Logical Constants between Inference and Reference - An Essay in the Philosophy of Logic
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.