This paper contains a joint study of two sentential logics that combine a many-valued character, namely tetravalence, with a modal character; one of them is normal and the other one quasinormal. The method is to study their algebraic counterparts and their abstract models with the tools of Abstract Algebraic Logic, and particularly with those of Brown and Suszko's theory of abstract logics as recently developed by Font and Jansana in their “A General Algebraic Semantics for Sentential Logics”. The logics studied here arise from the algebraic and lattice-theoretical properties we review of Tetravalent Modal Algebras, a class of algebras studied mainly by Loureiro, and also by Figallo. Landini and Ziliani, at the suggestion of the late Antonio Monteiro.

In [7] the author discussed the relative force —in the set theory ZF— of some representation theorems for ℓ-groups (lattice-ordered groups). One of the theorems not discussed in that paper is the Hahn representation theorem for abelian ℓ-groups. This result, originally proved by Hahn (see [8]) for totally ordered groups and half a century later by Conrad, Harvey and Holland for the general case (see [4]), states that any abelian ℓ-group can be embedded in a Hahn product of copies of R (the real line with its natural totally-ordered group structure). Both proofs rely heavily on Zorn's Lemma which is equivalent to AC (the axiom of choice).

The aim of this work is to point out the use of non-constructible axioms (i.e., AC and weaker forms of it) in the proofs. Working in the frame of ZFA, that is, the Zermelo-Fraenkel set theory where a non-empty set of atoms is allowed, we present alternative proofs which, in the totally ordered case, do not require the use of AC. For basic concepts and notation on ℓ-groups the reader can refer to [1] and [2]. For set theory, to [11].

The first example of a simultaneous inductive-recursive definition in intuitionistic type theory is Martin-Löfs universe à la Tarski. A set U0 of codes for small sets is generated inductively at the same time as a function T0, which maps a code to the corresponding small set, is defined by recursion on the way the elements of U0 are generated.

In this paper we argue that there is an underlying general notion of simultaneous inductive-recursive definition which is implicit in Martin-Löf's intuitionistic type theory. We extend previously given schematic formulations of inductive definitions in type theory to encompass a general notion of simultaneous induction-recursion. This enables us to give a unified treatment of several interesting constructions including various universe constructions by Palmgren, Griffor, Rathjen, and Setzer and a constructive version of Aczel's Frege structures. Consistency of a restricted version of the extension is shown by constructing a realisability model in the style of Allen.

We characterize Maharam spectra of Loeb probability spaces and give some applications of the results.

We extend in a natural way the operation of Turing machines to infinite ordinal time, and investigate the resulting supertask theory of computability and decidability on the reals. Every set. for example, is decidable by such machines, and the semi-decidable sets form a portion of the sets. Our oracle concept leads to a notion of relative computability for sets of reals and a rich degree structure, stratified by two natural jump operators.

A notion of an epistemic state is introduced as a generalization of common representations suggested for belief change. Based on it, a new kind of nonmonotonic inference relation corresponding to belief contractions is defined. A number of representation results is established that cover both traditional AGM contractions and contractions that do not satisfy recovery.

n + 1 nested k-ary fixed point operators are more expressive than n. This holds on finite structures for all sublogics of partial fixed point logic PFP that can express conjunction, existential quantification and deterministic transitive closure of binary relations using at most k-ary fixed point operators and that are closed against subformulas. Among those are a lot of popular fixed point logics.

In the paper we study the class of weakly algebraizable logics, characterized by the monotonicity and injectivity of the Leibniz operator on the theories of the logic. This class forms a new level in the non-linear hierarchy of protoalgebraic logics.

Rational Pavelka logic extends Łukasiewicz infinitely valued logic by adding truth constants r̄ for rationals in [0. 1]. We show that this is a conservative extension. We note that this shows that provability degree can be defined in Łukasiewicz logic. We also give a counterexample to a soundness theorem of Belluce and Chang published in 1963.

The use of elementary submodels has become a standard tool in set-theoretic topology and infinitary combinatorics. Thus, in studying some combinatorial objects, one embeds them in a set, M, which is an elementary submodel of the universe, V (that is, (M; Є) ≺ (V; Є)). Applying the downward Löwenheim-Skolem Theorem, one can bound the cardinality of M. This tool enables one to capture various complicated closure arguments within the simple “≺”.

However, in this paper, as in the paper [JT], we study the tool for its own sake. [JT] discussed various general properties of topological spaces in elementary submodels. In this paper, we specialize this consideration to the space of real numbers, ℝ. Our models M are not in general transitive. We will always have ℝ Є M, but not usually ℝ ⊆ M. We plan to study properties of the ℝ ⋂ M's. In particular, as M varies, we wish to study whether any two of these ℝ ⋂ M's are isomorphic as topological spaces, linear orders, or fields.

As usual, it takes some sleight-of-hand to formalize these notions within the standard axioms of set theory (ZFC), since within ZFC, one cannot actually define the notion (M;Є) ≺ (V;Є). Instead, one proves theorems about M such that (M;Є) ≺ (H(θ);Є), where θ is a “large enough” cardinal; here, H(θ) is the collection of all sets whose transitive closure has size less than θ.

Using projectivity groups, we classify some polygons with strongly minimal point rows and show in particular that no infinite quadrangle can have sharply 2-transitive projectivity groups in which the point stabilizers are abelian. In fact, we characterize the finite orthogonal quadrangles Q(4,2). Q− (5.2) and Q(4,3) by this property. Finally we show that the sets of points, lines and flags of any ℵ1-categorical polygon have Morley degree 1.

If G is an omega-stable group with a normal definable subgroup H, then the Sylow-2-subgroups G/H are the images of the Sylow-2-subgroups of G.

Zusammenfassung. Sei G eine omega-stabile Gruppe und H ein definierbarer Normalteiler von G. Dann sind die Sylow-2-Untergruppen von G/H Bilder der Sylow-2-Untergruppen von G.

Let Π be a complete théorie of pseudo-finite fields. In this article we prove that, in the langage of fields to which we add a unary predicate for a substructure, the theory of non trivial elementary pairs of models of Π has completions, that is, the maximum that could exist.

Preferential structures are probably the best examined semantics for nonmonotonic and deontic logics: in a wider sense, they also provide semantical approaches to theory revision and update, and other fields where a preference relation between models is a natural approach. They have been widely used to differentiate the various systems of such logics, and their construction is one of the main subjects in the formal investigation of these logics. We introduce new techniques to construct preferential structures for completeness proofs. Since our main interest is to provide general techniques, which can be applied in various situations and for various base logics (propositional and other), we take a purely algebraic approach, which can be translated into logics by easy lemmata, in particular, we give a clean construction via indexing by trees for transitive structures, this allows us to simplify the proofs of earlier work by the author, and to extend the results given there.

The relationship between killing ideals and adding reals by forcings is analysed.

In the present paper we prove that the poset of all extensions of the logic defined by a class of matrices whose sets of distinguished values are equationally definable by their algebra reducts is the retract, under a Galois connection, of the poset of all subprevarieties of the prevariety generated by the class of the algebra reducts of the matrices involved. We apply this general result to the problem of finding and studying all extensions of the logic of paradox (viz., the implication-free fragment of any non-classical normal extension of the relevance-mingle logic). In order to solve this problem, we first study the structure of prevarieties of Kleene lattices. Then, we show that the poset of extensions of the logic of paradox forms a four-element chain, all the extensions being finitely many-valued and finitely-axiomatizable logics. There are just two proper consistent extensions of the logic of paradox. The first is the classical logic that is relatively axiomatized by the Modus ponens rule for the material implication. The second extension, being intermediate between the logic of paradox and the classical logic, is the one relatively axiomatized by the Ex Contradictione Quodlibet rule.

An ω-categorical supersimple group is finite-by-abelian-by-finite, and has finite SU-rank. Every definable subgroup is commensurable with an acl(ø)-definable subgroup. Every finitely based regular type in a CM-trivial ω-categorical simple theory is non-orthogonal to a type of SU-rank 1. In particular, a supersimple ω-categorical CM-trivial theory has finite SU-rank.

One of the fundamental results of descriptive complexity theory, due to Immerman [13] and Vardi [18], says that a class of ordered finite structures is definable in fixed-point logic if, and only if, it is computable in polynomial time. Much effort has been spent on the problem of capturing polynomial time, that is, describing all polynomial time computable classes of not necessarily ordered finite structures by a logic in a similar way.

The most obvious shortcoming of fixed-point logic itself on unordered structures is that it cannot count. Immerman [14] responded to this by adding counting constructs to fixed-point logic. Although it has been proved by Cai, Fürer, and Immerman [1] that the resulting fixed-point logic with counting, denoted by IFP+C, still does not capture all of polynomial time, it does capture polynomial time on several important classes of structures (on trees, planar graphs, structures of bounded tree-width [15, 9, 10]).

The main motivation for such capturing results is that they may give a better understanding of polynomial time. But of course this requires that the logical side is well understood. We hope that our analysis of IFP+C-formulas will help to clarify the expressive power of IFP+C; in particular, we derive a normal form. Moreover, we obtain a problem complete for IFP+C under first-order reductions.

We describe a method for obtaining classical logic from intuitionistic logic which does not depend on any proof system, and show that by applying it to the most important implicational relevance logics we get relevance logics with nice semantical and proof-theoretical properties. Semantically all these logics are sound and strongly complete relative to classes of structures in which all elements except one are designated. Proof-theoretically they correspond to cut-free hypersequential Gentzen-type calculi. Another major property of all these logics is that the classical implication can faithfully be translated into them.

We address ZFC inequalities between some cardinal invariants of the continuum, which turned out to be true in spite of strong expectations given by [11].