Among the subtle shifts in scientific thinking that occurred in the mid 1800s, George Boole gave us a functional interpretation of logic, associating every propositional expression — at the most basic level of logic we now describe in terms of boolean algebras, boolean functions, propositional calculi, or Peirce’s alpha graphs — with a function from a universe of discourse to a domain of two values, say normally interpreted as logical values, false and true, respectively. This may seem like a small change so far as conceptual revolutions go but it made a big difference in the future development, growth, and power of our logical systems.
Among other things, the functional interpretation of logic enables the construction of a bridge from propositional logic, whose subject matter now consists of functions of the form to probability theory, that deals with probability distributions or probability densities of the form with values in the unit interval of the real number line This allows us to view propositional logic as a special case within the frame of a more general statistical theory. This turns out to be a very useful perspective in real-world research when it comes to moving back and forth between qualitative observations and the data given by quantitative measurement. And it gives us a bridge still further, connecting deductive and inductive reasoning, as Boole well envisioned.