Notes on the Model Theory of DeMorgan Logics (original) (raw)

On orderings of the family of all logics

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.

Hierarchies in transitive closure logic, stratified Datalog and infinitary logic

Annals of Pure and Applied Logic, 1996

formula with bounded number of universal quantifiers can express the negation of a transitive closure. This implies the solution of several open problems in finite model theory: On finite structures, positive transitive closure logic is not closed under negation. More generally the hierarchy defined by interleaving negation and transitive closure operators is strict. This proves a conjecture of Immerman. We also separate the expressive power of several extensions of Datalog, giving new insight in the fine structure of stratified Datalog. (FO + TC), denoted (FO + DTC). An important fragment of transitive closure logic is (FO + pos TC), which only allows positive applications of T C operators. In fact, Immerman's original result in [15] only identified (FO + pos TC) with NLOGSPACE. At that time most people believed that NLOGSPACE would not be closed under complementation, and therefore (FO + pos TC) not be equiva lent to (FO + TC). However, the closure of of NSPACE complexity classes under complementation [16,23] implied that on ordered structures, transitive closure logic collapses to its positive fragment. It has been one of the more notable open problems in finite model theory (posed in [15]) whether this is also true in the absence of order, i.e. whether (FO + pos TC) coincides with (FO + TC) on arbitrary finite

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, 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.

C T ] 1 0 M ay 2 01 4 Representation theory of logics : a categorial approach

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...

Theorems of Alternatives for Substructural Logics

Outstanding Contributions to Logic, 2021

A theorem of alternatives provides a reduction of validity in a substructural logic to validity in its multiplicative fragment. Notable examples include a theorem of Arnon Avron that reduces the validity of a disjunction of multiplicative formulas in the "R-mingle" logic RM to the validity of a linear combination of these formulas, and Gordan's theorem for solutions of linear systems over the real numbers, that yields an analogous reduction for validity in Abelian logic A. In this paper, general conditions are provided for axiomatic extensions of involutive uninorm logic without additive constants to admit a theorem of alternatives. It is also shown that a theorem of alternatives for a logic can be used to establish (uniform) deductive interpolation and completeness with respect to a class of dense totally ordered residuated lattices.

Comparing hierarchies of types in models of linear logic

Information and Computation/information and Control, 2004

We show that two models M and N of linear logic collapse to the same extensional hierarchy of types, when (1) their monoidal categories C and D are related by a pair of monoidal functors F : C D : G and transformations Id C ⇒ GF and Id D ⇒ FG, and (2) their exponentials ! M and ! N are related by distributive laws : ! N F ⇒ F ! M and Á : ! M G ⇒ G ! N commuting to the promotion rule. The key ingredient of the proof is a notion of back-and-forth translation between the hierarchies of types induced by M and N. We apply this result to compare (1) the qualitative and the quantitative hierarchies induced by the coherence (or hypercoherence) space model, (2) several paradigms of games semantics: error-free vs. error-aware, alternated vs. non-alternated, backtracking vs. repetitive, uniform vs. nonuniform. 203 The tensor product of two coherence spaces A = (|A|, A ) and B = (|B|, B ) is their product as graphs:

Inner models from extended logics: Part 1

Journal of Mathematical Logic, 2020

If we replace first-order logic by second-order logic in the original definition of Gödel’s inner model [Formula: see text], we obtain the inner model of hereditarily ordinal definable (HOD) sets [33]. In this paper, we consider inner models that arise if we replace first-order logic by a logic that has some, but not all, of the strength of second-order logic. Typical examples are the extensions of first-order logic by generalized quantifiers, such as the Magidor–Malitz quantifier [24], the cofinality quantifier [35], or stationary logic [6]. Our first set of results show that both [Formula: see text] and HOD manifest some amount of formalism freeness in the sense that they are not very sensitive to the choice of the underlying logic. Our second set of results shows that the cofinality quantifier gives rise to a new robust inner model between [Formula: see text] and HOD. We show, among other things, that assuming a proper class of Woodin cardinals the regular cardinals [Formula: see...