Uses and Misuses of Logic

The first philosopher. © 2002 by John Holden.

Introduction

We will use certain terms as scientists use them. For those not familiar with the language of science, we include here some fundamentals, so we'll all be starting with the same language.

• Fact. An isolated piece of information about nature. It can be simply a measurement. Sometimes related facts are called "data".
• Hypothesis. A proposition about nature that is testable, but not yet tested to the point of general acceptance.
• Law. A statement describing how some phenomenon of nature behaves. Laws are generalizations from data. They express regularities and patterns in the data. A law is usually limited in scope, to describe a particular process of nature.
• Theory. A model (usually mathematical) that links and unifies a broader range of phenomena, and that links and synthesizes the laws that describe those phenomena. In science we do not grant an idea the status of theory until its consequences have been very well tested and are generally accepted as correct by knowledgeable scientists. This meaning is very different from colloquial use of the word.

Induction and deduction

Theories and laws are required to be of such form that one can deductively proceed from theories to laws to data. The results of deduction must meet a stringent standard: they must agree with experiment and with observations of nature.

Mathematics is a process of deductive logic. Therefore it is ideally suited to be the language and the deductive link between theories and experimental facts. Because of this, some non-scientists think that mathematics and logic are used to "prove" scientific propositions, to deduce new laws and theories, and to establish laws and theories with mathematical certainty. This is false, as we shall see.

This diagram shows the relations between facts, laws, and theories, and the role of induction and deduction. It will take on more meaning as this essay progresses.

Quotes about logic

Formal logic, uses and misuses

So what went wrong? Aristotle understood that logic can be used to deduce true consequences from true premises. His error was his failure to realize that we have no absolutely true premises, except ones we define to be true (such as 2+2=4). Aristotle thought that the mind contains (from birth) some innate and absolutely true knowledge that can be used as premises for logical arguments. Medieval scholastics, who brought Aristotelian modes of thought to a height of absurdity, thought that absolutely true premises could be found in revelations from God, as recorded in the Bible.

Another error was to assume that the conclusions from a logical argument represent new truths. In fact, the deduced conclusions are just restatements and repackaging of the content contained in the premises. The conclusions may look new to us, because we hadn't thought through the logic, but they contain no more than the information contained in the premises. They are just cast in new form, a form that may seem to give us new insight and suggest new applications, but in fact no new information or truths are generated. This is especially noticeable in mathematics, for without considerable instruction in mathematics, the deductions from even a small set of premises are not at all obvious, and may take years to develop and understand.

The bottom line is that logic alone can tell us nothing new about the real world. Ditto for mathematics, as Albert Einstein observed: "Insofar as mathematics is exact, it does not apply to reality; and insofar as mathematics applies to reality, it is not exact."

So, of what use are logic and mathematics in science? Incalculable use, once we realize their strengths and limitations. In science we construct models and theories of nature. We test and use these by deriving their logical and mathematical consequences. Logic and mathematics are the cement that holds the scientific structure together, ensures its self-consistency, and helps us prevent errors of false inference. Logic and math do not, and cannot, generate new truths about nature. They only expose and reformulate the truths contained in our models, theories and laws. The conclusions aren't absolute truths, but `proximate' truths, since the raw materials upon which theories are built are based upon imperfect measurements and observations of nature. But when we know how good (how precise) the measurements are, we can also predict how good are the theories and the facts deduced from them.

Scientists do not arrive at models and theories by application of logic. They arrive at them by many processes lumped under the name 'induction'. Induction cannot be reduced to a set of logical rules (though many have tried). To see patterns (sometimes subtle and hidden ones) in data and observations requires creative ability. This is the ability to think ahead and say, "What model, set of statements (laws) or theoretical construct could I devise from which these observations and data might be deduced?"

We can't find, discover, or construct scientific laws and theories by mathematics and logic alone. But we can derive testable and useful results by application of mathematics and logic to laws and theories, and if those deduced results pass experimental tests, our confidence in the validity of the theory from which they were derived is strengthened.

In this context, logic and mathematics are reliable and essential tools. Outside of this context they are instruments of error and self-delusion. Whenever you hear a politician, theologian or evangelist casting verbal arguments in the trappings of logic, you can be pretty sure that person is talking moonshine. The quotes that open this essay reflect caution in accepting such misuses of logic.

"...philosophy gives us the means of speaking plausibly about all things, and of making ourselves admired by the less learned." — Rene Descartes

Misuse of logic is rampant in all fields, even academic ones. It is often used as a crutch to justify prejudices and as a club to smite those who hold opposing views. There are people who are thoroughly Aristotelian in their thinking, and do, indeed, believe in the profundity of empty logical arguments. Others, such as politicians and evangelists use logic cynically as an instrument for persuasion of those who don't realize that "There's a mighty big difference between good, sound reasons, and reasons that sound good." (Burton Hillis).

Just what is an 'empty' argument about the 'real world' of our experience?

• One kind is the argument that may have faultless logic but is based on premises that have not, or cannot, be experimentally verified. Another kind is based on premises that are not part of any well-established and accepted scientific theory.
• Some arguments are empty of content because they use words with no clear and unambiguous meaning, or words that cannot be related to anything real (experimentally unverifiable).
• The most seductive empty arguments build upon premises that are so emotionally appealing that we don't ask for verification, or which have appealing conclusions that blind us to the emptiness of the premises.

Mistrust of science.

It must be admitted at the outset that science is not in the business of finding absolute truths. Science proceeds as if there are no absolute truths, or if there are such truths, we can never know what they are. As the pre-Socratic skeptics observed: If we were to stumble upon an absolute truth, we'd have no way to be certain it is an absolute truth. The models and theories of science are approximations to nature—never perfect. But in most cases we know rather well how good they are. We can state quantitatively the limits of uncertainty of numeric results, and their range of applicability. Yet there's always the possibility that we may find exceptions to one of our accepted laws, or may even find alternative theories that do a better job than older ones.

Some critics of science attack this process of science, on the grounds that it cannot produce absolute truths. Theirs is a black/white view of the scientific process. Never mind that they have not proposed any other process that is capable of producing anything near the power and comprehensiveness of present science. They say that "Theory X" isn't perfect therefore it is "wrong".

The results and predictions of a theory, being well tested, will not crumble if the theory is someday modified, drastically changed, or even replaced with another theory. The results or predictions of a theory are not all suddenly rendered "wrong" when a theory is modified or replaced. These results and predictions may be improved in precision or scope. Sometimes the predictions of a new theory have greater scope than the old one, predicting things the old one didn't (and things that we never had observed or tested before). Very often a new theory is sought because the old one, while its predictions were mostly correct, predicted a few things that just weren't confirmed by good experiments. We'll need to say more about this later.

The fact that science claims no absolute truths is seized upon by people who hold strong religious beliefs and who dislike those conclusions of science that run counter to their emotional convictions. To them, if a thing is not absolutely and finally true, it is false, and therefore the methods used to formulate it must be flawed.

The futility of searching for absolutes.

Science has progressed by rejecting much of its past history, past practices and past theory. Though the sciences arose from a muddled mix of mysticism, magic and speculation, scientists eventually realized that those modes of thought were prone to error and simply not productive. So chemists reject the theories of the alchemists, astronomers reject the theory underlying astrology. Mathematicians reject the number-mysticism of the Pythagoreans. Physicists, when they bother to think about their discipline's roots, acknowledge the pre-scientific contributions of the ancient Greeks in mathematics, Democritus' view that nature is lawful, and also their attitude of seeking knowledge for its own sake. But they are embarrassed by the Greek teachings about physics, for most of these have all been consigned to the trash-heap of history.

Even those early ideas that happened to be in harmony with our present views seem based upon faulty methodology or were simply speculation. Sometimes a few of those guesses seemed surprisingly close to our modern views, at least superficially. But when examined in detail the similarity breaks down. Democritus' atomic theory, for example, was based on no hard evidence, had no historical connection with modern atomic theory, and its details bore no resemblance to what we now know about atoms. Once in a while, if you speculate wildly enough, you get lucky. Too many textbooks make a "big deal" out of such accidental similarities.

Scientific method.

How Science really works.

Through more precise and detailed study we found that many of these regularities can be modeled, often with mathematical models of great precision.

Sometimes these models break down when extended (extrapolated) beyond their original scope of validity. Sometimes extrapolation of a model beyond its original scope actually works. This warns us that we had better rigorously test each model for validity, and these tests should be capable of exposing any flaws in the model—flaws capable of demonstrating that the model isn't true.

Even when a model survives such testing, we should only grant it "provisional" acceptance, because cleverer people with more sophisticated measuring techniques may in the future expose some other deficiencies of the model.

When models are found to be incomplete or deficient, we often fix them by tweaking their details till they work well enough to agree with observations.

When rapid advances in experimental observations occur, a model may be found so seriously inadequate to accommodate the new data that we may scrap a large part of it and start over with a new model. Relativity and quantum mechanics are historical examples. These situations are often called "scientific revolutions."

When such upheavals occur, and old models are replaced with new ones, that doesn't mean the old ones were totally "wrong", nor does it mean their underlying concepts were invalid. They still work within their scope of applicability. Newton's physics wasn't suddenly wrong, nor were its predictions found unreliable or incorrect when we adopted Einstein's relativity. Relativity had greater scope than Newtonian physics, but it also rested on a different conceptual basis.

Past experience has shown that mathematical models of nature have tremendous advantages over the earlier, more appealing, models which used analogies to familiar everyday phenomena of our direct sensory experience. Mathematical models are less burdened with emotional baggage, being "pure" and abstract. Mathematics provides seemingly infinite adaptability and flexibility as a modeling structure. If a some natural phenomena can't be modeled by known mathematics, we invent new forms of mathematics to deal with them.

The history of science has been a process of finding successful descriptive models of nature. First we found the easy ones. As science progressed, scientists were forced to tackle the more subtle and difficult problems. So powerful are our models by now that we often delude ourselves into thinking that we must be very clever to have been able to figure out how nature "really" works. We may even imagine that we have achieved "understanding". But on sober reflection we realize that we have simply devised a more sophisticated and detailed description.

Whatever models or theories we use, they usually include some details or concepts that do not relate directly to observed or measurable aspects of nature. If the theory is successful we may think that these details are matched in nature, and are "real" even though they are not experimentally verifiable. Their reality is supposed to be demonstrated by the fact that the theory "works" to predict things we can verify and continue to verify. This is not necessarily so. Scientists often speak of energy, momentum, wave functions and force fields as if they were on the same status as objects of everyday experience such as rocks, trees and water. In a practical sense (for getting answers) this may not matter. But on another level, a change of scientific model may do away with a force field as an conceptual entity, but it wouldn't do away with a forest, mountain or lake.

Science progresses through trial and error, mostly error. Every new theory or law must be skeptically and rigorously tested before acceptance. Most fail, and are swept under the rug, even before publication. Others, like the luminiferous ether, flourish for a while, then their inadequacies accumulate till they are intolerable, and are quietly abandoned when something better comes along. Such mistakes will be found out. There's always someone who will delight in exposing them. Science progresses by making mistakes, correcting the mistakes, then moving on to other matters. If we stopped making mistakes, scientific progress would stop.

What do scientists really think about 'reality'?

The notion that we can find absolute and final truths is naive, but still appealing to many people, especially non-scientists. If there are any underlying "truths" of nature, our models are at best only close approximations to them—useful descriptions that "work" by correctly predicting nature's behavior. We are not in a position to answer the philosophical question "Are there any absolute truths?" We can't determine whether there is an underlying "reality" to be discovered. And, though our laws and models (theories) become better and better, we have no reason to expect they will ever be perfect. So we have no justification for absolute faith or belief in any of them. They may be replaced someday by something quite different in concept. At least they will be modified. But that won't make the old models "untrue". All this reservation and qualification about truth, reality, and belief, doesn't matter. It isn't relevant to doing science. We can do science quite well without 'answering' these questions, questions that may not have any answers. Science limits itself to more finite questions for which we can arrive at practical answers.

Also, we've learned that not all questions we can ask have answers that we can find. Any question that is in principle or in practice untestable, is not considered a valid scientific question. We like to think that scientists don't waste time on those, but they seem to pop up in discussion and in books quite often. (Many people think unanswerable questions are the most profound and important ones. Questions like "What is the meaning of it all," or "What jump-started the universe?" I think that scientists should set these aside for the philosophers to chew on, and get on with the business of answering more accessible questions.)

Aesthetic appeal of theories.

But there's no reason why nature's operations should be beautiful or appealing to us. There's no reason why nature's operations should even be fully comprehensible to us. It could be that when we achieve an even more successful theoretical description of nature it may turn out to be messy, difficult to understand and use, and totally devoid of emotional or aesthetic appeal. We may not be capable of devising more satisfying alternatives.

We've had a taste of this already. When quantum mechanics was being developed many physicists in the forefront of developing the theory didn't "like" it, and hoped that someday they'd find a different way to formulate the theory—one more to their liking. A couple of quotes illustrate this: