Uses and Misuses of Logic

As far as the propositions of mathematics refer to reality they are not certain, and so far as they are certain, they do not refer to reality. —Albert Einstein (1879-1955) U. S. physicist, born in Germany.

The first philosopher.
© 2002 by John Holden.


This chapter contains observations on the uses and misuses of logic, particularly in the sciences. Along the way we'll wander into the murky realms of absolute and proximate truths, deduction and induction and address the question of how we can have confidence in knowledge that is less than perfect.

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

Science proceeds from facts to laws to theories by a difficult-to-define process called induction. Induction includes pattern-recognition, brainstorming, tinkering, creative guessing and that elusive "insight". It is not a process of deductive logic.

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

Logic is the art of going wrong with confidence.
Joseph Wood Krutch

Logic: an instrument used for bolstering a prejudice.
Elbert Hubbard

It is always better to say right out what you think without trying to prove anything much: for all our proofs are only variations of our opinions, and the contrary-minded listen neither to one nor the other.
Johann Wolfgang von Goethe (1749-1832)

Most of our so-called reasoning consists in finding arguments for going on believing as we already do.
James Harvey Robinson

Logic is neither a science nor an art, but a dodge.
Benjamin Jowett

Logic, like whiskey, loses its beneficial effect when taken in too large quantities.
Lord Dunsany

He was in Logic a great critic,
Profoundly skill'd in Analytic;
He could distinguish, and divide
A hair 'twixt south and south-west side.
Samuel Butler, Hudibras.

We must beware of needless innovations, especially when guided by logic.
Sir Winston Churchill, Reply, House of Commons, Dec. 17, 1942.

...logic, the refuge of fools. The pedant and the priest have always been the most expert of logicians—and the most diligent disseminators of nonsense and worse.
H. L. Mencken. The American Mercury. p. 75.

Formal logic, uses and misuses

Formal logic was invented in Classical Greece and integrated into a `system' of thought by Aristotle. It was, for him, a tool for finding truth, but it didn't keep him from making the most profound errors of thought. Nearly every argument and conclusion he made about physical science was wrong and misguided. Any tool can be misused, and in these pre-scientific days logic was misused repeatedly.

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

The concern of this essay is with use and misuse of logic in science, and in discussions of the 'real world' of our experience. In the processes of science, mathematics holds a special place. While mathematics, being a subset of logic (or vice versa, you may argue), says nothing about the real world, it is the modeling tool we use for our knowledge of nature, providing the logical connection between our models and our measurements and observations. Without logic/mathematics, science as we know it is inconceivable. We would have no alternative way to integrate real-world knowledge into a unified and useful system.

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.

Some people are profoundly disturbed by the fact that reason alone can't generate truths. When the use of mathematics and logic in science is explained to them they respond, "If mathematics and logic can't produce absolute truths, then they produce only untruths or partial truths, and are therefore worthless." This sentence is itself an example of nonsense clothed in the appearance of logic.

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.

Though the philosophers of ancient Greece developed formal logic, and got a good start toward mathematics, they realized the limitations of logic and the futility of seeking absolutes. Here are a few comments about this dilemma.

Only one thing is certain—that is, nothing is certain. If this statement is true, it is also false.
Ancient paradox

The gods did not reveal from the beginning
All things to us; but in the course of time
Through seeking, men found that which is better.

But as for certain truth, no man has known it,
Nor will he know it; neither of the gods,
Nor yet of all the things of which I speak.
And even if by chance he were to utter
The final truth, he would himself not know it;
For all is but a woven web of guesses.

Xenophanes (c. 570-c. 480 BCE) Greek philosopher.

We know nothing in reality; for truth lies in an abyss.
Democritus, (c. 420 BCE) Greek philosopher.

None of us knows anything, not even whether we know or do not know, nor do we know whether not knowing and knowing exist, nor in general whether there is anything or not.
Metrodorus of Chios (c. 4th century BCE) Greek philosopher

This only is certain, that there is nothing certain; and nothing more miserable and yet more arrogant than man.
Pliny ("The Elder") (23-79) Roman naturalist. (Gaius Plinius Secundus).

All we know of the truth is that the absolute truth, such as it is, is beyond our reach.
Nicholas of Cusa (1401-64) German cardinal, mathematician, philosopher. De Docta Ignorantia (Learned Ignorance)

These folks who made these skeptical comments are not saying that "We can't know anything, so why bother?" They are saying that we can't "know" in the absolute sense, that we have no way to know if there are any absolute truths, and we wouldn't be able to prove the absoluteness of an absolute truth if we accidentally stumbled on one. Today we express it differently: "Science describes nature, it does not explain." Science attempts to answer "how" questions, but not "why" questions.

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 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, have no historical connection with modern atomic theory, and its details bear 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.

    —Donald E. Simanek, 1997, 1999, 2002.
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The Evolution Deniers.
Intelligent Design: The Glass is Empty.
Order from Disorder. Creation in Everyday Life.
Order and Disorder in Nature.
Is The Real World Really Real?
The Scientific Method.
Proofs of Unknowables. The Proof is Pudding.
Theory or Process?
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Why Not Angels?
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