This paper introduces valuation structures associated with preferential models. Based on KLM valuation structures, we present a canonical approach to obtain injective preferential models for any preferential relation satisfying the property INJ, and give uniform proofs of representation theorems for injective preferential relations appeared in the literature. In particular, we show that, in any propositional language (finite or infinite), a preferential inference relation satisfies INJ if and only if it can be represented by a standard preferential model. This conclusion generalizes the result obtained by Freund. In addition, we prove that, when the language is finite, our framework is sufficient to establish a representation theorem for any injective relation.

In the study of categories whose morphisms display a behaviour similar to that of partial functions, the concept of morphism domain is, obviously, central. In this paper an operation defined on morphisms describes those properties which are related to morphisms being regarded as abstractions of partial functions. This operation allows us to characterise the morphism domains directly, and gives rise to an algebra defined by a simple set of identities. No product-like categorical structures are needed therefore. We also develop the construction of topologies together with the notion of continuous morphism, in order to test the effectiveness of this approach. It is interesting to see how much of the computational character of the morphisms is translated into continuity.

We study the relationships between definitions of compactness in topological spaces and the roll the axiom of choice plays in these relationships.

The present report is a, somewhat lengthy, addendum to "A new deconstructive logic: linear logic," where the elimination of cuts from derivations in sequent calculus for classical logic was studied ‘from the point of view of linear logic’. To that purpose a formulation of classical logic was used, that - as in linear logic - distinguishes between multiplicative and additive versions of the binary connectives.

The main novelty here is the observation that this type-distinction is not essential: we can allow classical sequent derivations to use any combination of additive and multiplicative introduction rules for each of the connectives, and still have strong normalization and confluence of tq-reductions.

We give a detailed description of the simulation of tq-reductions by means of reductions of the interpre- tation of any given classical proof as a proof net of PN2 (the system of second order proof nets for linear logic), in which moreover all connectives can be taken to be of one type, e.g., multiplicative.

We finally observe that dynamically the different logical cuts, as determined by the four possible combinations of introduction rules, are independent: it is not possible to simulate them internally, i.e., by only one specific combination, and structural rules.

If T is a model complete theory with the strict order property, then the theory of the models of T with an automorphism has no model companion

We analyze the two cardinal properties of definable sets in homogeneous graphs.

We give a complete description of minimal groups infinitely definable in separably closed fields of finite degree of imperfection. In particular we answer positively the question of the existence of such a group with infinite transcendence degree (i.e., a minimal group with non thin generic).

A nonstandard set theory ∗ZFC is proposed that axiomatizes the nonstandard embedding ∗. Besides the usual principles of nonstandard analysis, all axioms of ZFC except regularity are assumed. A strong form of saturation is also postulated. ∗ZFC is a conservative extension of ZFC.

We show that equational independence in the sense of Srour equals local non-forking. We then examine so-called almost equational theories where equational independence is a symmetric relation.

We consider the problem of recognizing important properties of logical calculi and find complexity bounds for some decidable properties. For a given logical system $L$, a property $P$ of logical calculi is called decidable over $L$ if there is an algorithm which for any finite set $Ax$ of new axiom schemes decides whether the calculus $L + Ax$ has the property $P$ or not. In "Complexity of some problems in modal and superintuitionistic logics," the complexity of tabularity, pre-tabularity, and interpolation problems over the intuitionistic logic Int and over modal logic $S4$ was studied, also we found the complexity of amalgamation problems in varieties of Heyting algebras and closure algebras.

In the present paper we deal with positive calculi. We prove $NP$-completeness of tabularity, $DP$- hardness of pretabularity and PSPACE-completeness of interpolation problem over $Int^+$. In addition to above-mentioned properties, we consider Beth’s definability properties. Also we improve some complexity bounds for properties of superintuitionistic calculi.

A computably enumerable (c.e.) degree is a maximal contiguous degree if it is contiguous and no c.e. degree strictly above it is contiguous. We show that there are infinitely many maximal contiguous degrees. Since the contiguous degrees are definable, the class of maximal contiguous degrees provides the first example of a definable infinite anti-chain in the c.e. degrees. In addition, we show that the class of maximal contiguous degrees forms an automorphism base for the c.e. degrees and therefore for the Turing degrees in general. Finally we note that the construction of a maximal contiguous degree can be modified to answer a question of Walk about the array computable degrees and a question of Li about isolated formulas

We investigate the effect of a variant of Matet forcing on ultrafilters in the ground model and give a characterization of those $P$–points that survive such forcing, answering a question left open by Blass. We investigate the question of when this variant of Matet forcing can be used to diagonalize small filters without destroying $P$–points in the ground model. We also deal with the question of generic existence of stable ordered–union ultrafilters.