Logic in Philosophy vs. Mathematical Logic

Students majoring in philosophy take a course called "Logic in Philosophy" and there is also a course offered in the Math Department called "Mathematical Logic". Are these two distinct fields? If so, do they share common elements, concepts, and terminology/definitions?

Three points.

(A) Most philosophy students (at least those "majoring" in philosophy) are expected to do at least a Baby Formal Logic course, often in their very first year. Coverage will vary, but they should pick up an understanding of what makes a formally valid argument, of the truth-functional connectives, of the logic of quantifiers and identity, maybe one or two other things too like familiarity with set notation. This will usually be done very slowly, remembering that most philosophy students have no maths background and many will be symbol-phobic. There may be more formal logic taught later in later years in optional courses, and eventually (though less and less these days) they might be offered a mathematical logic course.

Beginning university mathematics students will at various points, in e.g. courses on discrete mathematics, pick up an understanding of the connectives, of quantifiers, and of course of set notation, as they go along. But they won't usually have a stand-alone course on Baby Formal Logic (which hasn't much mathematical excitement!). So usually the first stand-alone logic course they might be offered is at senior undergraduate/beginning postgraduate course, in which Baby Logic gets covered (again) very fast, but this time with added completeness proofs etc. etc., before going on to other stuff.

So one main difference between an intro course in formal logic for philosophers and the early lectures in a course in mathematical logic for mathematicians is simply speed and level of mathematical sophistication. As noted, by mathematical standards, the combinatorial ideas involved in the philosophers' Baby Formal Logic course are extremely easy and elementary (even if it doesn't always look that way to the students having to take the course!). The same ideas can be covered at a cracking pace at the beginning of a mathematical logic course.

(B) As I said, there may then be more formal logic offered to philosophers in later years in optional courses -- some of these courses covering various things of particular interest to philosophers, but not so much to (most) mathematicians, like modal logic. The second half of Sider's book indicates the sort of coverage typical of such courses. So here the focus of the coverage of courses in philosophy and mathematics department can indeed diverge.

(C) To take up some remarks in other answers, you need to sharply distinguish "Logic for Philosophers", meaning a treatment of some topics in formal logic that are of interest/use to philosophers, from "Philosophical Logic". The latter is a standard label for an area which is really a department of the broader philosophy of language, interested in questions about e.g. the nature of reference, the nature of logical necessity, the nature of a valid argument, the meaning of ordinary language conditionals, and so on. One technique for addressing such questions is to compare conditionals (to run with that example) in ordinary language with various constructed connectives in artificial languages of logic. So languages of formal logic get into the story as useful objects of comparison. And perhaps that's what unites the ragbag of topics in philosophical logic. But still, the real focus of courses on so-called philosophical logic is in issues that arise informally or pre-formally, not centrally in issues of formal logic.

The main difference between "Logic in Philosophy" and "Mathematical Logic" is that in the former case logic is used as a tool, while in the latter it is studied for its own sake.

A Logic class in a Philosophy degree will usually cover sentential, predicate and finally first-order logic (by order of increasing complexity and natural way of learning). At this point, you're able to give a formal expression of philosophical arguments, reason in terms of truth-value, validity and soundness, and you can detect and explain fallacies.

This is why (elementary, first-order) Logic is such an important tool in Philosophy: you need to be able to understand why the text you're reading is convincing, what the author's thoughts are, how he connects them together, how he draws conclusions from premises and so on. A frequent extension is modal logic, where you introduce the notions of necessity and possibility, and which is the basis for deontic logic - the language of Ethics.

On the other hand, "Mathematical Logic" uses the power of logic to formulate concepts in an unambiguous language where you can perform operations in order to lay the foundations of mathematics. If you study mathematical logic, you'll be introduced to much more complex types of logic, with which you'll then be able to define rigorously concepts such as theorems, deductions, axioms, sets, functions, graphs, representations, etc. - all of which form the basis for a meta-language that you know as "mathematics".