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Darllan Conceição Pinto

Publications and source records attributed to Darllan Conceição Pinto.

10 recordsLinked to original sources
arXiv2026

Some contributions to presheaf model theory, II -- back and forth

We discuss the back and forth technique in the context of presheaf model theory. The essence of the back and forth technique lies in showing the relationship between various hierarchies which calibrate similarity between two models and, more generally, between two pairs consisting of a model and a tuple from it. In this paper we define several such hierarchies for presheaf models (and tuples of sections from them): those based on the degree of extendibility of partial isomorphisms through literal back and forth conditions, on sharing specific, abstract invariants which we define (the $F^α_{M,\bar{a}}$ of §\ref{function_analysis} for example), on agreeing on the (truth) values of instantiations of formulae up to a given amount of quantifier completity, on the existence of winning strategies for player II in certain Ehrenfeucht-Fraïssé-type games and, finally, on satisfying certain infinitary sentences that arise in the construction of Scott sentences. We ultimately show that all of these hierarchies align.

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arXiv2026

Some contributions to presheaf model theory

This paper makes contributions to ``pure'' sheaf model theory, the part of model theory in which the models are sheaves over a complete Heyting algebra. We start by outlining the theory in a way we hope is readable for the non-specialist. We then give a careful treatment of the interpretation of terms and formulae. This allows us to prove various preservation results, including strengthenings of the results of \cite{BM14}. We give refinements of Miraglia's work on directed colimits, \cite{M88}, and an analogue of Tarski's theorem on the preservation of $\forall_2$-sentences under unions of chains. We next show various categories whose objects are (pairs of) presheaves and sheaves with various notions of morphism are accessible in the category theoretic sense. Together these ingredients allow us ultimately to prove that these categories are encompassed in the AECats framework for independence relations developed by Kamsma in \cite{K22}.

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arXiv2022

Filter pairs and natural extensions of logics

We adjust the notion of finitary filter pair, which was coined for creating and analyzing finitary logics, in such a way that we can treat logics of cardinality $κ$, where $κ$ is a regular cardinal. The corresponding new notion is called $κ$-filter pair. A filter pair can be seen as a presentation of a logic, and we ask what different $κ$-filter pairs give rise to a fixed logic of cardinality $κ$. To make the question well-defined we restrict to a subcollection of filter pairs and establish a bijection from that collection to the set of natural extensions of that logic by a set of variables of cardinality $κ$. Along the way we use $κ$-filter pairs to construct natural extensions for a given logic, work out the relationships between this construction and several others proposed in the literature, and show that the collection of natural extensions forms a complete lattice. In an optional section we introduce and motivate the concept of a general filter pair.

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arXiv2021

Congruence Filter Pairs, Adjoints and Leibniz Hierarchy

We review the notion of (finitary) filter pair as a tool for creating and analyzing logics. A filter pair can be seen as a presentation of a logic, given by presenting its lattice of theories as the image of a lattice homomorphism, with certain properties ensuring that the resulting logic is finitary and substitution invariant. Every finitary, substitution invariant logic arises from a filter pair. Particular classes of logics can be characterized as arising from special classes of filter pairs. We consider so-called congruence filter pairs, i.e. filter pairs for which the domain of the lattice homomorphism is a lattice of congruences for some quasivariety. We show that the class of logics admitting a presentation by such a filter pair is exactly the class of logics having an algebraic semantics. We study the properties of a certain Galois connection coming with such filter pairs. We give criteria for a congruence filter pair to present a logic in some classes of the Leibniz hierarchy by means of this Galois connection, and its interplay with the Leibniz operator. As an application, we show a bridge theorem, stating that the amalgamation property implies the Craig interpolation property, for a certain class of logics including non-protoalgebraic logics.

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arXiv2020

Connecting abstract logics and adjunctions in the theory of ($π$-)institutions: some theoretical remarks and applications

In the present work, a natural sequel to \cite{MaPi1}, we further discuss the existence of adjunctions between categories of institutions and of $π$-institutions. This is done at both a foundational and an applied level. Firstly, we reformulate and conceptually clarify such adjunctions in terms of the $2$-categorical data involved in the construction of categories of institution-like structures. More precisely, we remark that the process used for passing from rooms to institutions (\cite{Diac2}) can be extended, due to its $2$-functoriality, to more general room-like and institution-like structures in such a way that the aforementioned adjunctions are all seen to arise from simpler adjunctions at the room-like level. Secondly, and mostly independently, we provide some applications of such adjunctions to abstract logics, mainly to the setting of propositional logics and filter pairs (\cite{AMP1}); we also generalize the process of skolemization, a classical device from predicate logic, to the institutional setting.

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arXiv2016

An abstract approach to Glivenko's theorem

The aim of this work is to provide a special kind of conservative translation between abstract logics, namely an \textit{abstract Glivenko's theorem}. Firstly we define institutions on the categories of logic, algebraizable logics, and Lindenbaum algebraizable logic. In the sequel, we introduce the notion os Glivenko's context relating two algebraizable logics (respectively, Lindenbaum algebraizable logics) and we prove that for each Glivenko's context can be associated an institutions morphism between the corresponding logical institutions. As a consequence of the existence of such institutions morphisms, we have established abstract versions of Glivenko's theorem between those algebraizable logics (Lindenbaum algebraizable logics), generalizing the results presented in \cite{To}. In particular, considering the institutions of classical logic and of intuitionistic logic, we build a Glivenko's context and thus an abstract Glivenko's theorem that is exactly the traditional Glivenko's theorem. Finally we present a category of algebraizable logic with Glivenko's context as morphisms. We can interpret the results of this work as an evidence of the (virtually unexplored) relevance of institution theory in the study of propositional logic.

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arXiv2016

Algebraizable Logics and a functorial encoding of its morphisms

The present work presents some results about the categorial relation between logics and its categories of structures. A (propositional, finitary) logic is a pair given by a signature and Tarskian consequence relation on its formula algebra. The logics are the objects in our categories of logics; the morphisms are certain signature morphisms that are translations between logics (\cite{AFLM1},\cite{AFLM2},\cite{AFLM3} \cite{FC}). Morphisms between algebraizable logics (\cite{BP}) are translations that preserves algebraizing pairs (\cite{MaMe}): they can be completely encoded by certain functors defined on the quasi-variety canonically associated to the algebraizable logics. This kind of results will be useful in the development of a categorial approach to the representation theory of general logics (\cite{MaPi1}, \cite{MaPi2}, \cite{AJMP}).

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arXiv2014

Distributive abstract logics and the Esakia duality

In this paper we develop an almost general process to switch from abstract logics in the sense of Brown and Suszko to lattices. With this method we can establish dualities between some categories of abstract logics to the correspondent topological space categories. In more detail we will explain the duality between the category of abstract intuitionistic logics with intuitionistic morphisms and the category of Esakia spaces with the Esakia morphisms.

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arXiv2014

Representation theory of logics: a categorial approach

The major concern in the study of categories of logics is to describe condition for preservation, under the a method of combination of logics, of meta-logical properties. Our complementary approach to this field is study the "global" aspects of categories of logics in the vein of the categories $\Ss, \Ls, \cA_s$ studied in \cite{AFLM3}. All these categories have good properties however the category of logics $\cL$ does not allow a good treatment of the "identity problem" for logics (\cite{Bez}): for instance, the presentations of "classical logics" (e.g., in the signature $\{\neg, \vee\}$ and $\{\neg',\rightarrow'\}$) are not $\Ls$-isomorphic. In this work, we sketch a possible way to overcome this "defect" (and anothers) by a mathematical device: a representation theory of logics obtained from category theoretic aspects on (Blok-Pigozzi) algebraizable logics. In this setting we propose the study of (left and right) "Morita equivalence" of logics and variants. We introduce the concepts of logics (left/right)-(stably) -Morita-equivalent and show that the presentations of classical logics are stably Morita equivalent but classical logics and intuitionist logics are not stably-Morita-equivalent: they are only stably-Morita-adjointly related.

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