Department of Pure Mathematics (2000 - Present)
Pure Mathematics - Mathematical Logic
, Lion, France
Pure Mathematics
, Sharif University of Technology,
Pure Mathematics
, Sharif University of Technology,
I got my BSc and MSc in Pure Mathematics from Sharif University of Technology (1991 and 1993 respectively) and my PhD in mathematical logic from the University of Lyon (France) (1997). Now I am a permanent staff of Pure Math. Dept. in Tarbiat-Modares university.
I prove linear compactness and linear completeness for various forms of linear propositional logic where the value space is a module over a ring.
The ultraproduct construction is generalized to -ultramean constructions () by replacing ultrafilters with finitely additive measures. These constructions correspond to the linear fragments of continuous logic. A powermean variant of Keisler-Shelah isomorphism theorem is proved for . It is then proved that -sentences (and their approximations) are exactly those sentences of continuous logic which are preserved by such constructions. Some other applications are also given.
We combine continuous and integral logics and found a logical framework for metric measure spaces equipped with a family of continuous relations and operations. We prove the ultraproduct theorem and deduce compactness and other usual results. We also give applications of the compactness theorem in metric measure theory.Subjects: Logic (math. LO)Cite as: arXiv: 1910.00191 [math. LO](or arXiv: 1910.00191 v1 [math. LO] for this version)Submission historyFrom: Seyed-Mohammad Bagheri [view email][v1] Tue, 1 Oct 2019 03: 59: 20 UTC (16 KB)
The linear compactness theorem is a variant of the compactness theorem holding for linear formulas. We show that the linear fragment of continuous logic is maximal with respect to the linear compactness theorem and the linear elementary chain property. We also characterize linear formulas as those preserved by the ultramean construction.
We study a class of Kripke models with a naturally and well-behaved embedding relation. After proving a completeness theorem, we generalize some usual classical theorems of model theory into this framework. As an application, we give examples of non-classical ω-categorical theories which admit quantifier elimination.
We prove completeness of linear continuous logic introduced in and with respect to a linear system of axioms and rules.
This paper has two parts. First, we complete the proof of the Kolmogorov extension theorem for unbounded random variables using compactness theorem of integral logic which was proved for bounded case in [8]. Second, we give a proof of the Eberlein-Smulian compactness theorem by Ramsey's theorem and point out the correspondence between this theorem and a result in Shelah's classification theory.
We use the ultramean construction to prove linear compactness theorem. We also extend the Rudin-Keisler ordering to maximal probability charges and characterize it by embeddings of power ultrameans..
I study definability and types in the linear fragment of continuous logic. Linear variants of several definability theorems such as Beth, Svenonus and Herbrand are proved. At the end, a partial study of the theories of probability algebras, probability algebras with an aperiodic automorphism and AL-spaces is given.
Linear continuous logic is the fragment of continuous logic obtained by restricting connectives to addition and scalar multiplications. Most results in the full continuous logic have a counterpart in this fragment. In particular a linear form of the compactness theorem holds. We prove this variant and use it to deduce some basic preservation theorems.
A (linear) condition is an expression of the form ϕ⩽ ψ where ϕ, ψ are linear formulas. If ϕ, ψ are sentences, it is called a closed condition. The notion M⊨(ϕ⩽ ψ)(a) is defined in the obvious way. A collection of closed conditions is called a linear theory. M is a model of T, denoted M⊨ T, if M satisfies every condition in T.
We present some model theoretic results for Lukasiewicz predicate logic by using the methods of continuous model theory developed by Chang and Keisler. We prove compactness theorem with respect to the class of all structures taking values in the Lukasiewicz BL-algebra. We also prove some appropriate preservation theorems concerning universal and inductive theories. Finally, Skolemization and Morleyization in this framework are discussed and some natural examples of fuzzy theories are presented.
A quantified universe is a set M equipped with a Riesz space article amssymb empty A_n of real functions on Mn, for each n, and a second order operation article amssymb empty I:A→\mathbbR. Metric structures 4, graded probability structures 9 and many other structures in analysis are examples of such universes. We define ultraproduct of quantified universes and study properties preserved by this construction. We then discuss logics defined on the basis of classes of quantified universes which are closed under this construction.
We study model theory of random variables using finitary integral logic. We prove definability of some probability concepts such as having F(u) as distribution function, independence and martingale property. We then deduce Kolmogorov's existence theorem from the compactness theorem.
We prove an omitting types theorem and one direction of the related Ryll-Nardzewski theorem for semi-classical theories introduced in [2].
We define an ultraproduct of metric structures based on a maximal probability charge and prove a variant of Łoś theorem for linear metric formulas. We also consider iterated ultraproducts (? 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
A value space is a topological algebra B equipped with a non-empty family of continuous quantifiers: B∗→ B. We will describe first-order logic on the basis of B. Operations of B are used as connectives and its relations are used to define statements. We prove under some normality conditions on the value space that any theory in the new setting can be represented by a classical first-order theory.
Page 1. Annals of Pure and Applied Logic 161 (2010) iv Contents lists available at ScienceDirect Annals of Pure and Applied Logic journal homepage: www.elsevier.com/locate/apal Contents Special Issue The Proceedings of the IPM 2007 Logic Conference A. Enayat and I. Kalantari Preface 709 K. Aehlig and A. Beckmann On the computational complexity of cut-reduction 711 M. Ardeshir and R. Ramezanian The double negation of the intermediate value theorem 737 SM Bagheri, B. Poizat and M. Pourmahdian An arithmetical view to first-order logic 745 LD Beklemishev Kripke semantics for provability logic GLP 756 S. Barry Cooper Extending and interpreting Post's programme 775 F. Didehvar, K. Ghasemloo and M. Pourmahdian Effectiveness in RPL, with applicatio
We develop a model theoretic framework for studying algebraic structures equipped with a measure. The real line is used as a value space and its usual arithmetical operations as connectives. Integration is used as a quantifier. We extend some basic results of pure model theory to this context and characterize measurable sets in terms of zero-sets of formulas.
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