In Subversive Spinoza , Antonio Negri spells out the philosophical credo that inspired his radical renewal of Marxism and his compelling analysis of the modern state and the global economy by means of an inspiring reading of the challenging metaphysics of the seventeenth-century Dutch-Jewish philosopher Spinoza. For Negri, Spinoza's philosophy has never been more relevant than it is today to debates over individuality and community, democracy and resistance, modernity and postmodernity.

The paradox marking Spinoza’s reappearance in modernity is well known. If Mendelssohn wished to “give him new credence by bringing him closer to the philosophical orthodoxy of Leibniz and Wolff,” and Jacobi, “by presenting him as a heterodox figure in the literal sense of the term, wanted to do away with him definitively for modern Christianity”—well, “both failed in their goal, and it was the heterodox Spinoza who was rehabilitated.” The Mendelssohn-Jacobi debate can be grafted onto the crisis of a (...) specific philosophical model. It generates a figure of Spinoza capable of assuaging the exacerbated spiritual tension of that epoch, and of constituting the systematic preamble of the relation between power and substance—between subject and nature. Spinoza, the damned Spinoza, had a resurgence in modernity as a Romantic philosopher. Lessing won out by recognizing in Spinoza an idea of nature which was capable of balancing the relation between feeling and intellect, freedom and necessity, and history and reason. Herder and Goethe, against the subjective and revolutionary impatience of the Sturm und Drang, based themselves on this powerful image of synthesis and recomposed objectivity: Spinoza is not only the figure of Romanticism; he constitutes its grounding and its fulfillment. The omnipotence of nature was no longer to break off into the tragedy of feeling, but it was to triumph over it, by opposing it to a kingdom of completed forms. Spinoza’s first reception within Romanticism was thus an aesthetic reception, a perception of motion and perfection, of dynamism and forms. And it remained such, even when the general frame and the particular components of Romanticism were subjected to the labor of philosophical critique. Fichte, the real philosophical hero of Romanticism, considered both Spinoza’s and Kant’s systems to be “perfectly coherent,” in the incessant ontological movement of the I. For the Schelling of the 1790s, the assertion of a radical opposition between 1 critical philosophy and dogmatic philosophy—that is, between a philosophy of the absolute I founding itself on the critical philosophy and a dogmatic philosophy of the absolute object and Spinozism—was quickly resolved into an analysis of action which dialectically took on the weight of the objective. Far from becoming antinomial, the absolute position of the I composes itself into a necessary process which, above tragedy, exalts the “spiritual automatism” of the relation between subject and substance. The aesthetic dimension of this synthesis consists in ceaselessly and tirelessly bringing back power and substance, the productive element and the form of production, to perfection. Romanticism, according to Hegel, is characterized by a capacity to overcome the pure objectivity of the ideal and the natural as a true idea of beauty and truth, initially to destroy the union of the idea and its reality, and to locate the latter in the difference, so as then to bring to manifestation the inner world of absolute subjectivity and reconstruct its objectivity where the overcoming of sensibility is appeased in the absolute character of the result. The filiation of this process is still Lessingian, but the new dialectic expresses and articulates its motivations, while insisting on the propaedeutic of the beautiful along the path leading to the absolute. Spinoza, a certain Spinoza, becomes the central figure in this process. (shrink)

Geometric theories are presented as contraction- and cut-free systems of sequent calculi with mathematical rules following a prescribed rule-scheme that extends the scheme given in Negri and von Plato. Examples include cut-free calculi for Robinson arithmetic and real closed fields. As an immediate consequence of cut elimination, it is shown that if a geometric implication is classically derivable from a geometric theory then it is intuitionistically derivable.

Large portions of mathematics such as algebra and geometry can be formalized using first-order axiomatizations. In many cases it is even possible to use a very well-behaved class of first-order axioms, namely, what are called coherent or geometric implications. Such class of axioms can be translated to inference rules that can be added to a sequent calculus while preserving its structural properties. In this work, this fundamental result is extended to their infinitary generalizations as extensions of sequent calculi for both (...) classical and intuitionistic infinitary logic. As an application, a simple proof of the infinitary Barr’s theorem without the axioms of choice is shown. (shrink)

This paper provides a proof-theoretic study of quantified non-normal modal logics. It introduces labelled sequent calculi based on neighbourhood semantics for the first-order extension, with both varying and constant domains, of monotone NNML, and studies the role of the Barcan formulas in these calculi. It will be shown that the calculi introduced have good structural properties: invertibility of the rules, height-preserving admissibility of weakening and contraction and syntactic cut elimination. It will also be shown that each of the calculi introduced (...) is sound and complete with respect to the appropriate class of neighbourhood frames. In particular, the completeness proof constructs a formal derivation for derivable sequents and a countermodel for non-derivable ones, and gives a semantic proof of the admissibility of cut. (shrink)

Antonio Negri, one of the world's leading scholars on Baruch Spinoza (1632-1677) and his contemporary legacy, offers a straightforward explanation of the philosopher's elaborate arguments and a persuasive case for his ongoing relevance. Responding to a resurgent interest in Spinoza's thought and its potential application to contemporary global issues, Negri demonstrates the thinker's special value to politics, philosophy, and related disciplines. Negri's work is both a return to and an advancement of his initial affirmation of Spinozian thought (...) in _The Savage Anomaly_. He further defends his understanding of the philosopher as a proto-postmodernist, or a thinker who is just now, with the advent of the postmodern, becoming contemporary. Negri also connects Spinoza's theories to recent trends in political philosophy, particularly the reengagement with Carl Schmitt's "political theology," and the history of philosophy, including the argument that Spinoza belongs to a "radical enlightenment." By positioning Spinoza as a contemporary revolutionary intellectual, Negri addresses and effectively defeats twentieth-century critiques of the thinker waged by Jacques Derrida, Alain Badiou, and Giorgio Agamben. (shrink)

In a recent paper, Negri and Pavlović (Studia Logica 1–35, 2020) have formulated a decidable sequent calculus for the logic of agency, specifically for a deliberative see-to-it-that modality, or dstit. In that paper the adequacy of the system is demonstrated by showing the derivability of the axiomatization of dstit from Belnap et al. (Facing the future: agents and choices in our indeterminist world. Oxford University Press, Oxford, 2001). And while the influence of the latter book on the study of (...) logics of agency cannot be overstated, we note that this is not the only axiomatization of that modality available. In fact, an earlier (and arguably purer) one was offered in Xu (J Philosophical Logic 27(5):505–552, 1998). In this article we fill this lacuna by proving that this alternative axiomatization is likewise readily derivable in the system of Negri and Pavlović (Studia Logica 1–35, 2020). (shrink)

A philosopher and political thinker describes a new political grammar free of modernist assumptions. In 2004 and 2005, Antonio Negri held ten workshops at the Collège International de Philosophie in Paris to formulate a new political grammar of the postmodern. Biopolitics, biopowers, control, the multitude, people, war, borders, dependency and interdependency, state, nation, the common, difference, resistance, subjective rights, revolution, freedom, democracy: these are just a few of the themes Negri addressed in these experimental laboratories. Postmodernity, Negri (...) suggests, can be described as a “porcelain factory”: a delicate and fragile construction that could be destroyed through one clumsy act. Looking across twentieth century history, Negri warns that our inability to anticipate future developments has already placed coming generations in serious jeopardy. Describing the years 1917-1968 as the “short century,” Negri suggests that by the end of it, all of the familiar markers of modernity had lost their relevance. Confronted with an intolerable reality, indignation and the revolutionary will to transform the world have both taken new forms and must be understood anew, free of modernist assumptions. In the impassioned debates recounted in this book, Antonio Negri attempts to describe the formation of an alternative political horizon and looks for a way to define the practices and modes of expression that democracy could take.Antonio Negri is a philosopher and essay writer. A political and social activist in the 1960s and 1970s in Italy, he taught political sciences for many years and has written numerous books on political philosophy including Marx beyond Marx, The Savage Anomaly, Insurgencies ; and in collaboration with Michael Hardt, Empire and Multitude. (shrink)

A proof-theoretical analysis of elementary theories of order relations is effected through the formulation of order axioms as mathematical rules added to contraction-free sequent calculus. Among the results obtained are proof-theoretical formulations of conservativity theorems corresponding to Szpilrajn’s theorem on the extension of a partial order into a linear one. Decidability of the theories of partial and linear order for quantifier-free sequents is shown by giving terminating methods of proof-search.

Kan demokrati - folkets magt - realiseres. Forfatteren gennemgår dette på baggrund af den konflikt, der altid har været mellem den påtvungne magt og den valgte magt.

Structural proof theory is a branch of logic that studies the general structure and properties of logical and mathematical proofs. This book is both a concise introduction to the central results and methods of structural proof theory, and a work of research that will be of interest to specialists. The book is designed to be used by students of philosophy, mathematics and computer science. The book contains a wealth of results on proof-theoretical systems, including extensions of such systems from logic (...) to mathematics, and on the connection between the two main forms of structural proof theory - natural deduction and sequent calculus. The authors emphasize the computational content of logical results. A special feature of the volume is a computerized system for developing proofs interactively, downloadable from the web and regularly updated. (shrink)

This book continues from where the authors' previous book, Structural Proof Theory, ended. It presents an extension of the methods of analysis of proofs in pure logic to elementary axiomatic systems and to what is known as philosophical logic. A self-contained brief introduction to the proof theory of pure logic is included that serves both the mathematically and philosophically oriented reader. The method is built up gradually, with examples drawn from theories of order, lattice theory and elementary geometry. The aim (...) is, in each of the examples, to help the reader grasp the combinatorial behaviour of an axiom system, which typically leads to decidability results. The last part presents, as an application and extension of all that precedes it, a proof-theoretical approach to the Kripke semantics of modal and related logics, with a great number of new results, providing essential reading for mathematical and philosophical logicians. (shrink)

The axiomatic presentation of modal systems and the standard formulations of natural deduction and sequent calculus for modal logic are reviewed, together with the difficulties that emerge with these approaches. Generalizations of standard proof systems are then presented. These include, among others, display calculi, hypersequents, and labelled systems, with the latter surveyed from a closer perspective.

Ever since antiquity, attempts have been made at defining knowledge through belief augmented by additional properties such as truth and justification. These characterizations have been challenged by Gettier counterexamples and their variants. A modern proposal, what is known as defeasibility theory, characterizes knowledge through stability under revision of beliefs on the basis of true or arbitrary information. A formal investigation of such a proposal calls for the methods of dynamic epistemic logic: well developed semantic approaches to dynamic epistemic logic have (...) been given through the plausibility models, but a corresponding proof theory is still in its infancy. We shall recast plausibility models in terms of more general neighbourhood models and develop on their basis complete proof systems. An inferential treatment of epistemic and doxastic notions such as safe belief will give a new way to study their relationships; among these, the characterization of knowledge as belief stable under arbitrary revision will be grounded through formal labelled sequent calculus derivations. (shrink)

A sequent calculus is given in which the management of weakening and contraction is organized as in natural deduction. The latter has no explicit weakening or contraction, but vacuous and multiple discharges in rules that discharge assumptions. A comparison to natural deduction is given through translation of derivations between the two systems. It is proved that if a cut formula is never principal in a derivation leading to the right premiss of cut, it is a subformula of the conclusion. Therefore (...) it is sufficient to eliminate those cuts that correspond to detour and permutation conversions in natural deduction. (shrink)

This second volume of a new three-part series of Antonio Negri's work is focussed on the consequences of the rapid process of deindustrialisation that has occurred across the West in recent years. In this volume Negri investigates exactly what happens when the class subjects of industrial capitalism are demobilised and the factories close. Evidently capital continues to make profit, but how and where? According to Negri, the creation of value extends beyond the factory walls to embrace the (...) whole of society; the 'mass worker' of industrialism gives way to the 'socialised worker' (operaio sociale) and the terrain of exploitation now becomes the whole of human life. In postmodernity, the metropolis becomes the privileged arena of value extraction. We must therefore understand the global city, with its stratifications, its enclosures and its resistances. Old categories of the private and the public are inadequate to describe the new matrix of production, which is characterised rather by the 'common', the productive space of cognitive and immaterial labour. Today's metropolis can be defined as a space of antagonisms between forms of life produced, on the one hand, by finance capital (the capital that operates around rents), and on the other by the 'cognitive proletariat'. The central question is then how 'the common' of the latter can be mobilised for the destruction of capitalism. In an analysis that runs from the Italian workerism (operaismo) of the 1970s to the present day, From the Factory to the Metropolis offers readers valuable insight into the far-reaching impact of deindustrialisation, presenting both the challenges and opportunities. It will appeal to the many interested in the continuing development of Negri's project and to anyone interested in radical politics today. (shrink)

Superati gli ottant'anni, per la prima volta Toni Negri racconta la sua infanzia, la sua formazione politica e intellettuale, gli anni da militante e teorico della sinistra, in cui fu tra i fondatori di Potere Operaio e tra i leader di Autonomia Operaia.

This paper explores the question of whether it is possible to be communist without Marx. This entails encountering the ontological dimension of communism, that is, the material tenor of this ontology, its residual effectiveness, the desire of human beings to go beyond capital, and the reality of the episode of statism.

Stone representation theorems are a central ingredient in the metatheory of philosophical logics and are used to establish modal embedding results in a general but indirect and non-constructive way. Their use in logical embeddings will be reviewed and it will be shown how they can be circumvented in favour of direct and constructive arguments through the methods of analytic proof theory, and how the intensional part of the representation results can be recovered from the syntactic proof of those embeddings. Analytic (...) methods will also be used to establish the embedding of subintuitionistic logics into the corresponding modal logics. Finally, proof-theoretic embeddings will be interpreted as a reduction of classes of word problems. (shrink)

Writings by Negri on the brief thaw in the cold winter of neoliberalism, Thatcherism, Reaganomics, and counterrevolution. Automation and information technology have transformed the organization of labor to such an extent that the processes of exploitation have moved beyond the labor class and now work upon society as a whole. If this displacement has destroyed the political primacy of the labor class, it has not, however, eliminated exploitation; rather, it has broadened it, implanting it within the given conditions of (...) the most diverse spheres of society. —from The Winter Is Over In late 1995, in opposition to the conservative agenda of Jacques Chirac and his prime minister Alain Juppé and their proposed widespread welfare cuts, French students rose up against their government; public sector workers, together with all the major trade unions, went on strike. When railway workers and Paris Metro personnel joined in the protests, France's public transportation system came to a halt. These extensive social upheavals, the likes of which had not been seen in France since 1968, found widespread public support and fuelled the creation of many political organizations. Chirac backed down from restructuring the public retirement system. Antonio Negri's The Winter is Over comes out of the glimmer of optimism created by the events of 1995, when the long, cold season of neoliberalism, Thatcherism, Reaganomics, reaction, and counterrevolution appeared to have run its course. Published in Italian in 1996, The Winter is Over brings together a series of articles, speeches, and other documents written by Negri between 1989 and 1995 at the threshold of this thaw. It offers a revealing and wide-reaching account of those years of change and brink-of-change, focusing on such topics as the networks of social production, the decline of “limp thought,” the end of applied socialism, the Gulf War, and, finally, Italy's transition to its so-called “Second Republic,” as seen by an exile. (shrink)

The decision problem for positively quantified formulae in the theory of linearly ordered Heyting algebras is known, as a special case of work of Kreisel, to be solvable; a simple solution is here presented, inspired by related ideas in Gödel-Dummett logic.

Résumé Dans cet article, Toni Negri cherche à définir la façon dont se pose l’Europe à l’intérieur du processus constitutif du gouvernement impérial global. Pour ce faire il aborde successivement - par le haut - les schèmes constitutifs de la constitution européenne, pour analyser ensuite - par le bas-les tactiques constitutionnelles de mouvement et pour conclure il aborde les problèmes autour desquels il est immédiatement possible de faire ensemble l’Europe et le mouvement politique et social.

A uniform calculus for linear logic is presented. The calculus has the form of a natural deduction system in sequent calculus style with general introduction and elimination rules. General elimination rules are motivated through an inversion principle, the dual form of which gives the general introduction rules. By restricting all the rules to their single-succedent versions, a uniform calculus for intuitionistic linear logic is obtained. The calculus encompasses both natural deduction and sequent calculus that are obtained as special instances from (...) the uniform calculus. Other instances give all the invertibilities and partial invertibilities for the sequent calculus rules of linear logic. The calculus is normalizing and satisfies the subformula property for normal derivations. (shrink)

Is it possible to be Communist Without Marx ? It is evident that when Marxist communism achieved its actual realization the State became omnipotent and the Public falsified the Common. Do struggles for communism therefore have to start by eliminating Marx’s thought ? The answer is no. Communism needs Marx in order to root itself within Common praxis. Contrary to what a few contemporary philosophers think, without historical ontology there is no communism. Without a logic of production, the communist struggle (...) cannot become an “event”. (shrink)

In this work we show that every structure [MATHEMATICAL SCRIPT CAPITAL A] can be expanded to a partial structure [MATHEMATICAL SCRIPT CAPITAL A]* with universal functions for the class of polynomials on [MATHEMATICAL SCRIPT CAPITAL A]*. We can embed [MATHEMATICAL SCRIPT CAPITAL A]* monomorphically in a total structure [MATHEMATICAL SCRIPT CAPITAL A]º that preserves universal functions of [MATHEMATICAL SCRIPT CAPITAL A]* and that is universal among such structures, i.e. [MATHEMATICAL SCRIPT CAPITAL A]º can be homomorphically embedded in every total structure (...) that preserves universal functions of [MATHEMATICAL SCRIPT CAPITAL A]*. Universal functions are the starting point for developing recursion theoretic tools in an [MATHEMATICAL SCRIPT CAPITAL A]* that satisfies some simple additional conditions. (shrink)

A way is found to add axioms to sequent calculi that maintains the eliminability of cut, through the representation of axioms as rules of inference of a suitable form. By this method, the structural analysis of proofs is extended from pure logic to free-variable theories, covering all classical theories, and a wide class of constructive theories. All results are proved for systems in which also the rules of weakening and contraction can be eliminated. Applications include a system of predicate logic (...) with equality in which also cuts on the equality axioms are eliminated. (shrink)

Proofs and countermodels are the two sides of completeness proofs, but, in general, failure to find one does not automatically give the other. The limitation is encountered also for decidable non-classical logics in traditional completeness proofs based on Henkin’s method of maximal consistent sets of formulas. A method is presented that makes it possible to establish completeness in a direct way: For any given sequent either a proof in the given logical system or a countermodel in the corresponding frame class (...) is found. The method is a synthesis of a generation of calculi with internalized relational semantics, a Tait–Schütte–Takeuti style completeness proof, and procedures to finitize the countermodel construction. Finitizations for intuitionistic propositional logic are obtained through the search for a minimal derivation, through pruning of infinite branches in search trees by means of a suitable syntactic counterpart of semantic filtration, or through a proof-theoretic embedding into an appropriate provability logic. A number of examples illustrates the method, its subtleties, challenges, and present scope. (shrink)

_Factory of Strategy_ is the last of Antonio Negri's major political works to be translated into English. Rigorous and accessible, it is both a systematic inquiry into the development of Lenin's thought and an encapsulation of a critical shift in Negri's theoretical trajectory. Lenin is the only prominent politician of the modern era to seriously question the "withering away" and "extinction" of the state, and like Marx, he recognized the link between capitalism and modern sovereignty and the need (...) to destroy capitalism and reconfigure the state. Negri refrains from portraying Lenin as a ferocious dictator enforcing the proletariat's reappropriation of wealth, nor does he depict him as a mere military tool of a vanguard opposed to the Ancien Régime. Negri instead champions Leninism's ability to adapt to different working-class configurations in Russia, China, Latin America, and elsewhere. He argues that Lenin developed a new political figuration in and beyond modernity and an effective organization capable of absorbing different historical conditions. He ultimately urges readers to recognize the universal application of Leninism today and its potential to institutionally--not anarchically--dismantle centralized power. (shrink)

It is in Sein und Zeit that Heidegger decrees the end of the Geisteswissenschaften and their tradition, when, as he is commenting on the Briefwechsel [exchange of letters] between Dilthey and Yorck von Wartenburg, he pays homage to the latter for “his full understanding of the fundamental character of history as virtuality [...] [which he] owes to his knowledge of the character of being of human Dasein itself.” Consequently, Heidegger continues, “the interest of understanding historicality” is confronted with the task (...) of an elaboration of the “generic difference between the ontic and the historical.” But he must part ways with Yorck when the latter, after having clearly established that difference, moves from virtuality to mysticism. If, on the other hand, once separated from the ontic, “the question of historicality shows itself to be an ontological question which inquires into the constitution of being of historical being”, it is once again towards Dilthey that one must turn, in spite of his confused vitalism. Heidegger effects two operations at once. On the one hand, he expels the Geisteswissenschaften from the position they occupied at the heart of metaphysics, as the inheritors of the Enlightenment and the outcome of Hegelianism. On the other hand, he brings to fulfillment the critical labor which had precisely shown its value in Dilthey’s historicism —acritical labor which develops the search for the meaning of historicity and allows one to move from the theory of objectivity to that of expression, from the acknowledgment of historiography in the context of the critique of cognition to its definition at the heart of the transcendental schematism. Historicity is then posited as an ontological dimension and leaves only its ontic residue for historiography. (shrink)

Das Buch bietet die erste kritische Edition der Artikel XXV-XXVII der Summa (Quaestiones ordinariae) des Heinrich von Gent. Dabei leistet es einen Beitrag zur Geschichte der Formen und Pfade der Ideenvermittlung im Mittelalter und zur mittelalterlichen Buchkultur. Die Kollationierung der Handschriften der Artikel XXV-XXVII und die Untersuchung ihrer materiellen Überlieferung haben der Editorin erlaubt, den Prozess der Ausarbeitung, Publikation und Verbreitung einer Portion von Heinrichs Summa über einen längeren Zeitraum in großer Detailgenauigkeit zu rekonstruieren. Die hier edierten Artikel enthalten ein (...) Kernstück von Heinrichs theologischem und metaphysischem System. In der Behandlung von Einheit, Natur und Leben Gottes entfalten sie eine rationale Darstellung des ersten Prinzips. (shrink)