Teachers propagate a great many lies; not out of any malice for their students, but rather because the truth is complicated. One of these lies is the neat separation of academic work into various fields--the idea that there is a clearly-defined, principled distinction between mathematics and computer science, physics and chemistry, economics and political science. In truth, these separations are owed to accidents of history, a few theoretical commonalities, and a great deal of vaguely-felt family resemblance. The borders between academic subjects are permeable and much work freely crosses them. Nowhere is this more apparent than in logic, and, as a beginning student, this can be immensely frustrating. A brief review of logic textbooks, classes, graduate programs, or even professional conferences shows an astounding lack of consistency; lack of a common notation, lack of common definitions, even lack of a common vision for the subject itself. What follows is an attempt at giving a principled overview of this morass as well as some general advice for those looking to learn more. While the normative question of what the term 'logic' should refer to is an interesting one, I'll do my best to remain descriptive, to simply present various ways the term has come to be used.
For the neophyte, a lot of trouble can be avoided simply by remembering that there are many different conceptions of what logic is. Below I've listed what I take to be the three primary conceptions along with a name for each to ease the following discussion:
The logical consequence view of logic--unlike formal systems--gives logic a distinctly philosophical component, making the subject one of conceptual analysis and/or formalization, rather than simply the study and use of formal systems. This view is commonly expressed by philosophers and more philosophically-minded mathematicians. Of the three views, logical consequence is the least contentious and arguably the most historically accurate. Ironically, it is--in point of fact--the least active academically. For better or worse, the general consensus is that classical logic is correct (or at least correct in all the ways that really matter); while non-classical logics are a perennial favourite among undergraduates, very few logicians have found their study or development worthwhile. The only major divergence from this trend is modal logic, an area where debate and research remains lively. Those interested in logic as logical consequence are best suited for philosophy departments with logicians among the faculty or interdisciplinary programs with a philosophical component. For a very basic introductory text, consider Barwise and Etchemendy's Language, Proof, and Logic for classical logic. For modal logic, van Benthem's Modal Logic for Open Minds gives an introduction to propositional modal logic while Fitting and Mendelsohn's First-Order Modal Logic considers issues surrounding the move to a first-order modal system. Those interested in pursuing the topic further with regard to classical logic should work through John Etchemendy's The Concept of Logical Consequence alongside Alfred Tarski's more famous papers. For those interested in modal logic, pursuing citations from First-Order Modal Logic along with Stalnaker and van Benthem's work will provide plenty of additional readings in modal logic. If you happen to be a student at UC-Berkeley, I would strongly suggest enrolling in the typical logic courses (12A, 140A, 140B, 142, 143), then seriously considering the graudate-level seminars--especially those offered by Professors MacFarlane, Mancosu, and Holliday. Those considering graduate study in this area are also advised to refer to the logic texts recommended for formal systems; technical competence is a requirement for work beyond the undergraduate level and working through the initial portions of the suggested texts should provide it.
Finally, normative reasoning is--academically--the least prevalent understanding of 'logic' today; it's also the closest to the colloquial usage of the term. Normative reasoning contains logical consequence as a proper part, but expands the subject area beyond the premise-conclusion, deductive format. One can, for instance, reason about which of several actions to take under uncertain conditions (boardgames! real life!), as well as whether or not a given theory is probable relative to some set of evidence (science anyone?). Normative reasoning holds that providing a normative analysis of these other kinds of reasoning is properly logic as well. Under this view, decision theory, (normative) game theory, inductive logics, and formal epistemology among others topics either collapse wholesale into logic or possess significant overlap with the subject. Of all the views, normative reasoning is the most philosophical and broadest; those interested in this understanding of logic should take on the advice provided for logical consequence, but also explore the additional areas included in this conception. In particular, Resnik's Choices is a good introduction to decision theory while exploring the work of Joyce, Elga, and Hájek (along with those they cite) should provide ample additional material. Those wishing to pursue these studies at the graduate-level are best suited to an interdisciplinary program with a strong philosophical component or a purely philosophical program; in both cases, students should check that a decision theorist and a more traditional logician are present among the faculty. If you happen to be a Berkeley student, you should aim to not only exhaust the usual logic courses, but also any decision/game theory classes (e.g., 141) and seminars offered by Professor Buchak.
The diagram below is a simple visual representation of the three-way distinction given above:
Of course, while the picture above paints all these topics as distinct, most--if not all--are interrelated with at least one other, often in unexpected and surprising ways. Despite these relations, too much ground is covered for any single individual to be competent in more than a handful of the topics listed above. The most important piece of advice I have for those interested in pursuing logic by any definition is to determine what exactly in (or not in, as the case may be) the diagram above you find interesting; is it the complex mathematics in the formal systems, formalizing logical consequence, or giving normative analyses of reasoning under uncertainty? Perhaps it's simply using logic to clarify intuitive notions like truth or as a tool for semantics? What courses you should take and articles you should read ultimately hinges on this choice; logic itself happily runs the gamut from economics to linguistics, from philosophy to computer science.
I've found that compiling and editing notes helps me process academic material; over time, this has resulted in a collection of documents covering varying subjects, the longest of which I've posted below. A brief review of each has convinced me that they are embarrassingly resplendent in confusions and typos, and so no promises are made as to the veracity of the notes, nor their completeness. Nevertheless, I would appreciate being notified (hah!) of any serious errors. The .tex files for the pdfs below are available for academic use upon request (while I don't consider any of the below anywhere near polished enough for actual classroom use, you're welcome to copy-paste what you want and save yourself a great deal of time.
LATEX is a typesetting language capable of easily incorporating non-alphanumeric symbols (compare, for instance, to Microsoft Word). Its primary use is in academia by those pursuing symbol-laden subjects, e.g. logic, mathematics, computer science, and the hard sciences. In happy news, LATEX editors are usually open source, so you won't need to purchase anything. In less happy news, LATEX is a markup language (like html) which means there's a substantial learning curve, especially for those who have never done any programming. If you're planning to pursue graduate-level study (or really, really want completely-typed notes as an undergraduate) in one of the subjects listed above, it's probably worth your time to learn; if not, I wouldn't recommend bothering. To give you a better idea of what's entailed, I've attached a screen shot of a random page in my notes side-by-side with the LATEX input:
If you'd still like to learn, go ahead and install a LATEX distribution; I'd recommend MikTeX for Windows users and MacTeX for Mac users. Both are free and come with a built-in editor (TeXWorks and TeXShop respectively). For MikTeX, you'll want to allow it to download packages on the fly. Next, download and open (in your LATEX editor) this document and get comfortable with the basics; afterwards, take a look at this for some slightly more complicated constructions. If for some reason you don't like either of the included editors, Wikipedia maintains a page comparing alternatives. Finally, when--inevitably--you get stuck, remember that Google and Detexify are your friends.
In what has become a time-honored tradition (see Arc and Michael's thoughts), I have a few bits of advice on preparing for the mathematics prelim. My general sentiments on textbooks mirror those expressed by Michael. In my experience, Marker's text is the most cogent option; since the course has historically used Hodges', I would go so far as to say that you should read through Marker's text in parallel (Hodges occasionally takes a simple idea and makes it horrendously complex, e.g. his discussion of E-M models). For computability theory, I found it best to just read Soare's text from the first page. I've also posted my notes covering both subjects (with the material I consider important indicated in the preface and the .tex file available upon request). Beyond covering the basic material, I would highly advise spending considerable time simply doing problems from past prelims; it's good practice and familiarizes you with the topics that are usually tested. If for some reason you haven't found it yet, please check out the prelim Google doc for more resources generated by past students.
The group in logic has three distinct options for the qualifying exam: mathematics, philosophy, and special. Each has different requirements which you can review here. In addition to the material on the group website, several forms related to the exam and a handout with a brief overview of the process can be found in the mathematics department office. I myself opted for the philosophy option, and so my thoughts and documents may be misleading for other selections. The philosophy option is intended to mimic the qualifying exam for the philosophy department with the only major divergence between the two exams being a diversity requirement in the philosophy department's. The qualifying examination itself is an oral examination in three subject areas with one faculty member from the group advising each. I would suggest initially approaching a faculty member or two with whom you are comfortable to discuss possible topics, mock up a very basic syllabus, and discuss which other faculty members might make good additions to the committee. Approaching the remainder of the committee is the next step; I've found it most effective to first email faculty with whom I had only a passing acquaintance, then either set up a meeting (if they respond) or drop by their office hours (if they don't respond). You'll likely find that different faculty have very different attitudes toward the exam. Some will be relatively hands-off and allow you to select large portions of the syllabus unilaterally; others will have much more concrete ideas about what topics and readings should be covered. If you have a strong preference in this regard, you'll need to ask older graduate students about their experiences with particular faculty. In general, polishing your syllabus and picking readings will be an ongoing process and conversation throughout your preparation for the exam.
After you've set your faculty members (three primary from the group itself and one member outside the group) and a general subject area for each of the three primary members, you should begin reading and meeting with them. It's common (and strongly suggested) to take notes on the papers/books as you read while also meeting regularly with your three primary advisors to discuss what you've read. Typically, preparation for the exam takes place during a single term in which you take no other courses. Two months before your anticipated exam date, you'll want to schedule the exam itself (a three-hour block during normal business hours), designate one of your primary advisors as the chair of the examination committee (cannot be your eventual advisor!), and fill out this form. It's expected (although not explicitly written in the rules) that you type up a critical overview (1800-3000 words) of each of your subject areas and send them to your committee members a week or so before the exam. For the exam itself, you'll need to reserve the room yourself (see the philosophy or mathematics department offices). Finally, the format of the exam (at least for the philosophy option) is simply, by topic, each advisor asking you questions within that area, possibly with side-questions from the other committee members. In my experience, it's unnecessary to memorize intricate proofs, though you should have a firm grasp of general techniques and any straightforward results. Axioms, assumptions, and important examples are fair game and will likely come up, as will the general contours of any philosophical arguments. If you can find other students with the necessary background, arranging a mock examination can be very beneficial in fine-tuning your presentation of key ideas and results (at the very least I would practice aloud answering the most obvious questions). After the oral examination itself, you'll be asked to leave the room while the committee members discuss; if all goes well, you'll be called back in and informed that you've passed. If not, there's a brief wait period before you'll need to go through the oral exam again. Good luck!