Contemporary accounts of logic and language cannot give proper treatments of plural constructions of natural languages. They assume that plural constructions are redundant devices used to abbreviate singular constructions. This paper and its sequel, "The logic and meaning of plurals, II", aim to develop an account of logic and language that acknowledges limitations of singular constructions and recognizes plural constructions as their peers. To do so, the papers present natural accounts of the logic and meaning of plural constructions that result (...) from the view that plural constructions are, by and large, devices for talking about many things (as such). The account of logic presented in the papers surpasses contemporary Fregean accounts in its scope. This extension of the scope of logic results from extending the range of languages that logic can directly relate to. Underlying the view of language that makes room for this is a perspective on reality that locates in the world what plural constructions can relate to. The papers suggest that reflections on plural constructions point to a broader framework for understanding logic, language, and reality that can replace the contemporary Fregean framework as this has replaced its Aristotelian ancestor. (shrink)

A theory of syllogistic reasoning is proposed, derived from the medieval doctrine of 'distribution of terms'. This doctrine may or may not furnish an adequate ground for the logic of the syllogism but does appear to illuminate the psychological processes involved. Syllogistic thinking is shown to have its origins in the approach and avoidance behaviour of pre-verbal organisms and, in verbal (human) organisms, to bridge the gap between the intuitive grasp shown by most of us of the validity of simple (...) logical arguments and the failure of intuition in more complex arguments that require resort to calculation. Some difficulties are considered affecting the use of syllogisms as experimental material. These include failure on the part of the investigator to take account of the fact that a syllogism is always part of a continuing argument in which the topic of the argument is known to all parties and the possibility that subjects may find ways of appearing to solve syllogisms without actually doing so. (shrink)

The essence of the argument of the beard (so-called by some logic textbooks) is the tactic used by a respondent to reply to a proponent, "The criterion you used to define a key term in your argument is vague, therefore your use of this term in your argument is illegitimate, and your argument is refuted." This familiar kind of argument tactic is similar to the much more famous heap (sorites) argument of Eubulides, closely associated with the slippery slope argument. This (...) article provides a system of classification for sorting out these three arguments, and related types of argument of interest in informal logic. (shrink)

The aim of this investigation is to explore the role of argumentation schemes in enthymeme reconstruction. This aim is pursued by studying selected cases of incomplete arguments in natural language discourse to see what the requirements are for filling in the unstated premises and conclusions in some systematic and useful way. Some of these cases are best handled using deductive tools, while others respond best to an analysis based on defeasible argumentations schemes. The approach is also shown to work reasonably (...) well for weak arguments, a class of arguments that has always been difficult to analyze without the principle of charity producing a straw man. (shrink)

Defeasible reasoning is concerned with the logics of non-deductive argument. As is described in the literature, the study of this type of reasoning is considerably more involved than the study of deductive argument, even so that, in realistic applications, there is often a lack of resources to perform an exhaustive analysis. It follows that, in a theory of defeasible reasoning, the order and direction in which arguments are developed, i.e. theprocedure, is important. The aim of this article is to show (...) that debate is the most efficient procedure to argue in the presence of limited resources. To do so, there is first some general theory on defeasible argumentation, which is followed by an introduction to the problem of dialectical search. The problem of dialectical search is (or at least, should be) the essential issue in every theory on argumentation, and emerges at every occasion that involves adjudication on competing arguments. Starting with an example, it is explained that dialectical search can be best scheduled according to classical debating techniques, that work along well-tried methods. These methods (which include various forms of curtailment, interruption, and interpretation) have proven their value in keeping debating efforts within reasonable bounds. How they apply in a theory of formal argument, will be shown in this article. (shrink)

Jarmo Pulkkinen, The threat of logical mathematism. A study on the critique of mathematical logic in Germany at the turn of the 20th century. Frankfurt a.M:Peter Lang, 1994. Scandinavian University Studies in the Humanities and Social Sciences; 7). 186 pp. 24 DM. ISBN 3-631-47409-1.

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

We identify a class of paradoxes that is neither set-theoretical nor semantical, but that seems to depend on intensionality. In particular, these paradoxes arise out of plausible properties of propositional attitudes and their objects. We try to explain why logicians have neglected these paradoxes, and to show that, like the Russell Paradox and the direct discourse Liar Paradox, these intensional paradoxes are recalcitrant and challenge logical analysis. Indeed, when we take these paradoxes seriously, we may need to rethink the commonly (...) accepted methods for dealing with the logical paradoxes. (shrink)

In , Frank Ramsey separates paradoxes into two groups, now taken to be the logical and the semantical. But he also revises the logical system developed in Whitehead and Russellthe intensional paradoxess interest in these problems seriously, then the intensional paradoxes deserve more widespread attention than they have historically received.

Friedrich Albert Lange (1828-1875) author of a famous History of Materialism and Critique of Its Present Significance (1866, English transI. 1877-79, repr. 1925 with introduction by Bertrand Russell), was also interested in the epistemological foundations of formal logic. Part I of his intended two-volume Logische Studien was published posthumously in 1877 by Hermann Cohen, head of the Marburg school of neo-Kantianism. Lange, departing from Kant, claims that spatial intuition is the source of the apodeictic character not only of the truths (...) of mathematics, but also of the truths of logic. He aims at showing this by basing validity and invalidity of syllogistic inferences on an interpretation of the standard forms (of proposition in assertoric syllogistic) with the help of the five kinds of possible relations (in fact what is known today as the Gergonne-Euler relations) between extensions of concepts given to us as areas in a plane, i.e.in space. Generality is achieved by considering all possible variations within each type of spatial relation, exhibiting a connection between concept and intuition reminding Lange of the Kantian "schema". Lange is well aware of the contemporary English "algebraic" logic, but he considers its approach as the appropriate one for a logic of content (Inhaltslogik) and not for a logic of extension (Umfangslogik). Lange did not live to enjoy the recognition by some leading logicians (amongst them John Venn, to whose reference in 1881 to Lange's "admirable Logische Studien" the present paper owes it title), nor could he respond to the many critics of his proposed foundation of logic. Its radicality as well as its broad reception (and discussion up to at least 1959) seem to entitle Lange's Logische Studien to an, if modest, place in the history of logic in the 19th century. (shrink)

The introduction has a brief statement, sufficient for the purpose of this paper, which describes in general terms the notion of probability logic on which the paper is based. Contributions made in the eighteenth century by Leibniz, Jacob Bernoulli and Lambert, and in the nineteenth century by Bolzano, De Morgan, Boole, Peirce and MacColl are critically examined from a contemporary point of view. Historicity is maintained by liberal quotations from the original sources accompanied by interpretive explanation. Concluding the paper is (...) a summary. (shrink)

The approach used by Boole in Mathematical analysis of logic to develop propositional logic was based on the idea of ?cases? or ?conjunctures of circumstances?. But this was dropped in Laws of thought in favor of one which Boole considered to be more satisfactory, that of using the notion of ?time for which a proposition is true?. We show that, when suitable clarifications and corrections are made, the earlier approach? which accords with modern logic in eschewing the extraneous notion of (...) time?is indeed a viable one. (shrink)

Peter Geach describes the 'doctrine of distribution' as the view that a term is distributed if it refers to everything that it denotes, and undistributed if it refers to only some of the things that it denotes. He argues that the notion, so explained, is incoherent. He claims that the doctrine of distribution originates from a degenerate use of the notion of ?distributive supposition? in medieval supposition theory sometime in the 16th century. This paper proposes instead that the doctrine of (...) distribution occurs at least as early as the 12th century, and that it originates from a study of Aristotle's notion of a term's being ?taken universally?, and not from the much later theory of distributive supposition. A detailed version of the doctrine found in the Port Royal Logic is articulated, and compared with a slightly different modern version. Finally, Geach's arguments for the incoherence of the doctrine are discussed and rejected. (shrink)

Inference versus consequence , an invited lecture at the LOGICA 1997 conference at Castle Liblice, was part of a series of articles for which I did research during a Stockholm sabbatical in the autumn of 1995. The article seems to have been fairly effective in getting its point across and addresses a topic highly germane to the Uppsala workshop. Owing to its appearance in the LOGICA Yearbook 1997 , Filosofia Publishers, Prague, 1998, it has been rather inaccessible. Accordingly it is (...) republished here with only bibliographical changes and an afterword. (shrink)

The wider topic to which the content of this paper belongs is that of the relationship between formal logic and real argumentation. Of particular potential interest in this connection are held to be substantive arguments constructed by philosophers reputed equally as authorities in logical theory. A number of characteristics are tentatively indicated by the author as likely to be encountered in such arguments. The discussion centers afterwards, by way of specification, on a remarkable piece of argument quoted in Cicero’s dialog (...) On Divination and ascribed to Stoic thinkers. The Stoics’ formal theory of inference is summarily referred to in this context, with special emphasis on their basic deductive schemata (‘indemonstrables’), some of them recognizable as links in the overall structure of the quoted argument. The main lines of Cicero’s criticism of the Stoic argument are next commented upon, with emphasis on his implied view as to the requirements of a good argument. Towards the end of the paper, a few considerations are added on the changes in the prevailing style of argumentation conspicuous in the three famous Roman Stoics. (shrink)

We discuss external and internal graphical and linguistic representational systems. We argue that a cognitive theory of peoples' reasoning performance must account for (a) the logical equivalence of inferences expressed in graphical and linguistic form, and (b) the implementational differences that affect facility of inference. Our theory proposes that graphical representation limit abstraction and thereby aid “processibility”. We discuss the ideas of specificity and abstraction, and their cognitive relevance. Empirical support both comes from tasks which involve the manipulation of external (...) graphics and tasks that do not. For the former, we take Euler's (1772) circles, provide a novel computational reconstruction, show how it captures abstractions, and contrast it with earlier construals and with Johnson‐Laird's (1983) mental models representations. We demonstrate equivalence of the graphical Euler system, and the nongraphical mental models system. For tasks not involving manipulation of external graphics, we discuss text comprehension, and the mental performance of syllogisms. By positing an internal system with the same specificity as Euler's circles, we cover the mental models data, and generate new empirical predictions. Finally, we consider how the architecture of working memory explains why such specific representations are relatively easy to store. (shrink)

Whenever an adequate theory is found in science, we will still be left with two questions: why this theory rather than some other theory, and how should this theory be interpreted? I argue that these questions can be answered by a theory of system relations. The basic idea is that fundamental characteristics of systems, viz. those arising from the general systemic nature of those systems, cannot be comprehended with the aid of discipline-specific methods. The systems theory required should commence with (...) an analysis of the qualitatively different relations possible between systems, because it is precisely the nature of those relations that determines the basic structures of systems. That the theory of the fundamental system relations and their ontological and epistemological implications is indeed able to provide the answers sought is demonstrated in theoretical physics and Plessner's analysis of the basic structures of plant, animal and human being. (shrink)

The consensus among legal philosophers is probably that rule-based legal expert systems leave much to be desired as aids in legal decision-making. Why? What can we do about it? A bureaucrat administering some set of complex rules will ascertain the facts and apply the rules to them in order to discover their consequences for the case in hand. This process of deductive reasoning is characteristically bureaucratic.

Reply to Beaney: the closing of the historical mindIn his comments, Michael Beaney sets himself up as the arbiter of what is genuine history and what isn’t. While celebrating the outpouring of specialized scholarship on Frege, he has no patience with the enterprise outlined in the Précis, which attempts to construct a large-scale picture of the richness of the analytic tradition. That enterprise is one in which great figures of our recent past are challenged by aspects of contemporary thought, and (...) our current struggles are enriched by insights of theirs that haven’t been fully absorbed. Although some work of this sort can be done piecemeal, one historical figure at a time, there is much to be learned from a more encompassing perspective. While no picture constructed by a single author can be authoritative about everything, the attempt to give informative, connected assessments of major milestones in the tradition is our best hope of understanding who we are and where we have come from. (shrink)

There is a little known paradox the solution to which is a guide to a much more thoroughgoing solution to a whole range of classic paradoxes. This is shown in this paper with respect to Berry's Paradox, Heterologicality, Russell's Paradox, and the Paradox of Predication, also the Liar and the Strengthened Liar, using primarily the epsilon calculus. The solutions, however, show not only that the first-order predicate calculus derived from Frege is inadequate as a basis for a clear science, and (...) should be replaced with Hilbert and Bernays' conservative extension. Standard second-order logic, and quantified propositional logic also must be substantially modified, to incorporate, in the first place, nominalizations of predicates, and whole sentences. And further modifications must be made, so as to insist that predicates are parts of sentences rather than forms of them, and that truth is a property of propositions rather than their sentential expressions. In all, a thorough reworking of what has been called 'logic' in recent years must be undertaken, to make it more fit for use. (shrink)

Extensionalism, as I understand it here, is the view that physical reality consists exclusively of extensional entities. On this view, intensional entitities must either be eliminated in favor of an ontology of extensional entities, or be reduced to such an ontology, or otherwise be admitted as non-physical. In this paper I argue that extensionalism is a misguided philosophical doctrine. First, I argue that intensional phenomena are not confined to the realm of language and thought. Rather, the ontology of such phenomena (...) is intimately entwined with the ontology of properties. After providing some evidence to the popularity of extensionalism in contemporary analytic philosophy, I investigate the motivating reasons behind it. Considering several explanations, I argue that the main motivating reason is rooted in the identification of matter with extension, an identification which is one of the hallmarks of the mechanistic conception of nature inherited from the founding fathers of our modern scientific outlook. I then argue that such a conception is not only at odds with a robust ontology of properties but is also at odds with our best contemporary physics. Rather than vindicating extensionalism contemporary science undermines the position, and the lesson to be drawn from this surprising fact is that extensionalism needs no longer be espoused as a regulative ideal of naturalistic philosophy. I conclude by showing that the ontological approach to intensional phenomena advocated throughout the paper also gains support from an examination of the historical context within which ‘intension’ was first introduced as a semantic notion. (shrink)

La hipótesis que intentaremos defender en este trabajo es que en Sobre la filosofía Aristóteles hace una defensa dialéctica de su concepción de los principios, tomando como punto de partida algunas de las opiniones existentes al respecto. En ese sentido, el modo de proceder aristotélico en esta obra es consistente con el implementado con idénticos fines en las obras canónicas y contribuye a comprender el uso epistémico de la dialéctica en este pensador. Para demostrar esto, nos centraremos en el análisis (...) del fragmento 17. A tales efectos, dividiremos el trabajo en dos partes: en la primera, analizaremos el uso de las opiniones citadas en relación con el tema a discutir; en la segunda, examinaremos los recursos metodológicos por medio de los cuales Aristóteles da cuenta de sus propias concepciones. (shrink)

What all contemporary so-called aristotelian realists have in common has been identified by David Armstrong as the principle of instantiation. This principle has been put forward in different versions, but all of them have the following simple consequence in common: uninstantiated universals do not exist. Such entities are for the lotus-eating Platonist to countenance, but not for any sort of moderate realist. I shall argue that this principle, in any guise, is not the best way to differentiate aristotelianism from Platonism. (...) In its place, I shall suggest that the best way to differentiate the two versions of realism from each other is by means of a far more powerful idea: naturalism. And the surprising conclusion given this means of differentiation will be that contrary to the usual proclamations, Platonism will be the more naturalistic theory, whereas aristotelianism will come to be seen for what it really is, namely, non-naturalistic. (shrink)

How mathematicians conceive of the nature of mathematics is reflected in the metaphors they use to talk about it. In this paper I investigate a change in the use of metaphors in the late nineteenth and early twentieth centuries. In particular, I argue that the metaphor of mathematics as a tree was used systematically by Pasch and some of his contemporaries, while that of mathematics as a building was deliberately chosen by Hilbert to reflect a different view of mathematics. By (...) taking these metaphors seriously we attain a new vantage point for understanding historical changes in conceptions of mathematics. (shrink)

This paper explores the question of what makes diagrammatic representations effective for human logical reasoning, focusing on how Euler diagrams support syllogistic reasoning. It is widely held that diagrammatic representations aid intuitive understanding of logical reasoning. In the psychological literature, however, it is still controversial whether and how Euler diagrams can aid untrained people to successfully conduct logical reasoning such as set-theoretic and syllogistic reasoning. To challenge the negative view, we build on the findings of modern diagrammatic logic and introduce (...) an Euler-style diagrammatic representation system that is designed to avoid problems inherent to a traditional version of Euler diagrams. It is hypothesized that Euler diagrams are effective not only in interpreting sentential premises but also in reasoning about semantic structures implicit in given sentences. To test the hypothesis, we compared Euler diagrams with other types of diagrams having different syntactic or semantic properties. Experiment compared the difference in performance between syllogistic reasoning with Euler diagrams and Venn diagrams. Additional analysis examined the case of a linear variant of Euler diagrams, in which set-relationships are represented by one-dimensional lines. The experimental results provide evidence supporting our hypothesis. It is argued that the efficacy of diagrams in supporting syllogistic reasoning crucially depends on the way they represent the relational information contained in categorical sentences. (shrink)

Modern logicians have sought to unlock the modal secrets of Aristotle's Syllogistic by assuming a version of essentialism and treating it as a primitive within the semantics. These attempts ultimately distort Aristotle's ontology. None of these approaches make full use of tests found throughout Aristotle's corpus and ancient Greek philosophy. I base a system on Aristotle's tests for things that can never combine (polarity) and things that can never separate (inseparability). The resulting system not only reproduces Aristotle's recorded results for (...) the apodictic syllogistic in the Prior Analytics but it also generates rather than assumes Aristotle's distinctions among 'necessary', 'essential' and 'accidental'. By developing a system around tests that are in Aristotle and basic to ancient Greek philosophy, the system is linked to a history of practices, providing a platform for future work on the origins of logic. (shrink)

The work of Hugh MacColl (1837–1909) suffered the same fate after his death as before it:despite being vaguely alluded to and in part even commended, on the whole it has remained an unknown quantity. Even worse, those of his ideas which have played a decisive role in the history of logic have been credited to his successors; this is especially the case with the definition of strict implication and the first formal development of formal modal logic. This paper takes an (...) initial step towards a rectification of this unfortunate misrepresentation, presenting a bibliography of MacColl’s most significant publications with particular regard to their reception. (shrink)