continuous logic

From the beginning of the emergence of new logic, fundamental links have been established between logic and various branches of mathematics, which led to solving mathematical problems and, conversely, solving basic problems in logic itself. One of the challenges of the logical methods in the study of mathematical structures is the impossibility of studying some of the important structures of mathematics, including analytic structures, in the framework of the first-order language and logic. The main purpose of this paper is to provide a suitable logic for studying these structures and then solving problems in the analysis using logical tools. At the beginning of this article, we will briefly review some suitable logics for studying the structures in mathematical analysis, and will outline some of the most important uses of logic in analysis. Then we present and prove one of the recent achievements, which is an important application of logic in analysis. In particular, we study the concept of definability in logic and its relation with mathematical analysis.

Continuous logic is generalization of first order logic to a many valued logic with an infinitary truth value set. Many of the results of classic logic and it's model theory have been generalized to continuous logic. Continuous logic not only has many uses in the mathematical analysis and in the model theory of mathematical analysis structures, but also has created new attitudes in classical model theory. Firstly, the present paper study the development of continuous logic from Łukasiewicz logic. Then we have a review on some of the most important basic results of continuous logic, including the completeness of the proof system and the compactness theorem. Finally, according to the concept of continuity with respect to the truth value set, we will introduce a kind of continuous logic that is based on continuous t-norm based fuzzy logics. This will lead to the introduction of two kinds of continuous logics based on Gödel logic and product logic. Then we developed some of the results of continuous logic such as the compactness theorem for these two logics.