Arabic logicians in the thirteenth century discussed a set of arguments raised by the theologian Fakhr al-Dīn al-Rāzī (d. 1210) that in some respects closely resembles Carroll’s paradox. Roughly, the paradox states that we can never reach a conclusion from a set of premises without incurring an infinite regress. The present article presents and discusses Rāzī’s formulation of the problem with syllogistic deduction, his own solutions to the problem, and the contributions of Afḍal al-Dīn al-Khūnajī (d. 1248) (...) and Najm al-Dīn al-Kātibī (d. 1276). It is argued that the proposed solutions developed by these authors are best understood as tending in the direction of Gilbert Ryle’s distinction between knowing-that and knowing-how. (shrink)

Lewis Carroll, both logician and writer, suggested a logical paradox containing furthermore two connotations (connotations or metaphors are inherent in literature rather than in mathematics or logics). The paradox itself refers to implication demonstrating that an intermediate implication can be always inserted in an implication therefore postponing its ultimate conclusion for the next step and those insertions can be iteratively and indefinitely added ad lib, as if ad infinitum. Both connotations clear up links due to the shared formal (...) structure with other well-known mathematical observations: (1) the paradox of Achilles and the Turtle; (2) the transitivity of the relation of equality. Analogically to (1), one can juxtapose the paradox of the Liar (for Lewis Carroll’s paradox) and that of the arrow (for “Achilles and the Turtle”), i.e. a logical paradox, on the one hand, and an aporia of motion, on the other hand, suggesting a shared formal structure of both, which can be called “ontological”, on which basis “motion” studied by physics and “conclusion” studied by logic can be unified being able to bridge logic and physics philosophically in a Hegelian manner: even more, the bridge can be continued to mathematics in virtue of (2), which forces the equality (for its property of transitivity) of any two quantities to be postponed analogically ad lib and ad infinitum. The paper shows that Hilbert arithmetic underlies naturally Lewis Carroll’s paradox admitting at least three interpretations linked to each other by it: mathematical, physical and logical. Thus, it can be considered as both generalization and solution of his paradox therefore naturally unifying the completeness of quantum mechanics (i.e. the absence of hidden variables) and eventual completeness of mathematics as the same and isomorphic to the completeness of propositional logic in relation to set theory as a first-order logic (in the sense of Gödel (1930)’s completeness theorems). (shrink)

Hume argued that inductive inferences do not have rational justification. My aim is to reject Hume’s argument. The discussion is partly motivated by an analogy with Carroll’s Paradox, which concerns deductive inferences. A first radically externalist reply to Hume (defended by Dauer and Van Cleve) is that justified inductive inferences do not require the subject to know that nature is uniform, though the uniformity of nature is a necessary condition for having the justification. But then the subject does (...) not have reasons for believing what she believes. I defend a moderate externalist account that seeks to partly accommodate that objection to the radical externalist proposal. It is based on an extension of Peacocke’s theory of concepts: possession conditions for predicative concepts standing for natural properties include (fallible) dispositions to project them to new cases in accordance with inductive inferential patterns. (shrink)

The paper is a warning against an unreserved use of reasoning, which is designed to show that a premise in an argument is missing. The reasoning is susceptible to a common equivocation. As a result, it can be systematically misleading, making us judge that certain premises are missing where they are not. It is argued that the equivocation in question lies at the bottom of Lewis Carroll's paradox, a version of Hume's problem as well as some arguments in philosophy (...) of mind and ethics. The paper is first and foremost a warning because it turns out that there is some didactic value in playing on the equivocation. (shrink)

RésuméCet article démontre qu'un exemple cité par Ernest Adams pour montrer que l'implication matérielle n'est pas l'interprétation correcte de la sémantique de la conjonction de subordination si, n'est rien d'autre qu'un corollaire d'une observation d'jà faite par Lewis Carroll, il y a cent ans, dans l'exposition de son paradoxe du salon de coiffure.

Abstract Pavel Florensky solves Lewis Carroll’s ‘Barbershop’ paradox to support his reasoning in a previous chapter. Our discussion includes a) the problem (which we also refer to as the p paradox), b) Carroll’s solution, c) Bertrand Russell’s solution, d) Florensky’s solution and then e) a material example proffered by Florensky. Both Russell and Florensky disagree with Carroll’s solution, yet, (ostensibly) unbeknownst to themselves they offer the same solution, which is ‘p implies not-q’. Given Florensky’s material (...) example, the solution seems to tell us something about the logic of belief. We ask whether Florensky’s example has reverse implications for Russell’s solution. Content Type Journal Article Pages 1-10 DOI 10.1007/s11406-011-9333-6 Authors Michael Rhodes, Philosophy and Religious Studies (NPD), Chicago, IL, USA Journal Philosophia Online ISSN 1574-9274 Print ISSN 0048-3893. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Noël Carroll THE PARADOX OFJUNK FICTION Perhaps on your way to some academic conference, if you had no papers to grade, you stopped in die airport gift shop for something to read on the plane. You saw racks of novels authored by die likes of Mary Higgins Clark, Michael Crichton, John Grisham, Danielle Steele, Sidney Sheldon, Stephen King, Sue Grafton, Elmore Leonard, Sara Paretsky, Tom Clancy, and so (...) on. These are the kinds of novels that, when you lend them to friends, you don't care, unless you live in Bowling Green, Ohio, whether you ever get them back. They are mass, popular fictions. In another era, they would have been called pulp fictions. Following Thomas Roberts,1 I will call diem junk fictions, under which rubric I will also include things like Harlequin romances; sci-fi, horror, and mystery magazines; comic books; and broadcast narratives on either the radio or TV, as well as commercial movies. There are a number of interesting philosophical questions diat we may ask aboutjunk fiction. We could, for example, attempt to characterize its essential features. However, for the present, I will assume that the preceding examples are enough to provide you with a rough-andready notion of what I am calling junk fiction, and I will attempt to explore anotiier feature of the phenomenon, viz., what I call the paradox ofjunk fiction. Thejunk fictions that I have in mind are all narratives. Indeed, dieir story dimension is die most important thing about diem. Stephen King, for instance, makes diis point by saying that he is primarily a story teller rather than a writer. Junk fictions aspire to be page-turners—the blurb on the cover of Stillwatch by Mary Higgins Clark says that it is "designed to be read at breatiitaking speed"—and what motivates turning die page so quickly is our interest in what happens next. We do not dawdle Philosophy and Literature, © 1994, 18: 225-241 226Philosophy and Literature over Clark's diction as we might over Updike's nor do we savor die complexity of her sentence structure, as we do widi Virginia Woolfs. Radier, we read for story. Moreover, junk fictions are die sort of narratives that commentators are wont to call formulaic. That is, junk fictions generally belong to well-entrenched genres, which diemselves are typified by tiieir possession of an extremely limited repertoire of story-types. For example, as John Cawelti has pointed out, one such recurring Western narrative is diat of the recendy pacifist gunfighter, like Shane, who is forced by circumstances to take up his pistols again, widi altogedier devastating effect.2 Junk fictions tell diese generic stories again and again with minor variations. Sometimes diese variations may be quite clever and unexpected. Agatiia Christie was the master of this; she was able to use the conventions of the mystery genre in order to "hide" her murderers. In TheMurderofRogerAckroyd, she "secrets" the murderer in the personage of the narrator; in Ten Little Indians, die murderer is a "dead man"; while in Murder on the Orient Express, all the suspects did it. In each of these cases, Christie's brilliance hinges upon her playing (and preying) upon conventional expectations. Neverdieless, even diese surprising variations require a well-established background of narrative forms. That is, in order to appreciate diese variations, die reader must in some sense know die standard story already. And with junk fiction, it is generally fair to say that in some sense, the reader—or, at least, the reader who has read around in die genre before—knows in rough oudine how die story is likely to go. Readers and/or viewers of Jurassic Park surmised, once die dinosaur enclosures were described, diat in fairly short order die dinosaurs would trample them down and go on the rampage—after all we had already seen or read The Lost World, King Kong, and their progeny. So, junk fictions are formulaic. They rehearse certain narrative formats again and again. And, furthermore, in some very general sense, the audience already knows die story in question. But this knowledge on die part of die audience provokes a question, specifically, why if the reader, viewer or listener already... (shrink)

This double issue of the Carrollian, the journal of the Lewis Carroll Society, is entirely devoted to Lewis Carroll's famous short paper published in the journal Mind in 1895 under the title ‘What...

My main aim at this paper is to present Lewis Carrol’s Paradox on the justification of logical principles inasmuch as some attempts of solving it. This is important because if there are basic logical principles, it also seems necessary to exist some justification for them. By considering some observations from Ryle, Devitt and Kripke about the theme, we intend to briefly display their theories and their core critics among themselves and, mainly, the critics against adoption theory.

American health policy today faces dual problems of too little coverage at too high a cost. The mix of private and public financing leaves about one seventh of the population without any insurance coverage. At the same time, the coverage Americans do have costs an ever-larger share of our country’s productive capacity. The U.S. pays well above what other countries pay and what many people, health plans, businesses, and governments want to pay. This “paradox of excess and deprivation” results (...) from the incremental approach the U.S. has taken to promoting incompatible policy goals of increasing health insurance coverage and medical quality while trying to control costs, without squarely confronting tradeoffs. This essay examines the record of incremental developments and draws lessons for current efforts at reform. (shrink)

The self-visitation paradox is one paradox of time travel. As Ted Sider puts it, “Suppose I travel back in time and stand in a room with my sitting 10-year-old self. I seem to be both sitting and standing, but how can that be?” (2001, 101). So as not to beg any questions, let us label what is sitting B and what is standing C. The worry is about how B can be C in light of the looming contradiction (...) that this one person would be sitting and standing. Sider’s own approach is perdurantist, and holds that B is not C. My concern, though, is with solutions offered by, or on behalf of, endurantists–more 1 specifically, with solutions holding that B is C. The endurantist answer I shall criticize is a relativizer position maintaining that the sitting and the standing need to be relativized to the personal time or proper time of the time traveler. This manner of solution has been offered by Paul Horwich (1975, 433-435 ; 1987, 114-115) and also by Simon Keller and Michael Nelson (2001, 344). I will show that such a view has a linguistically suspect element and that there are three further reasons why relativizing only in this way falls short of solving the paradox. This will be enough to squash the relativizer position because it will not be clear how additional relativization could help, and furthermore any additional relativization would only make the linguistic matter worse. I will also present some considerations in favor of a non-contradiction endurantist alternative; this view eliminates the need for any relativization by denying that sitting and standing are contradictory properties. (shrink)

Some philosophers have tried to establish a connection between the normativity of instrumental rationality and the paradox presented by Lewis Carroll in his 1895 paper “What the Tortoise Said to Achilles.” I here examine and argue against accounts of this connection presented by Peter Railton and James Dreier before presenting my own account and discussing its implications for instrumentalism (the view that all there is to practical rationality is instrumental rationality). In my view, the potential for a Carroll-style regress (...) just shows us that since instrumental rationality involves a higher-order commitment to combine our willing an end with our taking the necessary means, it therefore cannot, on pain of regress, itself be added as a conjunct to one of the elements to be combined. This view does not support instrumentalism. (shrink)

Many arguments that show p to be enthymematic (in an argument for q) rely on claims like “if one did not believe that p, one would not have a reason for believing that q.” Such arguments are susceptible to the neg-raising fallacy. We tend to interpret claims like “X does not believe that p” as statements of disbelief (X's belief that not-p) rather than as statements of withholding the belief that p. This article argues that there is a tendency to (...) equivocate in arguments for the enthymematicity of arguments (e.g., Lewis Carroll's paradox, Hume's problem) as well as in arguments for the enthymematicity of action explanations (e.g., arguments for psychologism and for explanatory individualism). The article concludes with a warning, because the equivocation is often helpful in teaching and because neg-raising verbs include philosophically vital verbs: desire, want, intend, think, suppose, imagine, expect, feel, seem, appear. (shrink)

_Paradoxes from A to Z, Third edition_ is the essential guide to paradoxes, and takes the reader on a lively tour of puzzles that have taxed thinkers from Zeno to Galileo, and Lewis Carroll to Bertrand Russell. Michael Clark uncovers an array of conundrums, such as Achilles and the Tortoise, Theseus’ Ship, and the Prisoner’s Dilemma, taking in subjects as diverse as knowledge, science, art and politics. Clark discusses each paradox in non-technical terms, considering its significance and looking at (...) likely solutions. This third edition is revised throughout, and adds nine new paradoxes that have important bearings in areas such as law, logic, ethics and probability. Paradoxes from A to Z, Third edition is an ideal starting point for those interested not just in philosophical puzzles and conundrums, but anyone seeking to hone their thinking skills. (shrink)

I will use paradox as a guide to metaphysical grounding, a kind of non-causal explanation that has recently shown itself to play a pivotal role in philosophical inquiry. Specifically, I will analyze the grounding structure of the Predestination paradox, the regresses of Carroll and Bradley, Russell's paradox and the Liar, Yablo's paradox, Zeno's paradoxes, and a novel omega plus one variant of Yablo's paradox, and thus find reason for the following: We should continue to characterize (...) grounding as asymmetrical and irreflexive. We should change our understanding of the transitivity of grounding in a certain sense. We should require foundationality in a new, generalized sense, that has well-foundedness as its limit case. Meta-grounding is important. The polarity of grounding can be crucial. Thus we will learn a lot about structural properties of grounding from considering the various paradoxes. On the way, grounding will also turn out to be relevant to the diagnosis (if not the solution) of paradox. All the paradoxes under consideration will turn out to be breaches of some standard requirement on grounding, which makes uniform solutions of large groups of these paradoxes more desirable. In sum, bringing together paradox and grounding will be shown to be of considerable value to philosophy.1. (shrink)

This paper discusses a novel response to two closely related regress arguments from Bolzano’s Theory of Science and Carroll’s What the Tortoise Said to Achilles. Bolzano’s argument aims to refute the thesis that full grounds must include propositions involving notions such as entailment, grounding or lawhood which link the respective grounds to their groundee. This thesis is motivated, Bolzano’s argument is reconstructed, and a response based on self-referential linking propositions is developed and defended against objections concerning self-reference and Curry’s (...) paradox. Finally, the idea is applied to a reading of Carroll’s dialogue and a corresponding solution to the so-called infinite regress problem of inference is proposed. (shrink)

I discuss a neglected solution to the skeptical problem introduced by Lewis Carroll’s “What the Tortoise Said to Achilles” (1895) in terms of a self-citational inferential license. I then consider some responses to this solution. The most significant response on behalf of the skeptic utilizes the familiar distinction between two ways of accepting a rule: as action-guiding and as a mere truth. I argue that this is ultimately unsatisfactory and conclude by opting for an alternative conception of rules as (...) representations of behavior deployed for various purposes, some theoretical and others practical. This alternative conception does not allow the skeptical problem to get off the ground. (shrink)

Zeno's paradoxes of motion and the semantic paradoxes of the Liar have long been thought to have metaphorical affinities. There are, in fact, isomorphisms between variations of Zeno's paradoxes and variations of the Liar paradox in infinite-valued logic. Representing these paradoxes in dynamical systems theory reveals fractal images and provides other geometric ways of visualizing and conceptualizing the paradoxes.

This “fun, brain-twisting book... will make you think” as it explores more than 75 paradoxes in mathematics, philosophy, physics, and the social sciences (Sean Carroll, New York Times–bestselling author of Something Deeply Hidden) Paradox is a sophisticated kind of magic trick. A magician’s purpose is to create the appearance of impossibility, to pull a rabbit from an empty hat. Yet paradox doesn’t require tangibles, like rabbits or hats. Paradox works in the abstract, with words and concepts and (...) symbols, to create the illusion of contradiction. There are no contradictions in reality, but there can appear to be. In Sleight of Mind, Matt Cook and a few collaborators dive deeply into more than 75 paradoxes in mathematics, physics, philosophy, and the social sciences. As each paradox is discussed and resolved, Cook helps readers discover the meaning of knowledge and the proper formation of concepts—and how reason can dispel the illusion of contradiction. The journey begins with “a most ingenious paradox” from Gilbert and Sullivan’s Pirates of Penzance. Readers will then travel from Ancient Greece to cutting-edge laboratories, encounter infinity and its different sizes, and discover mathematical impossibilities inherent in elections. They will tackle conundrums in probability, induction, geometry, and game theory; perform “supertasks”; build apparent perpetual motion machines; meet twins living in different millennia; explore the strange quantum world—and much more. (shrink)

Expressing a widely-held view, David Hitchcock claims that "an enthymematic argument ... assumes at least the truth of the argument's associated conditional ... whose antecedent is the conjunction of the argument's explicit premises and whose consequent is the argument's conclusion." But even definitionally, this view is problematic, since an argument's being "enthymematic" or incomplete with respect to its explicit premises means that the conclusion is not implied by these premises alone. The paper attempts to specify the ways in which the (...) view is incorrect, as well as seemingly correct (e.g., the case of a Modus Ponens wherein the major premise is implicit). -/- . (shrink)

I propose that an irreducible property of physical space — consistency — is the origin of logic. I propose that an inconsistent space is inconceivable and that this inconceivability can be recognized as the force behind logical propositions. The implications of this argument are briefly explored and then applied to address two paradoxes: Zeno of Elea’s paradox regarding the race between Achilles and the Tortoise, and Lewis Carroll’s paradox regarding the Tortoise’s conversation with Achilles after the race. (...) I conclude that Achilles would have won on both accounts, in the race and the argument, by invoking the consistency of space as the foundation of both movement across space and logical argument. (shrink)

The sequencing of the human genome and decreasing costs of sequencing technology have led to the notion of ‘personalized medicine’. This has been taken by some authors to indicate that personalized medicine will provide individualized treatments solely based on one’s DNA sequence. We argue this is overly optimistic and misconstrues the notion of personalization. Such interpretations fail to account for economic, policy and structural constraints on the delivery of healthcare. Furthermore, notions of individualization based on genomic data potentially take us (...) down the road of genetic reductionism obscuring the role of environmental factors in disease and ill health. We propose that one should see personalized medicine as a way of using personal genomic information to stratify individuals into subpopulations and suggest that personalized medicine be seen within a broader idea of personalized healthcare, reflecting healthcare that integrates personal genomic data into cultural, environmental and personal contexts. (shrink)

G. E. Moore observed that to assert, 'I went to the pictures last Tuesday but I don't believe that I did' would be 'absurd'. Over half a century later, such sayings continue to perplex philosophers. In the definitive treatment of the famous paradox, Green and Williams explain its history and relevance and present new essays by leading thinkers in the area.

L'horreur en art vise à effrayer, bouleverser, dégoûter et terroriser. Puisque nous ne sommes pas normalement attirés par de ielles expériences, pourquoi quiconque s'exposerait-il délibérément a la fiction d'horreur? Noel Carroll soutient que le caractère constant du phénomène de l'horreur en art tient à certains plaisirs d'ordre cognitif, qui résultent de la satisfaction de notre curiosité naturelle à l'ègard des monstres. Je soutiens, quant è moi, que la solution cognitive de Carroll auparadoxe de l'horreur est profondément erronée, étant donné la (...) façon dont les monsters sont représentés dans la fiction d'horreur; j'explore brievement une approche plus prometteuse, qui traite les monstres comme des moyens pour acquerir la connaissance de soi. (shrink)

I will attempt to draw analogies between the problems of inductionand problems of deduction with Carroll’s paradox and Susan Haack’sarguments. These analogies either strengthen a justification of induction by showing that deduction faces similar problems, or weaken our justification of deduction by showing it is not entirely justified. I will show that although deduction does face similar problems any justification that can work for induction will also be applicable to deduction, but not vice versa. Finally, I will show (...) that Reichenbach’s system of vindicating induction will also provide vindication for deduction, and argue that deduction preserves truth to a greaterextent than induction. (shrink)

Abstract:Extrapolating from Carroll’s four domains of corporate social responsibility (1979) and Pyramid of CSR (1991), an alternative approach to conceptualizing corporate social responsibility (CSR) is proposed. A three-domain approach is presented in which the three core domains of economic, legal, and ethical responsibilities are depicted in a Venn model framework. The Venn framework yields seven CSR categories resulting from the overlap of the three core domains. Corporate examples are suggested and classified according to the new model, followed by a (...) discussion of limitations and teaching and research implications. (shrink)

As foreign direct investment in the U.S. continues to become both more visible and controversial, the general public remains skeptical about the corporate citizenship of these foreign affiliates. Four dimensions of corporate citizenship — orientations, organizational stakeholders, issues, and decision-making autonomy — were used to compare the inclinations of foreign affiliates with the domestic firms operating in the U.S. chemical industry. The only significant differences between the U.S. sample and those firms headquartered in other countries-of-origin were found in the area (...) of corporate citizenship decision making autonomy. (shrink)

It is impossible to hold patently contradictory beliefs in mind together at once. Why? Because we know that it is impossible for both to be true. This impossibility is a species of rational necessity, a phenomenon that uniquely characterizes the relation between one person's beliefs. Here, Eric Marcus argues that the unity of the rational mind--what makes it one mind--is what explains why, given what we already believe, we can't believe certain things and must believe certain others in this special (...) sense. What explains this is that beliefs, and the inferences by which we acquire them, are constituted by a particular kind of endorsement of those very states and acts. This, in turn, entails that belief and inference are essentially self-conscious: to hold a belief or to make an inference is at the same time to know that one does. An examination of the nature of belief and inference, in light of the phenomenon of rational necessity, reveals how the unity of the rational mind is a function of our knowledge of ourselves as bound to believe the true. Rational self-consciousness is the form of mental togetherness. (shrink)

Hempel's paradox of the ravens has to do with the question of what constitutes confirmation from a logical point of view; Wason 's selection task has been used extensively to investigate how people go about attempting to confirm or disconfirm conditional claims. This paper presents an argument that the paradox is resolved, and that people's typical performance in the selection task can be explained, by consideration of what constitutes an effective strategy for seeking evidence of the tenability of (...) universal or conditional claims in everyday life. (shrink)

The aim of this paper is to distinguish between, and examine, three issues surrounding Humphreys's paradox and interpretation of conditional propensities. The first issue involves the controversy over the interpretation of inverse conditional propensities — conditional propensities in which the conditioned event occurs before the conditioning event. The second issue is the consistency of the dispositional nature of the propensity interpretation and the inversion theorems of the probability calculus, where an inversion theorem is any theorem of probability that makes (...) explicit (or implicit) appeal to a conditional probability and its corresponding inverse conditional probability. The third issue concerns the relationship between the notion of stochastic independence which is supported by the propensity interpretation, and various notions of causal independence. In examining each of these issues, it is argued that the dispositional character of the propensity interpretation provides a consistent and useful interpretation of the probability calculus. (shrink)

The Tractatus rejects the sign of assertion as "logically meaningless", but the rejection of the sign did not lead Wittgenstein to reject the corresponding notion. I show the presence and the importance in the early Wittgenstein of a notion keenly similar to Fregean and Russellian logical assertion. I propose to call this notion "affirmation." The preparatory writings and the TLP present different theories about affirmation. The correct understanding of the nature and purpose of affirmation proves critical in order to confront (...) another issue about the Tractatus: the only partial similarity between the theory of pictures and the theory of propositions. (shrink)

There are three questions associated with Simpson’s paradox (SP): (i) Why is SP paradoxical? (ii) What conditions generate SP? and (iii) How to proceed when confronted with SP? An adequate analysis of the paradox starts by distinguishing these three questions. Then, by developing a formal account of SP, and substantiating it with a counterexample to causal accounts, we argue that there are no causal factors at play in answering questions (i) and (ii). Causality enters only in connection with (...) action. (shrink)

Shoemaker argues that a satisfactory resolution of Moore's paradox requires a _self-intimation thesis that posits a "constitutive relation between belief and believing that one believes." He claims that such a thesis is needed to explain the crucial fact that the assent conditions for '_P' entail those for '_I believe that P'. This paper argues for an alternative resolution of Moore's paradox that provides for an adequate explanation of the crucial fact without relying on the kind of necessary connection (...) between first and second-order beliefs that is posited by Shoemaker's self-intimation thesis. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:The Logic Pamphlets of Charles Lutwidge Dodgson and Related PiecesIrving H. AnellisFrancine F. Abeles, editor. The Logic Pamphlets of Charles Lutwidge Dodgson and Related Pieces. The Pamphlets of Lewis Carroll, 4. New York-Charlottesville-London: Lewis Carroll Society of North America-University Press of Virginia, 2010. Pp. xx + 271. Cloth, $75.00.Until William Bartley’s rediscovery and reconstruction of Dodgson’s lost Part II of Symbolic Logic, Lewis Carroll’s reputation in logic, (...) when taken seriously, rested upon his few published articles on logical paradoxes and puzzles, while his published books, The Game of Logic and Symbolic Logic, were regarded as suitable perhaps as pedagogical tools, but dismissed otherwise as amusements, on a par with the Alice works. Richard Braithwaite asserted (“Lewis Carroll as Logician”) that “Carroll regarded formal and symbolic logic not as a corpus of systematic knowledge about valid thought nor yet as an art for teaching a person to think correctly.” He suggested that Dodgson used his pseudonym for his popular and whimsical writings, reserving his legal name only for his serious mathematical products. Philosophers practicing linguistic analysis in their assault on nonsense deriving from misuse of language meanwhile developed an interest in Carroll (e.g. George Pitcher, “Wittgenstein, Nonsense, and Lewis Carroll”). Abeles, on the contrary, stresses the seriousness of Dodgson’s pedagogical and popularizing mission (195–99).Bartley’s discovery opened the door for reevaluating Dodgson as a serious logician, and Abeles has been in the forefront of that reevaluation. Her general introduction and introductions to the main divisions of this collection of Carroll’s publications (and additional material from his Nachlass and a handful of publications of others, composed in response to Carroll’s published articles) place his serious logical work and its significance in historical perspective, providing a deeper, more sophisticated view than found in Bartley’s “Editor’s Introduction” to Lewis Carroll’s Symbolic Logic. Abeles also offers brief introductions to the individual entries, which describe their physical characteristics and circumstances of composition.Dodgson aligned with the majority of those British logicians of his day, starting with George Boole, who understood formal logic to be Aristotelian syllogistic, and symbolic logic to be syllogistic logic expressed algebraically. His notational innovation was employing indices to terms so that, independently of the arrangement of terms, propositions could easily be read; thus, “xy0” can be read as “no x are y” or “no y are x” and “x1y0” as “all x are not y” or “no x are y”. In dealing with the classical elimination problem in the class calculus—to determine the maximum information, without duplications, obtainable from a given set of premises—Dodgson in his later work and unpublished letters anticipated several concepts of automated theorem proving.Central to both geometry and to formal logic are the rules of logical inference that establish the validity of arguments and guarantee that conclusions inferred from true [End Page 506] premises are true. The modern analytic tableaux, or tree method, is a fundamental tool in this regard, both for deriving theorems and for determining the validity or invalidity of the proofs for propositional calculus and first-order predicate calculus.Introducing his reconstruction of Part II of Symbolic Logic, Bartley noted that, along with the logic diagrams that Carroll devised, he also devised a tableau method that strongly resembled Beth’s semantic tableaux. Abeles went more deeply and convincingly into that resemblance in her earlier research. Beth’s tableaux, together with Hintikka’s model sets, were the direct ancestors of Smullyan’s analytic tableaux, which in turn are the theoretical basis for the Robinson resolution method that continues to be central to much work in programming logic.Abeles has also shown that the work in Studies in Logic by Charles Peirce and his logic students at Johns Hopkins University served as a source of inspiration for Dodgson. More critically, she demonstrated that Dodgson devised and employed the tree method for carrying out proofs of syllogisms and chains of syllogisms, or soriteses, in Part II of Symbolic Logic (see her “Lewis Carroll’s Method of Trees: Its Origins in Studies in Logic,” Modern Logic 1 [1990]: 25–34). The tree... (shrink)

There are three questions associated with Simpson’s Paradox (SP): (i) Why is SP paradoxical? (ii) What conditions generate SP?, and (iii) What should be done about SP? By developing a logic-based account of SP, it is argued that (i) and (ii) must be divorced from (iii). This account shows that (i) and (ii) have nothing to do with causality, which plays a role only in addressing (iii). A counterexample is also presented against the causal account. Finally, the causal and (...) logic-based approaches are compared by means of an experiment to show that SP is not basically causal. (shrink)

Bertrand Russell offered an influential paradox of propositions in Appendix B of The Principles of Mathematics, but there is little agreement as to what to conclude from it. We suggest that Russell's paradox is best regarded as a limitative result on propositional granularity. Some propositions are, on pain of contradiction, unable to discriminate between classes with different members: whatever they predicate of one, they predicate of the other. When accepted, this remarkable fact should cast some doubt upon some (...) of the uses to which modern descendente of Russell's paradox of propositions have been put in recent literature. (shrink)