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.