Shedding the New Light in the World of Logical Systems

We consider (finitary, propositional) logics through the original use of Category Theory: the study of the " sociology of mathematical objects", aligning us with a recent, and growing, trend of study logics through its relations with other logics (e.g. process of combinations of logics as fibring [Gab] and possible translation semantics [Car]). So will be objects of study the classes of logics, i.e. categories whose objects are logical systems (i.e., a signature with a Tarskian consequence relation) and the morphisms are related to (some concept of) translations between these systems. The present work provides the first steps of a project of considering categories of logical systems satisfying simultaneously certain natural requirements: it seems that in the literature ([AFLM1], [AFLM2], [AFLM3], [BC], [BCC1], [BCC2], [CG], [FC]) this is achieved only partially.

SSRN Electronic Journal, 2020

Category theory gives a mathematical characterization of naturality but not of canonicity. The purpose of this paper is to develop the logical theory of canonical maps based on the broader demonstration that the dual notions of elements & distinctions are the basic analytical concepts needed to unpack and analyze morphisms, duality, canonicity, and universal constructions in Sets, the category of sets and functions. The analysis extends directly to other Sets-based concrete categories (groups, rings, vector spaces, etc.). Elements and distinctions are the building blocks of the two dual logics, the Boolean logic of subsets and the logic of partitions. The partial orders (inclusion and re…nement) in the lattices for the dual logics de…ne morphisms. The thesis is that the maps that are canonical in Sets are the ones that are de…ned (given the data of the situation) by these two logical partial orders and by the compositions of those maps.

arXiv (Cornell University), 2021

Category theory gives a mathematical characterization of naturality but not of canonicity. The purpose of this paper is to develop the logical theory of canonical maps based on the broader demonstration that the dual notions of elements & distinctions are the basic analytical concepts needed to unpack and analyze morphisms, duality, canonicity, and universal constructions in Set, the category of sets and functions. The analysis extends directly to other Set-based concrete categories (groups, rings, vector spaces, etc.). Elements and distinctions are the building blocks of the two dual logics, the Boolean logic of subsets and the logic of partitions. The partial orders (inclusion and refinement) in the lattices for the dual logics define the canonical morphisms (where 'canonical' is always relative to the given data, not an absolute property of a morphism). The thesis is that the maps that are canonical in Set are the ones that are defined (given the data of the situation) by these two logical partial orders and by the compositions of those maps.

Logica Universalis, 2013

This volume of the Journal Logica Universalis collects together papers on categorical logic in an extended sense. The idea for this volume arose out of the workshop on categorical logic associated with the conference Unilog 2010 held in Lisbon in June 2010. We feel that many of the ongoing activities in this exciting and somehow under-represented research area should be presented to a wider audience in the form of a special journal issue. In view of the diversity of the topic a general call for papers was issued. Papers presented at the workshop as well as new papers have been submitted and reviewed, following the usual criteria of this journal.

Social Science Research Network, 2020

Category theory gives a mathematical characterization of naturality but not of canonicity. The purpose of this paper is to develop the logical theory of canonical maps based on the broader demonstration that the dual notions of elements & distinctions are the basic analytical concepts needed to unpack and analyze morphisms, duality, canonicity, and universal constructions in Sets, the category of sets and functions. The analysis extends directly to other Sets-based concrete categories (groups, rings, vector spaces, etc.). Elements and distinctions are the building blocks of the two dual logics, the Boolean logic of subsets and the logic of partitions. The partial orders (inclusion and re…nement) in the lattices for the dual logics de…ne morphisms. The thesis is that the maps that are canonical in Sets are the ones that are de…ned (given the data of the situation) by these two logical partial orders and by the compositions of those maps.

Logica Universalis, 2013

In this essay we analyse and elucidate the method to establish and clarify the scope of logic theorems offered within the theory of institutions. The method presented pervades a lot of abstract model theoretic developments carried out within institution theory. The power of the proposed general method is illustrated with the examples of (Craig) interpolation and (Beth) definability, as they appear in the literature of institutional model theory. Both case studies illustrate a considerable extension of the original scopes of the two classical theorems. Our presentation is rather narrative with the relevant logic and institution theory concepts introduced and explained gradually to the non-expert reader.

2011

The aim of these notes is to provide a succinct, accessible introduction to some of the basic ideas of category theory and categorical logic. The notes are based on a lecture course given at Oxford over the past few years. They contain numerous exercises, and hopefully will prove useful for self-study by those seeking a first introduction to the subject, with fairly minimal prerequisites. The coverage is by no means comprehensive, but should provide a good basis for further study; a guide to further reading is included.

Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1985

We introduce a generic notion of propositional categorical logic and provide a construction of an institution with proofs out of such a logic, following the Curry-Howard-Tait paradigm. We then prove logic-independent soundness and completeness theorems. The framework is instantiated with a number of examples: classical, intuitionistic, linear and modal propositional logics. Finally, we speculate how this framework may be extended beyond the propositional case.

Information Processing Letters, 2007

For conventional logic institutions, when one extends the sentences to contain open sentences, their satisfaction is then parameterized. For instance, in the first-order logic, the satisfaction is parameterized by the valuation of unbound variables, while in modal logics it is further by possible worlds. This paper proposes a uniform treatment of such parameterization of the satisfaction relation within the abstract setting of logics as institutions, by defining the new notion of stratified institutions. In this new framework, the notion of elementary model homomorphisms is defined independently of an internal stratification or elementary diagrams. At this level of abstraction, a general Tarski style study of connectives is developed. This is an abstract unified approach to the usual Boolean connectives, to quantifiers, and to modal connectives. A general theorem subsuming Tarski's elementary chain theorem is then proved for stratified institutions with this new notion of connectives. (M. Aiguier), razvan.diaconescu@imar.ro (R. Diaconescu). fact its most fundamental mathematical structure. An institution (Sig, Sen, Mod, |=) consists of (i) a category Sig, whose objects are called signatures, (ii) a functor Sen : Sig → Set, giving for each signature a set whose elements are called sentences over that signature, (iii) a functor Mod : Sig op → Cat giving for each signature Σ a category whose objects are called Σmodels, and whose arrows are called Σ -(model) homomorphisms, and (iv) a relation |= Σ ⊆ |Mod(Σ)|×Sen(Σ) for each Σ ∈ |Sig|, called Σ -satisfaction, such that for each morphism ϕ : Σ → Σ in Sig, the satisfaction condition M |= Σ Sen(ϕ)(e) iff Mod(ϕ)(M ) |= Σ e