Saturday, December 06, 2014

Knowledge and the Problem of Deduction

There are few things more debated in Philosophy than knowledge itself. What constitutes knowledge, how is it acquired, what process do we use to validate it. But in all of that morass what gets lost is that the reasoning process, philosophy in general and deduction in particular, have a deep and innate problem. A problem that is as old as the invention of the reasoning process, a problem that underlies all of philosophy and gave rise to the Rationalist, Skeptic, and Empiricist schools of philosophical thought.

For some context it is worth starting with the often quoted and misunderstood "Problem of Induction," which is generally and somewhat mistakenly ascribed to a 1748 treatise by Hume.

The Problem of Induction

As Wikipedia puts it:
The problem of induction is the philosophical question of whether inductive reasoning leads to knowledge understood in the classic philosophical sense, since it focuses on the lack of justification for either:
  1. Generalizing about the properties of a class of objects based on some number of observations of particular instances of that class (for example, the inference that "all swans we have seen are white, and therefore all swans are white," before the discovery of black swans) or
  2. Presupposing that a sequence of events in the future will occur as it always has in the past (for example, that the laws of physics will hold as they have always been observed to hold). Hume called this the principle of uniformity of nature.
Wikipedia continues with the rather important caveat:
The problem calls into question all empirical claims made in everyday life or through the scientific method and for that reason the philosopher C. D. Broad said that "induction is the glory of science and the scandal of philosophy...."
First thing to note is "knowledge understood in the classic philosophical sense" the second part to note is the "lack of justification" which means "no deductive justification." In short the problem of induction is that it is not deduction. Hume would be appalled that his name is associated with such a narrow-minded conception.

Hume himself, in "An Enquiry Concerning Human Understanding," made that distinction. Besides never using the term "induction" in his treatise, he was very clear that the majority of human knowledge about the world comes from innate forms of reasoning. His treatise was in part his attempt to shine some light on this wide gulf of philosophical understanding on what we now call induction and abduction. He was not calling into question induction itself, he was calling into question the capabilities of philosophy to understand how our mind worked. How could our mind narrow down the infinite possibilities into those that followed from reality. In his words:
...this operation of the mind, by which we infer like effects from like causes, and vice versa, is so essential to the subsistence of all human creatures, it is not probable, that it could be trusted to the fallacious deductions of our reason, which is slow in its operations, appears not during the first years of infancy, and at best is, in every age and period of human life, extremely liable to error and mistake. It is more conformable to the ordinary wisdom of nature to secure so necessary an act of the mind, by some instinct or mechanical tendency, which may be infallible in its operations...
So for Hume it is clear that this "problem" is only a problem for philosophy, not reality. Philosophy must be understood as one more of the sciences. It is the science of thinking about thinking within the confines of human language, knowledge, and understanding. But, as a science, philosophy is not above reality. What we "know" is not circumscribed to what philosophers claim to be its ill-defined "knowledge." Philosophy has its limitations, it is silly to ascribe those limitations to human knowledge. The reality is that brains are inductive machines, machines that rely almost exclusively on induction and abduction to ensure our survival. It can be reasonably claimed (and Hume does) that deduction itself, and therefore all of philosophy, is nothing more than the product of induction working via the evolution of ideas.

Evolution is an example of induction at work over a simple tautological truth: The fittest survive because these are the fittest to survive. An apparent circular reasoning that singularly defines evolution itself. If an idea or organism, out of the many random variations that are generated every day, is not fit to survive it won't survive. If it is fit to work inside the environment and the environment does not get rid of it, it will survive. Random variations funneled and guided by the environment, making the most fit organisms and ideas more probable and the less fit less probable.

That same tautology underlies Philosophy itself. Deduction works because deduction works (within its limitations). There cannot be any further justification for it except that it has worked in the past. That is, inductively it works. Deduction, the product of the evolution of ideas, is itself an inductive construct of our minds. From this it should be clear that induction, not deduction, is closer to being a fundamental natural law and it is just a problem in the sense of understanding how it works.

Once we place "the problem of induction" within this broader context, we would realize that it has long been solved by Bayesianism in general, and Solomonoff's Universal Induction in particular. But the solution is not trivially intuitive to those without a solid background in probability and computability theories. Probabilities and likelihood are the concepts that solidify and instrument our understanding of our natural induction process, in an attempt to explain how the machinery of the mind achieves its goals. Hume himself hinted at this:
...where different effects follow from causes, which are to appearance exactly similar, all these various effects must occur to the mind in transferring the past to the future, and enter into our consideration, when we determine the probability of the event. Though we give the preference to that which has been found most usual, and believe that this effect will exist, we must not overlook the other effects, but must assign to them a particular weight and authority, in proportion as we have found it to be more or less frequent...
That is the Bayesian subjective probability perspective coming from the very same treatise that has long been associated with "the problem of induction."

The Problem of Deduction

Instead of a Problem of Induction we should consider the long-standing Problem of Deduction which, as Hume himself said, we cannot trust it for our survival because of its being slow and extremely liable to error and mistake. This problem not only underlies all of philosophy at least since the age of enlightenment, but was deductively proven by Gödel with his incompleteness theorem, which can be restated as: Deductive systems are not only incomplete, but cannot prove their own consistency. (For those that consider that Gödel only applies to systems capable of arithmetic, multiple versions of the same basic idea have been proven that extend the conclusion to all reasonable deductive systems. Here is an example from doxastic logic, and another example from computability theory).

The problem of deduction, and philosophy in general, is a large part of what defined the empiricist movement starting in the mid 17th century , as Immanuel Kant put it:
How little cause have we to place trust in our reason if in one of the most important parts of our desire for knowledge it does not merely forsake us but even entices us with delusions and in the end betrays us!
All deductive systems, logic in particular and philosophy in general, rely on the truth of its axioms or premises. So the problem of deduction is really that it is impossible to know the truth of axioms without assuming some a priori "fountain of truth" on which to rely. While rationalism claims access to the truth of innate ideas or revelation, skepticism rightfully points out that such fountain of truth is unattainable.

Empiricism solves this problem, because as opposed to rationalism and skepticism, empiricism focuses on the role of experience, evidence, and the senses, over the notions of innate ideas, tradition, or revelation. That is, empiricism opens up the middle way by relying on induction to obtain truth, reliable enough, on which to set its foundations.

But the real problem of deduction is that most people fail to see anything wrong with it. Most ignore that whatever deductive process is followed, it is resting on probabilistic grounds. Probabilities do not combine gracefully. It is very easy to start with high-probability premises and end with very low probability outcomes. This is particularly true when we take into account the vagaries of language and even go as far as constructing logic systems to represent them.

When this expansion via language and probabilities is taken into account, deductive systems become less reliable the more complicated the deductive process is. The shakier the axioms, the faster it degrades; the more ill-defined the terminology, the shakier the conclusions will be; and the looser the logic the sillier the conclusions become. So the problem of deduction is that it is used with everyday language, while most people live under the illusion propagated by armchair philosophers that there is no problem with it.


When it comes to knowledge, that most debated of philosophical subjects, it is worth keeping in mind what Hume said:
When we entertain any suspicion that a philosophical term is employed without any meaning or idea (as is but too frequent), we need but enquire, from what impression is that supposed idea derived? And if it be impossible to assign any, this will serve to confirm our suspicion. By bringing ideas into so clear a light we might reasonably hope to remove all dispute, which may arise, concerning their nature and reality.
The traditional philosophical idea of knowledge is an ideal that has to resort to an axiomatic universal truth to survive. That view is a distortion that partially arises from the illusion of "truth" intrinsic to deductive reasoning. Skepticism has it right, if that is the idea of knowledge, then such knowledge is unattainable by humankind. Empiricism, being the cornerstone of science itself, is the golden standard for all of what we consider knowledge in the modern world.

Being an inductive construct that arises from reality makes knowledge tentative, a probabilistic and ever-improving body of information that approaches reality itself. Scientific knowledge can be superseded by further scientific knowledge. Theories get superseded by more precise theories. But note that it is a matter of degree, not of a discrete all or nothing "truth value" from some idealized unattainable construct. As induction itself, it gets closer and closer to the truth of reality the more information is gathered.

Newton's theory of gravitation has long been superseded by Einstein's General Relativity, but for our everyday reality Newton is much more than enough. An engineer does not use Einstein's equations when building a bridge or even a rocket, she would use Newton's equations. It is only when the additional precision is needed, for example in the clocks that drive the equations behind GPS satellites and receivers, that General Relativity is taken into account.

It is commonplace that multiple ideas of knowledge be conflated in language. People talk about theories being proven wrong as if that somehow invalidates the knowledge derived from them. As if somehow the fact that it is not an unattainable "universal truth" makes it any less true. Knowledge is, by necessity, tentative but ever-improving thanks to the machinery of induction operating over the continuous evolution of ideas.
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