Discusses how cultural and economic changes around the world have caused a shift in the concepts that shape modern politics and defined the new global order.

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)

A general method for generating contraction- and cut-free sequent calculi for a large family of normal modal logics is presented. The method covers all modal logics characterized by Kripke frames determined by universal or geometric properties and it can be extended to treat also Gödel-Löb provability logic. The calculi provide direct decision methods through terminating proof search. Syntactic proofs of modal undefinability results are obtained in the form of conservativity theorems.

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.

Constituent power : the concept of a crisis -- Virtue and fortune : the machiavellian paradigm -- The Atlantic model and the theory of counterpower -- Political emancipation in the American constitution -- The revolution and the constitution of labor -- Communist desire and the dialectic restored -- The constitution of strength.

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)

Various sources in the literature claim that the deduction theorem does not hold for normal modal or epistemic logic, whereas others present versions of the deduction theorem for several normal modal systems. It is shown here that the apparent problem arises from an objectionable notion of derivability from assumptions in an axiomatic system. When a traditional Hilbert-type system of axiomatic logic is generalized into a system for derivations from assumptions, the necessitation rule has to be modified in a way that (...) restricts its use to cases in which the premiss does not depend on assumptions. This restriction is entirely analogous to the restriction of the rule of universal generalization of first-order logic. A necessitation rule with this restriction permits a proof of the deduction theorem in its usual formulation. Other suggestions presented in the literature to deal with the problem are reviewed, and the present solution is argued to be preferable to the other alternatives. A contraction-and cut-free sequent calculus equivalent to the Hubert system for basic modal logic shows the standard failure argument untenable by proving the underivability of DA from A. (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)

Using labelled formulae, a cut-free sequent calculus for intuitionistic propositional logic is presented, together with an easy cut-admissibility proof; both extend to cover, in a uniform fashion, all intermediate logics characterised by frames satisfying conditions expressible by one or more geometric implications. Each of these logics is embedded by the Gödel–McKinsey–Tarski translation into an extension of S4. Faithfulness of the embedding is proved in a simple and general way by constructive proof-theoretic methods, without appeal to semantics other than in the (...) explanation of the rules. (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.

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.

That every first-order theory has a coherent conservative extension is regarded by some as obvious, even trivial, and by others as not at all obvious, but instead remarkable and valuable; the result is in any case neither sufficiently well-known nor easily found in the literature. Various approaches to the result are presented and discussed in detail, including one inspired by a problem in the proof theory of intermediate logics that led us to the proof of the present paper. It can (...) be seen as a modification of Skolem’s argument from 1920 for his “Normal Form” theorem. “Geometric” being the infinitary version of “coherent”, it is further shown that every infinitary first-order theory, suitably restricted, has a geometric conservative extension, hence the title. The results are applied to simplify methods used in reasoning in and about modal and intermediate logics. We include also a new algorithm to generate special coherent implications from an axiom, designed to preserve the structure of formulae with relatively little use of normal forms. (shrink)

Negri, a leading Marxist philosopher, has inspired anti-empire movements around the world through his writings and personal example. This work, which began as a conversation between Negri and literary critic Casarino, is the most complete review of the philosopher's work.

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)

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)

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)

Anti-realist epistemic conceptions of truth imply what is called the knowability principle: All truths are possibly known. The principle can be formalized in a bimodal propositional logic, with an alethic modality ${\diamondsuit}$ and an epistemic modality ${\mathcal{K}}$, by the axiom scheme ${A \supset \diamondsuit \mathcal{K} A}$. The use of classical logic and minimal assumptions about the two modalities lead to the paradoxical conclusion that all truths are known, ${A \supset \mathcal{K} A}$. A Gentzen-style reconstruction of the Church–Fitch paradox is presented (...) following a labelled approach to sequent calculi. First, a cut-free system for classical bimodal logic is introduced as the logical basis for the Church–Fitch paradox and the relationships between ${\mathcal {K}}$ and ${\diamondsuit}$ are taken into account. Afterwards, by exploiting the structural properties of the system, in particular cut elimination, the semantic frame conditions that correspond to KP are determined and added in the form of a block of nonlogical inference rules. Within this new system for classical and intuitionistic “knowability logic”, it is possible to give a satisfactory cut-free reconstruction of the Church–Fitch derivation and to confirm that OP is only classically derivable, but neither intuitionistically derivable nor intuitionistically admissible. Finally, it is shown that in classical knowability logic, the Church–Fitch derivation is nothing else but a fallacy and does not represent a real threat for anti-realism. (shrink)

A constructivisation of the cut-elimination proof for sequent calculi for classical, intuitionistic and minimal infinitary logics with geometric rules—given in earlier work by the second author—is presented. This is achieved through a procedure where the non-constructive transfinite induction on the commutative sum of ordinals is replaced by two instances of Brouwer’s Bar Induction. The proof of admissibility of the structural rules is made ordinal-free by introducing a new well-founded relation based on a notion of embeddability of derivations. Additionally, conservativity for (...) classical over intuitionistic/minimal logic for the seven (finitary) Glivenko sequent classes is here shown to hold also for the corresponding infinitary classes. (shrink)

A sequent calculus methodology for systems of agency based on branching-time frames with agents and choices is proposed, starting with a complete and cut-free system for multi-agent deliberative STIT; the methodology allows a transparent justification of the rules, good structural properties, analyticity, direct completeness and decidability proofs.

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)

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)

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)

The main result of this paper is a normalizing system of natural deduction for the full language of intuitionistic linear logic. No explicit weakening or contraction rules for -formulas are needed. By the systematic use of general elimination rules a correspondence between normal derivations and cut-free derivations in sequent calculus is obtained. Normalization and the subformula property for normal derivations follow through translation to sequent calculus and cut-elimination.

A proof-theoretical treatment of collectively accepted group beliefs is presented through a multi-agent sequent system for an axiomatization of the logic of acceptance. The system is based on a labelled sequent calculus for propositional multi-agent epistemic logic with labels that correspond to possible worlds and a notation for internalized accessibility relations between worlds. The system is contraction- and cut-free. Extensions of the basic system are considered, in particular with rules that allow the possibility of operative members or legislators. Completeness with (...) respect to the underlying Kripke semantics follows from a general direct and uniform argument for labelled sequent calculi extended with mathematical rules for frame properties. As an example of the use of the calculus we present an analysis of the discursive dilemma. (shrink)

Antonio Negri, a leading scholar 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 utility.

In this essential rereading of Spinoza's (1632-1677) philosophical and political writings, Negri positions this thinker within the historical context of the development of the modern state and its attendant political economy.

The logic of Conditional Beliefs has been introduced by Board, Baltag, and Smets to reason about knowledge and revisable beliefs in a multi-agent setting. In this article both the semantics and the proof theory for this logic are studied. First, a natural semantics forCDLis defined in terms of neighbourhood models, a multi-agent generalisation of Lewis’ spheres models, and it is shown that the axiomatization ofCDLis sound and complete with respect to this semantics. Second, it is shown that the neighbourhood semantics (...) is equivalent to the original one defined in terms of plausibility models, by means of a direct correspondence between the two types of models. On the basis of neighbourhood semantics, a labelled sequent calculus forCDLis obtained. The calculus has strong proof-theoretic properties, in particular admissibility of contraction and cut, and it provides a decision procedure for the logic. Furthermore, its semantic completeness is used to obtain a constructive proof of the finite model property of the logic. Finally, it is shown that other doxastic operators can be easily captured within neighbourhood semantics. This fact provides further evidence of the naturalness of neighbourhood semantics for the analysis of epistemic/doxastic notions. (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.

Contraction-free sequent calculi for intuitionistic theories of apartness and order are given and cut-elimination for the calculi proved. Among the consequences of the result is the disjunction property for these theories. Through methods of proof analysis and permutation of rules, we establish conservativity of the theory of apartness over the theory of equality defined as the negation of apartness, for sequents in which all atomic formulas appear negated. The proof extends to conservativity results for the theories of constructive order over (...) the usual theories of order. (shrink)

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.

A constructive definition of the continuum based on formal topology is given and its basic properties studied. A natural notion of Cauchy sequence is introduced and Cauchy completeness is proved. Other results include elementary proofs of the Baire and Cantor theorems. From a classical standpoint, formal reals are seen to be equivalent to the usual reals. Lastly, the relation of real numbers as a formal space to other approaches to constructive real numbers is determined.

In 1968, Orevkov presented proofs of conservativity of classical over intuitionistic and minimal predicate logic with equality for seven classes of sequents, what are known as Glivenko classes. The proofs of these results, important in the literature on the constructive content of classical theories, have remained somehow cryptic. In this paper, direct proofs for more general extensions are given for each class by exploiting the structural properties of G3 sequent calculi; for five of the seven classes the results are strengthened (...) to height-preserving statements, and it is further shown that the constructive and minimal proofs are identical in structure to the classical proof from which they are obtained. (shrink)

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)

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)

Translation of the authoro's Fabbrica della strategia.

With the economy deindustrialized and the working class decentralized, a call for alternative horizons for resistance: the university and the art world. What was once the factory is now the university. As deindustrialization spreads and the working class is decentralized, new means of social resistance and political activism need to be sought in what may be the last places where they are possible: the university and the art world. Gerald Raunig's new book analyzes the potential that cognitive and creative labor (...) has in these two arenas to resist the new regimes of domination imposed by cognitive capitalism. Drawing on Gilles Deleuze's concept of “modulation” as the market-driven imperative for the constant transformation and reinvention of subjectivity, in Factories of Knowledge, Industries of Creativity, Raunig charts alternative horizons for resistance. Looking at recent social struggles including the university strikes in Europe, the Spanish ¡Democracia real YA! organization, the Arab revolts, and the Occupy movement, Raunig argues for a reassessment of the importance of cultural and knowledge production. The central role of the university, he asserts, is not as a factory of knowledge but as a place of creative disobedience. (shrink)

Nine letters on art, written to friends from exile in France in the 1980s. Starting from earlier materialist approaches to art, Negri relates artistic production to the structures of social production characteristic of each historical era. This enables him to define the nature of both material and artistic production in the era of post-modernity and post-Fordism - the era Negri characterizes as that of immaterial labour. Negri then seeks to define artistic beauty in this new era, and this he does (...) in terms of concepts that have become fundamental to his thinking - singularity, multitude, abstraction, collective work, event, the biopolitical, the common. Art is living labour, and therefore invention of singularity, of singular figures and objects. But this expressive act only achieves beauty when the signs and language through which it expresses itself turn themselves into community, when they are contained within a common project. The beautiful is not the act of imagining, but an imagination that has become action. Art, in this sense, is multitude. (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.