Some characterization theorems for infinitary universal Horn logic without equality

1996 ◽  
Vol 61 (4) ◽  
pp. 1242-1260 ◽  
Author(s):  
Pilar Dellunde ◽  
Ramon Jansana
Keyword(s):  
Algebraic Logic ◽  
Point Of View ◽  
Abstract Algebraic Logic ◽  
Relation Symbol ◽  
Horn Logic ◽  
First Order ◽  
Order Language ◽  
Unary Relation ◽  
Theoretic Point ◽  
Horn Fragment

In this paper we mainly study preservation theorems for two fragments of the infinitary languages Lκκ, with κ regular, without the equality symbol: the universal Horn fragment and the universal strict Horn fragment. In particular, when κ is ω, we obtain the corresponding theorems for the first-order case.The universal Horn fragment of first-order logic (with equality) has been extensively studied; for references see [10], [7] and [8]. But the universal Horn fragment without equality, used frequently in logic programming, has received much less attention from the model theoretic point of view. At least to our knowledge, the problem of obtaining preservation results for it has not been studied before by model theorists. In spite of this, in the field of abstract algebraic logic we find a theorem which, properly translated, is a preservation result for the strict universal Horn fragment of infinitary languages without equality which, apart from function symbols, have only a unary relation symbol. This theorem is due to J. Czelakowski; see [5], Theorem 6.1, and [6], Theorem 5.1. A. Torrens [12] also has an unpublished result dealing with matrices of sequent calculi which, properly translated, is a preservation result for the strict universal Horn fragment of a first-order language. And in [2] of W. J. Blok and D. Pigozzi we find Corollary 6.3 which properly translated corresponds to our Corollary 19, but for the case of a first-order language that apart from its function symbols has only one κ-ary relation symbol, and for strict universal Horn sentences. The study of these results is the basis for the present work. In the last part of the paper, Section 4, we will make these connections clear and obtain some of these results from our theorems. In this way we hope to make clear two things: (1) The field of abstract algebraic logic can be seen, in part, as a disguised study of universal Horn logic without equality and so has an added interest. (2) A general study of universal Horn logic without equality from a model theoretic point of view can be of help in the field of abstract algebraic logic.

scholarly journals Completeness of two theories on ordered abelian groups and embedding relations

1980 ◽  
Vol 77 ◽  
pp. 33-39 ◽  
Author(s):  
Yuichi Komori
Keyword(s):  
Binary Relation ◽  
Abelian Groups ◽  
Function Symbol ◽  
Binary Function ◽  
Relation Symbol ◽  
Unary Function ◽  
First Order ◽  
Order Language ◽  
Unary Relation ◽  
Binary Relation Symbol

The first order language ℒ that we consider has two nullary function symbols 0, 1, a unary function symbol –, a binary function symbol +, a unary relation symbol 0 <, and the binary relation symbol = (equality). Let ℒ′ be the language obtained from ℒ, by adding, for each integer n > 0, the unary relation symbol n| (read “n divides”).

Relation algebra reducts of cylindric algebras and an application to proof theory

2002 ◽  
Vol 67 (1) ◽  
pp. 197-213 ◽  
Author(s):  
Robin Hirsch ◽  
Ian Hodkinson ◽  
Roger D. Maddux
Keyword(s):  
Proof Theory ◽  
Predicate Logic ◽  
Algebraic Logic ◽  
Relation Algebra ◽  
Relation Algebras ◽  
Cylindric Algebras ◽  
First Order ◽  
Order Language ◽  
Binary Relation Symbol ◽  
Made In

AbstractWe confirm a conjecture, about neat embeddings of cylindric algebras, made in 1969 by J. D. Monk, and a later conjecture by Maddux about relation algebras obtained from cylindric algebras. These results in algebraic logic have the following consequence for predicate logic: for every finite cardinal α ≥ 3 there is a logically valid sentence X, in a first-order language ℒ with equality and exactly one nonlogical binary relation symbol E, such that X contains only 3 variables (each of which may occur arbitrarily many times), X has a proof containing exactly α + 1 variables, but X has no proof containing only α variables. This solves a problem posed by Tarski and Givant in 1987.

QE commutative nilrings

1984 ◽  
Vol 49 (2) ◽  
pp. 644-651 ◽  
Author(s):  
D. Saracino ◽  
C. Wood
Keyword(s):  
Classification Problem ◽  
Commutative Rings ◽  
Point Of View ◽  
Negative Evidence ◽  
Characteristic P ◽  
Locally Finite ◽  
First Order ◽  
Order Language ◽  
Algebraic Description ◽  
Countably Infinite

If L is a first-order language, then an L-structure A is called quantifier-eliminable (QE) if every L-formula is equivalent in A to a formula without quantifiers.The classification problem for QE groups and rings has received attention in work by Berline, Boffa, Cherlin, Feigner, Macintyre, Point, Rose, the present authors, and others. In [1], Berline and Cherlin reduced the problem for rings of prime characteristic p to that for nilrings, but also constructed countable QE nilrings of characteristic p. Likewise, in [3], we constructed countable QE nil-2 groups. Both results can be viewed as “nonstructure theorems”, in that they provide negative evidence for any attempt at classification. In the present paper we show that the situation is equally bad (or rich, depending on one's point of view) for commutative rings:Theorem 1. For any odd prime p, there existcountable QE commutative nilrings of characteristic p.This solves a problem posed in [1]. We remark that the examples we produce are uniformly locally finite, hence ℵ0-categorical. A more algebraic description is that each of our rings R is uniformly locally finite (in fact, R3 = 0) and homogeneous, in the sense that any isomorphism of finitely generated subrings extends to an automorphism of R.Theorem 1 does not cover the case p = 2, and we show that for commutative rings this case is in fact exceptional:Theorem 2. There exist exactly two nonisomorphic countably infinite QE commutative nilrings of characteristic 2.

Provability with Finitely Many Variables

2002 ◽  
Vol 8 (3) ◽  
pp. 348-379 ◽  
Author(s):  
Robin Hirsch ◽  
Ian Hodkinson ◽  
Roger D. Maddux
Keyword(s):  
Predicate Logic ◽  
Algebraic Logic ◽  
Order Logic ◽  
First Order Logic ◽  
Alternative Semantics ◽  
Relation Symbol ◽  
First Order ◽  
Binary Predicate ◽  
Standard Set ◽  
Binary Relation Symbol

AbstractFor every finite n ≥ 4 there is a logically valid sentence φn with the following properties: φn contains only 3 variables (each of which occurs many times); φn contains exactly one nonlogical binary relation symbol (no function symbols, no constants, and no equality symbol); φn has a proof in first-order logic with equality that contains exactly n variables, but no proof containing only n − 1 variables. This result was first proved using the machinery of algebraic logic developed in several research monographs and papers. Here we replicate the result and its proof entirely within the realm of (elementary) first-order binary predicate logic with equality. We need the usual syntax, axioms, and rules of inference to show that φn has a proof with only n variables. To show that φn has no proof with only n − 1 variables we use alternative semantics in place of the usual, standard, set-theoretical semantics of first-order logic.

On countable fractions from an elementary class

1994 ◽  
Vol 59 (4) ◽  
pp. 1410-1413
Author(s):  
C. J. Ash
Keyword(s):  
Model Theory ◽  
Relation Symbol ◽  
First Order ◽  
Elementary Class ◽  
Free Variables ◽  
Order Language ◽  
Skolem Theorem ◽  
Elementary Result ◽  
Countable Models

The following fairly elementary result seems to raise possibilities for the study of countable models of a theory in a countable language. For the terminology of model theory we refer to [CK].Let L be a countable first-order language. Let L′ be the relational language having, for each formula φ of L and each sequence υ1,…,υn of variables including the free variables of φ, an n-ary relation symbol Pφ. For any L-structure and any formula Ψ(υ) of L, we define the Ψ-fraction of to be the L′-structure Ψ whose universe consists of those elements of satisfying Ψ(υ) and whose relations {Rφ}φϵL are defined by letting a1,…,an satisfy Rφ in Ψ if, and only if, a1,…, an satisfy φ in .An L-elementary class means the class of all L-structures satisfying each of some set of sentences of L. The countable part of an L-elementary class K means the class of all countable L-structures from K.Theorem. Let K be an L-elementary class and let Ψ(υ) be a formula of L. Then the class of countable Ψ-fractions of structures in K is the countable part of some L′-elementary class.Comment. By the downward Löwenheim-Skolem theorem, the countable Ψ-fractions of structures in K are the same as the Ψ-fractions of countable structures in K.Proof. We give a set Σ′ of L′-sentences whose countable models are exactly the countable Ψ-fractions of structures in K.

On the Craig-Lyndon interpolation theorem

1968 ◽  
Vol 33 (2) ◽  
pp. 271-274
Author(s):  
Arnold Oberschelp
Keyword(s):  
Interpolation Theorem ◽  
Relation Symbol ◽  
First Order ◽  
Order Language ◽  
Identity Symbol

In his paper [3] Henkin proved for a first order language with identity symbol but without operation symbols the following version of the Craig-Lyndon interpolation theorem:Theorem 1. If Γ╞Δ then there is a formula θ such that Γ ├Δand(i) any relation symbol with a positive (negative) occurrence in θ has a positive (negative) occurrence in some formula of Γ.

Semantics of the infinitistic rules of proof

1976 ◽  
Vol 41 (1) ◽  
pp. 121-138
Author(s):  
Krzysztof Rafal Apt
Keyword(s):  
Completeness Theorem ◽  
Recursion Theory ◽  
Point Of View ◽  
First Order ◽  
Order Language ◽  
The Subject ◽  
Order Arithmetic ◽  
Additional Rule ◽  
First Time ◽  
Further Development

This paper is devoted to the study of the infinitistic rules of proof i.e. those which admit an infinite number of premises. The best known of these rules is the ω-rule. Some properties of the ω-rule and its connection with the ω-models on the basis of the ω-completeness theorem gave impulse to the development of the theory of models for admissible fragments of the language . On the other hand the study of representability in second order arithmetic with the ω-rule added revealed for the first time an analogy between the notions of re-cursivity and hyperarithmeticity which had an important influence on the further development of generalized recursion theory.The consideration of the subject of infinitistic rules in complete generality seems to be reasonable for several reasons. It is not completely clear which properties of the ω-rule were essential for the development of the above-mentioned topics. It is also worthwhile to examine the proof power of infinitistic rules of proof and what distinguishes them from finitistic rules of proof.What seemed to us the appropriate point of view on this problem was the examination of the connection between the semantics and the syntax of the first order language equipped with an additional rule of proof.

scholarly journals The Road to Modern Logic—An Interpretation

2001 ◽  
Vol 7 (4) ◽  
pp. 441-484 ◽  
Author(s):  
José Ferreirós
Keyword(s):  
Order Logic ◽  
Point Of View ◽  
First Order Logic ◽  
Theoretical Point ◽  
Primary Role ◽  
Philosophical Discussion ◽  
Modern Logic ◽  
Core System ◽  
First Order ◽  
The Road

AbstractThis paper aims to outline an analysis and interpretation of the process that led to First-Order Logic and its consolidation as a core system of modern logic. We begin with an historical overview of landmarks along the road to modern logic, and proceed to a philosophical discussion casting doubt on the possibility of a purely rational justification of the actual delimitation of First-Order Logic. On this basis, we advance the thesis that a certain historical tradition was essential to the emergence of modern logic; this traditional context is analyzed as consisting in some guiding principles and, particularly, a set of exemplars (i.e., paradigmatic instances). Then, we proceed to interpret the historical course of development reviewed in section 1, which can broadly be described as a two-phased movement of expansion and then restriction of the scope of logical theory. We shall try to pinpoint ambivalencies in the process, and the main motives for subsequent changes. Among the latter, one may emphasize the spirit of modern axiomatics, the situation of foundational insecurity in the 1920s, the resulting desire to find systems well-behaved from a proof-theoretical point of view, and the metatheoretical results of the 1930s. Not surprisingly, the mathematical and, more specifically, the foundational context in which First-Order Logic matured will be seen to have played a primary role in its shaping.Mathematical logic is what logic, through twenty-five centuries and a few transformations, has become today. (Jean van Heijenoort)

Sign in / Sign up

Export Citation Format

Share Document