Carlo Nicolai

I am currently VENI NWO Research Fellow at the University of Utrecht. From September 2018, I'll be Lecturer in Philosophy (Logic) at King's College London. I obtained my DPhil from the University of Oxford in 2014. My research mostly focuses on the logic and the nature of the notions of truth, consequence, predication.


Research    Recent talks    Teaching    Manuale di Logica    Contact   

Research

"Iterated reflection over full disquotational truth" (with M. Fischer and L. Horsten), Journal fo Logic and Computation, online-first.
Abstract, Iterated reflection principles have been employed extensively to unfold epistemic commitments that are incurred by accepting a mathematical theory. Recently this has been applied to theories of truth. The idea is to start with a collection of Tarski biconditionals and arrive by iterated reflection at strong compositional truth theories. In the context of classical logic, it is incoherent to adopt an initial truth theory in which A and ‘A is true’ are inter-derivable. In this article, we show how in the context of a weaker logic, which we call Basic De Morgan Logic, we can coherently start with such a fully disquotational truth theory and arrive at a strong compositional truth theory by applying a natural uniform reflection principle a finite number of times. Preprint, DOI, BibTex. @article{fhn17,
author={Martin Fischer and Carlo Nicolai and Leon Horsten},
year={2017},
title={Iterated reflection over full disquotational truth}, publisher={Oxford University Press},
journal={Journal of Logic and Computation},
note={Online-First, DOI: https://doi.org/10.1093/logcom/exx023},
}

"On implicit commitment for arithmetical theories and the semantic core" (with M. Piazza), submitted.
Abstract, According to the implicit commitment thesis, once accepting a mathematical formal system S, one is implicitly committed to additional resources not immediately available in S. Traditionally, this thesis has been understood as entailing that, in accepting S, we are bound to accept reflection principles for S and therefore claims in the language of S that are not derivable in S itself. It has recently become clear, however, that such reading of the implicit commitment thesis cannot be compatible with well-established positions in the foundation of mathematics which consider a specific theory S as selfjustifying and doubt the legitimacy of any principle that is not derivable S: examples are Tait's finitism and the role played in it by Primitive Recursive Arithmetic, Isaacson's thesis and Peano Arithmetic, Nelson's ultrafinitism and sub-exponential arithmetical systems. This casts doubts on the very adequacy of the implicit commitment thesis for arithmetical theories. In the paper we show that such foundational standpoints are nonetheless compatible with the implicit commitment thesis. We also show that they can even be compatible with genuine soundness extensions of S with suitable forms of reflection. The analysis we propose is as follows: when accepting of system S, we are bound to accept a fixed set of principles extending S and expressing minimal soundness requirements for S, such as the fact that the non-logical axioms of S are true. We call this invariant component the semantic core of implicit commitment. But there is also a variable component of implicit commitment that crucially depends on the justification given for our acceptance of S in which, for instance, may or may not appear (proof-theoretic) reflection principles for S. We claim that the proposed framework regulates in a natural and uniform way our acceptance of different arithmetical theories. Preprint.

"On expressive power over arithmetic", forthcoming in M. Piazza and G. Pulcini (eds), Truth, Existence, and Explanation - FilMat Studies in the Philosophy of Mathematics.
Abstract, The paper is concerned with the ne boundary between expressive power and reducibility of semantic and intensional notions in the context of arithmetical theories. I will consider three notions of reduction of a theory characterizing a semantic or a modal notion to the underlying base theory – relative interpretability, speed up, conservativeness – and highlight a series of cases where moving between equally satisfactory base theories and keeping the semantic or modal principles xed yields incompatible results. I then consider the impact of the non-uniform behaviour of these reducibility relations on the philosophical signicance we usually attribute to them. Preprint.

"Provably true sentences across axiomatizations of Kripke's theory of truth", Studia Logica, online-first.
Abstract, We study the relationships between two clusters of axiomatizations of Kripke’s xed-point models for languages containing a self-applicable truth predicate. e rst cluster is represented by what we will call ‘PKF-like’ theories, originating in recent work Halbach and Horsten, whose axioms and rules (in Basic De Morgan Logic) are all valid in xed-point models; the second by ‘KF-like’ theories rst introduced by Solomon Feferman, that lose this property but reect the classicality of the metatheory in which Kripke’s construction is carried out. We show that to any natural system in one cluster – corresponding to natural variations on induction schemata – there is a corresponding system in the other proving the same sentences true, addressing a problem le open by Halbach and Horsten and accomplishing a suitably modied version of the project sketched by Reinhardt aiming at an instrumental reading of classical theories of self-applicable truth. Preprint, DOI, BibTex. @article{nic17,
author={Carlo Nicolai},
year={2017},
title={Provably true sentences across axiomatizations of Kripke's theory of truth}, publisher={Springer},
journal={Studia Logica},
note={Online-First, DOI: https://link.springer.com/article/10.1007/s11225-017-9727-y},
}

"Principles for object-linguistic consequence: from logical to irreflexive", Journal of Philosophical Logic, online-first.
Abstract, We discuss the principles for a primitive, object-linguistic notion of consequence proposed by [Beall and Murzi 2013] that yield a version of Curry’s paradox. We propose and study several strategies to weaken these principles and overcome paradox: all these strategies are based on the intuition that the object-linguistic consequence predicate internalizes whichever meta-linguistic notion of consequence we accept in the rst place. To these solutions will correspond dierent conceptions of consequence. In one possible reading of these principles, they give rise to a notion of logical consequence: we study the corresponding theory of validity (and some of its variants) by showing that it is conservative over a wide range of base theories: this result is achieved via a well-behaved form of local interpretation. The theory of logical consequence is based on a restriction of the introduction rule for the consequence predicate. To unrestrictedly maintain this principle, we develop a conception of object-linguistic consequence, which we call grounded consequence, that displays a restriction of the structural rule of reflexivity. This construction is obtained by generalizing Saul Kripke’s inductive theory of truth (strong Kleene version). Grounded validity will be shown to satisfy several desirable principles for a naïve, self-applicable notion of consequence. Preprint, DOI, BibTex. @article{niro17,
author={Carlo Nicolai and Lorenzo Rossi},
year={2017},
title={Principles for object-linguistic consequence: from logical to irreflexive}, publisher={Springer},
journal={Journal of Philosophical Logic},
note={Online-First, DOI: https://link.springer.com/article/10.1007/s11225-017-9727-y},
}

"On the costs of nonclassical logic", Journal of Philosophical Logic, online-first.
Abstract, Solutions to semantic paradoxes often involve restrictions of classical logic for semantic vocabulary. In the paper we investigate the costs of these restrictions in a model case. In particular, we fix two systems of truth capturing the same conception of truth: (a variant) of the system KF of [Feferman 1991] formulated in classical logic, and (a variant of) the system PKF of [Halbach & Horsten 2006], formulated in basic De Morgan logic. The classical system is known to be much stronger than the nonclassical one. We assess the reasons for this asymmetry by showing that the truth theoretic principles of PKF cannot be blamed: PKF with induction restricted to non-semantic vocabulary coincides in fact with what the restricted version of KF proves true. Preprint, DOI, BibTex. @article{niro17,
author={Volker Halbach and Carlo Nicolai},
year={2017},
title={On the costs of nonclassical logic}, publisher={Springer},
journal={Journal of Philosophical Logic},
note={Online-First, DOI: https://link.springer.com/article/10.1007/s10992-017-9424-3},
}

"Equivalences for truth predicates", The Review of Symbolic Logic, online-first.
Abstract, One way to study and understand the notion of truth is to examine principles thatwe are willing to associate with truth, often because they conform to a pre-theoretical or to a semi-formal characterization of this concept. In comparing different collections of such principles, one requires formally precise notions of inter-theoretic reduction that are also adequate to compare these conceptual aspects. In this work I study possible ways to make precise the relation of conceptual equivalence between notions of truth associated with collections of principles of truth. In doing so, I will consider refinements and strengthenings of the notion of relative truth-definability proposed by (Fujimoto 2010): in particular I employ suitable variants of notions of equivalence of theories considered in (Visser 2006; Friedman & Visser 2014) to show that there are better candidates than mutual truth-definability for the role of sufficient condition for conceptual equivalence between the semantic notions associated with the theories. In the concluding part of the paper, I extend the techniques introduced in the first and show that there is a precise sense in which ramified truth (either disquotational or compositional) does not correspond to iterations of comprehension. Preprint, DOI, BibTex. @article{niro17,
author={Carlo Nicolai},
year={2017},
title={Equivalences for truth predicates},
journal={The Review of Symbolic Logic},
note={Online-First, DOI: https://link.springer.com/article/10.1007/s10992-017-9424-3},
}

"Necessary Truths and Supervaluations", forthcoming in From arithmetic to metaphysics. A path through philosophical logic, De Gruyter, Philosophical Analysis.
Abstract, Starting with a trustworthy theory T, Galvan (1992) suggests to read offš, from the usual hierarchy of theories determined by consistency strength, a finer-grained hierarchy in which theories higher up are capable of ‘explaining’, though not fully justifying, our commitment to theories lower down. One way to ascend Galvan’s ‘hierarchy of explanation’ is to formalize soundness proofs: to this extent it often suffices to assume a full theory of truth for the theory T whose soundness is at stake. In this paper, we investigate the possibility of an extension of this method. Our ultimate goal will be to extend T not only with truth axioms, but with a combination of axioms for predicates for truth and necessity. We first consider two alternative strategies for providing possible worlds semantics for necessity as a predicate, one based on classical logic, the other on a supervaluationist interpretation of necessity. We will then formulate a deductive system of truth and necessity in classical logic that is sound with respect to the given (nonclassical) semantics. Preprint,

"More on systems of truth and predicative comprehension", in F. Boccuni and A. Sereni (eds), Philosophy of Mathematics: Objectivity, Cognition, and Proof, Boston Studies in the History and Philosophy of Science, chapter 14.
Abstract, Solutions to semantic paradoxes often involve restrictions of classical logic for semantic vocabulary. In the paper we investigate the costs of these restrictions in a model case. In particular, we fix two systems of truth capturing the same conception of truth: (a variant) of the system KF of [Feferman 1991] formulated in classical logic, and (a variant of) the system PKF of [Halbach & Horsten 2006], formulated in basic De Morgan logic. The classical system is known to be much stronger than the nonclassical one. We assess the reasons for this asymmetry by showing that the truth theoretic principles of PKF cannot be blamed: PKF with induction restricted to non-semantic vocabulary coincides in fact with what the restricted version of KF proves true. Preprint, DOI, BibTex. @incollection{nic16b,
author={Carlo Nicolai},
year={2016},
title={More on systems of truth and predicative comprehension}, booktitle = {Objectivity, Realism, and Proof. FilMat Studies in the Philosophy of Mathematics},
editor= {F. Boccuni and A. Sereni},
series = {Boston Studies in the Philosophy and History of Science},
chapter= {14},
publisher={Springer},
}

"A note on typed truth and consistency assertions", Journal of Philosophical Logic 45(1), 2016: pp. 89-119.
Abstract, In the paper we investigate typed (mainly compositional) axiomatizations of the truth predicate in which the axioms of truth come with a built-in, minimal and self-sufficient machinery to talk about syntactic aspects of an arbitrary base theory. Expanding previous works of the author and building on recent works of Albert Visser and Richard Heck, we give a precise characterization of these systems by investigating the strict relationships occurring between them, arithmetized model constructions in weak arithmetical systems and suitable set existence axioms. The framework considered will give rise to some methodological remarks on the construction of truth theories and provide us with a privileged point of view to analyze the notion of truth arising from compositional principles in a typed setting. Preprint, DOI, BibTex. @article{nic16a,
author={Carlo Nicolai},
year={2015},
title={A note on typed truth and consistency assertions},
journal={Journal of Philosophical Logic},
volume={45},
issue={1},
pages={89-119},
}

"Deflationary truth and the ontology of expressions", Synthese 192(12): pp. 4031-4055.
Abstract, The existence of a close connection between results on axiomatic truth and the analysis of truth-theoretic deflationism is nowadays widely recognized. The first attempt to make such link precise can be traced back to the so-called conservativeness argument due to Leon Horsten, Stewart Shapiro and Jeffrey Ketland: by employing standard Gödelian phenomena, they concluded that deflationism is untenable as any adequate theory of truth leads to consequences that were not achievable by the base theory alone. In the paper I highlight, as Shapiro and Ketland, the irreducible nature of truth axioms with respect to their base theories. But, I argue, this does not immediately delineate a notion of truth playing a substantial role in philosophical or scientific explanations. I first offer a refinement of Hartry Field’s reaction to the conservativeness argument by distinguishing between metatheoretic and object-theoretic consequences of the theory of truth and address some possible rejoinders. In the resulting picture, truth is an irreducible tool for metatheoretic ascent. How robust is this characterizaton? I test it by considering: (i) a recent example, due to Leon Horsten, of the alleged explanatory role played by the truth predicate in the derivation of Fitch’s paradox; (ii) an essential weakening of theories of truth analyzed in the first part of the paper. Preprint, DOI, BibTex. @article{nic15,
author={Carlo Nicolai},
year={2015},
title={Deflationary truth and the ontology of expressions},
journal={Synthese},
volume={192},
issue={12},
pages={4031-4055},
}

"Axiomatic Truth, Syntax, and Metatheoretic Reasoning" (with G.E. Leigh), The Review of Symbolic Logic 6(4), 2013: pp. 631-636.
Abstract, Following recent developments in the literature on axiomatic theories of truth, we investigate an alternative to the widespread habit of formalizing the syntax of the object-language into the object-language itself. We first argue for the proposed revision, elaborating philosophical evidences in favor of it. Secondly, we present a general framework for axiomatic theories of truth with ‘disentangled’ theories of syntax. Different choices of the object theory O will be considered. Moreover, some strengthenings of these theories will be introduced: we will consider extending the theories by the addition of coding axioms or by extending the schemas of O, if present, to the entire vocabulary of our theory of truth. Finally, we touch on the philosophical consequences that the theories described can have on the debate about the metaphysical status of the truth predicate and on the formalization of our informal metatheoretic reasoning. Preprint, DOI, BibTex. @article{nile13,
author={Graham Leigh and Carlo Nicolai},
year={2013},
title={Axiomatic Truth, Syntax, and Metatheoretic Reasoning},
journal={The Review of Symbolic Logic},
volume={6},
issue={4},
pages={613-636},
}


Some Recent Talks

Properties, Instantiation, and Paradox. Munich, "The structure of modal and semantic reasoning", 29 June, 2017.
Abductivism and the standard for logical revision. Bologna, "3rd SILFS Meeting", 23 June 2017.
Conceptual foundations: from semigroups to predicative analysis via reflection. Chieti, "Proof-Theory Afternoon", April 11, 2017.
A unified approach to truth and implication (with L. Rossi). Oxford, "Fourth New College Meeting", March 11, 2017.
What can we learn from reflecting on truth? Bristol, "Foundational Studies Seminar", March 8 2017.
Logical pluralism and abductive methodology: a case from semantics. Utrecht, TF-Seminar, March 3 2017.
Truth has to live up to its task. "Non-classical solutions to the paradoxes", Munich, February 27, 2017.


Teaching

Utrecht, 1st Block 2017, Topics in Philosophy of Language and Logic: The Status of Logical Principles (with Daniel Cohnitz).

- Week 1. Analyticity I
- Week 2. Analyticity II
- Week 3. Semantic Paradoxes
- Week 4. Vagueness (Due date for take-home exam)
- Week 5. Revision from Empirical Evidence
- Week 6. Models for revisions of logical beliefs

Manuale di Logica

I've recently translated Volker Halbach's Logic Manual into Italian. The Manuale di Logica is published by Mimesis Edizioni. Here's a short description.

Il Manuale di Logica è una breve introduzione alla logica per studenti di filosofia. Il testo offre un corso introduttivo completo, che accompagna il lettore attraverso i temi della logica che sono più rilevanti per lo studio della filosofia. Il Manuale introduce la logica proposizionale e predicativa con e senza identità e presenta una semantica di questi linguaggi introducendo le relative definizioni di verità e soddisfazione. Il sistema di deduzione scelto è la Deduzione Naturale. Volker Halbach introduce le nozioni essenziali sia attraverso esempi e spiegazioni informali, sia utilizzando definizioni più astratte e rigorose.

Exercises are only available in English on the Logic Manual's webpage maintained by Volker himself.


Contact

E-mail: carlonicolai6@gmail.com; c.nicolai6@uu.nl
Mail: Department of Philosophy and Religious Studies, Janskerkhof 13, 3512 Utrecht, The Netherlands.

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