This paper shows that Aristotle's De Interpretatione does not separate syntax from semantics. Linguistic sentences are not syntactic entities, and non-linguistic meanings are not semantic propositions expressed by linguistic sentences. In fact, Aristotle resorts to a mental conception of meaning, distinguishing linguistic meanings in a given language from non-linguistic mental contents in relation to actual things: while the former are not the same for all, the latter are shared by everyone. Aristotle is not a modern logician, like Boole, Frege, or (...) Russell, in so far as a mental conception of meaning does not reveal an abstract semantics for a syntactic language. (shrink)

Aristotelian causal theories incorporate some philosophically important features of the concept of cause, including necessity and essential character. The proposed formalization is restricted to one-place predicates and a finite domain of attributes (without individuals). Semantics is based on a labeled tree structure, with truth defined by means of tree paths. A relatively simple causal prefixing mechanism is defined, by means of which causes of propositions and reasoning with causes are made explicit. The distinction of causal and factual explanation are elaborated, (...) and examples of cyclic and convergent causation are given. Soundness and completeness proofs are sketched. (shrink)

Philosophers have had an interest in language from the earliest times but the twentieth century, with its so-called 'linguistic turn' in philosophy, has seen a huge expansion of work focused specifically on language and its foundations. No branch of philosophy has been unaffected by this shift of emphasis. It is timely at the end of the century to review and assess the vast range of issues that have been developed and debated in this central area. The distinguished international contributors present (...) a clear, accessible guide to the fundamental questions raised by the philosophers about language. Contributions include Graeme Forbes on necessity, Susan Haack on deviant logics, Paul Horwich on truth, Charles Travis on Wittgenstein, L.J. Cohen on linguistic philosophy, Ruth Kempson on semantics and syntax and Christopher Hookway on ontology, to name but a few. A wide range of topics are covered from the metaphysics and ontology of language, language and mind, truth and meaning, to theories or reference, speech act theory, philosophy of logic and formal semantics. There are also articles on key figures from the twentieth century and earlier. Based on the foundation provided by the award-winning Encyclopedia of Language and Linguistics this single volume provides a collection of articles that will be an invaluable reference tool for all those interested in the area of philosophy of language, and also to those in cognitive science and psychology. All the articles have been thoroughly revised and updated. This volume gives a unique survey of topics that are at the very core of contemporary philosophy. (shrink)

In his 1910 book On the principle of contradiction in Aristotle, Jan Łukasiewicz claims that syllogistic is independent of the principle of contradiction . He also argues that Aristotle would have defended such a thesis in the Posterior Analytics. In this paper, we first show that Łukasiewicz's arguments for these two claims have to be rejected. Then, we show that the thesis of the independence of assertoric syllogistic vis-à-vis PC is nevertheless true. For that purpose, we first establish that there (...) is no possibility to account for Aristotle's original views on syllogistic without using per impossibile deductions. At last, following an idea due to Smith 1983, we prove that there is a complete reconstruction of assertoric syllogistic using ecthesis instead of reductio ad absurdum. Contrary to what is claimed in Smith 1983, we show that Smith's system SE has to be improved for such a reconstruction. This results in an ecthetic system which is complete with respect to syllogistic arguments, not explosi.. (shrink)

This presentation of Aristotle's natural deduction system supplements earlier presentations and gives more historical evidence. Some fine-tunings resulted from conversations with Timothy Smiley, Charles Kahn, Josiah Gould, John Kearns,John Glanvillle, and William Parry.The criticism of Aristotle's theory of propositions found at the end of this 1974 presentation was retracted in Corcoran's 2009 HPL article "Aristotle's demonstrative logic".

Much of the last fifty years of scholarship on Aristotle’s syllogistic suggests a conceptual framework under which the syllogistic is a logic, a system of inferential reasoning, only if it is not a theory or formal ontology, a system concerned with general features of the world. In this paper, I will argue that this a misleading interpretative framework. The syllogistic is something sui generis: by our lights, it is neither clearly a logic, nor clearly a theory, but rather exhibits certain (...) characteristic marks of logics and certain characteristic marks of theories. In what follows, I will present a debate between a theoretical and a logical interpretation of the syllogistic. The debate centers on the interpretation of syllogisms as either implications or inferences. But the significance of this question has been taken to concern the nature and subject-matter of the syllogistic, and how it ought to be represented by modern techniques. For one might think that, if syllogisms are implications, propositions with conditional form, then the syllogistic, in so far as it is a systematic taxonomy of syllogisms, is a theory or a body of knowledge concerned with general features of the world. Furthermore, if the syllogistic is a theory, then it ought to be represented by an axiomatic system, a system deriving propositional theorems from axioms. On the other hand, if syllogisms are inferences, then the syllogistic is a logic, a system of inferential reasoning. And furthermore, it ought to be represented as a natural deduction system, a system deriving valid arguments by means of intuitively valid inferences. I will argue that one can disentangle these questions—are syllogisms inferences or implications, is the syllogistic a logic or a theory, is the syllogistic a body of worldly knowledge or a system of inferential reasoning, and ought we to represent the syllogistic as a natural deduction system or an axiomatic system—and that we must if we are to have a historically accurate understanding of Aristotle. (shrink)

Natural deduction is the type of logic most familiar to current philosophers, and indeed is all that many modern philosophers know about logic. Yet natural deduction is a fairly recent innovation in logic, dating from Gentzen and Jaśkowski in 1934. This article traces the development of natural deduction from the view that these founders embraced to the widespread acceptance of the method in the 1960s. I focus especially on the different choices made by writers of elementary textbooks—the standard conduits of (...) the method to a generation of philosophers—with an eye to determining what the ‘essential characteristics’ of natural deduction are. (shrink)

John Corcoran’s natural deduction system for Aristotle’s syllogistic is reconsidered.Though Corcoran is no doubt right in interpreting Aristotle as viewing syllogisms as arguments and in rejecting Lukasiewicz’s treatment in terms of conditional sentences, it is argued that Corcoran is wrong in thinking that the only alternative is to construe Barbara and Celarent as deduction rules in a natural deduction system.An alternative is presented that is technically more elegant and equally compatible with the texts.The abstract role assigned by tradition and Lukasiewicz (...) to Barbara and Celarent is retained.The two ‘ perfect syllogisms’ serve as ‘basic elements’ in the construction of an inductively defined set of valid syllogisms.The proposal departs from Lukasiewicz, and follows Corcoran, however, in construing the construction as one in natural deduction.The result is a sequent system with fewer rules and in which Barbara and Celarent serve as basic deductions.To compare the theory to Corcoran’s, his original is reformulated in current terms and generalized.It is shown to be equivalent to the proposed sequent system, and several variations are discussed.For all systems mentioned, a method of Henkin‐style completeness proofs is given that is more direct and intuitive than Corcoran’s original. (shrink)

I argue that Prior analyticsII.5?7, 8?10, and 1.45 actually contain studies of processes for transforming arguments into other arguments which Aristotle carried out before having completed the theory of perfecting syllogisms by reduction to first-figure moods as presented in Prior analytics1.4?7. This position rejects Ross's opinion that these passages are ?mental gymnastics?, and Patzig's view that some of these texts contain studies of alternative axiomatizations or other logical studies posterior to the completion of the basic theory of syllogisms.

Since Freges terms were meant to refer always to sets, that is, entities composed of individuals. Classical philosophy up to Leibniz and Kant had a different view on this questionBegriffes syntaxhighercorresponding to the idea which Leibniz used in the construction of his characteristic numbers. Thus, this paper is an addendum to Corcorans theory via predicate logic.

The Aristotelian syllogistic cannot account for the validity of certain inferences involving relational facts. In this paper, we investigate the prospects for providing a relational syllogistic. We identify several fragments based on (a) whether negation is permitted on all nouns, including those in the subject of a sentence; and (b) whether the subject noun phrase may contain a relative clause. The logics we present are extensions of the classical syllogistic, and we pay special attention to the question of whether reductio (...) ad absurdum is needed. Thus our main goal is to derive results on the existence (or nonexistence) of syllogistic proof systems for relational fragments. We also determine the computational complexity of all our fragments. (shrink)

The ancient Greeks were not only the founders of western philosophy, but the actual term "philosophy" is Greek in origin, most likely dating back to the late sixth century BC. Socrates, Plato, Aristotle, Pythagoras, Euclid, and Thales are but a few of the better-known philosophers of ancient Greece. During the amazingly fertile period running from roughly the middle of the first millennium BC to the middle of the first millennium AD, the world saw the rise of science, numerous schools of (...) thought, and—many believe—the birth of modern civilization. The Historical Dictionary of Ancient Greek Philosophy presents the history of Greek philosophy and the philosophers who made it famous. This is accomplished through a chronology, an introduction, a glossary, a bibliography, and hundreds of cross-referenced dictionary entries on important philosophers, concepts, issues, and events. (shrink)

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)

In 1934 a most singular event occurred. Two papers were published on a topic that had (apparently) never before been written about, the authors had never been in contact with one another, and they had (apparently) no common intellectual background that would otherwise account for their mutual interest in this topic.1 These two papers formed the basis for a movement in logic which is by now the most common way of teaching elementary logic by far, and indeed is perhaps all (...) that is known in any detail about logic by a number of philosophers (especially in North America). This manner of proceeding in logic is called ‘natural deduction’. And in its own way the instigation of this style of logical proof is as important to the history of logic as the discovery of resolution by Robinson in 1965, or the discovery of the logistical method by Frege in 1879, or even the discovery of the syllogistic by Aristotle in the fourth century BC. (shrink)

La définition du sullogismos est strictement identique dans les Topiques et dans les Premiers Analytiques, alors qu'on admet généralement que, dans ce dernier traité, le terme désigne spécifiquement la structure formelle appelée aujourd'hui encore « syllogisme » . Le mot peut avoir le même sens d'un bout à l'autre de l'Organon et du corpus aristotélicien : il désigne le moment de la joute dialectique où l'interrogateur récapitule une section de la discussion et se montre en mesure d'imposer une conclusion à (...) laquelle son interlocuteur ne peut pas se dérober. À la suite de Hintikka (1993), on suggère que cette notion est issue des dialogues de Platon. C'est à partir de cette conception dialectique originelle que la théorie aristotélicienne de la rhétorique, d'une part, et la « syllogistique » formelle, de l'autre, ont pu se développer. The definition of a sullogismos in Aristotle's Topics and Prior Analytics is strictly identical, although it is generally admitted that in the latter treatise this word refers to the distinct deductive structure that we call "syllogism". This paper aims at showing that the word sullogismos may be given the same meaning throughout the Organon and throughout the Aristotelian corpus and that it refers to a characteristic move in the course of a dialectical debate, i.e. when the questioner, summing up a whole sequence of the previous discussion, shows that he is in a position to force upon his interlocutor a conclusion which the latter cannot escape. Following a suggestion made by Hintikka (1993), this paper shows that the notion most probably originates from Plato's dialogues, and uses it as a leading thread to show how the Aristotelian concept of rhetoric, on the one hand, and Aristotle's formal "syllogistic", on the other, may have grown out of this original dialectical core. (shrink)

An investigation of Proclus' logic of the syllogistic and of negations in the Elements of Theology, On the Parmenides, and Platonic Theology. It is shown that Proclus employs interpretations over a linear semantic structure with operators for scalar negations (hypemegationlalpha-intensivum and privative negation). A natural deduction system for scalar negations and the classical syllogistic (as reconstructed by Corcoran and Smiley) is shown to be sound and complete for the non-Boolean linear structures. It is explained how Proclus' syllogistic presupposes converting the (...) tree of genera and species from Plato's diairesis into the Neoplatonic linear hierarchy of Being by use of scalar hyper and privative negations. (shrink)

Aristotle was the founder not only of logic but also of modal logic. In the Prior Analytics he developed a complex system of modal syllogistic which, while influential, has been disputed since antiquity--and is today widely regarded as incoherent. Combining analytic rigor with keen sensitivity to historical context, Marko Malink makes clear that the modal syllogistic forms a consistent, integrated system of logic, one that is closely related to other areas of Aristotle's philosophy. Aristotle's modal syllogistic differs significantly from modern (...) modal logic. Malink considers the key to understanding the Aristotelian version to be the notion of predication discussed in the Topics--specifically, its theory of predicables and the ten categories. The predicables introduce a distinction between essential and nonessential predication. In contrast, the categories distinguish between substantial and nonsubstantial predication. Malink builds on these insights in developing a semantics for Aristotle's modal propositions, one that verifies the ancient philosopher's claims of the validity and invalidity of modal inferences. While it acknowledges some limitations of this reconstruction, Aristotle's Modal Syllogistic brims with bold ideas, richly supported by close readings of the Greek texts. (shrink)

This Oxford Handbook covers the current state of the art in the philosophy of maths and logic in a comprehensive and accessible manner, giving the reader an overview of the major problems, positions, and battle lines. The 26 newly-commissioned chapters are by established experts in the field and contain both exposition and criticism as well as substantial development of their own positions. Select major positions are represented by two chapters - one supportive and one critical. The book includes a comprehensive (...) bibliography. (shrink)

This is a 1988 philosophical introduction to Aristotle, and Professor Lear starts where Aristotle himself starts. The first sentence of the Metaphysics states that all human beings by their nature desire to know. But what is it for us to be animated by this desire in this world? What is it for a creature to have a nature; what is our human nature; what must the world be like to be intelligible; and what must we be like to understand it (...) systematically? Through a consideration of these questions Professor Lear introduces us to the essence of Aristotle's philosophy and guides us through the central Aristotelian texts - selected from the Physics, Metaphysics, Ethics, Politics and from the biological and logical works. The book is written in a direct, lucid style which engages the reader with the themes in an active, participatory manner. (shrink)

This is the first full assessment of British philosophy in the 19th century. Specially written essays by leading experts explore the work of the key thinkers of this remarkable period in intellectual history, covering logic and scientific method, metaphysics, religion, positivism, the impact of Darwin, and ethical, social, and political theory.

In this radical reinterpretation of Aristotle's Metaphysics, Walter E. Wehrle demonstrates that developmental theories of Aristotle are based on a faulty assumption: that the fifth chapter of Categories is an early theory of metaphysics that Aristotle later abandoned.

I use the Corcoran–Smiley interpretation of Aristotle's syllogistic as my starting point for an examination of the syllogistic from the vantage point of modern proof theory. I aim to show that fresh logical insights are afforded by a proof-theoretically more systematic account of all four figures. First I regiment the syllogisms in the Gentzen–Prawitz system of natural deduction, using the universal and existential quantifiers of standard first-order logic, and the usual formalizations of Aristotle's sentence-forms. I explain how the syllogistic is (...) a fragment of my system of Core Logic. Then I introduce my main innovation: the use of binary quantifiers, governed by introduction and elimination rules. The syllogisms in all four figures are re-proved in the binary system, and are thereby revealed as all on a par with each other. I conclude with some comments and results about grammatical generativity, ecthesis, perfect validity, skeletal validity and Aristotle's chain principle. (shrink)

We extend the language of the classical syllogisms with the sentence-forms “At most 1 p is a q” and “More than 1 p is a q”. We show that the resulting logic does not admit a finite set of syllogism-like rules whose associated derivation relation is sound and complete, even when reductio ad absurdum is allowed.

In this article, the author attempts to explicate the notion of the best known Talmudic inference rule called qal wa-omer. He claims that this rule assumes a massive-parallel deduction, and for formalizing it, he builds up a case of massive-parallel proof theory, the proof-theoretic cellular automata, where he draws conclusions without using axioms.

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)

This book presents the first study of the development of the theory of modal syllogistic in the Middle Ages.

Each of the two major approaches to Aristotle--the unitarian, which understands his work as forming a single, unified system, and the developmentalist, which seeks a sequence of developing ideas--has inherent limitations. This book proposes a synthetic view of Aristotle that sees development as a change between systematic theories. Setting theories of the so-called logical works beside theories of the physical and metaphysical treatises, Graham shows that Aristotle's doctrines fall into two distinct systems of philosophies that are genetically related. This study--the (...) first major alternative to the unitarian approach since Jaeger pioneered the developmentalist method in 1923--provides a sweeping reappraisal of Aristotle's science and metaphysics and a new approach to the problem of substance presented in the Metaphysics. (shrink)

CONSTITUÉ d'analyses et de réflexions sur les méthodes pratiquées aujourd'hui dans l'écriture de l'histoire de la philosophie, cet ouvrage vise à rendre compte de la différence des pratiques existant selon les traditions et à interroger l'existence d'une démarche historique spécifique au domaine de la philosophie. L'enjeu fondamental est donc de savoir s'il existe une historiographie propre à la philosophie. Les interrogations portent en particulier sur les questions des critères d'exactitude, de vérification, de crédibilité qui permettraient de distinguer et de décider (...) entre les interprétations. Si le travail de l'interprétation est infini, toujours à refaire, reprendre et corriger, cela n'implique nullement l'abandon des exigences qui fondent la pertinence historique. Une introduction indispensable à toute pratique de l'histoire de la philosophie. L'ouvrage comporte notamment des textes de : P. Aubenque (Paris IV), J. Biard (Tours), B. Bourgeois (Paris I), C. Coutel (Artois), Th. De Koninck (Laval, Québec), D. Deleule (Paris X), D. Garber (Chicago), P. Magnard (Paris IV), J.-M. Narbonne (Laval, Québec), E. Nuzzo (Salerne), John Rogers (Keele), J. B. Schneewind (Johns-Hopkins, Baltimore), Tom Sorell (Essex), J.-L. Vieillard-Baron (Poitiers), Y. C. Zarka (CNRS, Paris), mais aussi de F. P. Adorno (Salerne, CNRS), P. Caye (CNRS), P. Destrée (FNRS, Belgique), T. Gontier (CNRS), J.-B. Gourinat (CNRS), E. Picavet (Paris I), I. Radrizzani (Munich), C. Viano (CNRS), A. Vasiliu (CNRS). (shrink)

This book provides a detailed exposition of one of the most practical and popular methods of proving theorems in logic, called Natural Deduction. It is presented both historically and systematically. Also some combinations with other known proof methods are explored. The initial part of the book deals with Classical Logic, whereas the rest is concerned with systems for several forms of Modal Logics, one of the most important branches of modern logic, which has wide applicability.

While logical principles seem timeless, placeless, and eternal, their discovery is a story of personal accidents, political tragedies, and broad social change. If A, Then B begins with logic's emergence twenty-three centuries ago and tracks its expansion as a discipline ever since. -/- The book treats logic as more than a tale of individual abstraction; it sees logic as also being a result of politics, economics, technology, and geography, because all these factors helped to generate an audience for the discipline (...) from the start. The book thus relates developments in logical theory to the social history of different historical periods. -/- The book defends a number of controversial philosophical theses: (1) that rationally persuasive arguments must always proceed from premises that are initially more convincing than the conclusions to be proved (a position derived from Aristotle), (2) that intuitive logical judgments, though sometimes mistaken, can also count as certain knowledge, independent of the techniques used to construct formal logical systems, (3) that Wittgenstein was mistaken in thinking that philosophy was riddled with unintelligible language and that the real problems, instead, have been ambiguity and pretentious diction, and (4) that Nelson Goodman's argument to the effect that deductive logic can be "virtuously circular" is founded on a non sequitur. -/- . (shrink)

Aristotle created logic and developed it to a level of great sophistication. There was nothing there before; and it took more than two millennia for something better to come around. The astonishment experienced by readers of the Prior Analytics, the most important of Aristotle's works that present the discipline, is comparable to that of an explorer discovering a cathedral in a desert. This article explains and evaluates some of Aristotle's views about propositions and syllogisms. The most important omission is the (...) difficult subject of syllogisms involving modalities. Aristotle distinguishes two relations of opposition that can obtain between propositions with the same subject- and predicate-expressions: contrariety and contradiction. In every canonical syllogism, one term is common to the two premises: it is called “middle term,” or simply “middle.” The remaining two terms of the premises are the only ones occurring in the conclusion: they are called “extreme terms,” or simply “extremes.”. (shrink)

Aristotle’s Prior Analytics identifies deductions simpliciter with inferential necessity, so that a deduced conclusion is necessarily inferred from some premises. Modern logical reconstructions claim that inferential necessity in Aristotle corresponds to logical validity. However, this logical reconstruction fails on two accounts. First, logical validity does not highlight Aristotle’s distinction between inferential necessity and predicative necessity, meaning that the inferential necessity of a deduction is not of the same kind as the predicative necessity of a non‑deductive argument. Second, logical validity does (...) not explain the relevance of Aristotle’s distinction between complete and incomplete deductions. Logicians speak of complete deduction by adding the term “obvious” or “transparent” to logical validity, and then criticize Aristotle’s view for being unclear. However, Aristotle’s position is not confronted with this difficulty. There is nothing to add to inferential necessity, which already means complete deducibility, as opposed to incomplete deducibility, deemed to be potentially complete. Accordingly, the Prior Analytics reduces the incomplete deductions to the complete deductions in order to prove the potential, inferential necessity of the incomplete deductions. Logical validity would have been faithful to Aristotle’s text, if it had been possible to coin a notion of potential validity, distinct from both validity and invalidity. (shrink)

The paper shows that in the Art of Thinking (The Port Royal Logic) Arnauld and Nicole introduce a new way to state the truth-conditions for categorical propositions. The definition uses two new ideas: the notion of distributive or, as they call it, universal term, which they abstract from distributive supposition in medieval logic, and their own version of what is now called a conservative quantifier in general quantification theory. Contrary to the interpretation of Jean-Claude Parienté and others, the truth-conditions do (...) not require the introduction of a new concept of ‘indefinite’ term restriction because the notion of conservative quantifier is formulated in terms of the standard notion of term intersection. The discussion shows the following. Distributive supposition could not be used in an analysis of truth because it is explained in terms of entailment, and entailment in terms of truth. By abstracting from semantic identities that underlie distribution, the new concept of distributive term is definitionally prior to truth and can, therefore, be used in a non-circular way to state truth-conditions. Using only standard restriction, the Logic’s truth-conditions for the categorical propositions are stated solely in terms of (1) universal (distributive) term, (2) conservative quantifier, and (3) affirmative and negative proposition. It is explained why the Cartesian notion of extension as a set of ideas is in this context equivalent to medieval and modern notions of extension. (shrink)

In the present article we attempt to show that Aristotle's syllogistic is an underlying logiC which includes a natural deductive system and that it isn't an axiomatic theory as had previously been thought. We construct a mathematical model which reflects certain structural aspects of Aristotle's logic. We examine the relation of the model to the system of logic envisaged in scattered parts of Prior and Posterior Analytics. Our interpretation restores Aristotle's reputation as a logician of consummate imagination and skill. Several (...) attributions of shortcomings and logical errors to Aristotle are shown to be without merit. Aristotle's logic is found to be self-sufficient in several senses: his theory of deduction is logically sound in every detail. (His indirect deductions have been criticized, but incorrectly on our account.) Aristotle's logic presupposes no other logical concepts, not even those of propositional logic. The Aristotelian system is seen to be complete in the sense that every valid argument expressible in his system admits of a deduction within his deductive system: every semantically valid argument is deducible. (shrink)

This book claims that Aristotle followed an aspect theory of predication.

Proceedings of an international research and development conference, Tuscon, Arizona, October 1985. One hundred and twenty-eight papers are presented in this hefty volume.

Polynomizing is a term that intends to describe the uses of polynomial-like representations as a reasoning strategy and as a tool for scientific heuristics. I show how proof-theory and semantics for classical and several non-classical logics can be approached from this perspective, and discuss the assessment of this prospect, in particular to recover certain ideas of George Boole in unifying logic, algebra and the differential calculus.