The Model Theory Of Dedekind Algebras

1998 ◽  
pp. 135-142
Author(s):  
George Weaver
Keyword(s):  
Model Theory ◽  
Existence And Uniqueness ◽  
Value Function ◽  
Similarity Transformation ◽  
Second Order ◽  
Order Logic ◽  
Isomorphism Type ◽  
Positive Integers ◽  
Second Order Logic

A Dedekind algebra is an ordered pair (B, h) where B is a non-empty set and h is a "similarity transformation" on B. Among the Dedekind algebras is the sequence of positive integers. Each Dedekind algebra can be decomposed into a family of disjointed, countable subalgebras which are called the configurations of the algebra. There are many isomorphic types of configurations. Each Dedekind algebra is associated with a cardinal value function called the confirmation signature which counts the number of configurations in each isomorphism type occurring in the decomposition of the algebra. Two Dedekind algebras are isomorphic if their configuration signatures are identical. I introduce conditions on configuration signatures that are sufficient for characterizing Dedekind algebras uniquely up to isomorphisms in second order logic. I show Dedekind's characterization of the sequence of positive integers to be a consequence of these more general results, and use configuration signatures to delineate homogeneous, universal and homogeneous-universal Dedekind algebras. These delineations establish various results about these classes of Dedekind algebras including existence and uniqueness.

An algebraic characterization of indistinguishable cardinals

1970 ◽  
Vol 35 (1) ◽  
pp. 97-104
Author(s):  
A. B. Slomson
Keyword(s):  
Second Order ◽  
Order Logic ◽  
Algebraic Characterization ◽  
Algebraic Criterion ◽  
Characterizable Cardinals ◽  
Second Order Logic ◽  
Interesting Theorem ◽  
Straightforward Proof

Two cardinals are said to beindistinguishableif there is no sentence of second order logic which discriminates between them. This notion, which is defined precisely below, is closely related to that ofcharacterizablecardinals, introduced and studied by Garland in [3]. In this paper we give an algebraic criterion for two cardinals to be indistinguishable. As a consequence we obtain a straightforward proof of an interesting theorem about characterizable cardinals due to Zykov [6].

Second-order logic, philosophical issues in

2018 ◽  
Author(s):  
Stewart Shapiro
Keyword(s):  
Mathematical Practice ◽  
Expressive Power ◽  
Deductive System ◽  
Second Order ◽  
Order Logic ◽  
Mathematical Structures ◽  
First Order ◽  
Correct Reasoning ◽  
Second Order Logic

Typically, a formal language has variables that range over a collection of objects, or domain of discourse. A language is ‘second-order’ if it has, in addition, variables that range over sets, functions, properties or relations on the domain of discourse. A language is third-order if it has variables ranging over sets of sets, or functions on relations, and so on. A language is higher-order if it is at least second-order. Second-order languages enjoy a greater expressive power than first-order languages. For example, a set S of sentences is said to be categorical if any two models satisfying S are isomorphic, that is, have the same structure. There are second-order, categorical characterizations of important mathematical structures, including the natural numbers, the real numbers and Euclidean space. It is a consequence of the Löwenheim–Skolem theorems that there is no first-order categorical characterization of any infinite structure. There are also a number of central mathematical notions, such as finitude, countability, minimal closure and well-foundedness, which can be characterized with formulas of second-order languages, but cannot be characterized in first-order languages. Some philosophers argue that second-order logic is not logic. Properties and relations are too obscure for rigorous foundational study, while sets and functions are in the purview of mathematics, not logic; logic should not have an ontology of its own. Other writers disqualify second-order logic because its consequence relation is not effective – there is no recursively enumerable, sound and complete deductive system for second-order logic. The deeper issues underlying the dispute concern the goals and purposes of logical theory. If a logic is to be a calculus, an effective canon of inference, then second-order logic is beyond the pale. If, on the other hand, one aims to codify a standard to which correct reasoning must adhere, and to characterize the descriptive and communicative abilities of informal mathematical practice, then perhaps there is room for second-order logic.

The relative expressive power of some logics extending first-order logic

1979 ◽  
Vol 44 (2) ◽  
pp. 129-146 ◽  
Author(s):  
John Cowles
Keyword(s):  
Model Theory ◽  
Expressive Power ◽  
Second Order ◽  
Order Logic ◽  
The Other ◽  
First Order Logic ◽  
Finite Sets ◽  
Limited Ability ◽  
First Order ◽  
Second Order Logic

In recent years there has been a proliferation of logics which extend first-order logic, e.g., logics with infinite sentences, logics with cardinal quantifiers such as “there exist infinitely many…” and “there exist uncountably many…”, and a weak second-order logic with variables and quantifiers for finite sets of individuals. It is well known that first-order logic has a limited ability to express many of the concepts studied by mathematicians, e.g., the concept of a wellordering. However, first-order logic, being among the simplest logics with applications to mathematics, does have an extensively developed and well understood model theory. On the other hand, full second-order logic has all the expressive power needed to do mathematics, but has an unworkable model theory. Indeed, the search for a logic with a semantics complex enough to say something, yet at the same time simple enough to say something about, accounts for the proliferation of logics mentioned above. In this paper, a number of proposed strengthenings of first-order logic are examined with respect to their relative expressive power, i.e., given two logics, what concepts can be expressed in one but not the other?For the most part, the notation is standard. Most of the notation is either explained in the text or can be found in the book [2] of Chang and Keisler. Some notational conventions used throughout the text are listed below: the empty set is denoted by ∅.

scholarly journals COMPLETENESS AND CATEGORICITY (IN POWER): FORMALIZATION WITHOUT FOUNDATIONALISM

2014 ◽  
Vol 20 (1) ◽  
pp. 39-79 ◽  
Author(s):  
JOHN T. BALDWIN
Keyword(s):  
Formal Methods ◽  
Model Theory ◽  
Second Order ◽  
Order Logic ◽  
First Order Logic ◽  
Specific Content ◽  
First Order ◽  
Complete Theories ◽  
The Stability ◽  
Second Order Logic

AbstractWe propose a criterion to regard a property of a theory (in first or second order logic) as virtuous: the property must have significant mathematical consequences for the theory (or its models). We then rehearse results of Ajtai, Marek, Magidor, H. Friedman and Solovay to argue that for second order logic, ‘categoricity’ has little virtue. For first order logic, categoricity is trivial; but ‘categoricity in power’ has enormous structural consequences for any of the theories satisfying it. The stability hierarchy extends this virtue to other complete theories. The interaction of model theory and traditional mathematics is examined by considering the views of such as Bourbaki, Hrushovski, Kazhdan, and Shelah to flesh out the argument that the main impact of formal methods on mathematics is using formal definability to obtain results in ‘mainstream’ mathematics. Moreover, these methods (e.g., the stability hierarchy) provide an organization for much mathematics which gives specific content to dreams of Bourbaki about the architecture of mathematics.

scholarly journals المكونات الاساسية والقواعد الاستنتاجية في منطق الرتبة الاولى

لارك ◽  
2019 ◽  
Vol 1 (22) ◽  
pp. 475-485
Author(s):  
ليث أثير يوسف
Keyword(s):  
Artificial Intelligence ◽  
Programming Language ◽  
Model Theory ◽  
Formal Language ◽  
Second Order ◽  
Order Logic ◽  
Logical Programming ◽  
Second Order Logic

كان الهدف من البحث بيان منطق الرتبة الاولى ومكوناته ورموزه وصيغه ومصطلحاته وكل مايتعلق به لما يمثل من اهمية في اوساط المنطق الرياضي وأهميته في حياتنا العملية فهذا المنطق له صيغ خاصة وطريقة في كتابة الرموز تختلف عن بقية حقول المنطق الرياضي فالمحمول هو من سيحدد شكل المصطلح وسيسلك في داخل الصيغة سلوك وظائف (دالية)( functional) والصيغة المعقدة تحتوي على اكثر من محمول فيها اضافة الى الاسوار التي ستحدد القضية من ناحية (الكم) ناهيك عن الية قيم صدق وكذب المصطلح والصيغ في هذا المنطق عن طريق منهج التفسير (interpretation) والحقيقة ان هذا النهج غريب وغير معروف في اوساط الاليات الرمزية والانساق المنطقية في منطق القضايا سواء بصيغه البسيطة أو المعقدة فقيم الصدق ( ثنائية القيم) هي ستحدد صدق الصيغ ، كما ان لمنطق الرتبة الاولى اساسات جعلت منه منطلقاً لاقامة منطق الرتبة الثانية (second order logic)[i] وهو النموذج المطور بالياته البرهانية وصيغه المعقدة عن الاول وكذلك اعتبر منطق الرتبة الاولى منطلقا لاقامة نظرية النماذج أو النمذجة ( models theory)[ii] تلك النظرية الرياضية التي تجمع مابين المجموعات الكلية في نظرية المجموعات والصيغ الجبرية بالاضافة الى ان منطق الرتبة الاولى ذا اهمية في تكوين لغات البرمجة المنطقية (logical programming language) ومنها لغة برولوغ (prolog) الشهيرة التي تعتمد بالاساس على هذا المنطق في بناء وتفسير صيغها وكذلك في مجال الذكاء الصناعي (artificial intelligence) . لذا كان من الضروري تقديم بيان ملخص ومفصل عن مكونات هذا النوع من المنطق الرياضي وشرح الياته الرياضية والمنطقية   [i] ) للاستزادة من الشرح المفصل حول موضوع (منطق الرتبة الثانية –second order logic) ينظر كتاب STEWART SHAPIRO, foundations without foundationalism, CLARENDON PRESS,Uk 1991,p.96 ) [ii] ) يعرفها جانغ (cc.chang) في كتابه (model theory) في مقدمته (ص1) الى ان هذه النظرية فرع من المنطق الرياضي وتتعامل مع اللغة الرمزية وتفسيراتها (formal language and its interpretations) أو مايسمى بالنماذج (Models)

scholarly journals Map Genus, Forbidden Maps, and Monadic Second-Order Logic

10.37236/1656 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
B. Courcelle ◽  
V. Dussaux
Keyword(s):  
Orientable Surface ◽  
Second Order ◽  
Order Logic ◽  
Minimal Genus ◽  
Planar Embeddings ◽  
Second Order Logic ◽  
Monadic Second Order Logic ◽  
Circular Order

A map is a graph equipped with a circular order of edges around each vertex. These circular orders represent local planar embeddings. The genus of a map is the minimal genus of an orientable surface in which it can be embedded. The maps of genus at most $g$ are characterized by finitely many forbidden maps, relatively to an appropriate ordering related to the minor ordering of graphs. This yields a "noninformative" characterization of these maps, that is expressible in monadic second-order logic. We give another one, which is more informative in the sense that it specifies the relevant surface embedding, in addition to stating its existence.

scholarly journals LOGICS FOR PROPOSITIONAL CONTINGENTISM

2017 ◽  
Vol 10 (2) ◽  
pp. 203-236 ◽  
Author(s):  
PETER FRITZ
Keyword(s):  
Model Theory ◽  
Possible Worlds ◽  
Second Order ◽  
Order Logic ◽  
Existential Quantifier ◽  
Philosophical Theory ◽  
Propositional Operator ◽  
Second Order Logic

AbstractRobert Stalnaker has recently advocated propositional contingentism, the claim that it is contingent what propositions there are. He has proposed a philosophical theory of contingency in what propositions there are and sketched a possible worlds model theory for it. In this paper, such models are used to interpret two propositional modal languages: one containing an existential propositional quantifier, and one containing an existential propositional operator. It is shown that the resulting logic containing an existential quantifier is not recursively axiomatizable, as it is recursively isomorphic to second-order logic, and a natural candidate axiomatization for the resulting logic containing an existential operator is shown to be incomplete.

Recognizable languages of arrows and cospans

2018 ◽  
Vol 28 (8) ◽  
pp. 1290-1332
Author(s):  
H. J. SANDER BRUGGINK ◽  
BARBARA KÖNIG
Keyword(s):  
Finite Automata ◽  
General Setting ◽  
Second Order ◽  
Order Logic ◽  
Straightforward Generalization ◽  
Simple Logic ◽  
Second Order Logic ◽  
Monadic Second Order Logic ◽  
Graph Languages

In this article, we generalize Courcelle's recognizable graph languages and results on monadic second-order logic to more general structures.First, we give a category-theoretical characterization of recognizability. A recognizable subset of arrows in a category is defined via a functor into the category of relations on finite sets. This can be seen as a straightforward generalization of finite automata. We show that our notion corresponds to recognizable graph languages if we apply the theory to the category of cospans of graphs.In the second part of the paper, we introduce a simple logic that allows to quantify over the subobjects of a categorical object. Again, we show that, for the category of graphs, this logic is equally expressive as monadic second-order graph logic (msogl). Furthermore, we show that in the more general setting of hereditary pushout categories, a class of categories closely related to adhesive categories, we can recover Courcelle's result that everymsogl-expressible property is recognizable. This is done by giving an inductive translation of formulas of our logic into automaton functors.

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