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 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.

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 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.

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.

§1. Introduction. The classical propositional calculus has an undeserved reputation among logicians as being essentially trivial. I hope to convince the reader that it presents some of the most challenging and intriguing problems in modern logic. Although the problem of the complexity of propositional proofs is very natural, it has been investigated systematically only since the late 1960s. Interest in the problem arose from two fields connected with computers, automated theorem proving and computational complexity theory. The earliest paper in the (...) subject is a ground-breaking article by Tseitin [62], the published version of a talk given in 1966 at a Leningrad seminar. In the three decades since that talk, substantial progress has been made in determining the relative complexity of proof systems, and in proving strong lower bounds for some restricted proof systems. However, major problems remain to challenge researchers. The present paper provides a survey of the field, and of some of the techniques that have proved successful in deriving lower bounds on the complexity of proofs. A major area only touched upon here is the proof theory of bounded arithmetic and its relation to the complexity of propositional proofs. The reader is referred to the book by Buss [10] for background in bounded arithmetic. The forthcoming book by Krajíček [40] also gives a good introduction to bounded arithmetic, as well as covering most of the basic results in complexity of propositional proofs. (shrink)

§1. Introduction. The classical propositional calculus has an undeserved reputation among logicians as being essentially trivial. I hope to convince the reader that it presents some of the most challenging and intriguing problems in modern logic. Although the problem of the complexity of propositional proofs is very natural, it has been investigated systematically only since the late 1960s. Interest in the problem arose from two fields connected with computers, automated theorem proving and computational complexity theory. The earliest paper in the (...) subject is a ground-breaking article by Tseitin [62], the published version of a talk given in 1966 at a Leningrad seminar. In the three decades since that talk, substantial progress has been made in determining the relative complexity of proof systems, and in proving strong lower bounds for some restricted proof systems. However, major problems remain to challenge researchers. The present paper provides a survey of the field, and of some of the techniques that have proved successful in deriving lower bounds on the complexity of proofs. A major area only touched upon here is the proof theory of bounded arithmetic and its relation to the complexity of propositional proofs. The reader is referred to the book by Buss [10] for background in bounded arithmetic. The forthcoming book by Krajíček [40] also gives a good introduction to bounded arithmetic, as well as covering most of the basic results in complexity of propositional proofs. (shrink)

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.

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)

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, 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)

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)

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)

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)

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)

Gould and Lewontin use San Marco, Venice, to criticise the adaptationist program in biology. Following their lead, the architectural term “spandrel” is now widely used in biology to denote a feature that is a necessary byproduct of other aspects of the organism. I review the debate over San Marco and argue that the spandrels are not necessary in the sense originally used by Gould and Lewontin. I conclude that almost all the claims that Gould makes about San Marco are wrong (...) and that it is reasonable to view the architectural spandrel as an adaptation. The spandrels example has not provided a good illustration of why adaptive explanations should be avoided. In fact, it can be used as an example of how adaptive explanations can be dismissed even when there is evidence in their favour. I also discuss the use of the concept of a spandrel in biology. (shrink)

Rolls discusses various adaptive explanations of physiological processes and the emotions. We give a critical analysis of some of these from the perspective of behavioural ecology. While agreeing with the approach adopted by Rolls, we identify topics that could have been better presented by making use of the existing literature.

John Maynard Smith was the founder of evolutionary game theory. He has also been the major influence on the direction of this field, which now pervades behavioural ecology and evolutionary biology. In its original formulation the theory had three components: a set of strategies, a payoff structure, and a concept of evolutionary stability. These three key components are still the basis of the theory, but what is assumed about each component is often different to the original assumptions. We review modern (...) approaches to these components. We emphasis that if a game is considered in isolation, and arbitrary payoffs are assumed, then the payoffs may not be consistent with other components of the system which are not modelled. Modelling the whole system, including not only the focal game, but also the future behaviour of the players and the behaviour of other population members, allows a consistent model to be constructed. We illustrate this in the case of two models of parental care, showing how linking a focal game to other aspects of the system alters what is predicted. (shrink)

I discuss various theoretical issues concerning maximizing, matching, and melioration. The model of addiction based on melioration has the key feature that a reduction in drag use increases both the value of drag-taking and the value of conventional activities. I found Heyman's target article stimulating. I don't feel competent to comment on drags but I do have some thoughts, both general and specific, on the theoretical issues.

In the theory outlined in the target article, an animal forages continuously, making sequential decisions in a world where the amount of food and its uncertainty are fixed, but delays are variable. These assumptions contrast with the risk-sensitive foraging theory and create a problem for comparing the predictions of this model with many laboratory experiments that do not make these assumptions.

We concentrate on two aspects of the article by Pothos & Busemeyer (P&B): the relationship between classical and quantum probability and quantum probability as a basis for rational decisions. We argue that the mathematical relationship between classical and quantum probability is not quite what the authors claim. Furthermore, it might be premature to regard quantum probability as the best practical rational scheme for decision making.

A general framework for analysing the effects of variability and the effects of interruptions on foraging is presented. The animal is characterised by its level of energetic reserves, x. We consider behaviour over a period of time [0,T]. A terminal reward function R(x) determines the expected future reproductive success of an animal with reserves x at time T. For any state x at a time in the period, we give the animal a choice between various options and then constrain it (...) to follow a background strategy. The best option is the one that maximizes expected future reproductive success. Using this framework, we show that sensitivity to variability in amount of energy gained is logically distinct from sensitivity to variability in the time at which food is obtained. We also show that incorporating interruptions results in both a preference for variability in time and a preference for a reward followed by a delay as opposed to the same delay before the reward. (shrink)

Gigerenzer, Todd, and the ABC Research Group give an interesting account of simple decision rules in a variety of contexts. I agree with their basic idea that animals use simple rules. In my commentary I concentrate on some aspects of their treatment of decision rules in behavioural ecology.

Behavioral ecologists often assume that natural selection will produce organisms that make optimal decisions. In the context of information processing, this means that the behavior of animals will be consistent with models from fields such as signal detection theory and Bayesian decision theory. We discuss work that applies such models to animal behavior and use the case of Bayesian updating to make the distinction between a description of behavior at the level of optimal decisions and a mechanistic account of how (...) decisions are made. The idea of ecological rationality is that natural selection shapes an animal's decision mechanisms to suit its environment. As a result, decision-making mechanisms may not perform well outside the context in which they evolved. Although the assumption of ecological rationality is plausible, we argue that the exact nature of the relationship between ecology and cognitive mechanism may not be obvious. (shrink)

In unraveling the long-hidden issues of the most famous free speech case of all time, noted author I.F. Stone ranges far and wide over Roman as well as Greek history to present an engaging and rewarding introduction to classical antiquity and its relevance to society today. The New York Times called this national best-seller an "intellectual thriller.".

Hebbian mechanisms are justified according to their functional utility in an evolutionary sense. The selective advantage of correlating content-contingent stimuli reflects the putative common cause of temporally or spatially contiguous inputs. The selective consequences of such correlations are discussed by using examples from the evolution of signal form in sexual selection and model-mimic coevolution. We suggest that evolutionary justification might be considered in addition to neurophysiology plansibility when constructing representational models.

It is shown that in a range of models, the probability that a forager dies from starvation is, to a good approximation, an exponential function of energy reserves. Using a time and energy budget for a 19g passerine, we explore the consequences, in terms of starvation and predation, of various levels of energy reserves. It is shown that there exists an optimal level L of reserves at which total mortality (starvation plus predation) is minimized. L increases when the environment deteriorates (...) as a result of a decrease in either temperature or mean gross gain or an increase in the mean search time. The effect of combined deteriorations is greater than the sum of their individual effects. At L, the probability of predation is much higher than the probability of starvation. A simple analytic model suggests that this result will be fairly general, but also indicates conditions under which the result might not hold. (shrink)

Islamic ethics; Muslims; conduct of life.