Type space functors and interpretations in positive logic

We construct a 2-equivalence CohTheoryop≃TypeSpaceFunc. Here CohTheory is the 2-category of positive theories and TypeSpaceFunc is the 2-category of type space functors. We give a precise definition of interpretations for positive logic, which will be the 1-cells in CohTheory. The 2-cells are definable homomorphisms. The 2-equivalence restricts to a duality of categories, making precise the philosophy that a theory is ‘the same’ as the collection of its type spaces (i.e. its type space functor). In characterising those functors that arise as type space functors, we find that they are specific instances of (coherent) hyperdoctrines. This connects two different schools of thought on the logical structure of a theory. The key ingredient, the Deligne completeness theorem, arises from topos theory, where positive theories have been studied under the name of coherent theories

Closure Hyperdoctrines

(Pre)closure spaces are a generalization of topological spaces covering also the notion of neighbourhood in discrete structures, widely used to model and reason about spatial aspects of distributed systems. In this paper we present an abstract theoretical framework for the systematic investigation of the logical aspects of closure spaces. To this end, we introduce the notion of closure (hyper)doctrines, i.e. doctrines endowed with inflationary operators (and subject to suitable conditions). The generality and effectiveness of this concept is witnessed by many examples arising naturally from topological spaces, fuzzy sets, algebraic structures, coalgebras, and covering at once also known cases such as Kripke frames and probabilistic frames (i.e., Markov chains). By leveraging general categorical constructions, we provide axiomatisations and sound and complete semantics for various fragments of logics for closure operators. Hence, closure hyperdoctrines are useful both for refining and improving the theory of existing spatial logics, and for the definition of new spatial logics for new applications

Proof-theoretic Semantics and Tactical Proof

The use of logical systems for problem-solving may be as diverse as in proving theorems in mathematics or in figuring out how to meet up with a friend. In either case, the problem solving activity is captured by the search for an \emph{argument}, broadly conceived as a certificate for a solution to the problem. Crucially, for such a certificate to be a solution, it has be \emph{valid}, and what makes it valid is that they are well-constructed according to a notion of inference for the underlying logical system. We provide a general framework uniformly describing the use of logic as a mathematics of reasoning in the above sense. We use proof-theoretic validity in the Dummett-Prawitz tradition to define validity of arguments, and use the theory of tactical proof to relate arguments, inference, and search.Comment: submitte

On Logical connectives and quantifiers as adjoint functors

This thesis deals with the issue of treating logical connectives, quantifiers and equality in categorical terms, by means of adjoint functors combined into the notion of hyperdoctrine, introduced by Francis William Lawvere in 1969. After proving the general Theorem of Soundness and Completeness for the intuitionistic predicate logic with equality with respect to hyperdoctrines, we formulate instances of such categorical models by using H-valued sets and Kleene realizability, in order to produce easily models and countermodels for logical formulas

Hyperdoctrines and the Ontology of Stratified Semantics

I present a version of Kit Fine's stratified semantics for the logic RWQ and define a natural family of related structures called RW hyperdoctrines. After proving that RWQ is sound with respect to RW hyperdoctrines, we show how to construct, for each stratified model, a hyperdoctrine that verifies precisely the same sentences. Completeness of RWQ for hyperdoctrinal semantics then follows from completeness for stratified semantics, which is proved in an appendix. By examining the base category of RW hyperdoctrines, we find reason to be worried about the ontology of stratified models

Tripos models of Internal Set Theory

This thesis provides a framework to make sense of models of E. Nelson’s Internal Set Theory (and hence of nonstandard analysis) in elementary toposes by exploiting the technology of tripos theory and Lawvere’s hyperdoctrines. A new doctrinal account of nonstandard phenomena is described, which avoids a few key restrictions in Nelson’s approach: chiefly, the dependence on Set Theory (which is done by replacing a model of set theory with a topos as the starting point) and reliance on an internally defined notion of standard element. From the new perspective, validity of the schemes of Idealisation, Standardisation, and Transfer correspond to the existence of certain relationships between hyperdoctrines, leading to the new notion of a tripos model of IST. After discussing the properties of such models that make them a suitable abstraction of classic IST we explore situations in which such structures arise, leading to two distinct main classes of models: what we refer to as Nelson models, which are those for which there is a well-behaved `predicate of standard elements’ (providing thus a close approximation of classic IST valid for toposes), and models obtained by elaborating on the work started by A. Kock and C. J. Mikkelsen on the categorification of Transfer. We then elaborate upon one particular kind of model of each type: the ultrapower models, which are Nelson models obtained from a choice of adequate ultrafilter and that mirror the constructions of classical Robinson nonstandard analysis, and the localic models for Transfer and Standardisation, which are obtained from any given open surjection of locales (e.g universal covers of topological spaces) via the Kock-Mikkelsen construction at the level of sheaf toposes.This study was funded by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) (process n∘ 8881.128278/2016-01)

A General Framework for the Semantics of Type Theory

We propose an abstract notion of a type theory to unify the semantics of various type theories including Martin-L\"{o}f type theory, two-level type theory and cubical type theory. We establish basic results in the semantics of type theory: every type theory has a bi-initial model; every model of a type theory has its internal language; the category of theories over a type theory is bi-equivalent to a full sub-2-category of the 2-category of models of the type theory