In the course of the long debate on the nature and the classification of signs, from Boethius to Ockham, there are at least three lines of thought: the Stoic heritage, that influences Augustine, Abelard, Francis Bacon; the Aristotelian tradition, stemming from the commentaries on De Interpretatione; the discussion of the grammarians, from Priscian to the Modistae. Modern interpreters are frequently misled by the fact that the various authors regularly used the same terms. Such a homogeneous terminology, however, covers profound theoretical (...) differences. The aim of these essays is to show that the medieval theory of signs does not represent a unique body of semiotic notions: there are diverse and frequently alternative semiotic theories. This book thus represents an attempt to encourage further research on the still unrecognized variety of the semiotic approaches offered by the medieval philosophies of language. (shrink)

A comprehensive examination of the interfaces of logic, computer science, and game theory, drawing on twenty years of research on logic and games.

J'examine plusieurs sources selon lesquelles Swyneshed (malgré les prétentions d'Angel D'Ors dans ses articles récents) donne une nova responsio en partie sous forme de la règle « On peut nier une proposition conjonctive après avoir concédé ses deux parties. » Je montre que cette nova responsio est liée à un rejet de la règle « Chaque proposition qui suit de l'ensemble de propositions déjà concédées doit être concédée », et j'attribue ce rejet à une théorie selon laquelle une inférence se (...) base sur le rapport logique entre seulement deux propositions. I examine a number of sources according to which Swyneshed (despite the claims made by Angel D'Ors in his recent articles) does give a nova responsio partly in the form of the rule « One can deny a conjunction whose conjuncts have already been granted. » I show that this nova responsio is linked to a rejection of the rule « Every proposition following from a set of propositions which have already been granted must be granted », and I attribute this rejection to a theory whereby an inference is based on the logical relations between just two propositions. (shrink)

In this article, we discuss the notion of merely confused supposition as it arose in the medieval theory of suppositio personalis. The context of our analysis is our formalization of William of Ockham's theory of supposition sketched in Mind 86 (1977), 109-13. The present paper is, however, self-contained, although we assume a basic acquaintance with supposition theory. The detailed aims of the paper are: to look at the tasks that supposition theory took on itself and to use our formalization to (...) relate them to more modern ideas; to explain the notion of merely confused supposition and to defend it against certain criticisms; and to discuss two issues closely related to the idea of merely confused supposition which we could not broach in a shorter article: the mode of supposition of terms in intensional contexts, and the possible existence of a fourth mode, often called suppositio copulatim. (shrink)

THE PREVALENCE TODAY of "semiotics" as the preferred linguistic form for designating the study of signs in its various aspects already conceals a history, a story of the ways in which, layer by layer, the temporal achievement we call human understanding builds, through public discourse, ever new levels of common acceptance each of which presents itself as, if not self-evident, at least the common wisdom. Overcoming such present-mindedness is not the least of the tasks faced by the awakening of semiotic (...) consciousness. (shrink)

In this paper, I distinguish different kinds of pluralism about logical consequence. In particular, I distinguish the pluralism about logic arising from Carnap’s Principle of Tolerance from a pluralism which maintains that there are different, equally “good” logical consequence relations on the one language. I will argue that this second form of pluralism does more justice to the contemporary state of logical theory and practice than does Carnap’s more moderate pluralism.

It is a commonplace that in the development of modern logic towards its actual shape at least two directions or traditions have to be distinguished. These traditions may be called, following the mo...

This paper defends the thesis that the mediaeval genre of logical treatises De obligatiombus contained a theoretical account of counterfacutal reasoning, perhaps the first such account in the history of philosophy. This interpretation helps to explain some of the theoretical disputes in the obligationes literature in the first half of the fourteenth century. Section 1 is introductory. Section 2 presents Walter Burley's theory, while section 3 argues for the counterfactual interpretation of obligationes and section 4 discusses difficulties with Burley's theory. (...) Section 5 presents the textual basis for Richard Kilvington's theory, and section 6 outlines that theory. Section 7 discusses Roger Swyneshed's theory. Section 8 contains a summary and conclusion. (shrink)

The history of the mediaeval obligationes-literature has only recently begun to be studied. Two important treatises in this literature, one by Walter Burley and the other attributed to William of Sherwood, have been edited by Romuald Green in a forthcoming book. But there is considerable doubt concerning the authenticity of the text attributed to Sherwood. The correct attribution and dating of this treatise is crucial for our understanding of the history of this literature. In this paper, we argue that the (...) treatise is not as early as William of Sherwood, that on the contrary it dates no earlier than the very end of the thirteenth century, and that, although there is some contrary evidence, the treatise may be an early work by Walter Burley himself. (shrink)

We provide for the first time an exact translation into English of the Polish version of Alfred Tarski's classic 1936 paper, whose title we translate as ?On the Concept of Following Logically?. We also provide in footnotes an exact translation of all respects in which the German version, used as the basis of the previously published and rather inexact English translation, differs from the Polish. Although the two versions are basically identical, to an extent that is even uncanny, we note (...) more than 400 differences. Several dozen of these are substantive differences due to revisions by Tarski to the Polish version which he did not incorporate in the German version. With respect to these revisions the Polish version should be regarded as more authoritative than the German. Hence scholars limited to an English translation should use ours. (shrink)

I. SUPPOSITION AND SIGNIFICATIONIn an insightful article on the medieval theory of supposition, Elizabeth Karger noted a remarkable development in the characterization of the material mode of supposition between William of Ockham and his contemporaries in the early fourteenth century and Paul of Venice and others at the turn of the fifteenth century.1. E. Karger, “La Supposition Materielle comme Supposition Significative: Paul de Venise, Paul de Pergula,” in A. Maierú, ed., English Logic in Italy in the 14th and 15th Centuries (...) (Naples, 1982), pp. 331–41. For William, material supposition is explicitly ‘non-significative’, while for Paul, every mode of supposition is a kind of signification.2. Paul of Venice, Logica Magna (Tractatus de Suppositionibus), ed. and trans. A. Perreiah (St. Bonaventure, 1971), p. 4: “nam omnis suppositio est significatio.” In particular, we can set these definitions in opposition. (shrink)

This chapter contains sections titled: I II III IV V VI VII VIII IX X.

An implication is a proposition, a consequence is a relation between propositions, and an inference is act of passage from certain premise-judgements to another conclusion-judgement: a proposition is true, a consequence holds, whereas an inference is valid. The paper examines interrelations, differences, refinements and linguistic renderings of these notions, as well as their history. The truth of propositions, respectively the holding of consequences, are treated constructively in terms of verification-objects. The validity of an inference is elucidated in terms of the (...) existence of a chain of immediately evident steps linking premises and conclusion. (shrink)

REFERENCES Barwise, J. & R. Cooper (1981) — 'Generalized Quantifiers and Natural Language', Linguistics and Philosophy 4:2159-219. Van Benthem, J. (1983a) — ' Five Easy Pieces', in Ter Meulen (ed.), 1-17. Van Benthem, J. (1983b) ...

Available for the first time in 20 years, here is the Rudolf Carnap's famous principle of tolerance by which everyone is free to mix and match the rules of ...

Stephen Cole Kleene was one of the greatest logicians of the twentieth century and this book is the influential textbook he wrote to teach the subject to the next generation. It was first published in 1952, some twenty years after the publication of Godel's paper on the incompleteness of arithmetic, which marked, if not the beginning of modern logic. The 1930s was a time of creativity and ferment in the subject, when the notion of computable moved from the realm of (...) philosophical speculation to the realm of science. This was accomplished by the work of Kurt Gode1, Alan Turing, and Alonzo Church, who gave three apparently different precise definitions of computable. When they all turned out to be equivalent, there was a collective realization that this was indeed the right notion. Kleene played a key role in this process. One could say that he was there at the beginning of modern logic. He showed the equivalence of lambda calculus with Turing machines and with Godel's recursion equations, and developed the modern machinery of partial recursive functions. This textbook played an invaluable part in educating the logicians of the present. It played an important role in their own logical education.". (shrink)

v. 1. On the twelfth century theories of fallacy.--v. 2. The origin and early development of the theory of supposition.

Several of these essays have been printed whole in journals; others are in varying degrees new. Two main themes run through them. One is the problem of meaning, particularly as involved in the notion of an analytic statement. The other is the notion of ontological, commitment, particularly as involved in the problem of universals.

EDITOR'S INTRODUCTION Throughout his life Bolzano's interest was divided between ethics and mathematics, between his will to reform the religion of the ...

The central contention of this book is that second-order logic has a central role to play in laying the foundations of mathematics. In order to develop the argument fully, the author presents a detailed description of higher-order logic, including a comprehensive discussion of its semantics. He goes on to demonstrate the prevalence of second-order concepts in mathematics and the extent to which mathematical ideas can be formulated in higher-order logic. He also shows how first-order languages are often insufficient to codify (...) many concepts in contemporary mathematics, and thus that both first- and higher-order logics are needed to fully reflect current work. Throughout, the emphasis is on discussing the associated philosophical and historical issues and the implications they have for foundational studies. For the most part, the author assumes little more than a familiarity with logic comparable to that provided in a beginning graduate course which includes the incompleteness of arithmetic and the Lowenheim-Skolem theorems. All those concerned with the foundations of mathematics will find this a thought-provoking discussion of some of the central issues in the field today. (shrink)

I propose to examine the issue of whether the ancient tradition in logic continued to be developed in the later medieval period from the vantage point of the relations between two specific groups of theories, namely the medieval theories of supposition and the (originally) ancient theories of fallacies. More specifically, I examine whether supposition theories absorbed and replaced theories of fallacies, or whether the latter continued to exist, with respect to one particular author, William of Ockham. I compare different parts (...) of Ockham's Summa Logicae, namely III-4 (on fallacies), and the final chapters of part I and first chapters of part II (on supposition). I conclude that there is overlap of conceptual apparatus and of goals (concerning propositions that must be distinguished) in Ockham's theories of supposition and of fallacies, but that the respective conceptual apparatuses also present substantial dissimilarities. Hence, theories of supposition are better seen as an addition to the general logical framework that medieval authors had inherited from ancient times, rather than the replacement of an ancient tradition by a medieval one. Indeed, supposition theories and fallacy theories had different tasks to fulfil, and in this sense both had their place in fourteenth century logic. (shrink)

The “dragon” that graces the cover of this volume has a story that goes with it. In the summer of 1980, I was on the teaching staff of the Summer Institute on Medieval Philosophy held at Cornell University under the direction of Norman Kretzmann and the auspices of the Council for Philosophical Studies and the National Endowment for the Humanities. While I was giving a series of lectures there (lectures that contribute to this volume, as it turns out), I went (...) to my office one morning, and there under the door some anonymous wag from the Institute had slid the pen and ink drawing you see in the picture. It represents “Supposition” as a dragon, making a rude face at the viewer. The tail of the dragon is divided — not entirely accurately, as it turns out — into the various branches and subbranches of supposition. If the details are not altogether correct, the spirit is certainly understandable. (shrink)

What I want to talk about here is a puzzle for historians of philosophy who, like me, have spent a fair amount of time studying the history of mediaeval logic and semantic theory. I don’t know how to solve it, but in various forms it has come up repeatedly in my own work and in the work of colleagues I have talked with about it. I would like to share it with you now.

Much philosophy of logic is shaped, explicitly or implicitly, by the thought that logic is distinctively formal and abstracts from material content. The distinction between formal and material does not appear to coincide with the more familiar contrasts between a priori and empirical, necessary and contingent, analytic and synthetic—indeed, it is often invoked to explain these. Nor, it turns out, can it be explained by appeal to schematic inference patterns, syntactic rules, or grammar. What does it mean, then, to say (...) that logic is distinctively formal? (shrink)

After a brief flirtation with logicism around 1917, David Hilbertproposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays andWilhelm Ackermann, throughout the 1920s. The two technical pillars of the project were the development of axiomatic systems for everstronger and more comprehensive areas of mathematics, and finitisticproofs of consistency of these systems. Early advances in these areaswere made by Hilbert (and Bernays) in a series of lecture courses atthe (...) University of Göttingen between 1917 and 1923, and notably in Ackermann's dissertation of 1924. The main innovation was theinvention of the -calculus, on which Hilbert's axiom systemswere based, and the development of the -substitution methodas a basis for consistency proofs. The paper traces the developmentof the ``simultaneous development of logic and mathematics'' throughthe -notation and provides an analysis of Ackermann'sconsistency proofs for primitive recursive arithmetic and for thefirst comprehensive mathematical system, the latter using thesubstitution method. It is striking that these proofs use transfiniteinduction not dissimilar to that used in Gentzen's later consistencyproof as well as non-primitive recursive definitions, and that thesemethods were accepted as finitistic at the time. (shrink)

This paper examines the question of the extensional correctness of Tarskian definitions of logical truth and logical consequence. I identify a few different informal properties which are necessary for a sentence to be an informal logical truth and look at whether they are necessary properties of Tarskian logical truths. I examine arguments by John Etchemendy and Vann McGee to the effect that some of those properties are not necessary properties of some Tarskian logical truths, and find them unconvincing. I stress (...) the point that since the hypothesis that Tarski's definitions are extensionally correct is deeply entrenched, the burden of proof is still on the shoulders of Tarski's critics, who have not lifted the burden. (shrink)

Quote marks, I claim, serve to select from the multiple ostensions that are produced whenever any expression is uttered; they act to constrain pragmatic ambiguity or indeterminacy. My argument proceeds by showing that the proffered account fares better than its rivals-the Name, Description, Demonstrative, and Identity Theories. Along the way I shall need to explain and emphasize that quoting is not simply the same thing as mentioning. Quoting, but not mentioning, relies on the use of conventional devices.