## The Difference That Makes A Difference That Peirce Makes : 8

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 $X$ to a domain of two values, say $\mathbb{B} = \{ 0, 1 \},$ 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 $f : X \to \mathbb{B},$ to probability theory, that deals with probability distributions or probability densities of the form $p : X \to [0, 1],$ with values in the unit interval $[0, 1]$ of the real number line $\mathbb{R}.$  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.

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