Babel is an important and lively contiribution to the current discussion about mora;lity and public life. While the book focuses primarily on issues central to moral philosophy, its arguments have broad implications for debate on topics of importance to public life more generally. Stout'.s work can be seen as an exemplification, expansion, and justification of the four points identified in the previous pM"agraph. But the book is much more than that: it is an extended, sustained, and persuasive argument that pluralism iin public discourse about moral issues need not lead to 1skepticism, nihilism, or relativism. Stout steers a steady oourse between the cultural pessimists who decry our current state as one of hopeless mor:al fragmentation and the intellectual foundationafists who seek to dispel our cultural malaise through some philosophical sleight-of-hand. Ethics After Babel is divided into three major sections. The first," Spectres of MoI"al Diversity," tackles the issue of truthclaiming in moral discouvse and defends non-foundational ethics against the charge of relativism. The -second, " The Eclipse of Religious Ethics," deals with the relation between reJigion and moraJity, reruson and tradition, and with the question of the future of a public theology. In the final section, "Moral Discourse in P·lur:aiistic Society," Stout defends aversion of pragmatic liiberalism and distinguishes his own po.s:i.tion from that of Alasdair Maicintyre and Richaro Rorty. In the·last chapter Stout shows how many of the concepts Macintyre has introduced into the philosopmcal discussion can be used in service of Li:beral social criticism. The reader gilimpses in 1 It is unlikely that the Divinity School would have been represented in any such gathering as recently as five years ago. The fact that no one from the School of Education was invited to join this group is evidence of the continuing marginality of certain crucial professions within American higher education. IN PRAISE OF PLURALISM 491 these final chapters the :beginnings of Stout's own constructive oontriibution to the issues of morality and pwblic life. Stout seeks in this v:olume to move the discussion in poJitical philosophy and ethics beyond the "liberal vs. oommuntarian " issues that have dominruted the recent litemture. In opposition to the individualism and foundationalism that characterize the classic modern... (shrink)

The title of this and proposed second volume presents the basic idea which unifies the wide variety of topics developed and investigated by the principal authors, major contributing authors, J. M. Dunn and Robert K. Meyer, and eleven other contributors. The other contributors are: J. R. Chidgey, J. A. Coffa, Dorthy L. Grover, Bas van Fraassen, H. Leblanc, Storrs McCall, A. Parks, G. Pottinger, R. Routley, A. Urquhart, and R. G. Wolf. From both the useful analytic table of contents (...) and from the section titles it is clear which contributing author has written each section. Chapter I, "The Pure Calculus of Entailment," suggests how logics of entailment are logics of relevance and necessity. When we explain why a large number of systems are investigated, it will be clear why we write of logics of entailment instead of the logic of entailment. The principal authors build a case in Chapter I, and elsewhere, that a genuinely valid formal inference, viz., a formal entailment, from a formula A to a formula B requires that A be relevant to B in addition to it being impossible that there be an interpretation making A true and B false. So, a claim that A entails B tells us, explicitly or implicitly, that A is relevant to B and that "if A then B" is necessarily true, i.e., A strictly implies B. Consequently since a logic of entailment presents allegedly unfalsifiable claims, i.e., theses, using entailment claims, a logic of entailment presents theses about relevance between antecedents and consequents and necessity. It should be noted that the systems of logic developed are for formal entailments; they may not be logics for material entailments such as: being a cube entails having six surfaces. (shrink)

The lattices of the title generalize the concept of a De Morgan lattice. A representation in terms of ordered topological spaces is described. This topological duality is applied to describe homomorphisms, congruences, and subdirectly irreducible and free lattices in the category. In addition, certain equational subclasses are described in detail.

We present a general framework for analyzing the contribution to reproductive success of a behavioural action. An action may make a direct contribution to reproductive success, but even in the absence of a direct contribution it may make an indirect contribution by changing the animal's state. We consider actions over a period of time, and define a reward function that characterizes the relationship between the animal's state at the end of the period and its future reproductive success. Working back from (...) the end of the period using dynamic programming, the optimal action as a function of state and time can be found. The procedure also yields a measure of the cost, in terms of future reproductive success, of a suboptimal action. These costs provide us with a common currency for comparing activities such as eating and drinking, or eating and hiding from predators. The costs also give an indication of the robustness of the conclusions that can be drawn from a model. We review how our framework can be used to analyze optimal foraging decisions in a stochastic environment. We also discuss the modelling of optimal daily routines and provide an illustration based on singing to attract a mate. We use the model to investigate the features that can produce a dawn song burst in birds. State is defined very broadly so that it includes the information an animal has about its environment. Thus, exploration and learning can be included within the framework. (shrink)

Craig's interpolation theorem fails for the propositional logics E of entailment, R of relevant implication and T of ticket entailment, as well as in a large class of related logics. This result is proved by a geometrical construction, using the fact that a non-Arguesian projective plane cannot be imbedded in a three-dimensional projective space. The same construction shows failure of the amalgamation property in many varieties of distributive lattice-ordered monoids.

This paper defines a category of bounded distributive lattice-ordered grupoids with a left-residual operation that corresponds to a weak system in the family of relevant logics. Algebras corresponding to stronger systems are obtained by adding further postulates. A duality theoey piggy-backed on the Priestley duality theory for distributive lattices is developed for these algebras. The duality theory is then applied in providing characterizations of the dual spaces corresponding to stronger relevant logics.

Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes.

Henry M. Sheffer is well known to logicians for the discovery (or rather, the rediscovery) of the ?Sheffer stroke? of propositional logic. But what else did Sheffer contribute to logic? He published very little, though he is known to have been carrying on a rather mysterious research program in logic; the only substantial result of this research was the unpublished monograph The General Theory of Notational Relativity. The main aim of this paper is to explain, as far as possible (given (...) the scanty evidence), the nature of Sheffer's program, and the reasons for its failure. The paper concludes with a discussion of Sheffer's only true logical descendant, C.H. Langford, and his contributions to model theory. (shrink)

The prelims comprise: Introduction Time and Space in Computation Hierarchies and Reducibility NP‐completeness and Beyond Parallel Computation Complexity and Philosophy.

The first part of the paper is devoted to surveying the remarks that philosophers and mathematicians such as Maddy, Hardy, Gowers, and Zeilberger have made about mathematical depth. The second part is devoted to the question of whether we can make the notion precise by a more formal proof-theoretical approach. The idea of measuring depth by the depth and bushiness of the proof is considered, and compared to the related notion of the depth of a chess combination.

This note defines Ehrenfeucht-Fraïssé games where identity is not present in the basic language. The formulation is applied to show that there is no elementary theory in the language of one binary relation that exactly characterizes models in which the relation is the identity relation.

First published in 1994. Routledge is an imprint of Taylor & Francis, an informa company.

An Ockham lattice is defined to be a distributive lattice with 0 and 1 which is equipped with a dual homomorphic operation. In this paper we prove: (1) The lattice of all equational classes of Ockham lattices is isomorphic to a lattice of easily described first-order theories and is uncountable, (2) every such equational class is generated by its finite members. In the proof of (2) a characterization of orderings of with respect to which the successor function is decreasing is (...) given. (shrink)

This article presents a simplified proof of the result that bounded depth propositional proofs of the pigeonhole principle are exponentially large. The proof uses the new techniques for proving switching lemmas developed by Razborov and Beame. A similar result is also proved for some examples based on graphs.

This paper investigates the depth of resolution proofs, that is to say, the length of the longest path in the proof from an input clause to the conclusion. An abstract characterization of the measure is given, as well as a discussion of its relation to other measures of space complexity for resolution proofs.

This latest volume in the series Cambridge Studies in the History and Theory of Politics is much more than a reassessment of humanist themes in Karl Marx. It is a rereading of the entire Marxian corpus from the viewpoint of alienation taken to be core concept of Marx's thought at every stage of its development. By underscoring the conceptual primacy of "the acting and acted-upon individual" in capitalist society throughout Marx's writings, Ollman counters Feuer, Fromm and others who defend what (...) Alasdair MacIntyre has called "the myth of the plastic figure," i.e., the image of a young Marx, Hegelian and humanist, who later abandons his concern with alienation for socio-economics. In using the Grundrisse to prove his point, Ollman joins those scholars who insist that these writings, virtually unavailable until recently, are indispensable for a balanced view of Marx's thought. But Ollman's principal contribution to current discussion lies in his locating the theory of alienation in Marx's metaphysic of internal relations. This position justifies a redescription of Marx's other theories and concepts in terms of "alienation." This approach does clarify several ambiguous terms in the Marxian vocabulary, and the novelty of the interpretation restates questions which, in the West at least, had been put aside as settled. The labor theory of value, for example, is revived and given a new meaning: "Value is the relations of alienated labor, transmitted by such labor, as they appear in the product." The basic unity of Marx's and Engels' thought is likewise defended, and the dialectic of nature is exhibited as an instance of their fundamentally relational ontology. Even if one is not moved to accept all or any of these interpretations, he should not ignore the challenge which Ollman's scholarly volume presents. It is an important contribution to the ongoing debate on the meaning of Marx.--T. R. F. (shrink)