Follows: What comes after, what depends on, is conditioned or caused by, or is preceeded by something. In logic: B follows from A if there is a valid proof or argument that if A is true, then B is true.
Following is an extremely general concept, and many intuitions about it somehow involve time and sequences. Also, the notions of dependency, conditoning and causing have had various explanations and all involve some unclarities.
In any case, that such-and-such somehow follows (probably or certainly) from so-and-so is a fundamental idea or kind of fact or statement, temporally, logically, semantically, prooftheoretically, probabilistically, and physically, and has found various formalizations, explanations and analyses.
The common logical kernel in the various notions of following (that may be quite complicated) seems to be a reflexive and transitive relation conforming to <=.
It is an interesting - technical - fact that at present (2007) there are quite a few programs available that make or check mathematical proofs, and also systems that are capable of some sort of heuristical reasoning. Those who are interested in this should check the internet, and should realize that these systems generally require some understanding of logic, mathematics and programming (the last often in the form of Lisp).
See also: Causation, Consequence, Entailment, Induction, Logic, Logical Terms
Adams 1 and 2, Burks, Bochenski, Gazdar, Hasenjäger, Hume, Goldblatt, Grize, Hooker, W. Johnson, Kneale, D. Lewis, Lyons, Schopenhauer