What is Logic?
Teachers propagate a great many lies. They do so 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 clear 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:
- Formal Systems - Logic is the mathematical study of first-order logic and variations thereupon.
- Logical Consequence - Logic is the study of logical consequence, of a conclusion necessarily following from some set of premises (in virtue of its form).
- Normative Reasoning - Logic is the normative study of reasoning.
The first and narrowest view--formal systems--understands logic as a purely formal subject, a mathematical enterprise akin to algebra or analysis. Unsurprisingly, this usage and view is generally propagated by mathematicians and computer scientists. Courses and textbooks on mathematical logic, model theory, and computability theory (sometimes also called 'recursion theory') commonly take this tact, and significant opportunities for graduate-level work exist here. Those interested in logic in this sense are generally mathematicians and should aim to take both mathematical logic (model theory, computability / recursion theory, proof theory, set theory) and other pure mathematics courses (algebra, topology, analysis). Graduate programs in logic housed in mathematics departments as well as interdisciplinary programs with a mathematics component should be a good fit for interested students. For an introductory text, see Chiswell and Hodges' Mathematical Logic. For more advanced reading, working through either Hodges' A Shorter Model Theory or Marker's Model Theory: An Introduction along with Soare's Recursively Enumerable Sets and Degrees should give ample preparation for graduate study. For those with computer science inclinations, mathematical logic and theoretical work in computer science feel very similar to one another, especially computability / recursion theory and computational complexity theory. This said, computer scientists tend to make infrequent use of formal systems with heavy investment occurring only with graduate-level work on particular subjects.
The logical consequence view of logic---unlike formal systems---gives logic a distinctly philosophical component, making the subject one of conceptual analysis and formalization rather than simply the study of a particular kind of formalism. 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 also the least active academically. For better or worse, the general consensus is that classical logic is correct (or, more accurately, correct in all the ways that really matter). While non-classical logics are a perennial favorite among undergraduates, relatively few logicians have found their study or development worthwhile. The only major divergence (and it is a 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) and then seriously considering the graduate-level seminars---especially those offered by Professors MacFarlane, Mancosu, and Holliday. Those considering graduate study in this area are also advised to study at least the first few chapters of 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 deduction. One can, for instance, reason about which of several actions to take under uncertain conditions as well as whether or not a given theory is probable relative to some set of evidence. Normative reasoning holds that providing a normative analysis of these other kinds of reasoning is properly logic as well. Under this view, decision theory, game theory, confirmation theory, formal epistemology, and philosophy of science either collapse wholesale into logic or possess significant overlap with the subject. Of all the views, normative reasoning is both the broadest and the most philosophically-oriented; 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, actual research often draws freely from across the spectrum (remember the convenient lie!). Despite this, 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 initial choice; logic itself happily runs the gamut from economics to linguistics, from philosophy to computer science.