## About

60

Publications

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162

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Introduction

My research interests lay in set theory, logic, philosophy of mathematics and philosophy of language: theory of forcing, modal logic, justification of new axioms, arbitrary objects and speech acts. I am also interested in the historical aspects of the origin of the axiomatic method and the foundational work of David Hilbert.

Additional affiliations

July 2015 - July 2016

**IHPST**

Position

- Visiting fellow

March 2014 - January 2017

October 2009 - February 2014

## Publications

Publications (60)

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...

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 Propositi...

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-a...

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 wil...

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 c...

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 conjecture...

In this paper, we describe a tableau system for reasoning about ecumenical propositional logic, and introduce the central definitions of its implementation in the Coq proof assistant.

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 set...

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.

In this paper, we unify the study of classical and non-classical algebra-valued models of set theory, by studying variations of the interpretation functions for = and ∈. Although, these variations coincide with the standard interpretation in Boolean-valued constructions, nonetheless they extend the scope of validity of ZF to new algebra-valued mode...

We present recent results on the model companions of set theory, placing them in the context of the 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...

In this paper, we investigate the family L S0.5 of many-valued modal logics L S0.5 's. We prove that the modalities and ♦ of the logics L S0.5 's capture well-defined bivalent concepts of logical validity and logical consistency. We also show that these modalities can be used as recovery operators.

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 dimensi...

In this short paper we analyze whether assuming that mathematical objects are "thin" in Linnebo's sense simplifies the epistemol-ogy of mathematics. Towards this end we introduce the notion of transparency and we show that not all thin objects are transparent. We end by arguing that, far from being a weakness of thin objects, the lack of transparen...

2 هياكش و هلدم حِر ٌَلهناًة, ًة هؼر ٕوضاء ا يف هخور اك ٕسِام ا امللال ُذا يف احؽ ىر َط: املَخ هخور اك دفـت اًيت اًفَسفِة فاكر ٔ ال و اًخلٌَة اث الاجناز التخداع وحـَِا اجملموؿاث ًة هؼر ملدوةل. ملدمة: ا اترخي ّ ٕن ا املخـددت توحوُِا هناًة َ اًل هت اك ًع...

We offer a novel picture of mathematical language from the perspective of speech act theory. There are distinct speech acts within mathematics (not just assertions), 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...

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 t...

We present a generalization of the algebra-valued models of ZF 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 ZF.

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 propos...

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 th...

In this paper we argue that squeezing arguments à la Kreisel fail to univocally capture an informal or intuitive notion of validity. This suggests a form of logical pluralism, at a conceptual level, not only among but also within logical systems.

In this brief article we present the following paradox: one cannot assume that mathematicians are trustworthy when they express their mathematical (dis)beliefs, while also maintaining four basic theses about natural and mathematical language. We carefully present the very natural hypotheses on which this paradox is based and then we show how to ded...

At the end of the 19th century, genericity took an important step toward mathematical analysis, due to the developments promoted by the Italian school of algebraic geometry. However, its origins can be traced back to ancient mathematics in the work of prominent philosophers and mathematicians, such as Plato and Euclid. In this article, we will try...

In this paper we analyze the connection between reflexive-insensitive modal logics, logics of provability, and the modal logic of forcing. Because of the inter-definability of the $\circ$-operator that characterizes the reflexive-insensitive logics and Boolos's $\boxdot$-operator, characterization results for the reflexive-insensitive logics can be...

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 a...

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 ZF. Then, we build lattice-valued models of full ZF...

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.

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.

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 first 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 all universally Baire sets of reals.
We conclude our analysis with some basic conditions granting the model complet...

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 set...

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

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.

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 a...

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 a...

In this article we compare the notions of genericity and arbitrariness on the basis of the realist import of the method of forcing. We argue that Cohen's Theorem, similarly to Cantor's Theorem, can be considered a meta-theoretical argument in favor of the existence of uncountable collections. Then we discuss the effects of this meta-theoretical per...

In this article we review Cantor's contribution to the construction of a theory of infinity. We will present and discuss the technical achievements and the philosophical ideas that brought Cantor to the creation of set theory and to its justification.

In this article we analyze the notion of natural axiom in set theory. To this aim we review the intrinsic-extrinsic dichotomy, finding both the- oretical and practical difficulties in its use. We will describe and discuss a theoretical framework, that we will call conceptual realism, where the standard justification strategy is usually placed. In o...

Review of "David Hilbert's lectures on the foundations of arithmetic and logic (1917--1933)'', edited by William Ewald and Wilfried Sieg.

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 issu...

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 dis...

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, specifically the conditions under which sound axiomatizations of these logics...

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 gen...

We show how to force, with finite conditions, the forcing axiom PFA(
T
), a relativization of PFA to proper forcing notions preserving a given Suslin tree
T
. The proof uses a Neeman style iteration with generalized side conditions consisting of models of two types, and a preservation theorem for such iterations. The consistency of this axiom was p...

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 H...

We analyze a class of modal logics rendered insensitive to reflexivity by way
of a modification to the semantic definition of the modal operator. We explore
the extent to which these logics can be characterized, and prove a general
completeness theorem on the basis of a translation between normal modal logics
and their reflexive-insensitive counter...

In mathematical literature, it is quite common to make reference to an informal notion of naturalness: axioms or definitions may be defined as “natural,” and part of a proof may deserve the same label (i.e., “in a natural way…”). Our aim is to provide a philosophical account of these occurrences. The paper is divided in two parts. In the first part...

We show how to force, with finite conditions, the forcing axiom PFA(T), a
relativization of PFA to proper forcing notions preserving a given Souslin tree
T. The proof uses a Neeman style iteration with generalized side conditions
consisting of models of two types, and a preservation theorem for such
iterations. The consistency of this axiom was pre...

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...

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 condition...

In these notes we present the method introduced by Neeman of generalized side
conditions with two types of models. We then discuss some applications: the
Friedman-Mitchell poset for adding a club in \omega_2 with finite conditions,
Koszmider's forcing construction of a strong chain of length \omega_2 of
functions from \omega_1 to \omega_1, and the...

In these notes we present the method introduced by Neeman of generalized side conditions with two types of models. We then discuss some applications: the Friedman-Mitchell poset for adding a club in \omega_2 with finite conditions, Koszmider's forcing construction of a strong chain of length \omega_2 of functions from \omega_1 to \omega_1, and the...

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 notion of saturation of Aronszajn trees considered by Koenig, Larson, Moore and Velickovic and simplifies the original proof of Moore.