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 Logic and the Generative Power of Autosegmental Phonology

2014 ◽  
Vol 1 (1) ◽  
Author(s):  
Adam Jardine
Keyword(s):  
Formal Language ◽  
Second Order ◽  
Order Logic ◽  
Language Theory ◽  
Preliminary Conclusion ◽  
Autosegmental Phonology ◽  
Finite State ◽  
Regular Sets ◽  
Second Order Logic ◽  
Monadic Second Order Logic

<p>Autosegmental Phonology is studied in the framework of Formal Language Theory, which classifies the computational complexity of patterns. In contrast to previous computational studies of Autosegmental Phonology, which were mainly concerned with finite-state implementations of the formalism, a methodology for a model-theoretic study of autosegmental diagrams with monadic second-order logic is introduced. Monadic second order logic provides a mathematically rigorous way of studying autosegmental formalisms, and its complexity is well understood. The preliminary conclusion is that autosegmental diagrams which conform to the well-formedness constraints defined here likely describe at most regular sets of strings.</p>

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 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.

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.

Categoricity and the sets

2018 ◽  
Author(s):  
Tim Button ◽  
Sean Walsh
Keyword(s):  
Set Theory ◽  
Second Order ◽  
Order Logic ◽  
First Order ◽  
The Status ◽  
Second Order Logic ◽  
Philosophy Of Set Theory ◽  
Order Set

In this chapter, the focus shifts from numbers to sets. Again, no first-order set theory can hope to get anywhere near categoricity, but Zermelo famously proved the quasi-categoricity of second-order set theory. As in the previous chapter, we must ask who is entitled to invoke full second-order logic. That question is as subtle as before, and raises the same problem for moderate modelists. However, the quasi-categorical nature of Zermelo's Theorem gives rise to some specific questions concerning the aims of axiomatic set theories. Given the status of Zermelo's Theorem in the philosophy of set theory, we include a stand-alone proof of this theorem. We also prove a similar quasi-categoricity for Scott-Potter set theory, a theory which axiomatises the idea of an arbitrary stage of the iterative hierarchy.

Algorithmic Logic with Stacks and Its Model-Theoretical Properties

1984 ◽  
Vol 7 (4) ◽  
pp. 391-428
Author(s):  
Wiktor Dańko
Keyword(s):  
Second Order ◽  
Order Logic ◽  
Algorithmic Logic ◽  
Second Order Logic

In this paper we propose to transform the Algorithmic Theory of Stacks (cf. Salwicki [30]) into a logic for expressing and proving properties of programs with stacks. We compare this logic to the Weak Second Order Logic (cf. [11, 15]) and prove theorems concerning axiomatizability without quantifiers (an analogon of Łoś-Tarski theorem) and χ 0 - categoricity (an analogon of Ryll-Nardzewski’s theorem).

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