In the Tractatus, it is stated that questions about logical formatting cannot be meaningfully formulated, since it is precisely the application of logical rules which enables the formulation of a question whatsoever; analogously, Wittgenstein’s celebrated infinite regress argument on rule-following seems to undermine any explanation of deduction, as relying on a logical argument. On the other hand, some recent mathematical developments of the Curry-Howard bridge between proof theory and type theory address the issue of describing the “subjective” side of logic, (...) that is, the concrete manipulation of rules and proofs in space and time. It is advocated that such developments can shed some light on the question of logical formatting and its apparently unintelligible paradoxes, thus reconsidering Wittgenstein’s verdict. (shrink)

In the contemporary philosophy of set theory, discussion of new axioms that purport to resolve independence necessitates an explanation of how they come to be justified. Ordinarily, justification is divided into two broad kinds: intrinsic justification relates to how `intuitively plausible' an axiom is, whereas extrinsic justification supports an axiom by identifying certain `desirable' consequences. This paper puts pressure on how this distinction is formulated and construed. In particular, we argue that the distinction as often presented is neither well-demarcated nor (...) sufficiently precise. Instead, we suggest that the process of justification in set theory should not be thought of as neatly divisible in this way, but should rather be understood as a conceptually indivisible notion linked to the goal of explanation. (shrink)

In this paper, we argue for an instrumental form of existence, inspired by Hilbert’s method of ideal elements. As a case study, we consider the existence of contradictory objects in models of non-classical set theories. Based on this discussion, we argue for a very liberal notion of existence in mathematics.

How does our language relate to reality? This is a question that is especially pertinent in set theory, where we seem to talk of large infinite entities. Based on an analogy with the use of models in the natural sciences, we argue for a threefold correspondence between our language, models, and reality. We argue that so conceived, the existence of models can be underwritten by a weak notion of existence, where weak existence is to be understood as existing in virtue (...) of language. (shrink)

We present recent results on the model companions of set theory, placing them in the context of a current debate in the philosophy of mathematics. We start by describing the dependence of the notion of model companionship on the signature, and then we analyze this dependence in the specific case of set theory. We argue that the most natural model companions of set theory describe (as the signature in which we axiomatize set theory varies) theories of $H_{\kappa ^+}$, as $\kappa (...) $ ranges among the infinite cardinals. We also single out $2^{\aleph _0}=\aleph _2$ as the unique solution of the continuum problem which can (and does) belong to some model companion of set theory (enriched with large cardinal axioms). While doing so we bring to light that set theory enriched by large cardinal axioms in the range of supercompactness has as its model companion (with respect to its first order axiomatization in certain natural signatures) the theory of $H_{\aleph _2}$ as given by a strong form of Woodin’s axiom $(*)$ (which holds assuming $\mathsf {MM}^{++}$ ). Finally this model-theoretic approach to set-theoretic validities is explained and justified in terms of a form of maximality inspired by Hilbert’s axiom of completeness. (shrink)

We analyze axioms and postulates as speech acts. After a brief historical appraisal of the concept of axiom in Euclid, Frege, and Hilbert, we evaluate contemporary axiomatics from a linguistic perspective. Our reading is inspired by Hilbert and is meant to account for the assertive, directive, and declarative components of modern axiomatics. We will do this by describing the constitutive and regulative roles that axioms possess with respect to the linguistic practice of mathematics.

In this paper we study a new operator of strong modality ⊞, related to the non-contingency operator ∆. We then provide soundness and completeness theorems for the minimal logic of the ⊞-operator.

In this paper we extend to non-classical set theories the standard strategy of proving independence using Boolean-valued models. This extension is provided by means of a new technique that, combining algebras (by taking their product), is able to provide product-algebra-valued models of set theories. In this paper we also provide applications of this new technique by showing that: (1) we can import the classical independence results to non-classical set theory (as an example we prove the independence of $\mathsf {CH}$ ); (...) and (2) we can provide new independence results. We end by discussing the role of non-classical algebra-valued models for the debate between universists and multiversists and by arguing that non-classical models should be included as legitimate members of the multiverse. (shrink)

In this article we analyze the method of forcing from a more philosophical perspective. After a brief presentation of this technique we outline some of its philosophical imports in connection with realism. We shall discuss some philosophical reactions to the invention of forcing, concentrating on Mostowski’s proposal of sharpening the notion of generic set. Then we will provide an overview of the notions of multiverse and the related philosophical debate on the foundations of set theory. In conclusion, we connect this (...) modern debate and Mostowski’s proposal, suggesting a way to analyze the notion of genericity within the framework of a multiverse structure. (shrink)

In this article we present a technique for selecting models of set theory that are complete in a model-theoretic sense. Specifically, we will apply Robinson infinite forcing to the collections of models of ZFC obtained by Cohen forcing. This technique will be used to suggest a unified perspective on generic absoluteness principles.

This is an introductory paper to a series of results linking generic absoluteness results for second and third order number theory to the model theoretic notion of model companionship. Specifically we develop here a general framework linking Woodin’s generic absoluteness results for second order number theory and the theory of universally Baire sets to model companionship and show that (with the required care in details) a $$\Pi _2$$ -property formalized in an appropriate language for second order number theory is forcible (...) from some $$T\supseteq \mathsf {ZFC}+$$ large cardinals if and only if it is consistent with the universal fragment of T if and only if it is realized in the model companion of T. In particular we show that the first order theory of $$H_{\omega _1}$$ is the model companion of the first order theory of the universe of sets assuming the existence of class many Woodin cardinals, and working in a signature with predicates for $$\Delta _0$$ -properties and for all universally Baire sets of reals. We will extend these results also to the theory of $$H_{\aleph _2}$$ in a follow up of this paper. (shrink)

We present a generalization of the algebra-valued models of \ where the axioms of set theory are not necessarily mapped to the top element of an algebra, but may get intermediate values, in a set of designated values. Under this generalization there are many algebras which are neither Boolean, nor Heyting, but that still validate \.

We offer tableaux systems for logics of essence and accident and logics of non-contingency, showing their soundness and completeness for Kripke semantics. We also show an interesting parallel between these logics based on the semantic insensitivity of the two non-normal operators by which these logics are expressed.

We propose a novel, ontological approach to studying mathematical propositions and proofs. By “ontological approach” we refer to the study of the categories of beings or concepts that, in their practice, mathematicians isolate as fruitful for the advancement of their scientific activity (like discovering and proving theorems, formulating conjectures, and providing explanations). We do so by developing what we call a “formal ontology” of proofs using semantic modeling tools (like RDF and OWL) developed by the computer science community. In this (...) article, (i) we describe this new approach and, (ii) to provide an example, we apply it to the problem of the identity of proofs. We also describe open issues and further applications of this approach (for example, the study of purity of methods). We lay some foundations to investigate rigorously and at large scale intellectual moves and attitudes that underpin the advancement of mathematics through cognitive means (carving out investigationally valuable concepts and techniques) and social means (like communication, collaboration, revision, and criticism of specific categories, inferential patterns, and levels of analysis). Our approach complements other types of analysis of proofs such as reconstruction in a deductive system and examination through a proof-assistant. (shrink)

In this paper, we examine the logics of essence and accident and attempt to ascertain the extent to which those logics are genuinely formalizing the concepts in which we are interested. We suggest that they are not completely successful as they stand. We diagnose some of the problems and make a suggestion for improvement. We also discuss some issues concerning definability in the formal language.

In this paper, we investigate the family LS0.5 of many-valued modal logics LS0.5's. We prove that the modalities of necessity and possibility of the logics LS0.5's capture well-defined bivalent concepts of logical validity and logical consistency. We also show that these modalities can be used as recovery operators.

This article outlines a semantic approach to the logics of unknown truths, and the logic of false beliefs, using neighborhood structures, giving results on soundness, completeness, and expressivity. Relational semantics for the logics of unknown truths are also addressed, specically the conditions under which sound axiomatizations of these logics might be obtained from their normal counterparts, and the relationship between refexive insensitive logics and logics containing the provability operator as the primary modal operator.

It is prima facie uncontroversial that the justification of an assertion amounts to a collection of other (inferentially related) assertions. In this paper, we point at a class of assertions, i.e. mathematical assertions, that appear to systematically flout this principle. To justify a mathematical assertion (e.g. a theorem) is to provide a proof—and proofs are sequences of directives. The claim is backed up by linguistic data on the use of imperatives in proofs, and by a pragmatic analysis of theorems and (...) their proofs. Proofs, we argue, are sequences of instructions whose performance inevitably gets one to truth. It follows that a felicitous theorem, i.e. a theorem that has been correctly proven, is a persuasive theorem. When it comes to mathematical assertions, there is no sharp distinction between illocutionary and perlocutionary success. (shrink)

ABSTRACT We present and discuss a change in the introduction of Hilbert’s Grundlagen der Geometrie between the first and the subsequent editions: the disappearance of the reference to the independence of the axioms. We briefly outline the theoretical relevance of the notion of independence in Hilbert’s work and we suggest that a possible reason for this disappearance is the discovery that Hilbert’s axioms were not, in fact, independent. In the end we show how this change gives textual evidence for the (...) connection between the notions of independence and simplicity. (shrink)

In this article I propose to look at set theory not only as a founda­tion of mathematics in a traditional sense, but as a foundation for mathemat­ical practice. For this purpose I distinguish between a standard, ontological, set theoretical foundation that aims to find a set theoretical surrogate to every mathematical object, and a practical one that tries to explain mathematical phenomena, giving necessary and sufficient conditions for the proof of mathematical propositions. I will present some example of this use (...) of set theoretical methods, in the context of mainstream mathematics, in terms of independence proofs, equiconsistency results and discussing some recent results that show how it is possible to “complete” the structures H(ℵ1) and H(ℵ2). Then I will argue that a set theoretical foundation of mathematics can be relevant also for the philosophy of mathematical practice, as long as some axioms of set theory can be seen as explanations of mathematical phenomena. In the end I will propose a more general distinction between two different kinds of foundation: a practical one and a theoretical one, drawing some examples from the history of the foundation of mathematics. (shrink)

In this article I propose to look at set theory not only as a founda­tion of mathematics in a traditional sense, but as a foundation for mathemat­ical practice. For this purpose I distinguish between a standard, ontological, set theoretical foundation that aims to find a set theoretical surrogate to every mathematical object, and a practical one that tries to explain mathematical phenomena, giving necessary and sufficient conditions for the proof of mathematical propositions. I will present some example of this use (...) of set theoretical methods, in the context of mainstream mathematics, in terms of independence proofs, equiconsistency results and discussing some recent results that show how it is possible to “complete” the structures H(ℵ1) and H(ℵ2). Then I will argue that a set theoretical foundation of mathematics can be relevant also for the philosophy of mathematical practice, as long as some axioms of set theory can be seen as explanations of mathematical phenomena. In the end I will propose a more general distinction between two different kinds of foundation: a practical one and a theoretical one, drawing some examples from the history of the foundation of mathematics. (shrink)

: In this article we analyze the key concept of Hilbert's axiomatic method, namely that of axiom. We will find two different concepts: the first one from the period of Hilbert's foundation of geometry and the second one at the time of the development of his proof theory. Both conceptions are linked to two different notions of intuition and show how Hilbert's ideas are far from a purely formalist conception of mathematics. The principal thesis of this article is that one (...) of the main problems that Hilbert encountered in his foundational studies consisted in securing a link between formalization and intuition. We will also analyze a related problem, that we will call "Frege's Problem", form the time of the foundation of geometry and investigate the role of the Axiom of Completeness in its solution. (shrink)

This paper aims to show how the mathematical content of Hilbert's Axiom of Completeness consists in an attempt to solve the more general problem of the relationship between intuition and formalization. Hilbert found the accordance between these two sides of mathematical knowledge at a logical level, clarifying the necessary and sufficient conditions for a good formalization of geometry. We will tackle the problem of what is, for Hilbert, the definition of geometry. The solution of this problem will bring out how (...) Hilbert's conception of mathematics is not as innovative as his conception of the axiomatic method. The role that the demonstrative tools play in Hilbert's foundational reflections will also drive us to deal with the problem of the purity of methods, explicitly addressed by Hilbert. In this respect Hilbert's position is very innovative and deeply linked to his modern conception of the axiomatic method. In the end we will show that the role played by the Axiom of Completeness for geometry is the same as the Axiom of Induction for arithmetic and of Church-Turing thesis for computability theory. We end this paper arguing that set theory is the right context in which applying the axiomatic method to mathematics and we postpone to a sequel of this work the attempt to offer a solution similar to Hilbert's for the completeness of set theory. (shrink)

We offer a novel picture of mathematical language from the perspective of speech act theory. There are distinct speech acts within mathematics, and, as we intend to show, distinct illocutionary force indicators as well. Even mathematics in its most formalized version cannot do without some such indicators. This goes against a certain orthodoxy both in contemporary philosophy of mathematics and in speech act theory. As we will comment, the recognition of distinct illocutionary acts within logic and mathematics and the incorporation (...) of illocutionary force indicators in the formal language for both goes back to Frege’s conception of these topics. We are, therefore, going back to a Fregean perspective. This paper is part of a larger project of applying contemporary speech act theory to the scientific language of mathematics in order to uncover the varieties and regular combinations of illocutionary acts present in it. For reasons of space, we here concentrate only on assertive and declarative acts within mathematics, leaving the investigation of other kinds of acts for a future occasion. (shrink)

We review Linnebo's Philosophy of Mathematics, briefly describing the content of the book.

This article investigates the connections between the logics of being wrong, introduced in Steinsvold (2011, Notre Dame J. Form. Log., 52, 245–253), and factive ignorance, presented in Kubyshkina and Petrolo (2021, Synthese, 198, 5917–5928). The first part of the paper provides a sound and complete axiomatization of the logic of factive ignorance that corrects errors in Kubyshkina and Petrolo (2021, Synthese, 198, 5917–5928) and resolves questions about the expressivity of the language. In the second half, it is shown that the (...) relationship between the two logics suggests an alternative axiomatization for the logic of factive ignorance, the adequacy of which is easily proved via straightforward translations. (shrink)

In this paper we argue, against a somewhat standard view, that pragmatic phenomena occur in mathematical language. We provide concrete examples supporting this thesis.

We present a direct proof of the consistency of the existence of a five element basis for the uncountable linear orders. Our argument is based on the approach of König, Larson, Moore and Veličković and simplifies the original proof of Moore.

Ecumenical logic aims to peacefully join classical and intuitionistic logic systems, allowing for reasoning about both classical and intuitionistic statements. This paper presents a semantic tableau for propositional ecumenical logic and proves its soundness and completeness concerning Ecumenical Kripke models. We introduce the Ecumenical Propositional Tableau ( $$E_T$$ ) and demonstrate its effectiveness in handling mixed statements.

In this short paper, we analyse whether assuming that mathematical objects are “thin” in Linnebo's sense simplifies the epistemology of mathematics. Towards this end, we introduce the notion of transparency and show that not all thin objects are transparent. We end by arguing that, far from being a weakness of thin objects, the lack of transparency of some thin objects is a fruitful characteristic mark of abstract mathematics.

In this book, one of Italy's most important and original contemporary philosophers considers the status of art in the modern era. He takes seriously Hegel's claim that art has exhausted its spiritual vocation. He argues, however, that Hegel by no means proclaimed the 'death of art' (as many still imagine) but proclaimed rather the indefinite continuation of art in a 'self-annulling' mode. With astonishing breadth and originality, he probes the meaning, aesthetics, and historical consequences of that self-annulment. He argues that (...) the birth of modern aesthetics is the result of a series of schisms - such as between artist and spectator, genius and taste, and form and matter - that are manifestations of the deeper, self-negating yet self-perpetuating movement of irony. He offers an imaginative reinterpretation of the history of aesthetic theory from Kant to Heidegger. The final chapter offers a dazzling interpretation of Dürer's Melancholia. (shrink)

The work of Giorgio Agamben, one of Italy's most important and original philosophers, has been based on an uncommon erudition in classical traditions of philosophy and rhetoric, the grammarians of late antiquity, Christian theology, and modern philosophy. Recently, Agamben has begun to direct his thinking to the constitution of the social and to some concrete, ethico-political conclusions concerning the state of society today, and the place of the individual within it. In Homo Sacer, Agamben aims to connect the problem (...) of pure possibility, potentiality, and power with the problem of political and social ethics in a context where the latter has lost its previous religious, metaphysical, and cultural grounding. Taking his cue from Foucault's fragmentary analysis of biopolitics, Agamben probes with great breadth, intensity, and acuteness the covert or implicit presence of an idea of biopolitics in the history of traditional political theory. He argues that from the earliest treatises of political theory, notably in Aristotle's notion of man as a political animal, and throughout the history of Western thinking about sovereignty, a notion of sovereignty as power over "life" is implicit. The reason it remains merely implicit has to do, according to Agamben, with the way the sacred, or the idea of sacrality, becomes indissociable from the idea of sovereignty. Drawing upon Carl Schmitt's idea of the sovereign's status as the exception to the rules he safeguards, and on anthropological research that reveals the close interlinking of the sacred and the taboo, Agamben defines the sacred person as one who can be killed and yet not sacrificed—a paradox he sees as operative in the status of the modern individual living in a system that exerts control over the collective "naked life" of all individuals. (shrink)

Why has power in the West assumed the form of an "economy," that is, of a government of men and things? If power is essentially government, why does it need glory, that is, the ceremonial and liturgical apparatus that has always accompanied it? In the early centuries of the Church, in order to reconcile monotheism with God's threefold nature, the doctrine of Trinity was introduced in the guise of an economy of divine life. It was as if the Trinity amounted (...) to nothing more than a problem of managing and governing the heavenly house and the world. Agamben shows that, when combined with the idea of providence, this theological-economic paradigm unexpectedly lies at the origin of many of the most important categories of modern politics, from the democratic theory of the division of powers to the strategic doctrine of collateral damage, from the invisible hand of Smith's liberalism to ideas of order and security. But the greatest novelty to emerge from _The Kingdom and the Glory _ is that modern power is not only government but also glory, and that the ceremonial, liturgical, and acclamatory aspects that we have regarded as vestiges of the past actually constitute the basis of Western power. Through a fascinating analysis of liturgical acclamations and ceremonial symbols of power—the throne, the crown, purple cloth, the Fasces, and more—Agamben develops an original genealogy that illuminates the startling function of consent and of the media in modern democracies. With this book, the work begun with _Homo Sacer_ reaches a decisive point, profoundly challenging and renewing our vision of politics. (shrink)

In this highly topical book, Agamben ultimately arrives at original ideas about the future of democracy and casts a new light on the hidden relationship that ties law to violence.

The end of human history is an event that has been foreseen or announced by both messianics and dialecticians. But who is the protagonist of that history that is coming—or has come—to a close? What is man? How did he come on the scene? And how has he maintained his privileged place as the master of, or first among, the animals? In The Open, contemporary Italian philosopher Giorgio Agamben considers the ways in which the “human” has been thought of (...) as either a distinct and superior type of animal, or a kind of being that is essentially different from animal altogether. In an argument that ranges from ancient Greek, Christian, and Jewish texts to twentieth-century thinkers such as Heidegger, Benjamin, and Kojève, Agamben examines the ways in which the distinction between man and animal has been manufactured by the logical presuppositions of Western thought, and he investigates the profound implications that the man/animal distinction has had for disciplines as seemingly disparate as philosophy, law, anthropology, medicine, and politics. (shrink)

Giorgio Agamben's Homo Sacer was one of the seminal works of political philosophy in recent decades. It was also the beginning of a series of interconnected investigations of staggering ambition and scope, investigating the deepest foundations of Western politics and thought. The Use of Bodies represents the ninth and final volume in this twenty-year undertaking, breaking considerable new ground while clarifying the stakes and implications of the project as a whole. It comprises three major sections. The first uses Aristotle's (...) discussion of slavery as a starting point for radically rethinking notions of selfhood; the second calls for a complete reworking of Western ontology; and the third explores the enigmatic concept of "form-of-life," which is in many ways the motivating force behind the entire Homo Sacer project. Interwoven between these major sections are shorter reflections on individual thinkers, while the epilogue pushes toward a new approach to political life that breaks with the destructive deadlocks of Western thought. The Use of Bodies represents a true masterwork by one of our greatest living philosophers. (shrink)

This volume constitutes the largest collection of writings by the Italian philosopher Giorgio Agamben hitherto published in any language and all but one appear in English for the first time. The essays consider figures in the history of philosophy (Plato, Plotinus, Spinoza, Hegel) and twentieth-century thought (Walter Benjamin, Heidegger, Derrida, Deleuze, the historian Aby Warburg, and the linguist J.-C. Milner). They also examine several central concerns of Agamben: the relation of linguistic and metaphysical categories; messianism in Islamic, Jewish, and (...) Christian theology; and the state and future of contemporary politics. Despite the diversity of the texts collected here, they show a consistent concern for a set of overriding philosophical themes concerning language, history, and potentiality. (shrink)

Arguing that Western power is both "government" and "glory," this book reveals the "theological-economic" paradigm at the origin of several of the most important components of modern politics and illuminates the function of consent and the ...

We can no longer speak of a state of war in any traditional sense, yet there is currently no viable theory to account for the manifold internal conflicts, or civil wars, that increasingly afflict the world's populations. Meant as a first step toward such a theory, Giorgio Agamben's latest book looks at how civil war was conceived of at two crucial moments in the history of Western thought: in ancient Athens (from which the political concept of stasis emerges) and (...) later, in the work of Thomas Hobbes. It identifies civil war as the fundamental threshold of politicization in the West, an apparatus that over the course of history has alternately allowed for the de-politicization of citizenship and the mobilization of the unpolitical. The arguments herein, first conceived of in the immediate aftermath of 9/11, have become ever more relevant now that we have entered the age of planetary civil war. (shrink)

What is an apparatus? -- The friend -- What is the contemporary?

This paper contributes to the generalization of lattice-valued models of set theory to non-classical contexts. First, we show that there are infinitely many complete bounded distributive lattices, which are neither Boolean nor Heyting algebra, but are able to validate the negation-free fragment of \. Then, we build lattice-valued models of full \, whose internal logic is weaker than intuitionistic logic. We conclude by using these models to give an independence proof of the Foundation axiom from \.

In this follow-up to The Kingdom and the Glory and The Highest Poverty, Agamben investigates the roots of our moral concept of duty in the theory and practice of Christian liturgy. Beginning with the New Testament and working through to late scholasticism and modern papal encyclicals, Agamben traces the Church's attempts to repeat Christ's unrepeatable sacrifice. Crucial here is the paradoxical figure of the priest, who becomes more and more a pure instrument of God's power, so that his own motives (...) and character are entirely indifferent as long as he carries out his priestly duties. In modernity, Agamben argues, the Christian priest has become the model ethical subject. We see this above all in Kantian ethics. Contrasting the Christian and modern ontology of duty with the classical ontology of being, Agamben contends that Western philosophy has unfolded in the tension between the two. This latest installment in the study of Western political structures begun in Homo Sacer is a contribution to the study of liturgy, an extension of Nietzsche's genealogy of morals, and a reworking of Heidegger's history of Being. (shrink)