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Carlo Nicolai
King's College London
  1.  44
    On the Costs of Nonclassical Logic.Volker Halbach & Carlo Nicolai - 2018 - Journal of Philosophical Logic 47 (2):227-257.
    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: of the system KF of Feferman formulated in classical logic, and the system PKF of Halbach and Horsten, formulated in basic De Morgan logic. The classical system is known to be much stronger than the nonclassical one. We assess the (...)
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  2.  19
    Principles for Object-Linguistic Consequence: From Logical to Irreflexive.Carlo Nicolai & Lorenzo Rossi - 2018 - Journal of Philosophical Logic 47 (3):549-577.
    We discuss the principles for a primitive, object-linguistic notion of consequence proposed by ) 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 first place. To these solutions will correspond different conceptions of consequence. In one possible reading of these principles, they give rise to a notion (...)
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  3.  37
    A Note on Typed Truth and Consistency Assertions.Carlo Nicolai - 2016 - Journal of Philosophical Logic 45 (1):89-119.
    In the paper we investigate typed 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 (...)
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  4.  39
    Axiomatic Truth, Syntax and Metatheoretic Reasoning.Graham E. Leigh & Carlo Nicolai - 2013 - Review of Symbolic Logic 6 (4):613-636.
    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 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 (...)
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  5.  25
    Provably True Sentences Across Axiomatizations of Kripke’s Theory of Truth.Carlo Nicolai - 2018 - Studia Logica 106 (1):101-130.
    We study the relationships between two clusters of axiomatizations of Kripke’s fixed-point models for languages containing a self-applicable truth predicate. The first cluster is represented by what we will call ‘\-like’ theories, originating in recent work by Halbach and Horsten, whose axioms and rules are all valid in fixed-point models; the second by ‘\-like’ theories first introduced by Solomon Feferman, that lose this property but reflect the classicality of the metatheory in which Kripke’s construction is carried out. We show that (...)
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  6. Hypatia's Silence.Martin Fischer, Leon Horsten & Carlo Nicolai - 2021 - Noûs 55 (1):62-85.
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  7.  40
    Deflationary Truth and the Ontology of Expressions.Carlo Nicolai - 2015 - Synthese 192 (12):4031-4055.
    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, (...)
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  8.  28
    Equivalences for Truth Predicates.Carlo Nicolai - 2017 - Review of Symbolic Logic 10 (2):322-356.
    One way to study and understand the notion of truth is to examine principles that we 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 (...)
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  9.  19
    The Implicit Commitment of Arithmetical Theories and Its Semantic Core.Carlo Nicolai & Mario Piazza - 2019 - Erkenntnis 84 (4):913-937.
    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 (...)
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  10. A Theory of Implicit Commitment for Mathematical Theories.Mateusz Łełyk & Carlo Nicolai - manuscript
    The notion of implicit commitment has played a prominent role in recent works in logic and philosophy of mathematics. Although implicit commitment is often associated with highly technical studies, it remains so far an elusive notion. In particular, it is often claimed that the acceptance of a mathematical theory implicitly commits one to the acceptance of a Uniform Reflection Principle for it. However, philosophers agree that a satisfactory analysis of the transition from a theory to its reflection principle is still (...)
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  11. Hypatia's Silence. Truth, Justification, and Entitlement.Martin Fischer, Leon Horsten & Carlo Nicolai - manuscript
    Hartry Field distinguished two concepts of type-free truth: scientific truth and disquotational truth. We argue that scientific type-free truth cannot do justificatory work in the foundations of mathematics. We also present an argument, based on Crispin Wright's theory of cognitive projects and entitlement, that disquotational truth can do justificatory work in the foundations of mathematics. The price to pay for this is that the concept of disquotational truth requires non-classical logical treatment.
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  12.  12
    Nonclassical Truth with Classical Strength.Martin Fischer, Carlo Nicolai & Pablo Dopico - forthcoming - Review of Symbolic Logic:1-26.
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  13. On Logical and Scientific Strength.Luca Incurvati & Carlo Nicolai - manuscript
    The notion of strength has featured prominently in recent debates about abductivism in the epistemology of logic. Following Williamson and Russell, we distinguish between logical and scientific strength and discuss the limits of the characterizations they employ. We then suggest understanding logical strength in terms of interpretability strength and scientific strength as a special case of logical strength. We present applications of the resulting notions to comparisons between logics in the traditional sense and mathematical theories.
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  14. Cut Elimination for Systems of Transparent Truth with Restricted Initial Sequents.Carlo Nicolai - manuscript
    The paper studies a cluster of systems for fully disquotational truth based on the restriction of initial sequents. Unlike well-known alternative approaches, such systems display both a simple and intuitive model theory and remarkable proof-theoretic properties. We start by showing that, due to a strong form of invertibility of the truth rules, cut is eliminable in the systems via a standard strategy supplemented by a suitable measure of the number of applications of truth rules to formulas in derivations. Next, we (...)
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  15.  55
    Fix, Express, Quantify. Disquotation After its Logic.Carlo Nicolai - forthcoming - Mind:fzaa054.
    Truth-theoretic deflationism holds that truth is simple, and yet that it can fulfil many useful logico-linguistic roles. Deflationism focuses on axioms for truth: there is no reduction of the notion of truth to more fundamental ones such as sets or higher-order quantifiers. In this paper I argue that the fundamental properties of reasonable, primitive truth predicates are at odds with the core tenets of classical truth-theoretic deflationism that I call fix, express, and quantify. Truth may be regarded as a broadly (...)
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  16.  1
    More on Systems of Truth and Predicative Comprehension.Carlo Nicolai - 2016 - In Francesca Boccuni & Andrea Sereni (eds.), Objectivity, Realism, and Proof. Filmat Studies in the Philosophy of Mathematics. Springer Verlag.
    In the paper we survey the known connections between theories that extend a common base theory with typed truth axioms on the one hand and predicative set-existence assumptions on the other. How general can the mutual reductions between truth and comprehension be taken to be? In trying to address this question, we consider classical, positive truth and predicative comprehension as operations on theories.
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  17. Modes of Truth: The Unified Approach to Modality, Truth, and Paradox.Carlo Nicolai (ed.) - forthcoming
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  18.  13
    Necessary Truths and Supervaluations.Carlo Nicolai - 2018 - In Alessandro Giordani & Ciro de Florio (eds.), From Arithmetic to Metaphysics: A Path Through Philosophical Logic. De Gruyter. pp. 309-330.
    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 (...)
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  19. Systems for Non-Reflexive Consequence.Carlo Nicolai & Lorenzo Rossi - manuscript
    Substructural logics and their application to logical and semantic paradoxes have been extensively studied, but non-reexive systems have been somewhat neglected. Here, we aim to (at least partly) ll this lacuna, by presenting a non-reexive logic and theory of naïve consequence (and truth). We also investigate the semantics and the proof-theory of the system. Finally, we develop a compositional theory of truth (and consequence) in our non-reexive framework.
     
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  20. The Modal Logics of Kripke-Feferman Truth.Carlo Nicolai & Johannes Stern - manuscript
    We determine the modal logic of fixed-point models of truth and their axiomatizations by Solomon Feferman via Solovay-style completeness results.
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  21.  8
    The Modal Logics of Kripke-Feferman Truth.Carlo Nicolai & Johannes Stern - forthcoming - Journal of Symbolic Logic:1-31.
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