We review the logical basis of inference as distinct from deduction, and show that measurements in general, and dimensional metrology in particular, are best viewed as exercises in probable inference: reasoning from incomplete information. The result of a measurement is a probability distribution that provides an unambiguous encoding of one's state of knowledge about he measured quantity. We stress the distinction between probability (being a numerical measure of the degree of rational belief) and statistics (being the mathematics of collecting, organizing, and manipulating numerical data). We show how simple requirements for rationality, consistency, and accord with common sense lead to a set of unique rules for combining probabilities and thus to an algebra of inference. Methods of assigning probabilities and application to measurement and calibration are discussed.