A short proof of Glivenko theorems for intermediate predicate logics

We give a simple proof-theoretic argument showing that Glivenko’s theorem for propositional logic and its version for predicate logic follow as an easy consequence of the deduction theorem, which also proves some Glivenko type theorems relating intermediate predicate logics between intuitionistic and classical logic. We consider two schemata, the double negation shift (DNS) and the one consisting of instances of the principle of excluded middle for sentences (REM). We prove that both schemata combined derive classical logic, while each one of them provides a strictly weaker intermediate logic, and neither of them is derivable from the other. We show that over every intermediate logic there exists a maximal intermediate logic for which Glivenko’s theorem holds. We deduce as well a characterization of DNS, as the weakest (with respect to derivability) scheme that added to REM derives classical logic.     

A minimal classical sequent calculus free of structural rules

Gentzen’s classical sequent calculus has explicit structural rules for contraction and weakening. They can be absorbed (in a right-sided formulation) by replacing the axiom P,¬P by Γ,P,¬P for any context Γ, and replacing the original disjunction rule with Γ,A,B implies Γ,A∨B.This paper presents a classical sequent calculus which is also free of contraction and weakening, but more symmetrically: both contraction and weakening are absorbed into conjunction, leaving the axiom rule intact. It uses a blended conjunction rule, combining the standard context-sharing and context-splitting rules: Γ,Δ,A and Γ,Σ,B implies Γ,Δ,Σ,A∧B. We refer to this system as minimal sequent calculus.We prove a minimality theorem for the propositional fragment : any propositional sequent calculus S (within a standard class of right-sided calculi) is complete if and only ifS contains (that is, each rule of is derivable in S). Thus one can view as a minimal complete core of Gentzen’s .     

The concept of proof in the light of mathematical work

ZDM – Mathematics Education - Our article aims to show how illuminating mathematical work as a concept from didactics of mathematics is useful in understanding issues relating to proving and...     

The perturbation classes problem in Fredholm theory

We give a negative answer to the perturbation classes problem: the perturbation class of the upper semi-Fredholm operators contains properly the strictly singular operators, and the perturbation class of the lower semi-Fredholm operators contains properly the strictly cosingular operators.     

Decomposition of two classes of structural models

The conditional independence structure of a common probability measure is a structural model. In this paper, we solve an open problem posed by Studeny [Probabilistic Conditional Independence Structures, Theme 9, p. 206]. A new approach is proposed to decompose a directed acyclic graph and its optimal properties are studied. We interpret this approach from the perspective of the decomposition of the corresponding conditional independence model and provide an algorithm for identifying the maximal prime subgraphs in a directed acyclic graph.     

Some classes of matrices in linear complementarity theory

The linear complementarity problem is the problem of finding solutionsw, z tow = q + Mz, w0,z0, andw ^T z=0, whereq is ann-dimensional constant column, andM is a given square matrix of dimensionn. In this paper, the author introduces a class of matrices such that for anyM in this class a solution to the above problem exists for all feasibleq, and such that Lemke's algorithm will yield a solution or demonstrate infeasibility. This class is a refinement of that introduced and characterized by Eaves. It is also shown that for someM in this class, there is an even number of solutions for all nondegenerateq, and that matrices for general quadratic programs and matrices for polymatrix games nicely relate to these matrices.Research partially supported by National Science Foundation Grant NSF-GP-15031.     

On the global theory of some classes of mappings

This paper is devoted to the study of the global theory of certain mappings between Riemannian manifolds. We generalize results by Vilms, Yano and Ishihara, and study in detail projective, umbilical and harmonic maps.The work of the author was supported by RBRF, grant 94-01-01595 (Russia).     

A proof of Pasch's axiom in the absolute theory of oriented parallelity

Trapezium and Parallelogram Configurations in the theory of oriented parallelity are considered. The Pasch theorem is proved.     

Continuity, proof systems and the theory of transfinite computations

 We use the theory of domains with totality to construct some logics generalizing ω-logic and β-logic and we prove a completenes theorem for these logics. The key application is E-logic, the logic related to the functional ³ E. We prove a compactness theorem for sets of sentences semicomputable in ³ E. Received: 21 January 1998 / Published online: 2 September 2002     

Order-enriched categorical models of the classical sequent calculus

It is well-known that weakening and contraction cause naïve categorical models of the classical sequent calculus to collapse to Boolean lattices. Starting from a convenient formulation of the well-known categorical semantics of linear classical sequent proofs, we give models of weakening and contraction that do not collapse. Cut-reduction is interpreted by a partial order between morphisms. Our models make no commitment to any translation of classical logic into intuitionistic logic and distinguish non-deterministic choices of cut-elimination. We show soundness and completeness via initial models built from proof nets, and describe models built from sets and relations.     

Some remarks on nonstandard theory of classes

In the paper, we discuss several hitherto open problems concerning classes in nonstandard set theory. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 5, pp. 233–255, 2005.     

Lattice theory for certain classes of rings

Conclusion We observe that every Boolean ring is aP ₁-ring, where each element is its own minimal idempotent duplicator; a Boolean ring with unity is also aP ₂-ring, where each element is the maximal idempotent annihilator of its complement; and thus,P ₁-rings andP ₂-rings are generalisations of a Boolean ring in terms of its lattice structure. Then, it is only natural that these two classes of rings have such close affiliations with the lattice structure of a Boolean ring.