In a recent paper Johan van Benthem reviews earlier work done by himself and colleagues on ‘natural logic’. His paper makes a number of challenging comments on the relationships between traditional logic, modern logic and natural logic. I respond to his challenge, by drawing what I think are the most significant lines dividing traditional logic from modern. The leading difference is in the way logic is expected to be used for checking arguments. (...) For traditionals the checking is local, i.e. separately for each inference step. Between inference steps, several kinds of paraphrasing are allowed. Today we formalise globally: we choose a symbolisation that works for the entire argument, and thus we eliminate intuitive steps and changes of viewpoint during the argument. Frege and Peano recast the logical rules so as to make this possible. I comment also on the traditional assumption that logical processing takes place at the top syntactic level, and I question Johan’s view that natural logic is ‘natural’. (shrink)

I read Stefan Collini’s What are Universities For? last week with very mixed feelings. In the past, I’ve much admired his polemical essays on the REF, “impact”, the Browne Report, etc. in the London Review of Books and elsewhere: they speak to my heart. If you don’t know those essays, you can get some of their flavour from his latest article in the Guardian yesterday. But I found the book a disappointment. Perhaps the trouble is that Collini is too decent, (...) too high-minded, has too unrealistically exalted a view about what actually happens in universities which is too coloured by attitudes appropriate to the traditional humanities. And he is optimistic to the point of fantasy if he thinks that people are so susceptible to “the romance of ideas and the power of beauty” that they will want, or can be brought to want, lots and lots of universities in order to promote the ideas (as if they would suppose that the task of “conserving, understanding, extending and handing on to subsequent generations the intellectual, scientific, and artistic heritage of mankind” was clearly ill-served when there were only forty universities in England, as opposed to a hundred and whatever). (shrink)

In his Introduction to Logical Theory, Strawson argues that Aristotelian logic can be given a successful interpretation into ordinary English, but not into the symbolism of Principia Mathematica, on the grounds that Aristotelian logic and ordinary English share something absent in PM, namely, the doctrine of presupposition. It is argued that Strawson is mistaken. PM does justice to the logical rules of Aristotelian logic and also has a fully articulated doctrine of presupposition.

It is shown that all those theses of traditional logic which were rejected by Russell in terms of a preferred interpretation of 'all' and 'some', in fact lead to inconsistency in any formal system of traditional logic satisfying certain minimal conditions. Hence, Russell's refutation is ultimately independent of his interpretation. Further, the derivation of each of the refutable theses depends crucially on the Bochenski/Lukasiewicz postulate 'Some _A are _A'. If this postulate is removed, the theses which (...) remain are exactly those which translate into theses of quantification theory. (shrink)

This paperback is a programed text designed for teaching introductory logic, either in conjunction with a standard text based upon traditional logic or as a do-it-yourself supplement for students taking courses stressing symbolic logic. The student learns logical theory by answering a variety of short answer, objective type exercises. The correct answer is given directly below each question or exercise, and the student is required to cover the answer while working the exercise; the purpose of this (...) immediate access to the answer is to enable the student to determine quickly whether or not he comprehends the material. The content of the book concentrates primarily upon categorical statements, presenting both the existential and hypothetical interpretations of the square of opposition, but unfortunately continues to promulgate the historically inaccurate terminology of "Aristotelian" and "Boolean." [Categorical statements were first expressed as existential statements by Franz Brentano in 1874.] Categorical statements not in standard form are standardized, e.g., "No roaches feel despair" is standardized by being rewritten as "No roaches are entities who feel despair." In addition to considering the logical relations of opposition with respect to A, E, I, O statements, these same relations are extended to non-categorical statements, e.g., "George Washington did not die in Europe" and "George Washington did not die in Asia" are subcontraries, while "There is philosophical activity on Venus" and "There is philosophical activity in the universe" are alterns. Determining validity of syllogisms is based upon the distribution of terms, and also by the use of Venn diagrams. The traditional figures and moods of the syllogism are ignored, as is the distinction between major and minor terms, with these latter two being lumped together as end terms. Immediate inferences and non-syllogistic arguments are treated by Venn diagrams. One-, two-, three- and four-term Venn diagrams are utilized throughout a substantial part of the book.--T. G. N. (shrink)

There have been and continue to be disagreements about how to consider the traditional square of opposition and the traditional inferences of obversion, conversion, contraposition and inversion from the perspective of contemporary quantificational logic. Philosophers have made many different attempts to save traditional inferences that are invalid when they involve empty classes. I survey some of these attempts and argue that the only satisfactory way of saving all the traditional inferences is to make the existential (...) assumption that both the subject and predicate classes and their complement classes are non-empty for all the propositions we admit. I briefly indicate the room for continued controversy over how properly to interpret Aristotle's statements regarding these inferences, but find some plausibility in the views of Manley Thompson and A.N.Prior that Aristotle had in mind a particular arrangement of existential import unfamiliar to most contemporary logicians. (shrink)

Uses ideas of mathematical logic to clarify and to a certain extent defend a controverted and rather obscure notion of traditional syllogistic.

Then some Pharisees and scribes came to Jesus from Jerusalem, saying, "Why do your disciples transgress the tradition of the elders?" .... And he answered and said to them, "And why do you yourselves transgress the commandment of God for the sake of your tradition?" [Matthew 15:1 3].

The subject of metaphysics and traditional logic of confrontation is the subject of alienation - the object opposing the inevitable consequences of the theory, the tension between the two depends on the maintenance of the spirit of the times since the Enlightenment, that is the subject of support for the abstract. Once the main body of the abstract is dissolved, the main myth was exposed, the alienation of the natural transgression of ontological existence. Only out of Marx's historical (...) dialectic to the main body of subject-object dichotomy of alienation logic of confrontation with the traditional paradox. In the "Economic and Philosophical Manuscripts of 1844", Marx has been the traditional logic of alienation made ​​the initial breakthrough. The antinomy between the traditional theory of alienation and the metaphysics of subjectivity is the inevitable theoretic aftermath rooted into the dichotomy of subject-object, whose tension depends on the pure subjectivity-so called the spirit of time that initiated from the era of enlightenment. Once the pure subjectivity was cleared up and the myth of subject was exposed, the alienation must arrogate the reality. Only ground on the historical creation and transcendent of the capitalism mode of production exposed by Marx can we walk out the paradox of the dichotomy of subject- object. (shrink)

This paper is concerned with De Morgan’s explanation of the validity of arguments that involve relational notions. It discusses De Morgan’s expansion of traditional logic aimed at accommodating those inferences, and makes the point that his endeavour is not successful in that the rules that made up his new logic are not sound. Nevertheless, the most important scholarly work on De Morgan’s logic, and contrary to that De Morgan’s mistake is not beyond repair. The rules that (...) determine his new logic are in fact monotonie replacement rules. And provided they are restricted in the correct way, these rules are demonstrably sound. (shrink)

Officially, for Kant, judgments are analytic iff the predicate is "contained in" the subject. I defend the containment definition against the common charge of obscurity, and argue that arithmetic cannot be analytic, in the resulting sense. My account deploys two traditional logical notions: logical division and concept hierarchies. Division separates a genus concept into exclusive, exhaustive species. Repeated divisions generate a hierarchy, in which lower species are derived from their genus, by adding differentia(e). Hierarchies afford a straightforward sense of (...) containment: genera are contained in the species formed from them. Kant's thesis then amounts to the claim that no concept hierarchy conforming to division rules can express truths like '7+5=12.' Kant is correct. Operation concepts ( ) bear two relations to number concepts: and are inputs, is output. To capture both relations, hierarchies must posit overlaps between concepts that violate the exclusion rule. Thus, such truths are synthetic. (shrink)

Originally published in 1937. A short account of the traditional logic, intended to provide the student with the fundamentals necessary for the specialized study. Suitable for working through individualy, it will provide sufficient knowledge of the elements of the subject to understand materials on more advanced and specialized topics. This is an interesting historic perspective on this area of philosophy and mathematics.

There is a clash between some people's positive logical intuitions about traditional or Aristotelian logic and the assessment ofthat logic made by modem logic. In response to the clash, four sorts of reasons that might be given for referring one logic to the other are considered, but it is argued that none of them provides a decisive reason in favor of one rather than the other. A reformist and a radical response to the apparent inability (...) to give reasons to prefer one logic to the other are considered and reasons given for preferring the radical response. (shrink)

Some interesting exceptions notwithstanding, the traditional logic of economic efficiency has long favored hierarchical forms of organization and disfavored democracy in business. What does the balance of arguments look like, however, when values besides efficient revenue production are brought into the picture? The question is not hypothetical: In recent years, an ever increasing number of corporations have developed and adopted socially responsible behaviors, thereby hybridizing aspects of corporate businesses and social organizations. We argue that the joint pursuit of (...) financial and social objectives warrants significant rethinking of organizational democracy’s merits compared both to hierarchy and to non-democratic alternatives to hierarchy. In making this argument, we draw on an extensive literature review to document the relative lack of substantive discussion of organizational democracy since 1960. And we draw lessons from political theory, suggesting that the success of political democracy in integrating diverse values offers some grounds for asserting parallel virtues in the business case. (shrink)

Unlike Dickens’s novel, this is not a tale of light and darkness, order and chaos, good and evil… It is, though, a story worth to be told about two standpoints about games and sports, teaching and research, physical education simply put, that have pursued similar interests on parallel tracks for too long, despite their apparent closeness and expected shared cultural grounds. The objective of this conceptual analysis is to try and reconcile two perspectives, namely motor praxeology and teaching games for (...) understanding (TGfU), born in the last third of the XX century in France and England with the intention to rethink the foundations of physical education (PE) and sports teaching. Pierre Parlebas, from the French side of the English Channel, claimed in 1967 that sports make part of PE, that team sports must be considered from a specific, sociomotor point of view, and that motor conducts (i.e. the significative organization of motor behaviour), not sports techniques, are the corner-stone of PE and sports coaching. In the early 1980s, from the English side of La Manche, Almond, Thorpe, and Bunker made a plea for a shift in the way to teach games (sporting collective duels mostly), deeply concerned by the negative impact of the traditional technics-centred approach on motivation, competence and attained level of the least able in school situations. Our conclusion is that TGfU, or game-based approaches to sports coaching and teaching, can take great advantage of the motor-praxeological rationale for three reasons: firstly, because concepts like understanding, game sense and action principles are operatively, semiotically linked to the reality of the playing process; secondly, because the inner structures of the games that constrain players and guide their motor conducts, permit to integrate games in the general system of sporting games, no matter their level of institutionalisation; finally, because the intervention practice is better thought of and more systematically developed upon the operational concepts of internal logic and expected practical effects of game playing. This time, Paris could be the place to go to in search of solutions, not the city to run away from in hope of consolation. (shrink)

Computability and Logic has become a classic because of its accessibility to students without a mathematical background and because it covers not simply the staple topics of an intermediate logic course, such as Godel's incompleteness theorems, but also a large number of optional topics, from Turing's theory of computability to Ramsey's theorem. This 2007 fifth edition has been thoroughly revised by John Burgess. Including a selection of exercises, adjusted for this edition, at the end of each chapter, it (...) offers a simpler treatment of the representability of recursive functions, a traditional stumbling block for students on the way to the Godel incompleteness theorems. This updated edition is also accompanied by a website as well as an instructor's manual. (shrink)

Early moderns such as Locke and Descartes thought we could guarantee the justification of our beliefs, even in worlds most hostile to their truth, if only we form those beliefs with sufficient care. That is, they thought it possible for us to be impeccable with respect to justification. This principle has traditionally been used to argue for internalism. By placing all of the normatively relevant conditions in our minds, we ensure reflective access to what those norms require of us and (...) so sustain the possibility of impeccability (unlike externalism). However, recent challenges to transparency leave this reasoning vulnerable. In response, I show how impeccability can be sustained without requiring transparency. The account only works if we define internal states as those directly accessible to our rational belief forming systems. I argue that this sort of causal internalism, while somewhat revisionary, preserves traditional motivations for internalism while avoiding problems faced by other varieties. The result is a renewed argument for internalism that simultaneously moves us away from access internalism and towards a species of mentalism. (shrink)

In §93 of The Principles of Mathematics, Bertrand Russell observes that “the variable is a very complicated logical entity, by no means easy to analyze correctly”. This assessment is borne out by the fact that even now we have no fully satisfactory understanding of the role of variables in a compositional semantics for first-order logic. In standard Tarskian semantics, variables are treated as meaning-bearing entities; moreover, they serve as the basic building blocks of all meanings, which are constructed out (...) of variable assignments. But this has disquieting consequences, including Fine’s antinomy of the variable and an undue dependence of meanings on language. Here I develop an alternative, Fregean version of predicate logic that uses the traditional quantifier–variable apparatus for the expression of generality, possesses a fully compositional, non-representational semantics, and is not subject to the antinomy of the variable. The advantages of Fregean over Tarskian predicate logic are due to the former’s treating variables not as meaningful lexical items, but as mere marks of punctuation, similar to parentheses. I submit that this is indeed how the variables of predicate logic should be construed. (shrink)

This is the second edition of an important introduction to Leibniz's philosophy of logic and language first published in 1972. It takes issue with several traditional interpretations of Leibniz (by Russell amongst others) while revealing how Leibniz's thought is related to issues of great interest in current logical theory. For this new edition, the author has added new chapters on infinitesimals and conditionals as well as taking account of reviews of the first edition.

Abstract. Traditional epistemology of knowledge and belief can be succinctly characterized as JTB-epistemology, i.e., it is characterized by the thesis that knowledge is justified true belief. Since Gettier’s trail-blazing paper of 1963 this account has become under heavy attack. The aim of is paper is to study the Gettier problem and related issues in the framework of topological epistemic logic. It is shown that in the framework of topological epistemic logic Gettier situations necessarily occur for most topological (...) models of knowledge and belief. On the other hand, there exists a special class of topological models (based on so called nodec spaces) for which traditional JTB-epistemology is valid. Further, it is shown that for each topological model of Stalnaker’s combined logic KB of knowledge and belief a canonical JTB-model (its JTB-doppelganger) can be constructed that shares many structural properties with the original model but is free of Gettier situations. The topological model and its JTB-doppelganger both share the same justified belief operator and have very similar knowledge operators. Seen from a somewhat different perspective, the JTB-account of epistemology amounts to a simplification of a more general epistemological account of knowledge and belief that assumes that these two concepts may differ in some cases. The JTB-account of knowledge and belief assumes that the epistemic agent’s cognitive powers are rather large. Thereby in the JTB-epistemology Gettier cases do not occur. Eventually, it is shown that for all topological models of Stalnaker’s KB-logic Gettier situations are topologically characterized as nowhere dense situations. This entails that Gettier situations are epistemologically invisible in the sense that they can neither be known nor believed with justification with respect to the knowledge operator and the belief operator of the models involved. (shrink)

First published in 1943, and revised for this 1952 edition, this book was intended for use by students of philosophy and as such traditional and modern developments in logic have been combined in a unified treatment. The author envisaged this volume as filling a gap for a simple, introductory text on formal logic, written from a modern point of view, unencumbered by traditional doctrine. This title provides a thorough introduction and grounding in the philosophy of (...) class='Hi'>logic, and was later revised after the author’s death to correct a number of logical errors — making this edition the most complete version of the work. (shrink)