Articles
'Choice and Iteration'
(with Beatriz Souza)
Philosophia Mathematica (Accepted)
(with Beatriz Souza)
Philosophia Mathematica (Accepted)
Abstract
The iterative conception is arguably the best worked out conception of set available. What is the status of the Axiom of Choice under this conception? Boolos argues that it is not justified by it. We show that Boolos's argument overgenerates. For, if cogent, it would imply that none of the axioms of ZFC which Boolos took to be justified by the iterative conception is so justified. We furthermore show that, to the extent that they are consequences of a plural formulation of Boolos's stage theory, all those axioms are justified by the iterative conception of set - Choice included.
The iterative conception is arguably the best worked out conception of set available. What is the status of the Axiom of Choice under this conception? Boolos argues that it is not justified by it. We show that Boolos's argument overgenerates. For, if cogent, it would imply that none of the axioms of ZFC which Boolos took to be justified by the iterative conception is so justified. We furthermore show that, to the extent that they are consequences of a plural formulation of Boolos's stage theory, all those axioms are justified by the iterative conception of set - Choice included.
'Counterparts as Near-Equals' | Link
(with Bruno Dinis)
Logic and Logical Philosophy (2025), Online First, pp. 1-23.
(with Bruno Dinis)
Logic and Logical Philosophy (2025), Online First, pp. 1-23.
Abstract
This paper offers an account of the ship of Theseus paradox along the lines of the so-called nonstandard primitivism about vagueness. This account is inspired by a model of the ship of Theseus paradox offered by Dinis that consideres near-equality, in the context of Nonstandard Analysis, as the proper way to model the 'same as' relation. The output is a class of models which unifies the semantic account of vague gradable adjectives recently proposed by Dinis and Jacinto with that of the 'same as' relation. It does so by taking both paradoxes to arise from a confusion between relations of marginal difference between vague degrees and "small" precise relations between the things that have those degrees.
This paper offers an account of the ship of Theseus paradox along the lines of the so-called nonstandard primitivism about vagueness. This account is inspired by a model of the ship of Theseus paradox offered by Dinis that consideres near-equality, in the context of Nonstandard Analysis, as the proper way to model the 'same as' relation. The output is a class of models which unifies the semantic account of vague gradable adjectives recently proposed by Dinis and Jacinto with that of the 'same as' relation. It does so by taking both paradoxes to arise from a confusion between relations of marginal difference between vague degrees and "small" precise relations between the things that have those degrees.
'A Theory of Marginal and Large Difference' | Link
(with Bruno Dinis)
Erkenntnis (2025), 90: 517-544.
(with Bruno Dinis)
Erkenntnis (2025), 90: 517-544.
Abstract
We propose a new theory based on the notions of marginal and large difference which has natural models in the context of nonstandard mathematics. We introduce the notion of finite marginality and show a representation result which ensures, for finitely marginal countable models, the existence of a homomorphism of the structure of marginal and large difference into a nonstandard model of the natural numbers, and show the extent to which any such homomorphism is unique. Finally, we show that our theory constitutes part of the underlying abstract structure of three distinct philosophical theories of vagueness: Dean’s neofeasibilism, Itzhaki’s theory of nonstandard heuristics, and our own initial sketch of a nonstandard primitivism about vagueness.
We propose a new theory based on the notions of marginal and large difference which has natural models in the context of nonstandard mathematics. We introduce the notion of finite marginality and show a representation result which ensures, for finitely marginal countable models, the existence of a homomorphism of the structure of marginal and large difference into a nonstandard model of the natural numbers, and show the extent to which any such homomorphism is unique. Finally, we show that our theory constitutes part of the underlying abstract structure of three distinct philosophical theories of vagueness: Dean’s neofeasibilism, Itzhaki’s theory of nonstandard heuristics, and our own initial sketch of a nonstandard primitivism about vagueness.
Abstract
This paper proposes and partially defends a novel philosophy of arithmetic -- Finitary Upper Logicism. According to it, the natural numbers are finite cardinalities - conceived of as properties of properties - and arithmetic is nothing but higher-order modal logic. Finitary Upper Logicism is furthermore essentially committed to the logicality of Finitary Plenitude, the principle according to which every finite cardinality could have been instantiated. Among other things, it is proved in the paper that second-order Peano arithmetic is interpretable, on the basis of the finite cardinalities' conception of the natural numbers, in a weak modal type theory consisting of the modal logic K, negative free quantified logic, a contingentist-friendly comprehension principle, and Finitary Plenitude. By replacing Finitary Plenitude for the Axiom of Infinity this result constitutes a significant improvement on Russell and Whitehead's interpretation of second-order Peano arithmetic, itself based on the finite cardinalities' conception of the natural numbers.
This paper proposes and partially defends a novel philosophy of arithmetic -- Finitary Upper Logicism. According to it, the natural numbers are finite cardinalities - conceived of as properties of properties - and arithmetic is nothing but higher-order modal logic. Finitary Upper Logicism is furthermore essentially committed to the logicality of Finitary Plenitude, the principle according to which every finite cardinality could have been instantiated. Among other things, it is proved in the paper that second-order Peano arithmetic is interpretable, on the basis of the finite cardinalities' conception of the natural numbers, in a weak modal type theory consisting of the modal logic K, negative free quantified logic, a contingentist-friendly comprehension principle, and Finitary Plenitude. By replacing Finitary Plenitude for the Axiom of Infinity this result constitutes a significant improvement on Russell and Whitehead's interpretation of second-order Peano arithmetic, itself based on the finite cardinalities' conception of the natural numbers.
'Bridge Principles and Epistemic Norms' | Link
(with Claire Field)
Erkenntnis (2024), 89: 1629-1681.
(with Claire Field)
Erkenntnis (2024), 89: 1629-1681.
Abstract
Is logic normative for belief? A standard approach to answering this question has been to investigate bridge principles relating claims of logical consequence to norms for belief. Although the question is naturally an epistemic one, bridge principles have typically been investigated in isolation from epistemic debates over the correct norms for belief. In this paper we tackle the question of whether logic is normative for belief by proposing a Kripkean model theory accounting for the interaction between logical, doxastic, epistemic and deontic notions and using this model theory to show which bridge principles are implied by epistemic norms that we have independent reason to accept, for example, the Knowledge norm and the Truth norm. We propose a preliminary theory of the interaction between logical, doxastic, epistemic and deontic notions that has among its commitments bridge principles expressing how logic is normative for belief. We also show how our framework suggests that logic is exceptionally normative.
Is logic normative for belief? A standard approach to answering this question has been to investigate bridge principles relating claims of logical consequence to norms for belief. Although the question is naturally an epistemic one, bridge principles have typically been investigated in isolation from epistemic debates over the correct norms for belief. In this paper we tackle the question of whether logic is normative for belief by proposing a Kripkean model theory accounting for the interaction between logical, doxastic, epistemic and deontic notions and using this model theory to show which bridge principles are implied by epistemic norms that we have independent reason to accept, for example, the Knowledge norm and the Truth norm. We propose a preliminary theory of the interaction between logical, doxastic, epistemic and deontic notions that has among its commitments bridge principles expressing how logic is normative for belief. We also show how our framework suggests that logic is exceptionally normative.
'Quineanism, Noneism and Metaphysical Equivalence' | Link
(with Javier Belastegui)
Studia Logica (2023), 112: 1019-1090.
(with Javier Belastegui)
Studia Logica (2023), 112: 1019-1090.
Abstract
In this paper we propose and defend the Synonymy account , a novel account of metaphysical equivalence which draws on the idea (Rayo, 2013) that part of what it is to formulate a theory is to lay down a theoretical hypothesis concerning logical space. Roughly, two theories are synonymous - and so, in our view, equivalent - just in case i) they take the same propositions to stand in the same entailment relations, and ii) they are committed to the truth of the same propositions. Furthermore, we put our proposal to work by showing that it affords a better and more nuanced understanding of the debate between Quineans and noneists. Finally we show how the Synonymy account fares better than some of its competitors, specifically, McSweeney’s (2016) epistemic account and Miller’s (2017) hyperintensional account.
In this paper we propose and defend the Synonymy account , a novel account of metaphysical equivalence which draws on the idea (Rayo, 2013) that part of what it is to formulate a theory is to lay down a theoretical hypothesis concerning logical space. Roughly, two theories are synonymous - and so, in our view, equivalent - just in case i) they take the same propositions to stand in the same entailment relations, and ii) they are committed to the truth of the same propositions. Furthermore, we put our proposal to work by showing that it affords a better and more nuanced understanding of the debate between Quineans and noneists. Finally we show how the Synonymy account fares better than some of its competitors, specifically, McSweeney’s (2016) epistemic account and Miller’s (2017) hyperintensional account.
'A Nonstandard view on vagueness' | Link
(with Bruno Dinis)
Proceedings of the 13th Panhellenic Logic Symposium (2021), Eds. G. Barmpalias and K. Tsaprounis
(with Bruno Dinis)
Proceedings of the 13th Panhellenic Logic Symposium (2021), Eds. G. Barmpalias and K. Tsaprounis
Abstract
We propose a new theory of vagueness based on the notions of marginal and large differences in the context of nonstandard mathematics. We apply this theory to an explanation of the seductiveness of the Sorites paradox by coupling it with Fara's interest-relative theory of vagueness.
We propose a new theory of vagueness based on the notions of marginal and large differences in the context of nonstandard mathematics. We apply this theory to an explanation of the seductiveness of the Sorites paradox by coupling it with Fara's interest-relative theory of vagueness.
'Models for Hylomorphism' | Link
(with Aaron J. Cotnoir)
Journal of Philosophical Logic (2019), 48: 909-955.
(with Aaron J. Cotnoir)
Journal of Philosophical Logic (2019), 48: 909-955.
Abstract
In a series of papers, Fine develops his hylomorphic theory of embodiments. In this article, we argue that the theory faces cardinality problems on broadly Cantorian grounds. We then stabilize the theory by supplying a formal semantics that is adequate to the principles laid down for it in Fine (1999). We examine the resulting mereology, showing that it is surprisingly robust: it is a partially ordered, atomistic, weakly supplemented, junky, but non-extensional mereology that satisfies (versions of) unrestricted composition. We draw some philosophical lessons from the formal semantics, and in particular respond to Koslicki's (2008) main objection to Fine's theory.
In a series of papers, Fine develops his hylomorphic theory of embodiments. In this article, we argue that the theory faces cardinality problems on broadly Cantorian grounds. We then stabilize the theory by supplying a formal semantics that is adequate to the principles laid down for it in Fine (1999). We examine the resulting mereology, showing that it is surprisingly robust: it is a partially ordered, atomistic, weakly supplemented, junky, but non-extensional mereology that satisfies (versions of) unrestricted composition. We draw some philosophical lessons from the formal semantics, and in particular respond to Koslicki's (2008) main objection to Fine's theory.
'Serious Actualism and Higher-Order Predication' | Link
Journal of Philosophical Logic (2019), 48: 471-499.
Journal of Philosophical Logic (2019), 48: 471-499.
Abstract
Serious actualism is the prima facie plausible thesis that things coudn't have been related while being nothing. The thesis plays an important role in a number of arguments in metaphysics, e.g., in Plantinga's argument (1983) for the claim that propositions do not ontologically depend on the things that they are about and in Williamson's argument (2002) for the claim that he, Williamson, is necessarily something. Salmon (1987) has put forward that which is, arguably, the most pressing challenge to serious actualists. Salmon's objection is based on a scenario intended to elicit the judgment that merely possible entities may nonetheless be actually referred to, and so may actually have properties.
Pace Salmon, it is shown that predicativism, the thesis that names are true of their bearers, provides the resources for replying to Salmon's objection. In addition, an argument for serious actualism based on (Stephanou 2007) is offered. Finally, it is shown that once serious actualism is conjoined with some minimal assumptions it implies property necessitism, the thesis that necessarily, all properties are necessarily something, as well as a strong comprehension principle for higher-order modal logic, according to which for every condition there necessarily is the property of being a thing satisfying that condition.
Serious actualism is the prima facie plausible thesis that things coudn't have been related while being nothing. The thesis plays an important role in a number of arguments in metaphysics, e.g., in Plantinga's argument (1983) for the claim that propositions do not ontologically depend on the things that they are about and in Williamson's argument (2002) for the claim that he, Williamson, is necessarily something. Salmon (1987) has put forward that which is, arguably, the most pressing challenge to serious actualists. Salmon's objection is based on a scenario intended to elicit the judgment that merely possible entities may nonetheless be actually referred to, and so may actually have properties.
Pace Salmon, it is shown that predicativism, the thesis that names are true of their bearers, provides the resources for replying to Salmon's objection. In addition, an argument for serious actualism based on (Stephanou 2007) is offered. Finally, it is shown that once serious actualism is conjoined with some minimal assumptions it implies property necessitism, the thesis that necessarily, all properties are necessarily something, as well as a strong comprehension principle for higher-order modal logic, according to which for every condition there necessarily is the property of being a thing satisfying that condition.
'Strongly Millian Second-Order Modal Logics' | Link
Review of Symbolic Logic (September 2017), 10(3): 397-454.
Review of Symbolic Logic (September 2017), 10(3): 397-454.
Abstract
The most common first- and second-order modal logics either have as theorems every instance of the Barcan and Converse Barcan formulae and of their second-order analogues, or else fail to capture the actual truth of every theorem of classical first- and second-order logic. In this paper we characterise and motivate sound and complete first- and second-order modal logics that successfully capture the actual truth of every theorem of classical first- and second-order logic and yet do not possess controversial instances of the Barcan and Converse Barcan formulae as theorems, nor of their second-order analogues. What makes possible these results is an understanding of the individual constants and predicates of the target languages as strongly Millian expressions, where a strongly Millian expression is one that has an actually existing entity as its semantic value. For this reason these logics are called 'strongly Millian'. It is shown that the strength of the strongly Millian second-order modal logics here characterised afford the means to resist an argument by Timothy Williamson for the truth of the claim that necessarily, every property necessarily exists.
The most common first- and second-order modal logics either have as theorems every instance of the Barcan and Converse Barcan formulae and of their second-order analogues, or else fail to capture the actual truth of every theorem of classical first- and second-order logic. In this paper we characterise and motivate sound and complete first- and second-order modal logics that successfully capture the actual truth of every theorem of classical first- and second-order logic and yet do not possess controversial instances of the Barcan and Converse Barcan formulae as theorems, nor of their second-order analogues. What makes possible these results is an understanding of the individual constants and predicates of the target languages as strongly Millian expressions, where a strongly Millian expression is one that has an actually existing entity as its semantic value. For this reason these logics are called 'strongly Millian'. It is shown that the strength of the strongly Millian second-order modal logics here characterised afford the means to resist an argument by Timothy Williamson for the truth of the claim that necessarily, every property necessarily exists.
'General-Elimination Stability' | Link
(with Stephen Read)
Studia Logica (April 2017), 105(2):361-405
(with Stephen Read)
Studia Logica (April 2017), 105(2):361-405
Abstract
General-elimination harmony articulates Gentzen's idea that the elimination-rules are justified if they infer from an assertion no more than can already be inferred from the grounds for making it. Dummett described the rules as not only harmonious but stable if the E-rules allow one to infer no more and no less than the I-rules justify. Pfenning and Davies call the rules locally complete if the E-rules are strong enough to allow one to infer the original judgement. A method is given of generating harmonious general-elimination rules from a collection of I-rules. We show that the general-elimination rules satisfy Pfenning and Davies' test for local completeness, but question whether that is enough to show that they are stable. Alternative conditions for stability are considered, including equivalence between the introduction- and elimination-meanings of a connective, and recovery of the grounds for assertion, finally generalizing the notion of local completeness to capture Dummett's notion of stability satisfactorily. We show that the general-elimination rules meet the last of these conditions, and so are indeed not only harmonious but also stable.
General-elimination harmony articulates Gentzen's idea that the elimination-rules are justified if they infer from an assertion no more than can already be inferred from the grounds for making it. Dummett described the rules as not only harmonious but stable if the E-rules allow one to infer no more and no less than the I-rules justify. Pfenning and Davies call the rules locally complete if the E-rules are strong enough to allow one to infer the original judgement. A method is given of generating harmonious general-elimination rules from a collection of I-rules. We show that the general-elimination rules satisfy Pfenning and Davies' test for local completeness, but question whether that is enough to show that they are stable. Alternative conditions for stability are considered, including equivalence between the introduction- and elimination-meanings of a connective, and recovery of the grounds for assertion, finally generalizing the notion of local completeness to capture Dummett's notion of stability satisfactorily. We show that the general-elimination rules meet the last of these conditions, and so are indeed not only harmonious but also stable.
'Knowing Who: How Perspectives and Contexts Interact' | Link
(with Maria Aloni)
Epistemology, Context and Formalism (2014), Eds. F. Lihoreau and M. Rebuschi
(with Maria Aloni)
Epistemology, Context and Formalism (2014), Eds. F. Lihoreau and M. Rebuschi
Abstract
In this article we investigate how conceptual perspectives and context interact in the determination of the truth of sentences in which 'knowing-wh' constructions occur.
In this article we investigate how conceptual perspectives and context interact in the determination of the truth of sentences in which 'knowing-wh' constructions occur.
Encyclopedia Entries & Outreach
'Normatividade da Lógica' | Link
(with Francisca Silva)
Online Companion to Problems of Analytic Philosophy (2022), Eds. P. Galvão and R. Santos
(with Francisca Silva)
Online Companion to Problems of Analytic Philosophy (2022), Eds. P. Galvão and R. Santos
Abstract
Several authors have questioned what is the relationship between logic’s object of study – what are the valid argument forms – and the way in which we ought to reason – how should our beliefs be updated and revised. Some of these authors think that logic is intimately connected to how we ought to think, whereas others reject this view – or at least that logic is normative in any way that distinguishes it from other descriptive sciences. In this paper we consider some of the main questions, debates and views on the normativity of logic. In particular, we analyse characterisations of validity in terms of its supposed normative role; we inquire into the relationship between logical laws and reasoning; we consider the influential collapse objection to logical pluralism, based on the normativity of logic; we discuss what are the logical constraints underlying minimal rationality; we investigate challenges to the view that logic is a source of normativity for our beliefs; we distinguish several bridge principles between logical principles and norms on how to reason correctly; we inquire as to whether these bridge principles are founded on more general epistemic norms; and we discuss whether, if logic is normative at all, it is autonomously normative or exceptionally normative in comparison to the descriptive sciences.
Several authors have questioned what is the relationship between logic’s object of study – what are the valid argument forms – and the way in which we ought to reason – how should our beliefs be updated and revised. Some of these authors think that logic is intimately connected to how we ought to think, whereas others reject this view – or at least that logic is normative in any way that distinguishes it from other descriptive sciences. In this paper we consider some of the main questions, debates and views on the normativity of logic. In particular, we analyse characterisations of validity in terms of its supposed normative role; we inquire into the relationship between logical laws and reasoning; we consider the influential collapse objection to logical pluralism, based on the normativity of logic; we discuss what are the logical constraints underlying minimal rationality; we investigate challenges to the view that logic is a source of normativity for our beliefs; we distinguish several bridge principles between logical principles and norms on how to reason correctly; we inquire as to whether these bridge principles are founded on more general epistemic norms; and we discuss whether, if logic is normative at all, it is autonomously normative or exceptionally normative in comparison to the descriptive sciences.
'Possible Worlds' | Link
Online Companion to Problems of Analytic Philosophy (2013), Eds. J. Branquinho and R. Santos
Online Companion to Problems of Analytic Philosophy (2013), Eds. J. Branquinho and R. Santos
Abstract
Possible worlds' semantics for modal logic has proven to be theoretically useful. But talk of possible worlds is puzzling. After all, what are possible worlds? This essay provides an overview of two of the main theories on the nature of possible worlds, namely, Lewis's Extreme Realism and Plantinga and Stalnaker's Moderate Realism. The essay also explores the merits and shortcomings of both theories.
Possible worlds' semantics for modal logic has proven to be theoretically useful. But talk of possible worlds is puzzling. After all, what are possible worlds? This essay provides an overview of two of the main theories on the nature of possible worlds, namely, Lewis's Extreme Realism and Plantinga and Stalnaker's Moderate Realism. The essay also explores the merits and shortcomings of both theories.
'O que raio são conjuntos?' | Link
Seminário de Jovens Cientistas, Academia das Ciências de Lisboa (2023)
Seminário de Jovens Cientistas, Academia das Ciências de Lisboa (2023)
