Balance and Duality

There is a commonly used device in literature that big, important events start small.  I don’t know if that’s true.  I don’t know if small things are heralds of momentous things but I do know that I received a fairly big shock from a small, almost ignorable footnote in a book.

I was reading through Theory and Problems in Logic, by John Nolt and Dennis Rohatyn, when I discovered the deadly aside.  But before I explain what surprised me so, let me say a few words about the work itself.  This book, for those who don’t know, is a Schaum’s Outline.  Despite that, it is actually a well-constructed outline on Logic.  The explanations and examples are quite useful and the material is quite comprehensive.  I think that the study of logic lends itself quite nicely to the whole approach of Schaum’s since examples seem to be heart of learning logic and the central place where logicians tangle is over some controversial argument or curious sentence like ‘this sentence is false’.

As I was skimming Nolt and Rohatyn’s discussion about how to evaluate arguments I came across this simple exercise

Their answer (which is the correct one) is that the argument above is not deductive.  At the heart of their explanation for why it isn’t deductive is the fact that while it is highly unlikely that anyone 3 months old or younger could read The Wall Street Journal, nothing completely rules it out.  Since the concept of probability enters into the argument, it cannot be deductive.

So far so good.  Of course, this is an elementary argument so I didn’t expect any surprises.

Nolt and Rohaytn go on to say that this example can be made to be deductive by the inclusion of an additional premise.  This is the standard fig-leaf of logicians, mathematicians, and, to a lesser extent, scientists the world over.  If at first your argument doesn’t succeed, redefine success by axiomatically ruling out all the stuff you don’t like.  Not that that approach is necessarily bad; it is a standard way of making problems more manageable but usually causes confusion in those not schooled in the art.

For their particular logical legerdemain, they amend the argument to read

This argument is now deductive because they refuse to allow the possibility (no matter how low in probability) that those amongst us who are 3 months old are younger cannot read The Wall Street Journal. They elevate to metaphysical certitude the idea that youngsters such as they can’t by simple pronouncement.

Again there are really no surprises here and this technique is a time honored one.  It works pretty well when groping one’s way through a physical theory where one may make a pronouncement that nature forbids or allows such and such, and then one looks for the logical consequences of such a pronouncement.  But a caveat is in order.  This approach is most applicable when a few variables have been identified and/or isolated as being the major cause of the phenomenon that is being studied.  Thus it works better the simpler the system under examination is.  It is more applicable to the study of the electron than it is to the study of a molecule.  It is more applicable to the study of the molecule than to an ensemble of molecules and so on.  By the time we are attempting to apply it to really complex systems (like a 3-month old) its applicability is seriously in doubt.

Imagine then, my surprise by the innocent, little footnote associated with this exercise that reads

Note the language used by the pair of logicians.  Not that the deductivism school of thought wants to minimize the use of inductive arguments or maximize the use of deductive ones.  Not that its adherents want to limit the abuses that occur in inductive arguments.  Nothing so cautious as that.  Rather the blanket statement that “there are no genuine inductive arguments.”

A few minutes of exploring on the internet led me to slightly deeper understanding of the school of deductivism but only marginally so.  What could be meant by no genuine arguments?  A bit more searching led me to some arguments due to Karl Popper (see the earlier column on Black Swan Science).

These arguments, as excerpted from Popper’s The Logic of Scientific Discovery, roughly summarized, center on his uneasiness with inductive methods as applied to the empirical sciences.  In his view, an inference is called inductive if it proceeds from singular statements to universal statements.  As his example, we again see the black-swan/white-swan discussion gliding to the front.  His concern is for the ‘problem of induction’ defined as

Under his analysis, Popper finds that any ‘principle of induction’ that would solve the problem of induction is doomed to failure since it would necessarily be a synthetic statement, not an analytic one.  From this observation, one would then need a ‘meta principle of induction’ to justify the principle of induction and a ‘meta-meta principle of induction’ to justify that one and so on, to an infinite regress.

Having established this initial work, Popper jumps into his argument for deductivism with the very definite statement

And off he goes. By the end, he has constructed an argument that banishes inductive logic from the scientific landscape, using what, in my opinion, amounts to a massive redefinition of terms.

I’ll not try to present anymore of his argument.  The interested reader can follow the link above and read the excerpt in its entirety.  I would like to try to ask a related but, in my view, more human question.  To what end is all this work leading?  I recognize that it is important to understand how well a scientific theory is supported.  It is also important to understand the limits of knowledge and logic.  But surely, human understanding and knowledge are not limited by our scientific theories nor are they adequate described by formal logic.  Somehow, human understanding is a balance between intuition and logic, between deduction and induction.

Popper’s critiques sound too much like the sounds of someone obsessing over getting the thinking just so without stopping to ask if such a task is worth it.  Scientific discovery happens without the practitioners knowing exactly how it happens and what to call each step.  Should that be enough?

Of course, objectors to my point-of-view will be quick to point out all the missteps that logicians can see in the workings of science – all the black swans that fly in the face of a white-swan belief.  My retort is simply “so what?”

Human existence is not governed solely by logic nor should it be.  If it were, a part of the population would be frozen in indecision because terms were not defined properly, another part would be stuck in an infinite loop, and the last part would be angrily arguing with itself over the proper structure.  There is a duality between induction and deduction that works for the human race – a time to generalize from the specific to the universal and a time to deduce from the universal to the specific.

Perhaps someday, someone will perfect deductivism in such a way so that scientific discovery can happen efficiently without all the drama and controversy and uncertainty.  Maybe… but I doubt it.  After all, we know that we humans aren’t perfect – why should we expect one of our enterprises to be perfectible?