The Poset of All Logics III: Finitely Presentable Logics

arXiv: Logic, 2020

A Leibniz class is a class of logics closed under the formation of term-equivalent logics, compatible expansions, and non-indexed products of sets of logics. We study the complete lattice of all Leibniz classes, called the Leibniz hierarchy. In particular, it is proved that the classes of truth-equational and assertional logics are meet-prime in the Leibniz hierarchy, while the classes of protoalgebraic and equivalential logics are meet-reducible. However, the last two classes are shown to be determined by Leibniz conditions consisting of meet-prime logics only.

The Journal of Symbolic Logic

A notion of interpretation between arbitrary logics is introduced, and the poset $\mathsf {Log}$ of all logics ordered under interpretability is studied. It is shown that in $\mathsf {Log}$ infima of arbitrarily large sets exist, but binary suprema in general do not. On the other hand, the existence of suprema of sets of equivalential logics is established. The relations between $\mathsf {Log}$ and the lattice of interpretability types of varieties are investigated.

Archive for Mathematical Logic, 1980

The purpose of this paper is to examine the structural complexity of the sublogic relation between abstract logics. Let ~(") denote n 'h order logic. Then cp(,) is a proper subtogic of A °("÷ 1~ for each n < co, and we have the chain ~(1)<~(2)< ... <~cp(,)< .... The question naturally arises, what other kinds of chains or partial orderings we can have among sufficiently regular abstract logics. Can one have a family of logics ordered in the order type of the rationals (or perhaps the reals)? In Chapter 2 we prove that any distributive lattice, and therefore any partial ordering, can be embedded into the sublogic relation among what we call normal logics. Chapter 3 is devoted to a sublogic relation defined using PC-classes rather than elementary classes. In the last chapter we restrict ourselves to the very special logics of the form ~(Q1,-, Q,)-The situation becomes more problematic but we can still prove that any countable partial ordering is embeddable into the sublogic relation of these logics. We confine ourselves to single-sorted structures, but many of the results are equally true of many-sorted structures.

2018

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,As studied in [AFLM3]. All these categories have good properties however the category of logics L does not allow a good treatment of the ”identity problem” for logics ([Bez]): for instance, the presentations of ”classical logics” (e.g., in the signature {¬,∨} and {¬,→}) 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 a...

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, As studied in [AFLM3]. All these categories have good properties however the category of logics L does not allow a good treatment of the "identity problem" for logics ([Bez]): for instance, the presentations of "classical logics" (e.g., in the signature {¬, ∨} and {¬ ′ , → ′ }) 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.

For every consequence (or closure) operator Cn on a set S, the family C of all Cn-closed sets, partially ordered by set inclusion, forms a complete lattice, called the lattice of logics. If the lattice C is distributive, then, it forms a Heyting algebra, since it has the zero element Cn(0) and is complete. Logics determined by this Heyting algebra is studied in the second part. In Part I it is shown (for Cn finitary or compact) that the lattice C is distributive iff its dual space is topological. Moreover a representation theorem for lattices of logics is given.

Slogica, 1993

The intermediate logics have been dassified into slices (cf. Hosoi [I]), but the detailed structure of slices has been studied only for the first two slices (cL IIosoi and Ono [2]). In order to study the structure of slices, we give a method of a finer dassification of slices ~ (n ~> 3). Here we treat.only the third slice as an example, but the method can be extended to other slices in an obvious way. It is proved that each subs]ice contains continuum of logics. A characterization of logics in each subslice is given in terms of the form of models. 1. Classification of the third slice By LJ, we mean the intuitionistic propositional logic. A partially ordered set (POS) model, which is a Kripke type model modified for the intermediate logics, is defined in Ono [6] with the notions such as validity, height, the Cartesian product, the pile operation M T N and so on. Hereafter, simply by model, we mean a POS model. The one element model with the canonical order relation is named as a. The model a 1" a is referred to by 2a, and na is understood similarly. By L(M), we mean the logic defined as the set of valid formulas in the model M.

Publications of the Research Institute for Mathematical Sciences, 1972

Studia Logica, 1978

Archive for Mathematical Logic, 2002

In this paper we present a method to reduce the decision problem of several infinite-valued propositional logics to their finite-valued counterparts. We apply our method to Łukasiewicz, Gödel and Product logics and to some of their combinations. As a byproduct we define sequent calculi for all these infinite-valued logics and we give an alternative proof that their tautology problems are in co-NP.