Articles
Securing Arithmetical Determinacy, Ergo - An Open Access Journal of Philosophy, forthcoming.
The existence of non-standard models of first-order Peano-Arithmetic (PA) threatens to undermine the claim of the moderate mathematical realist that non-mysterious access to the natural number structure is possible on the basis of our best arithmetical theories. The move to logics stronger than FOL is denied to the moderate realist on the grounds that it merely shifts the indeterminacy ‘one level up’ into the meta-theory by, illegitimately, assuming the determinacy of the notions needed to formulate such logics. This paper argues that the challenge can be met. We show how the quantifier “there are infinitely many” can be uniquely determined in a naturalistically acceptable fashion and thus be used in the formulation of a theory of arithmetic. We compare the approach pursued here with Field’s justification of the same device and the popular strategy of invoking a second-order formalism, and argue that it is more robust than either of the alternative proposals.
Logical Constants, in: Oxford Handbook of Philosophy of Logic, E. Brendel, M. Carrara, F. Ferrari, O. Hjortland, G. Sagi, G. Sher, F. Steinberger (eds.), Oxford University Press, forthcoming.
An argument is logically valid if all arguments of the same logical form are valid. The logical form of an argument, in turn, is determined by the occurrence of members of a distinguished set of expressions: the logical constants. To distinguish the logical constants of a language from its non-logical expressions is therefore essential for an appropriate classification of its arguments and an adequate explication of its relation of logical consequence. The current chapter investigates the demarcation problem of the logical constants for formal languages. It does so by comparing and assessing criteria of logicality - mathematically precise and philosophically informative sets of principles that aim to distinguish the logical from the non-logical expressions. It considers criteria from the semantic/model-theoretic tradition, providing a classification on the basis of properties of the referents of expressions, as well as from the inferentialist/proof-theoretic tradition, which bases assessments of logicality on features of the inferential behaviour of expressions. Further topics discussed include: criteria combining semantic and inferential considerations in a determination of logicality, extensions of the delineation to alternative, non-classical logics, and general criticisms of the very starting point of the demarcation project.
Logical Constants and Arithmetical Forms, Logic and Logical Philosophy 32.3, 2023, 495-510. [here]
This paper reflects on the limits of logical form set by a novel criterion of logicality proposed in (Bonnay and Speitel, 2021). The interest stems from the fact that the delineation of logical terms according to the criterion exceeds the boundaries of standard first-order logic. Among ‘novel’ logical terms is the quantifier “there are infinitely many”. Since the structure of the natural numbers is categorically characterisable in a language including this quantifier we ask: does this imply that arithmetical forms have been reduced to logical forms? And, in general, what other conditions need to be satisfied for a form to qualify as “fully logical”? We survey answers to these questions.
The Ways of Logicality: Invariance and Categoricity (with D. Bonnay), in: The Semantic Conception of Logic: Essays on Consequence, Invariance, and Meaning, G. Sagi & J. Woods (eds.), Cambridge University Press 2021, 55-80. [here]
After reviewing issues arising with purely invariance-based criteria of logicality we propose to supplement invariance with inferential constraints, aiming to secure the categoricity of logical notions. We discuss and criticize Feferman’s (2015) Semantical-Inferential Necessary Criterion of Logicality before formulating and defending our own Combined Criterion of Logicality. We explore the scope of the proposed criterion by considering expressions from the category of generalized quantifiers and conclude with some questions concerning the bounds of logic it sets.
In Progress & Under Review
Logic and Evolution (with G. Sbardolini), under review.
A paper about the possibility of gradual change in logic and ways of modelling logical meanings in light of such changes.
Carnap’s (Categoricity) Problem - A Survey of Solutions, in progress.
A survey and comparison of solution strategies put forward to overcome Carnapian underdetermination of logical semantics by syntax.
Invariance in Non-Classical Logics (with C. Caret and E. Stei), in progress.
A paper investigating the results and motivations of extending classical invariance-based criteria of logicality to alternative, non-classical settings.
Carnap’s Problem for Languages L(Q1, …, Qn) (with D. Westerståhl), in progress.
An investigation of Carnap’s Problem for languages enriched with various generalized quantifiers.
Other
Logical Constants between Inference and Reference - An Essay in the Philosophy of Logic, PhD Dissertation, UC San Diego Electronic Theses and Dissertations, UC San Diego 2020. [here]
At the foundations of contemporary mathematical logic lies Tarski's model-theoretic definition of logical consequence. Underlying this definition is a division of all expressions of a language into logical and extra-logical. Drawing such a distinction by mere enumeration, as is common in familiar logical languages, means proceeding on an arbitrary basis. To fully secure logical consequence against skeptical attacks it is necessary to devise a criterion of logicality, a mathematically precise and philosophically informative set of principles, to demarcate the class of logical constants.
At the center of this thesis is the development of a combined criterion of logicality - involving both model- and proof-theoretic elements. From the semantic tradition it adopts the idea that logic must be formal and that to exhibit this property logical notions must display a high degree of invariance. From the proof-theoretic tradition it takes up the insight that logical expressions must be categorical, in the sense of being uniquely determined by their inferential roles. Together, these conditions delineate a robust core of logical expressions extending the class of standard operators of the first-order predicate calculus. We explore the consequences of the criterion and develop a theory of the notion of Carnap-categoricity, the unique determinability of formal notions.
The very possibility of a combined criterion of the kind pursued here is threatened by a set of underdetermination phenomena, collectively referred to as Carnap's Problem. The thesis presents a comprehensive and systematic examination of the issues pertaining to the underdetermination of the semantics of logical expressions by their syntax and explores the extent and degree of the underdetermination of generalised quantifiers by their inferential roles.
Carnap's problem has a profound impact on adequate formulations of a proper notion of unique determinability of meaning by inference. Due to the failure of the most natural and naive way of understanding what it means for (model-theoretic) meaning to be uniquely determined by inference we develop and defend a new notion of what it means for meaning to be uniquely determined in such a setting and draw out its consequences.
Note: Feel free to email me for a draft of any of these papers.