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97Thin Objects Are Not TransparentTheoria 89 (3): 314-325. 2023.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.
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13Fine’s Postulationism, Objectivity, and Mathematical CreationNoesis 38 123-137. 2024.We analyse Kit Fine’s proposal of a procedural Postulationism for mathematics. From a linguistic perspective, we argue that Postulationism is better understood in terms of declarative speech acts. Based on this observation, we argue in favor of a form of Declarationism able to account for both the objectivity of mathematics and its creative dimension.
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Naturalness in MathematicsIn Giorgio Venturi, Marco Panza & Gabriele Lolli (eds.), From Logic to Practice: Italian Studies in the Philosophy of Mathematics, Springer International Publishing. 2014.
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21At least one black sheep: Pragmatics and the language of mathematicsJournal of Pragmatics 1 (160): 114-119. 2020.In this paper we argue, against a somewhat standard view, that pragmatic phenomena occur in mathematical language. We provide concrete examples supporting this thesis.
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42Modelling Afthairetic ModalityJournal of Philosophical Logic 53 (4). 2024.Despite their controversial ontological status, the discussion on arbitrary objects has been reignited in recent years. According to the supporting views, they present interesting and unique qualities. Among those, two define their nature: their assuming of values, and the way in which they present properties. Leon Horsten has advanced a particular view on arbitrary objects which thoroughly describes the earlier, arguing they assume values according to a sui generis modality, which he calls afth…Read more
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23Ecumenical Propositional TableauStudia Logica 1-28. forthcoming.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.
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10The Palgrave Companion to the Philosophy of Set Theory (edited book)Palgrave. 2023.This volume showcases some of the up-and-coming voices of an emerging field - the philosophy of set theory - which in recent years has gained prominence in the philosophy of mathematics. The chapters in this volume both present new topics and propose solutions to old problems. It contains a broad picture of the philosophy of set theory, examining questions from epistemology and ontology, whilst touching on the use of formal theories in the study of mathematical infinity. Key features of this vol…Read more
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21How to make (mathematical) assertions with directivesSynthese 202 (5): 1-16. 2023.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 t…Read more
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42Neighborhood Semantics for Logics of Unknown Truths and False BeliefsAustralasian Journal of Logic 14 (1). 2017.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 operat…Read more
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51Axioms and Postulates as Speech ActsErkenntnis 89 (8): 3183-3202. 2024.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.
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42Tableaux for essence and contingencyLogic Journal of the IGPL 29 (5): 719-738. 2021.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.
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68Speech acts in mathematicsSynthese 198 (10): 10063-10087. 2020.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 a…Read more
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26What Model Companionship Can Say About the Continuum ProblemReview of Symbolic Logic 17 (2): 546-585. 2024.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 th…Read more
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56Formal Ontology and Mathematics. A Case Study on the Identity of ProofsTopoi 42 (1): 307-321. 2023.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) dev…Read more
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36Logics of Ignorance and Being WrongLogic Journal of the IGPL 30 (5): 870-885. 2022.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 s…Read more
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415How 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.
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30Second order arithmetic as the model companion of set theoryArchive for Mathematical Logic 62 (1): 29-53. 2023.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 ord…Read more
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27Many-Valued Logics and Bivalent ModalitiesLogic and Logical Philosophy 1-26. forthcoming.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.
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48Ideal Objects for Set TheoryJournal of Philosophical Logic 51 (3): 583-602. 2022.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.
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41$$\mathrm {ZF}$$ ZF Between Classicality and Non-classicalityStudia Logica 110 (1): 189-218. 2021.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 \.
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36Independence Proofs in Non-Classical Set TheoriesReview of Symbolic Logic 16 (4): 979-1010. 2023.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 indepen…Read more
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47On Negation for Non-classical Set TheoriesJournal of Philosophical Logic 50 (3): 549-570. 2020.We present a case study for the debate between the American and the Australian plans, analyzing a crucial aspect of negation: expressivity within a theory. We discuss the case of non-classical set theories, presenting three different negations and testing their expressivity within algebra-valued structures for ZF-like set theories. We end by proposing a minimal definitional account of negation, inspired by the algebraic framework discussed.
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31A Non-Standard Kripke Semantics for the Minimal Deontic LogicLogic and Logical Philosophy 1. forthcoming.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.
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906On Forms of Justification in Set TheoryAustralasian Journal of Logic 17 (4): 158-200. 2020.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 …Read more
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32Non-classical Models of ZFStudia Logica 109 (3): 509-537. 2020.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…Read more
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21Forcing, Multiverse and RealismIn Francesca Boccuni & Andrea Sereni (eds.), Objectivity, Realism, and Proof. FilMat Studies in the Philosophy of Mathematics, Springer International Publishing. 2016.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 conc…Read more
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69A note on logics of essence and accidentLogic Journal of the IGPL 28 (5): 881-891. 2020.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.
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31Book review: Linnebo, ø., philosophy of mathematics (review)Manuscrito 42 (2): 113-119. 2019.We review Linnebo's Philosophy of Mathematics, briefly describing the content of the book.