On True And False Theories

What makes a theory true? Bad question, that. Conflates too easily why a theory is true with our knowledge whether a theory is true. Both are subjects of great interest, but they are not the same thing.

It’s a hard question, too. Here’s a simpler: how do we know a theorem is true? Notice y has been swapped with em.

Knowledge of true theorems are deductions from compound propositions where each compound proposition is itself, taken together, known to be true—knowledge which itself is based on other, usually simpler, compound propositions, themselves also known to be true, and so on in a chain tied to indubitable propositions, which themselves are known via certain kinds of inductions made from sense impressions. Mathematics, then, is a giant web of true propositions all tied together.

Even in math the why is different than the how (we know). We do not know why, for instance, π takes the values it does. Something caused π to be what it is (I discuss this more fully in Uncertainty). Yes, we can know lots of things about π that give hints to its cause, such as π equals this or that infinite series, and isn’t it curious how this infinite series makes use of other fundamental mathematical truths, etc. But why God chose the universe to be such that π took the value it does versus some other (in a continuous infinity of choices) we do not know.

Let’s return to our original question in its epistemological sense: how do we know this theory is true? Well, in just the same way we know how this theorem is true. Theories like theorems are deduced from compound propositions. We know theorems are true because we learn no mistakes have been made in the deductions and because the propositions on which it is based are themselves truth. We check the truth of theories in the same way.

The twist is that, with theorems, we are dealing with strict truth and falsity, whereas with theories we often have uncertainty. Not all the propositions from which a theory is deduced are known with certainty to be true; neither, then, can we know the compound proposition is true, even though we can know the deduction from the assumed-true compound proposition is itself true (supposing no mistakes have been made, which as a compound proposition grows becomes less and less believable; have you seen the computer code for our “best”, say, climate models?).

We’re done. That’s the answer. It’s not yet satisfying, though. Examples are necessary.

Let’s risk a dice example (though I once had a paper rejected in part because the reviewer was understandably sick up to his gills with dice examples). These aren’t real dice. They’re fictional. In fact, (as I do in Uncertainty) let’s make the dice states of an interocitor, a device manufactured on the planet Metaluna. We know—it is true—that all interocitors must take only one of n states. We also know—it is also true—that this is an interocitor before us. We deduce from these two true propositions that this interocitor must take only one of n states.

Our theory, which we deduce from true premises, is that this interocitor will be state s has probability 1/n (where, of course, s can be only one of the allowed states).

Our theory is therefore true. We know it is true because it is based on a true compound proposition, and because the deduction is valid and sound. The theory is probabilistic and that it is true means we cannot have better understanding or make superior predictions based on this theory given only the information of the number of possible states. That point must be stressed, and stressed again.

The theory is true but limited; it is limited because we do not know why the states will be what they will be, and we cannot predict with perfection. That is, we do not understand the causes of the states. If we did, it would mean we had a whole new set of premises which allow us to deduce the states via the causes. We’d have a second true theorem, or even a true theory.

A true theory therefore does not mean a perfect theory. A true theory is one deduced from true premises, even true observational premises. These are universally true theories. Contrast these with locally true theories. A locally true theory (like any local or conditional truth) posits its own premises, which may merely be supposeds or guesses or even fictions, but where the conclusions are deduced (without error) from these premises.

Most statistical models fall into this category because most are built on ad hoc premises. The models are locally true, and may even be true if and only if the ad hoc premises themselves turn out to be universally true. They are only false—but still locally true (assuming no calculation errors)—if is it known one of the model premises is itself false.

This means the quip, heard everywhere, that “all models are false but some are useful” is itself false. But it also means we have a hint about a land between truth and falsity, where we know a theory isn’t universally true, but where we also know it isn’t universally false. Models and theories can themselves be probable.

How this all works out we’ll save in our discussion of non-empirical confirmation of theories. Coming soon!