Łoś's theorem and the axiom of choice

In set theory without the Axiom of Choice ( $\mathsf{AC}$), we investigate the problem of the placement of ?o?'s Theorem ( $\mathsf{LT}$) in the hierarchy of weak choice principles, and answer several open questions from the book Consequences of the Axiom of Choice by Howard and Rubin, as well as an open question by Brunner. We prove a number of results summarised in § 3.     

A Łoś type theorem for linear metric formulas

We define an ultraproduct of metric structures based on a maximal probability charge and prove a variant of ?o? theorem for linear metric formulas. We also consider iterated ultraproducts (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Corrigendum to “Categorical abstract algebraic logic: The criterion for deductive equivalence”

We give a correction to the paper [2] mentioned in the title. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fuzzy logic in commercial expert systems — Results and prospects

This paper surveys the use of fuzzy methods in commercial applications of the technology of expert systems and in commercial products which aim to support such applications. It attempts to evaluate the utility of such an approach to uncertainty management compared to other well known methods of handling uncertainty in expert systems. Starting from this base it attempts an evaluation of the prospects for fuzzy expert systems in the medium term. As a survey it attempts to list applications and commercial products as comprehensively as is practical and includes an extensive bibliography on the topic.     

Extending possibilistic logic over Gödel logic

In this paper we present several fuzzy logics trying to capture different notions of necessity (in the sense of possibility theory) for Gödel logic formulas. Based on different characterizations of necessity measures on fuzzy sets, a group of logics with Kripke style semantics is built over a restricted language, namely, a two-level language composed of non-modal and modal formulas, the latter, moreover, not allowing for nested applications of the modal operator N. Completeness and some computational complexity results are shown.     

Categorical abstract algebraic logic: Gentzen π ‐institutions and the deduction‐detachment property

Given a π ‐institution I , a hierarchy of π ‐institutions I ^{(n )} is constructed, for n ≥ 1. We call I ^{(n )} the n‐th order counterpart of I . The second‐order counterpart of a deductive π ‐institution is a Gentzen π ‐institution, i.e. a π ‐institution associated with a structural Gentzen system in a canonical way. So, by analogy, the second order counterpart I ⁽²⁾ of I is also called the “Gentzenization” of I . In the main result of the paper, it is shown that I is strongly Gentzen , i.e. it is deductively equivalent to its Gentzenization via a special deductive equivalence, if and only if it has the deduction‐detachment property . (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The logic of pseudo-S-integers

Let {n _i } be a sequence of natural numbers and let {p _i } be a listing of rational primes. Then an abelian groupG={x ∈ √| ord _pi x ≥ −n _i } is called a group of pseudo-integers. We investigate the logical properties of such groups of pseudo-integers and the counterparts of such groups in global fields in the case the number of primes allowed to appear in the denominator is infinite. We show that, while the addition problem of any recursive group of pseudo-integers is decidable, the Diophantine problem for some recursive groups of pseudo-integers with infinite number of primes allowed in the denominator, is not decidable. More precisely, there exist recursive groups of pseudo-integers, where infinite number of primes are allowed to appear in the denominator, such that there is no uniform algorithm to decide whether a polynomial equation over ℤ in several variables has solutions in the group. This result is obtained by giving a Diophantine definition of ℤ over these groups. The proof is based on the strong Hasse norm principal. The research for this paper has been partially supported by NSA grant MDA904-96-1-0019.     

The logic of integration

We develop a model theoretic framework for studying algebraic structures equipped with a measure. The real line is used as a value space and its usual arithmetical operations as connectives. Integration is used as a quantifier. We extend some basic results of pure model theory to this context and characterize measurable sets in terms of zero-sets of formulas.      

The simplest protoalgebraic logic

The logic is the sentential logic defined in the language with just implication → by the axiom of reflexivity or identity “” and the rule of Modus Ponens “from φ and to infer ψ”. The theorems of this logic are exactly all formulas of the form . We argue that this is the simplest protoalgebraic logic, and that in it every set of assumptions encodes in itself not only all its consequences but also their proofs. In this paper we study this logic from the point of view of abstract algebraic logic, and in particular we use it as a relatively natural counterexample to settle some open problems in this theory. It appears that this logic has almost no properties: it is neither equivalential nor weakly algebraizable; it does not have an algebraic semantics; it does not satisfy any form of the Deduction Theorem, other than the most general parameterized and local one that all protoalgebraic logics satisfy; it is not filter‐distributive; and so on. It satisfies some forms of the interpolation property but in a rather trivial way. Very few things are known about its algebraic counterpart, save that its intrinsic variety is the class of all algebras of the similarity type.     

The complexity of plane hyperbolic incidence geometry is ∀∃∀∃

We show that plane hyperbolic geometry, expressed in terms of points and the ternary relation of collinearity alone, cannot be expressed by means of axioms of complexity at most ???, but that there is an axiom system, all of whose axioms are ???? sentences. This remains true for Klingenberg's generalized hyperbolic planes, with arbitrary ordered fields as coordinate fields. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modal‐type orthomodular logic

In this paper we enrich the orthomodular structure by adding a modal operator, following a physical motivation. A logical system is developed, obtaining algebraic completeness and completeness with respect to a Kripkestyle semantic founded on Baer*‐semigroups as in [22] (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The 1‐2‐3‐Conjecture for Hypergraphs

A weighting of the edges of a hypergraph is called vertex‐coloring if the weighted degrees of the vertices yield a proper coloring of the graph, i.e. every edge contains at least two vertices with different weighted degrees. In this article, we show that such a weighting is possible from the weight set for all hypergraphs with maximum edge size and not containing edges solely consisting of identical vertices. The number is best possible for this statement.     

The ∞‐Besov capacity problem

A theory of ∞‐Besov capacities is developed and several applications are provided. In particular, we solve an open problem in the theory of limits of the ∞‐Besov semi‐norms, we obtain new restriction‐extension inequalities, and we characterize the pointwise multipliers acting on the ∞‐Besov spaces.     

The modal logic of forcing

A set theoretical assertion is forceable or possible, written , if holds in some forcing extension, and necessary, written , if holds in all forcing extensions. In this forcing interpretation of modal logic, we establish that if is consistent, then the ZFC-provable principles of forcing are exactly those in the modal theory .