What is Quantum Mechanics?

So far, I've told you a little about where I believe quantum theory comes from. To briefly recap, information-theoretic incompleteness, a feature of every universal system (where 'universal' is to be understood in the sense of 'computationally universal'), introduces the notion of complementarity. This can be interpreted as the impossibility for any physical system to answer more than finitely many questions about its state -- i.e. it furnishes an absolute restriction on the amount of information contained within any given system. From this, one gets to quantum theory via either a deformation of statistical mechanics (more accurately, Liouville mechanics, i.e. statistical mechanics in phase space), or, more abstractly, via introducing the possibility of complementary propositions into logic. In both cases, quantum mechanics emerges as a generalization of ordinary probability theory. Both points of view have their advantages -- the former is more intuitive, relying on little more than an understanding of the notions of position and momentum; while the abstractness of the latter, and especially its independence from the concepts of classical mechanics, highlights the fundamental nature of the theory: it is not merely an empirically adequate description of nature, but a necessary consequence of dealing with arbitrary systems of limited information content. For a third way of telling the story of quantum mechanics as a generalized probability theory see this lecture by Scott Aaronson, writer of the always-interesting Shtetl-Optimized.
But now, it's high time I tell you a little something about what, actually, this generalized theory of probability is, how it works, and what it tells us about the world we're living in. First, however, I'll tell you a little about the mathematics of waves, the concept of phase, and the phenomenon of interference.

The Origin of the Quantum, Part III: Deviant Logic and Exotic Probability

Classical logic is a system concerned with certain objects that can attain either of two values (usually interpreted as propositions that may be either true or false, commonly denoted 1 or 0 for short), and ways to connect them. Though its origins can be traced back in time to antiquity, and to the Stoic philosopher Chrysippus in particular, its modern form was essentially introduced by the English mathematician and philosopher George Boole (and is thus also known under the name Boolean algebra) in his 1854 book An Investigation of the Laws of Thought, and intended by him to represent a formalization of how humans carry out mental operations. In order to do so, Boole introduced certain connectives and operations, intended to capture the ways a human mind connects and operates on propositions in the process of reasoning.
An elementary operation is that of negation. As the name implies, it turns a proposition into its negative, i.e. from 'it is raining today' to 'it is not raining today'. If we write 'it is raining today' for short as p, 'it is not raining today' gets represented as ¬p, '¬' thus being the symbol of negation.
Two propositions, p and q, can be connected to form a third, composite proposition r in various ways. The most elementary and intuitive connectives are the logical and, denoted by ˄, and the logical or, denoted ˅.
These are intended to capture the intuitive notions of 'and' and 'or': a composite proposition r, formed by the 'and' (the conjunction) of two propositions p and q, i.e. r = p ˄ q, is true if both of its constituent propositions are true -- i.e. if p is true and q is true. Similarly, a composite proposition s, formed by the 'or' (the disjunction) of two propositions p and q, i.e. s = p ˅ q, is true if at least one of its constituent propositions is true, i.e. if p is true or q is true. So 'it is raining and I am getting wet' is true if it is both true that it is raining and that you are getting wet, while 'I am wearing a brown shirt or I am wearing black pants' is true if I am wearing either a brown shirt or black pants -- but also, if I am wearing both! This is a subtle distinction to the way we usually use the word 'or': typically, we understand 'or' to be used in the so-called exclusive sense, where we distinguish between two alternatives, either of which may be true, but not both; however, the logical 'or' is used in the inclusive sense, where a composite proposition is true also if both of its constituent propositions are true.