What are the relations between logic as an area of (modern) philosophy and mathematical logic.
The world "modern" refers to 20th century and later, and I am curious mainly about the second half of the 20th century.
Background and motivation
Logic is an ancient area of philosophy which, while extensively beein studied in Universities for centuries, not much happened (unlike other areas of philosophy) from ancient times until the end of the 19th century. The development of logic in the first part of the 20th century since Frege, Russell and others is a turning point both in logic as an area of philosophy and in mathematical logic. In the first half of the 20th century there were close connections between the development of logic as an area of philosophy and the development of mathematical logic. Later, in addition to its interest for mathematicians and philosophers logic became a central applied field in computer science.
My question is about relations between logic as part of philosophy and mathematical logic from the second half of the 20th century when its seems that connections between these two areas have weakened. So I am asking about formal models developed in philosophy that had become important in mathematical logic and about works in philosophical logic that were motivated or influenced by developments in mathematical logic.
I am quite curious also about the reasons for the much weaker connections between mathematical logic and formal models developed by philosophers at the later part of the 20th century. (This makes this question much less board than it seems. Another thing I am curious about is to what extent for the applications to computer science formal models described by philosophers turned out to be useful.
Update: To complement the excellent answers already given I will try to ask some additional researchers in relevant fields to contribute directly or through me.
Update While it was clear in some of the answers let me make it explicit that I refer also to relations between philosophy and set theory.
Related MO question: In what ways did Leibniz's philosophy foresee modern mathematics?
Has philosophy ever clarified mathematics?
Best Answer
I agree with the commentators that the question is rather too broad, but here's an attempt to answer it anyway.
Readers of MO will likely have less familiarity with non-mathematical logic, so it might help to begin by skimming the tables of contents of the 18-volume (!) Handbook of Philosophical Logic to get some feeling for what people mean by "philosophical logic." [Edit: The preceding link no longer works; one can find some content using Google Books and the Wayback Machine.] It includes many topics that will likely be unfamiliar to mathematicians, such as temporal logic, multi-modal logic, non-monotonic reasoning, labelled deductive systems, and fallacy theory.
Roughly speaking, philosophical logic is the general study of reasoning and related topics. As in other areas of philosophy, this study is not necessarily formal. However, the success of formal methods in mathematical logic has led philosophers to try to formalize many other kinds of reasoning. Formalized modal logics are perhaps the best known of these. These are not always classified as "mathematical logic" because in mathematics one does not typically reason formally about concepts such as possibility, necessity, belief, etc. On the other hand, once a system of logic has been made sufficiently formal, it can of course be subject to mathematical study. Thus the boundary between (for example) formal modal logic and traditional mathematical logic is somewhat blurry. A notable example of the cross-fertilization that is possible here is Fitting and Smullyan's book on Set Theory and the Continuum Problem, which develops the (highly mathematical) subject of forcing from the perspective of modal logic, providing a fresh and completely rigorous approach to a now-classical mathematical subject.
If I had to summarize in one sentence, I would say that mathematical logic is the subfield of philosophical logic devoted to logical systems that have been sufficiently formalized to admit mathematical study. This is a slightly broader definition of mathematical logic than is customary, but I think it's a good definition in the context of this MO question, which tacitly seems to be asking if mathematicians have anything to learn from so-called "philosophical logic."