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.

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)

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)

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)

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)

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)

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 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)

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)

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.

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)

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.

In a fragment entitled Elementa Nova Matheseos Universalis Leibniz writes “the mathesis [...] shall deliver the method through which things that are conceivable can be exactly determined”; in another fragment he takes the mathesis to be “the science of all things that are conceivable.” Leibniz considers all mathematical disciplines as branches of the mathesis and conceives the mathesis as a general science of forms applicable not only to magnitudes but to every object that exists in our imagination, i.e. that is (...) possible at least in principle. As a general science of forms the mathesis investigates possible relations between “arbitrary objects”. It is an abstract theory of combinations and relations among objects whatsoever. In 1810 the mathematician and philosopher Bernard Bolzano published a booklet entitled Contributions to a Better-Grounded Presentation of Mathematics. There is, according to him, a certain objective connection among the truths that are germane to a certain homogeneous field of objects: some truths are the “reasons” of others, and the latter are “consequences” of the former. The reason-consequence relation seems to be the counterpart of causality at the level of a relation between true propositions. Arigorous proof is characterized in this context as a proof that shows the reason of the proposition that is to be proven. Requirements imposed on rigorous proofs seem to anticipate normalization results in current proof theory. The contributors of Mathesis Universalis, Computability and Proof, leading experts in the fields of computer science, mathematics, logic and philosophy, show the evolution of these and related ideas exploring topics in proof theory, computability theory, intuitionistic logic, constructivism and reverse mathematics, delving deeply into a contextual examination of the relationship between mathematical rigor and demands for simplification. (shrink)

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)

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)

We give a direct proof of admissibility of cut and contraction for the contraction-free sequent calculus G4ip for intuitionistic propositional logic and for a corresponding multi-succedent calculus: this proof extends easily in the presence of quantifiers, in contrast to other, indirect, proofs. i.e., those which use induction on sequent weight or appeal to admissibility of rules in other calculi.

The contraction-free sequent calculus G4 for intuitionistic logic is extended by rules following a general rule-scheme for nonlogical axioms. Admissibility of structural rules for these extensions is proved in a direct way by induction on derivations. This method permits the representation of various applied logics as complete, contraction- and cut-free sequent calculus systems with some restrictions on the nature of the derivations. As specific examples, intuitionistic theories of apartness and order are treated.

We give a direct proof of admissibility of cut and contraction for the contraction-free sequent calculus G4ip for intuitionistic propositional logic and for a corresponding multi-succedent calculus: this proof extends easily in the presence of quantifiers, in contrast to other, indirect, proofs. i.e., those which use induction on sequent weight or appeal to admissibility of rules in other calculi.

Τι είναι τα Κίτρινα Γιλέκα; Δεν μπορούμε εμείς, ως συντακτική της Βαβυλωνίας να μιλήσουμε εκ μέρους κάποιου κινήματος ενόσω ακόμη κινείται. Ούτε, βέβαια, μπορούμε να το αγνοήσουμε και να μην μιλήσουμε γι' αυτό. Χρειάζεται να αναστοχαστούμε αυτά τα κινήματα, να δούμε ποια είναι η σύνδεσή τους και η σημασία τους για εμάς. Σκοπός του παρόντος τόμου είναι να θέσει τη βάση αυτού του αναστοχασμού σε έναν διάλογο με τα κείμενα των ανθρώπων που ζουν αυτό το κίνημα, που μετέχουν σε αυτό, (...) που βιώνουν τη συλλογική πράξη που το αναδημιουργεί σε κάθε βήμα. Οι εξεγέρσεις του 21ου αιώνα δεν δίνουν απαντήσεις, αντίθετα φέρνουν στο φως τα πραγματικά ερωτήματα. Παρακολουθούμε τα Κίτρινα Γιλέκα όχι σαν θεατές ενός αγώνα από την κερκίδα, αλλά ως ενεργοί συνένοχοι στην προσπάθεια της κοινωνίας να απεγκλωβιστεί από το κράτος και το κεφάλαιο, να σπάσει τα δεσμά της. Συζητώντας για τα Κίτρινα Γιλέκα συζητάμε για το παρόν και το μέλλον μας. Και πριν μιλήσουμε, αυτή τη φορά, είναι σωστό πρώτα να ακούσουμε. (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)

The issue of whether the writings of Thomas Aquinas show internal contradictions has not only stirred readers from his earliest, often critical, reception, but also led to the emergence of a literary genre that has crucial relevance to the history of medieval Thomism. Concordances were drawn up which listed Thomas' contradictory statements and, in most cases, tried to disguise the appearance of contradiction by exegesis. But what was at stake in this interpretive endeavor? What role did the concordances play in (...) shaping Thomism? What tensions did they reveal in the works of Thomas? The book aims to investigate these questions and puts the concordance of Peter of Bergamo (1482), which represents the most important example of this type of text, at the center of the investigation. Contributors are Marieke Abram, Kent Emery, Jr., Maarten J.F.M. Hoenen, Isabel Iribarren, Thomas Jeschke, Catherine König-Pralong, Mario Meliadò, Silvia Negri, Zornitsa Radeva, and Peter Walter. (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.

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.

_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)

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)

The development of bacterial resistance to antibiotics is one of the best documented examples of contemporary biological evolution. Variability in the mechanisms of resistance depends on the diversity of genotypes in the huge bacterial populations, and also on the diversity of selective pressures that are produced along the antibiotic concentration gradients formed in the highly compartmentalized human body during therapy. These antibiotic gradients can be conceived as comprising selective compartments, each one of them defined as the concentration able to select (...) a particular genetic variant. In vitro experimental models confirm that some antibiotic resistant variants are selected only at certain selective concentrations of antibiotics. The correspondence between selective compartments and selectable variants could offer a way of describing more accurately the antibiotic selective landscapes and for taking measures to prevent the development of a major threat to the future of modern medicine. (shrink)

The development of bacterial resistance to antibiotics is one of the best documented examples of contemporary biological evolution. Variability in the mechanisms of resistance depends on the diversity of genotypes in the huge bacterial populations, and also on the diversity of selective pressures that are produced along the antibiotic concentration gradients formed in the highly compartmentalized human body during therapy. These antibiotic gradients can be conceived as comprising selective compartments, each one of them defined as the concentration able to select (...) a particular genetic variant. In vitro experimental models confirm that some antibiotic resistant variants are selected only at certain selective concentrations of antibiotics. The correspondence between selective compartments and selectable variants could offer a way of describing more accurately the antibiotic selective landscapes and for taking measures to prevent the development of a major threat to the future of modern medicine. (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)