The infinite regress of Carroll’s ‘What the Tortoise said to Achilles’ is interpreted as a problem in the epistemology of mathematical proof. An approach to the problem that is both diagrammatic and non-logical is presented with respect to a specific inference of elementary geometry.

John Venn and Charles L. Dodgson (Lewis Carroll) created systems of logic diagrams capable of representing classes (sets) and their relations in the form of propositions. Each is a proof method for syllogisms, and Carroll's is a sound and complete system. For a large number of sets, Carroll diagrams are easier to draw because of their self-similarity and algorithmic construction. This regularity makes it easier to locate and thereby to erase cells corresponding with classes destroyed by the premises of an (...) argument, a particularly difficult task in Venn diagrams for more than four sets. Carroll diagrams can represent existential propositions easily, so they are capable of clearly representing more complex problems than Venn's system can. Finally, both Carroll and Venn diagrams are maximal, in the sense that no additional logic information like inclusive disjunctions is able to be represented by them. Carroll's logic diagrams and logic trees constitute his visual logic system. (shrink)

Lewis Carroll published a system of logic in the symbolic tradition that developed in his time. Carroll’s readers may be puzzled by his system. On the one hand, it introduced innovations, such as his logic notation, his diagrams and his method of trees, that secure Carroll’s place on the path that shaped modern logic. On the other hand, Carroll maintained the existential import of universal affirmative Propositions, a feature that is rather characteristic of traditional logic. The object of (...) this paper is to untangle this dilemma by exploring Carroll’s guidelines in the design of his logic, and in particular his theory of existential import. It will be argued that Carroll’s view reflected his belief in the social utility of symbolic logic. (shrink)

Some Carrollian posthumous manuscripts reveal, in addition to his famous ‘logical diagrams’, two mysterious ‘logical charts’. The first chart, a strange network making out of fourteen logical sentences a large 2D ‘triangle’ containing three smaller ones, has been shown equivalent—modulo the rediscovery of a fourth smaller triangle implicit in Carroll's global picture—to a 3D tetrahedron, the four triangular faces of which are the 3+1 Carrollian complex triangles. As it happens, such an until now very mysterious 3D logical shape—slightly deformed—has been (...) rediscovered, independently from Carroll and much later, by a logician , a mathematician and a linguist studying the geometry of the ‘opposition relations’, that is, the mathematical generalisations of the ‘logical square’. We show that inside what is called equivalently ‘n-opposition theory’, ‘oppositional geometry’ or ‘logical geometry’, Carroll's first chart corresponds exactly, duly reshap.. (shrink)

It is of common use in modern Venn diagrams to mark a compartment with a cross to express its non-emptiness. Modern scholars seem to derive this convention from Charles S. Peirce, with the assumption that it was unknown to John Venn. This paper demonstrates that Venn actually introduced several methods to represent existentials but felt uneasy with them. The resistance to formalize existentials was not limited to diagrammatic systems, as George Boole and his followers also failed to provide a satisfactory (...) symbolic representation for them. This difficulty points out issues that are inherent to the very nature of existentials. This paper assesses the various methods designed for the representation of existential statements with Venn diagrams. First, Venn’s own attempts are discussed and compared with other solutions proposed by his contemporaries and successors, notably Lewis Carroll and Peirce. Since disjunctives hold an important role in an effective representation of existentials, their representation is also discussed. Finally, recent methods for the diagrammatic representation of existing individuals, rather than mere existence, are surveyed. (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)

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)

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)

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)

Introduction: liberal-rationalism and the progress model i This study stands primarily as an essay in morals. It is governed by Nietzsche's contention that ...

The lattice of r.e. equivalence relations has not been carefully examined even though r.e. equivalence relations have proved useful in logic. A maximal r.e. equivalence relation has the expected lattice theoretic definition. It is proved that, in every pair of r.e. nonrecursive Turing degrees, there exist maximal r.e. equivalence relations which intersect trivially. This is, so far, unique among r.e. submodel lattices.

This essay argues that the main lesson of Lewis Carroll's Regress is that arguments are constitutively presuppositional.

In this paper, I address a key argument in favour of logical expressivism, the view that knowing a logical principle such as Modus Ponens is not a cognitive state but a pro-attitude towards drawing certain types of conclusions from certain types of premises. The argument is that logical expressivism is the only view that can take us out of Lewis Carroll's Regress – which suggests that elementary deductive reasoning is impossible. I show that the argument does not hold scrutiny and (...) that logical cognitivism can be vindicated. In the course of the discussion, I draw substantially on a comparison with a similar argument in meta-ethics, for moral expressivism. (shrink)

“Noverim me, noverim te.” – Saint Augustine, Confessions, 10.1.1. (397-400 AD). -/- What would and will an urban commons look like that is slowly and incrementally being re-socialized? How would that affect urban planning “now” and in times of crisis? How do we prepare for the likelihood of rolling similar crises with an eye on returning the urban commons to citizens? -/- There is the old adage that under capitalism, risk is always socialized and profit is always privatized. We are (...) seeing it now, under the COVID-19 crisis. The huge bailouts launched by governments are symptomatic of the crisis in political economy, just as they were post-2008. “Too big to fail” has sponsored monsters that refuse to back off without threatening the collapse of the entire system. Francisco Goya’s “The sleep of Reason produces monsters …” comes to mind. -/- Physical and immaterial culture, in our current Western civilisation, are intimately linked. Yet the focus for urban design is generally on the material or physical side, with the immaterial left to its own devices. Increasingly, urban design measures are merely ameliorative and aesthetic, with the larger share shaped by a political economy dictated by market ideology or “politicalology.” What transpires, nonetheless, is an immaterial commons that constitutes a public or private intellectual commons – often a mix of the two; but, in the case of domination by market ideology, the privatization of “general intellect” proceeds by abject appropriation. In such a technocratically driven model, subjective states become increasingly important. As Indian architect Balkrishna Vithaldas Doshi once said, “Smart cities are smart people.” -/- How might these two otherwise contiguous and synchronous systems be brought back into a properly civic-minded rapport with or without crisis-driven change? Are there alternate models for the urban commons? What measures might be put in place in advance, or as provisional intercessions? (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.

On an internalist account of logical inference, we are warranted in drawing conclusions from accepted premises on the basis of our knowledge of logical laws. Lewis Carroll's regress challenges internalism by purporting to show that this kind of warrant cannot ground the move from premises to conclusion. Carroll's regress vindicates a repudiation of internalism and leads to the espousal of a standpoint that regards our inferential practice as not being grounded on our knowledge of logical laws. Such a standpoint can (...) take two forms. One can adopt either a broadly externalist model of inference or a sceptical stance. I will attempt, in what follows, to defend a version of internalism which is not affected by the regress. The main strategy will be to show that externalism and scepticism are not satisfying standpoints to adopt with regard to our inferential practice, and then to suggest an internalist alternative. (shrink)

Rationality requires us to have certain propositional attitudes (beliefs, intentions, etc.) given certain other attitudes that we have. Carroll’s Tortoise repeatedly shows up in this discussion. Following up on Brunero (Ethical Theory Moral Pract 8:557–569, 2005), I ask what Carroll-style considerations actually prove. This paper rejects two existing suggestions, and defends a third.

This is a review of Marc Lange's _Natural Laws in Scientific Practice<D>.

On an internalist account of logical inference, we are warranted in drawing conclusions from accepted premises on the basis of our knowledge of logical laws. Lewis Carroll’s regress challenges internalism by purporting to show that this kind of warrant cannot ground the move from premises to conclusion. Carroll’s regress vindicates a repudiation of internalism and leads to the espousal of a standpoint that regards our inferential practice as not being grounded on our knowledge of logical laws. Such a (...) standpoint can take two forms. One can adopt either a broadly externalist model of inference or a sceptical stance. I will attempt, in what follows, to defend a version of internalism which is not affected by the regress. The main strategy will be to show that externalism and scepticism are not satisfying standpoints to adopt with regard to our inferential practice, and then to suggest an internalist alternative. (shrink)

This study has been designed to investigate whether Corporate Social Responsibility (CSR) orientations have shifted in their priority in response to society's changing expectations. For this sample of U.S.-based multinational chemical subsidiaries, it appears that the top priority continues to be economic responsibilities, followed closely by legal responsibilities. A socially accountable corporation ... must be a thoughtful institution, able to rise above economic interest to anticipate the impact of its actions on all individuals and groups, from shareholders to employees to (...) customers, to fellow-breathers of the air and fellow-sharers of the land. A successful business organization must possess a moral sense as well as an economic sense (Thornton Bradshaw, President of Atlantic Richfield Co. inBusiness and Society: Strategies for the 1980's, 1980, p. xiv). (shrink)

Numbers. Natural numbers -- Integers -- Real numbers -- Irrational and transcendental numbers -- Funny numbers. Zero -- e : the unnatural natural number -- [Phi] : the golden ratio -- i : the imaginary number -- Writing numbers. Roman numerals -- Egyptian fractions -- Continued fractions -- Logic. Mr. Spock is not logical -- Proofs, truth, and trees : oh my! -- Programming with logic -- Temporal reasoning -- Sets. Cantor's diagonalization : infinity isn't just infinity -- Axiomatic set (...) theory : keep the good, dump the bad -- Models : using sets as the LEGOs of the math world -- Transfinite numbers : counting and ordering infinite sets -- Group theory : finding symmetries with sets -- Mechanical math. Finite state machines : simplicity goes far -- The turing machine -- Pathology and the heart of computing -- Calculus : no, not that calculus- [lambda] calculus -- Numbers, booleans, and recursion -- Types, types, types : modeling [lambda] calculus -- The halting problem. (shrink)

I show that in our theoretical representations of argument, vicious infinite regresses of self-reference may arise with respect to each of the three usual, informal criteria of argument cogency: the premises are to be relevant, sufficient, and acceptable. They arise needlessly, by confusing a cogency criterion with argument content. The three types of regress all are structurally similar to Lewis Carroll’s famous regress, which involves quantitative extravagance with no explanatory power. Most attention is devoted to the sufficiency criterion, including (...) its relation to the view au courant that inferring necessarily involves the thinker taking her premises to support her conclusion. I contend that this view is mistaken and likewise that arguments make no such assumption or inference claim as that the premises support the conclusion. The core of the positive alternative model I propose is that there is commitment to, but not claiming, the proposition that the premises support the conclusion. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Philosophy and Literature 26.1 (2002) 199-206 [Access article in PDF] Notes and FragmentsArt and Friendship Noël Carroll YASMINA REZA'S PLAY Art is about one man, Serge, who buys a painting, and the reactions of his friends, Marc and Yvan, to his purchase. 1 Marc's response is quite volcanic; for him, Serge's purchase of the painting threatens to wreck their friendship. Yvan tries to mediate the disaffection between Serge and (...) Marc, often at the cost of redirecting their hostilities at himself.As the play opens, Marc addresses the audience directly. He says: My friend Serge has bought a painting. It's a canvas about five foot by four: white. The background is white and if you screw up your eyes, you can make out some fine white diagonal lines. Serge is one of my oldest friends. He's done very well for himself, he's a dermatologist and he's keen on art. On Monday, I went to see the painting; Serge had actually got hold of it on the Saturday, but he's been lusting after it for several months. This white painting with white lines. 2This prelude explicitly connects the two major themes of the play—art and friendship (Serge is one of Marc's oldest friends), and the relation thereof. After this brief exposition, then, there is a flashback to the scene where Marc first sees Serge's painting, a painting by Antrios, a famous artist. At first Marc reacts warily, tentatively, but, in almost no time at all, he denounces the painting as shit, despite its whiteness. Marc tries to pass off his vituperation as humor and invites Serge to [End Page 199] laugh along with him. But Serge finds nothing funny about the situation, which deteriorates throughout the play as everyone's nerves fray and angers mount.The action of the play raises an immediate question. Why is Marc's reaction to the painting by Antrios so intense, so violent? Why would a painting endanger a friendship? Why do Marc and Serge seem to be willing to split apart after fifteen years over a matter of taste? Can't they just agree to disagree and leave it at that?But Marc, at least, cannot. He says, "It's a complete mystery to me, Serge buying this painting. It's unsettled me, it's filled me with some undefinable unease." And that undefinable unease is enough to motivate Marc to attack Serge and Yvan savagely to the point where the friendship among them seems no longer possible. But what is the nature of this undefinable unease and what does its existence tell us about the relation of art to friendship?Marc and Serge obviously see the painting very differently. Marc keeps referring to the Antrios as white, prompting Serge to correct him—pointing out that it has diagonals as well as bits of various other colors in it. Perhaps this is author Yasmina Reza's way of signaling that Serge, so to speak, sees something in the painting—sees that there's something to it—whereas for Marc, it's a blank, it's empty, it's nothing, it's worthless. Marc continues to be amazed that Serge has paid two hundred thousand francs for the painting. But it's not the money, as such, that bothers Marc; it is what the money symbolizes: that Serge sees something worthy in that which Marc finds worthless. And this threatens their friendship. Why?One of the earliest conceptions of friendship in Western thought is Aristotle's notion of what he calls the friendship of character or character friendship. This is not the only type of friendship there is—there are also friendships based on such things as expediency—but, for Aristotle, character friendship is the highest sort. This is the type of friendship that obtains between equals—people of equal virtue and excellence.Now you might think that people who are virtuous and excellent already should have little need of friends. They already have it all. 3 But Aristotle suggests that without friends—friends who are... (shrink)

The philosophy of Spinoza can hardly be said to have been in the fore-front of recent developments in the philosophy of mind. Notwithstanding, Stuart Hampshire has put himself on record as saying “that in the philosophy of mind he [Spinoza] is nearer to the truth at certain points than any other philosopher ever has been.” The purpose of this paper is to get even nearer the truth with Spinoza’s leading. The idea of the body is, however, a confused idea; so (...) confused, in fact, that Spinoza was never able to form an adequate idea of the idea. I wish to argue that Spinoza has even worse reasons than Descartes for taking the idea of the body to be the idea of something that “actually exists.” Lacking any appeal to the divine veracity, Spinoza’s contention that the body does exist constitutes a standing petitio for the whole of the philosophy of mind. (shrink)

This essay reads Alice’s Adventures in Wonderland (1865) alongside influential mid-century Victorian psychology studies—paying special attention to those that Carroll owned—in order to trace the divergence of Carroll’s literary representations of the “dream child” from its prevailing medical association with mental illness. The goals of this study are threefold: to trace the medico-historical links between dream-states and childhood, to investigate the medical reasons behind the pathologization of dream-states, and to understand how Alice’s Adventures in Wonderland contributed to Victorian interpretations (...) of the child’s mind. (shrink)

While contemporary feminist philosophical discussions focus on the oppressiveness of universality which obliterates “difference,” the complete demise of universality might hamper feminist philosophy in its political project of furthering the well-being of all women. Dewey's thoroughly functionalized, relativized, and fallibilized understanding of universality may help us cut universality down to size while also appreciating its limited contribution. Deweyan universality may signify the ongoing search for a genuinely common language in the midst of difference.

While contemporary feminist philosophical discussions focus on the oppressiveness of universality which obliterates "difference," the complete demise of universality might hamper feminist philosophy in its political project of furthering the well-being of all women. Dewey's thoroughly functionalized, relativized, and fallibilized understanding of universality may help us cut universality down to size while also appreciating its limited contribution. Deweyan universality may signify the ongoing search for a genuinely common language in the midst of difference.