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.

Forcing and reducibilities. III. Forcing in fragments of set theory

1983 ◽  
Vol 48 (4) ◽  
pp. 1013-1034
Author(s):  
Piergiorgio Odifreddi
Keyword(s):  
Set Theory ◽  
Recursion Theory ◽  
Second Order ◽  
Analytic Hierarchy ◽  
Second Order Arithmetic ◽  
First Order ◽  
Order Language ◽  
Order Arithmetic ◽  
Definition Of ◽  
Generic Set

We conclude here the treatment of forcing in recursion theory begun in Part I and continued in Part II of [31]. The numbering of sections is the continuation of the numbering of the first two parts. The bibliography is independent.In Part I our language was a first-order language: the only set we considered was the (set constant for the) generic set. In Part II a second-order language was introduced, and we had to interpret the second-order variables in some way. What we did was to consider the ramified analytic hierarchy, defined by induction as:A0 = {X ⊆ ω: X is arithmetic},Aα+1 = {X ⊆ ω: X is definable (in 2nd order arithmetic) over Aα},Aλ = ⋃α<λAα (λ limit),RA = ⋃αAα.We then used (a relativized version of) the fact that (Kleene [27]). The definition of RA is obviously modeled on the definition of the constructible hierarchy introduced by Gödel [14]. For this we no longer work in a language for second-order arithmetic, but in a language for (first-order) set theory with membership as the only nonlogical relation:L0 = ⊘,Lα+1 = {X: X is (first-order) definable over Lα},Lλ = ⋃α<λLα (λ limit),L = ⋃αLα.

α-degrees of α-theories

1972 ◽  
Vol 37 (4) ◽  
pp. 677-682 ◽  
Author(s):  
George Metakides
Keyword(s):  
Initial Segment ◽  
Recursion Theory ◽  
The Other ◽  
Infinite Length ◽  
Infinitary Logic ◽  
Admissible Set ◽  
Recursively Enumerable ◽  
Limit Ordinal ◽  
The Subject ◽  
Further Development

Let α be a limit ordinal with the property that any “recursive” function whose domain is a proper initial segment of α has its range bounded by α. α is then called admissible (in a sense to be made precise later) and a recursion theory can be developed on it (α-recursion theory) by providing the generalized notions of α-recursively enumerable, α-recursive and α-finite. Takeuti [12] was the first to study recursive functions of ordinals, the subject owing its further development to Kripke [7], Platek [8], Kreisel [6], and Sacks [9].Infinitary logic on the other hand (i.e., the study of languages which allow expressions of infinite length) was quite extensively studied by Scott [11], Tarski, Kreisel, Karp [5] and others. Kreisel suggested in the late '50's that these languages (even which allows countable expressions but only finite quantification) were too large and that one should only allow expressions which are, in some generalized sense, finite. This made the application of generalized recursion theory to the logic of infinitary languages appear natural. In 1967 Barwise [1] was the first to present a complete formalization of the restriction of to an admissible fragment (A a countable admissible set) and to prove that completeness and compactness hold for it. [2] is an excellent reference for a detailed exposition of admissible languages.

scholarly journals Analysis of Accessibility to Family Health Centers in Antalya Using GIS

2022 ◽  
Vol 9 (sp) ◽  
pp. 2442-2448
Author(s):  
Orhun Soydan
Keyword(s):  
Social Ties ◽  
Family Health ◽  
Point Of View ◽  
Health Centers ◽  
Walking Distance ◽  
Physical Conditions ◽  
The Social ◽  
The Subject ◽  
Physical Features ◽  
First Time

Family health centers in Turkey started to be implemented for the first time in Düzce in 2004 years within the scope of Law No. 5258. While determining the physical conditions of the places where family health centers are built, the first item in the regulation is that the building should be easily accessible. This situation shows the importance of the subject in terms of accessibility. While determining the features of the places where FHCs will be made, environmental characteristics are also taken into consideration. Environmental features are effective in determining the FHCs location in different ways. These impacts are divided into two groups: the physical features that pavements, roads and parks can include, and the social, cultural and institutional features of neighborhoods that include local social ties and collective activities. From this point of view, the importance of the location of family health centers relative to roads and houses is understood. The aim of this study is to examine the accessibility of Family Health Centers in Konyaaltı, Antalya, on a neighborhood basis using Geographic Information Systems. Konyaaltı has 21 Family Health Centers. As a result of the analyses, it was determined that most of the neighborhoods had problems in terms of accessibility, while a very few of them did not experience problems in terms of accessibility. In terms of the total number of buildings, the ratio of buildings that are 500 meters walking distance from any family health center by using highways is 35.56%. With these rates, 3,634 of the 10,2018 buildings remain within the limits of the regulation. Finally; suggestions were made to increase accessibility to these areas.

scholarly journals Functioning of the verb’s reflexive voice in the modern Udmurt language

2021 ◽  
Vol 13 (4) ◽  
pp. 358-368
Author(s):  
Ksenia G. Kostina
Keyword(s):  
Semantic Content ◽  
Point Of View ◽  
Grammatical Structure ◽  
Passive Voice ◽  
Continuous Sampling ◽  
The Press ◽  
The Subject ◽  
Reflexive Verbs ◽  
First Time ◽  
Colloquial Speech

Introduction. Any language’s verb system has many resources for denoting various actions of people. The relations of the action or state of the subject to its object are determined by the grammatical category of the voice, represented in the Udmurt language by the pairs of causative – non-causative, reflexive – non-reflexive forms of voices. The article considers the functioning of the verb’s reflexive voice in the modern Udmurt language, including the etymology of the voice’s affix, the grammatical meanings of reflexive verbs. Materials and Methods. The main material of the research is based on the Udmurt-Russian Dictionary (2008) and the texts of Udmurt writers included into the National Corpus of the Udmurt Language. The article used a set of such research methods as descriptive, continuous sampling, contextual analysis, taking into account the situational conditioning of the verb voice. On specific examples, the use of these methods makes it possible to consider the structure, dynamics and features of the functioning of the reflexive voice of the verb in the Udmurt language. Results and Discussion. As a result of the research, for the first time, among the reflexive voice’s groups we include verbs of passive voice. The reason of it is the low probability of using passive constructions in colloquial speech. The frequent cases of using passive meanings of verbs in the literature and in the press are defined by the calcified translation of foreign-language constructions. Conclusion. The grammatical structure of the Udmurt language is represented by two binary voice’s forms: reflexive/non-reflexive voice and causative/non-causative voice. Specific indicators of reflexive voice are affixes -ськ(ы)-/-ск(ы), -иськ(ы)-/-üськ(ы)-. From the point of view of semantic content, five semantic groups of returnable pledges are distinguished: reflexive, medial, reciprocal, impersonal, passive. The proposed classification is determined by the specifics of the relations between the subject and the object of action.

Noninitial segments of the α-degrees

1973 ◽  
Vol 38 (3) ◽  
pp. 368-388 ◽  
Author(s):  
John M. Macintyre
Keyword(s):  
Distributive Lattice ◽  
Initial Segment ◽  
Elementary Theory ◽  
Recursion Theory ◽  
Partial Ordering ◽  
First Order ◽  
Order Language ◽  
Suitable Notion ◽  
The Universe ◽  
Degrees Of Unsolvability

Let α be an admissible ordinal and let L be the first order language with equality and a single binary relation ≤. The elementary theory of the α-degrees is the set of all sentences of L which are true in the universe of the α-degrees when ≤ is interpreted as the partial ordering of the α-degrees. Lachlan [6] showed that the elementary theory of the ω-degrees is nonaxiomatizable by proving that any countable distributive lattice with greatest and least members can be imbedded as an initial segment of the degrees of unsolvability. This paper deals with the extension of these results to α-recursion theory for an arbitrary countable admissible α > ω. Given α, we construct a set A with α-degree a such that every countable distributive lattice with greatest and least member is order isomorphic to a segment of α-degrees {d ∣ a ≤αd≤αb} for some α-degree b. As in [6] this implies that the elementary theory of the α-degrees is nonaxiomatizable and hence undecidable.A is constructed in §2. A is a set of integers which is generic with respect to a suitable notion of forcing. Additional applications of such sets are summarized at the end of the section. In §3 we define the notion of a tree and construct a particular tree T0 which is weakly α-recursive in A. Using T0 we can apply the techniques of [6] and [2] to α-recursion theory. In §4 we reduce our main results to three technical lemmas concerning systems of trees. These lemmas are proved in §5.

On generalized quantifiers in arithmetic

1982 ◽  
Vol 47 (1) ◽  
pp. 187-190 ◽  
Author(s):  
Carl Morgenstern
Keyword(s):  
Axiom System ◽  
Peano Arithmetic ◽  
Order Logic ◽  
First Order Logic ◽  
Generalized Quantifiers ◽  
First Order ◽  
Order Language ◽  
Truth Definition ◽  
Order Arithmetic ◽  
Infinite Set

In this note we investigate an extension of Peano arithmetic which arises from adjoining generalized quantifiers to first-order logic. Markwald [2] first studied the definability properties of L1, the language of first-order arithmetic, L, with the additional quantifer Ux which denotes “there are infinitely many x such that…. Note that Ux is the same thing as the Keisler quantifier Qx in the ℵ0 interpretation.We consider L2, which is L together with the ℵ0 interpretation of the Magidor-Malitz quantifier Q2xy which denotes “there is an infinite set X such that for distinct x, y ∈ X …”. In [1] Magidor and Malitz presented an axiom system for languages which arise from adding Q2 to a first-order language. They proved that the axioms are valid in every regular interpretation, and, assuming ◊ω1, that the axioms are complete in the ℵ1 interpretation.If we let denote Peano arithmetic in L2 with induction for L2 formulas and the Magidor-Malitz axioms as logical axioms, we show that in we can give a truth definition for first-order Peano arithmetic, . Consequently we can prove in that is Πn sound for every n, thus in we can prove the Paris-Harrington combinatorial principle and the higher-order analogues due to Schlipf.

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.

Scandinavian pancake constructions as a family of constructions

2014 ◽  
Vol 1 (2) ◽  
pp. 171-196 ◽  
Author(s):  
Tor Arne Haugen ◽  
Hans-Olav Enger
Keyword(s):  
Classical Problem ◽  
Point Of View ◽  
Theoretical Point ◽  
Specific Level ◽  
The Subject ◽  
Corpus Data ◽  
Construction Type ◽  
Empirical Point ◽  
Plural Subject ◽  
First Time

This paper deals with a classical problem in Scandinavian grammar, so-called ‘pancake sentences’, nicknamed after examples like Pannekaker er godt ‘Pancakes are good’ where there seemingly is disagreement between the plural subject and the predicative adjective in the neuter singular. Our aim is twofold. From the theoretical point of view, we shall argue that there are advantages with a construction-based approach, and that such an approach is superior to previous analyses within various generative frameworks.The main reason for this is that the data require generalizations over combinations of subjects and predicative adjectives at a rather specific level. From a more empirical point of view, we shall argue that Scandinavian displays a range of different, but related pancake constructions. For the first time, corpus data are brought into the debate. We show that a construction type that has not received much attention previously is in fact the most frequent type, namely constructions where the subject is a deverbal noun.

scholarly journals HEGEL ON THE DUAL, ABSTRACT-SPECIFIC NATURE OF THE PERSONALITY AS A SUBJECT OF LAW: NATURAL LAW AND THEOLOGICAL FOUNDATIONS OF GENESIS

2021 ◽  
Vol 17 (3(65)) ◽  
pp. 44-58
Author(s):  
Виктор Петрович САЛЬНИКОВ ◽  
Дмитрий Владимирович МАСЛЕННИКОВ ◽  
Александр Александрович МАКСИМОВ
Keyword(s):  
Natural Law ◽  
Scientific Work ◽  
Point Of View ◽  
Dual Nature ◽  
Modern System ◽  
Theological Foundations ◽  
The Subject ◽  
Historical Comparative ◽  
Further Development ◽  
Qualitative Characteristics

Hegel’s doctrine in the XXI century remains relevant for the Russian politico-legal science due to the deep development of the categorical apparatus and methodological tools for expressing the value nature of law and state. Hegel's doctrine of the subject of law, especially presented in his early works, is one of the little-studied, but at the same time very significant part of his scientific work, the appeal to which should contribute to the further development of the modern system of legal categories. Purpose: to carry out a comparative analysis of the various versions of the Hegelian doctrine of the personality as a subject of law and its genesis, taking into account the peculiarities of natural law and theological pproaches. Methods: the authors use logical, historical-genetic, historical-comparative, systemic method of cognition, as well as the methodology of theoretical and legal comparative studies. Results: in the late philosophical system, Hegel develops the doctrine of the dual nature of the personality as a subject of law: as a specific subject, a person mediates his relationship with other subjects through a single objective thing (qualitative characteristics of the subject), as an abstract subject, a person bases his relations with them on recognition as equal subjects of freedom (quantitative characteristics of the subject). The way to resolve this conflict is through morality. During various periods of his scientific work, Hegel considers the genesis of the personality as a subject of law from a natural law and theological point of view, including the idea of God, while  dentifying new substantive aspects of the concept of the subject of law, related to the the historical development of the state and the formation of the rule-of-law state.

scholarly journals Islamic psychology abroad: the state and development prospects

2019 ◽  
Vol 11 (4) ◽  
pp. 850-865 ◽  
Author(s):  
O. S. Pavlova ◽  
N. Yu. Bariyeva ◽  
Z. M. Bairova
Keyword(s):  
Future Development ◽  
International Association ◽  
The State ◽  
Point Of View ◽  
Present Stage ◽  
Inaugural Meeting ◽  
The Subject ◽  
Islamic Psychology ◽  
Further Development ◽  
Development Prospects

The paper presents summaries of the talks given at the Inaugural Meeting of the International Association of Islamic Psychology (IAIP): “Evolving Islamic Psychology: Past, Present & Future”. It also shows different approaches to the content and methods of Islamic psychology. The talks were arranged chronologically. In the day one were given talks on the origins of Islamic psychology as reflected in the works by Al-Ghazali, Al-Balkhi and others. In the day two the present stage of the discipline received a thorough analysis. It has been approached from the point of view of theory, psychology, and psychotherapeutics. The agenda of the day three contained the suggested ways of the future development of this discipline. The meeting has shown a considerable interest to the subject by scholars from different countries, the diversity of approaches to the scholarly discipline as well as enthusiasm and optimism about its further development.

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