Logic is the science of rational thinking or reasoning. As such, it includes foundation of the scientific method including the hypothetico-deductive account of science. So, logic is not only the foundation of mathematics, but also of physics and other branches of science. In analogy with mathematical physics, a large portion of logic has been already formalized by using mathematics. Hence, mathematical logic is an ever growing part of logic, namely that part or "skeleton" which has been understood and clarified enough for admitting mathematical treatment.


Logic includes both the theory of deductive reasoning and something that could be roughly called inductive reasoning or theory formation. The deductive part of logic is the better understood part and it has been discussed in the literature more extensively. Especially, the mathematical foundation for deductive logic or briefly mathematics of deductive logic has been elaborated more extensively. In particular, most of the mathematical logic textbooks address the deductive part of logic. All the same, the inductive part is just as important, especially in the methodology of sciences. The inductive part concerns theory formation. For example, it studies the procedure when we start out with some observations or facts and search for a set of hypotheses called axioms which would explain all these observations. This means that the set of axioms searched for should prove via deductive logic all the observations and at the same time they should satisfy certain criteria e.g. "coherence", "elegance", "economy". What these words really mean is already a subject matter of inductive logic. Actually, finding satisfactory definitions for these criteria is still an unfinished issue or, more bluntly, open problem of inductive logic. Very roughly speaking, we are looking for axioms which prove all the observations and prove as little more as possible. Of course, this statement is an oversimplification, it is included here to give a vague idea of what the purpose of inductive logic is.


All we have said so far sounds as if we were imprisoned within the framework of a single language or vocabulary. Logic also studies the change and interaction of various languages, forming a so-called logical system. An example for a logical system is classical first-order logic. Another example is many-sorted classical first-order logic. Logic also studies the connections between various logical systems and methods for choosing the suitable one for a particular task.


A logical system incorporates many logical theories and each theory has a particular language. Examples for logical theories are Peano's Arithmetic, Zermelo-Fraenkel Set Theory and Euclidean geometry (say, Tarski's version of the latter). The deductive part of logic works usually in the framework of a single theory. However, theories are perhaps the most important building blocks in logic and an important part of logic concerns how we can put theories together in order to form new theories. Algebraic logic is the part of logic which concerns itself with the method of putting theories together to form new "world-views" or new understandings of some parts of reality. Cf. [Burstall-Goguen: Putting theories together] and [Andreka-Nemeti-Sain: Algebraic Logic, in Handbook of Philosophical Logic] or [Andreka-Nemeti-Sain: Universal Algebraic Logic (Dedicated to the Unity of Science), Studies in Universal Logic, Birkhauser]. The "putting theories together" part of logic can be used for the study of theory formation and more generally for studying the dynamics of knowledge. This dynamic trend of logic has been emphasized by the recent Amsterdam-Budapest-London cooperation called the "dynamic trend in logic" [van Benthem: Exploring Logical Dynamics]. In the present sense, the putting theories part of logic points into the inductive direction, namely it studies how new theories are built up during the process of knowledge acquisition or, more generally, of reasoning. The same putting theories together direction can be used for analyzing a given theory e.g. as the co-limit of certain simpler theories in the category of all theories. An important development in the inductive direction of logic is called abduction. Non-monotonic logic also can be regarded as such.


Besides the inductive-deductive distinction in logic, there are other important distinctions like the logical theory of meaning called model theory contrasted with e.g. proof theory. In the present entry we concentrated on the inductive aspect of logic including theory change and language change because, especially in the foundation of sciences, the appreciation of these aspects of logic is very important for assessing the relevance of logic for the given branch of science. E.g. in physics it is an important part of the methodology to emphasize the empirical aspects of physics besides its deductive (or mathematical) aspects. Now if one is not aware of the inductive parts of logic, then one might get the misguided impression that logic would be relevant only to a part of physics, namely to its deductive part. As it was outlined above, this is not the case, though.