The Pragmatic Maxim and Information Theory

1. What "hard" means

What does it mean to say a diamond is hard? In the same year he set out abduction, Peirce used this question to show his answer. To be hard is to not be scratched by many other substances. Beyond that, there is nothing left in the word. An argument over whether a diamond that has never been scratched is hard or soft makes no difference to the world whichever way it's settled, so it is a quarrel over words.

This is the pragmatic maxim. In Peirce's words: consider what effects, that might conceivably have practical bearings, we conceive the object of our conception to have. Then, our conception of these effects is the whole of our conception of the object.

The maxim is a blade for splitting propositions. If not a single conceivable effect changes whether a proposition is true or false, that proposition has said nothing. However plausible its form, it doesn't pass the maxim.

2. Shannon's ruler

In 1948 Claude Shannon cut a scale into the same blade. In that paper he defined information as the amount by which uncertainty is reduced. Hearing the result of a fair coin toss reduces your uncertainty by one bit. In twenty questions, a single yes-or-no question gives at most one bit. As a formula: H = -Σ p(x) log p(x).

Apply this to a proposition and you get a verdict. If the list of possible states is not one bit shorter after you hear a proposition than before, that proposition carries zero information. A proposition that rules out no state is not information, however deep it sounds.

Peirce's maxim and Shannon's measure gauge the same thing from different sides. The maxim asks whether effects change; information asks whether possible states shrink. Both are criteria for filtering out propositions with zero information. Peirce laid down the principle of the judgment, and Shannon put numbers on it.

3. Meaning can be dropped, the goal cannot

At the start of his paper Shannon explicitly set meaning aside. What a message means, he said, is irrelevant to the engineering problem. But he could not set aside the goal.

Before any calculation starts, three things have to be decided. What counts as a symbol has to be fixed before probabilities can be defined. What is signal and what is noise has to be fixed; the same waveform is noise in communications and signal in astronomy. How close counts as a faithful copy has to be fixed too; the answer differs depending on whether it has to match bit for bit or just be audible.

All three are settled before the calculation, and the goal is what settles them. They are not derived inside the theory. Meaning is attached after the cutting, so the calculation runs without it; the goal is what does the cutting, so without it nothing gets cut.

4. Meaningful cognition has a purpose

Lay the maxim and information on top of each other and one proposition stands up: every meaningful cognitive operation has a purpose.

The maxim asks whether effects change. But the maxim alone can't say what has to change to count as a difference. Information asks whether possible states shrink. But Shannon alone can't say what to count as a state. Both criteria demand, before any verdict, an answer to what the splitting is for. So a cognitive operation that passes the maxim — one whose information is not zero — cannot help carrying a purpose. An operation that runs without a purpose splits nothing, and an operation that splits nothing yields no information.

This applies directly to abduction. Abduction opens spaces, but purpose decides which way to open them. Abduction without a purpose opens spaces at random, and a space opened that way has no criterion to be judged by, so it never becomes information.