Michał Krynicki

The Ha¨rtig quantifier: a survey

A fundamental notion in a large part of mathematics is the notion of equicardinality. The language with Hartig quantifier is, roughly speaking, a first-order language in which the notion of equicardinality is expressible. Thus this language, denoted by LI, is in some sense very natural and has in consequence special interest. Properties of LI are studied in many papers. In [BF, Chapter VI] there is a short survey of some known results about LI. We feel that a more extensive exposition of these results is needed. The aim of this paper is to give an overview of the present knowledge about the language LI and list a selection of open problems concerning it. After the Introduction (?1), in ??2 and 3 we give the fundamental results about LI. In ?4 the known model-theoretic properties are discussed. The next section is devoted to properties of mathematical theories in LI. In ?6 the spectra of sentences of LI are discussed, and ?7 is devoted to properties of Llwhich depend on set-theoretic assumptions. The paper finishes with a list of open problem and an extensive bibliography. The bibliography contains not only papers we refer to but also all papers known to us containing results about the language with Hartig quantifier.

Hierarchies of Partially Ordered Connectives and Quantifiers

Connections between partially ordered connectives and Henkin quantifiers are considered. It is proved that the logic with all partially ordered connectives and the logic with all Henkin quantifiers coincide. This implies that the hierarchy of partially ordered connectives is strongly hierarchical and gives several nondefinability results between some of them. It is also deduced that each Henkin quantifier can be defined by a quantifier of the form what is a strengthening of the Walkoe result. MSC: 03C80.

An axiomatization of the logic with the rough quantifier

We consider the language L(Q), where L is a countable first-order language and Q is an additional generalized quantifier. A weak model for L(Q) is a pair , where W is a first-order structure for L and q is a family of subsets of its universe. In case that q is the set of classes of some equivalence relation the weak model is called a partition model. The interpretation of Q in partition models was studied by Szczerba [3], who was inspired by Pawlaks paper [2]. The corresponding set of tautologies in L(Q) is called rough logic. In the following we will give a set of axioms of rough logic and prove its completeness. Rough logic is designed for creating partition models.

A note on syntactical and semantical functions

We say that a semantical function Г is correlated with a syntactical function F iff for any structure A and any sentence ϕ we have A ⊧ Fϕ ↔ ΓA ⊧ ϕ.It is proved that for a syntactical function F there is a semantical function Г correlated with F iff F preserves propositional connectives up to logical equivalence. For a semantical function Г there is a syntactical function F correlated with Г iff for any finitely axiomatizable class X the class Г−1X is also finitely axiomatizable (i.e. iff Г is continuous in model class topology).

Decidability problems in languages with Henkin quantifiers

Abstract Krynicki, M. and M. Mostowski, Decidability problems in languages with Henkin quantifiers, Annals of Pure and Applied Logic 58 (1992) 149–172. We consider the language L (Hω) with all Henkin quantifiers H n defined as follows: H n x 1 ... x n y 1 ... y n φ( x 1 ,..., x n , y 1 ,..., y n ) iff ∃ f 1 ... f n ∀ x 1 . .. x n φ( x 1 ,..., x n , f 1 ( x 1 ), ..., f n ( x n )). We show that the theory of equality in L (Hω) is undecidable. The proof of this result goes by interpretation of the word problem for semigroups. Henkin quantifiers are strictly related to the function quantifiers F n defined as follows: F n x 1 ... x n y 1 ... y n φ( x 1 ,..., x n , y 1 ,..., y n ) iff ∃ f ∀ x 1 ... x n φ( x 1 ,..., x n , f ( x 1 ),..., f ( x n )). In contrast with the first result we show that the theory of equality with all quantifiers F n is decidable. We also consider decidability problems for other theories in languages L (F 2 ) and L (H 2 ).

On simplicity of formulas

Simple formula should contain only few quantifiers. In the paper the methods to estimate quantity and quality of quantifiers needed to express a sentence equivalent to given one.

Quantifiers, Some Problems and Ideas

The word quantifier comes from the latin quantitas (quantity) as contrasted with qualitas (quality). This intuition does not mean that the notion of quantifier is understood as an arithmetical concept. We understand quantifiers rather as qualifiers for distributive concepts (such as countable nouns) unlike those applicable to nondistributive concepts.

Theories of initial segments of standard models of arithmetics and their complete extensions

We investigate families of finite initial segments of standard models for various arithmetics. We give an axiomatization of the theory of sentences true in almost all finite models with addition. We also characterize its complete extensions and relate its infinite models to models of Presburger arithmetic. We also estimate the complexity of complete extensions of the arithmetic with addition and multiplication.

On the Logic with Rough Quantifier

The main aim of this paper is to present a survey of results on the logic with rough quantifier. Besides, a classification of simplicity of formulas of the logic with rough quantifier is defined and a criterion for placing a formula on the exact simplicity level is given.

The non-definability notion and first order logic

The theorem to the effect that the languageLΔ introduced in [2] is mutually interpretable with the first order language is proved. This yields several model-theoretical results concerningLΔ.