By Basil Gala, Ph.D.
In Search of Meaning
My mother, short and plump, usually followed her intuition and religion. When she succeeded getting something she wanted, she would say: “I followed my instinct.” My father, tall and thin, was an atheist. He liked to reason, plan, and act accordingly. My dad often said: “This is the logical thing to do.” Once, his bank employers offered him the management of a branch in a different town, a big promotion. Next morning, my mother said to him, “You can’t accept this job. An angel in a dream warned me against this move.” “You are being silly; my career demands that I accept the offer,” my father argued. “Well then, go on alone. I’ll stay here with the kids,” my mother said. The outcome of this argument proved, that although logic is superior to intuition, in the end intuition wins when wielded by a woman.
Philosophers say logic or reason is that faculty which most clearly distinguishes humans from all other animals, and they often define a human as a rational animal. Well, philosophers deal with reason, that’s their basic tool, so we expect them to praise reason as the supreme human faculty. What can we say about intuition? Intuition is defined as the ability to sense or to know things immediately without reasoning. On the other hand, logic uses those rules of reasoning men have established as valid to prove the truth of statements. When I say men I don’t include women. Aristotle, the father of logic was a man. Aristotle’s Organon treatise held sway in the West unchallenged until the nineteenth century. Then in 1853 came George Boole’s “The Laws of Thought” and binary algebra, used to design logical circuits and computers today. Binary logic was based on “true” or “false”, “yes” or “no”, zero or one, (on or off with switches), and Boolean logic allowed no “maybe” state, where women like to dwell. Boole was a man, and a Scot. Frege and Russell-Whitehead, after them Carnap, followed Boole’s lead in symbolic logic, all of them men. No wonder men like to say women are not logical; women follow feminine intuition, which serves the needs of women well, while keeping women subservient to men. If men are superior to women, then logic is superior to intuition.
Aptitude tests conducted over many decades show that men perform better in quantitative, mathematical, and logical tasks; women on the other hand are better at verbal skills. Evolution has fashioned the female brain differently from the male brain, so “maybe,” “perhaps,” “negative yes,” and “positive no” are useful concepts for the survival of women. When men prove the truth of a statement they do so by using previous statements, and they continue this process until they cannot go any further; the last statement on which they are stuck is called an axiom, a self-evident truth to men. The definitions of words in statements are also given using other definitions, until men arrive at elemental or primitive definitions, which they accept on faith. The rules followed in proving one statement from others and the elemental definitions are based on agreements deemed valid among gentlemen, not ladies. To find out how the minds of ladies work, look into feminist texts, such as Betty Friedan’s “The Feminine Mystique,” and Germaine Greer’s “The Female Eunuch.”
A logic akin to female thinking was published in 1965 by Lotfi Zadeh of UC, Berkeley, entitled “Fuzzy Logic;” fuzzy logic allows intermediate states between 0 and 1. Fuzzy logic has been implemented into some electronic systems in Japan to handle problems of adaptation and the recognition of ambiguous patterns. Based on fuzzy set theory, fuzzy logic is different from probability, because it represents membership in vaguely defined sets, not likelihood of some event or condition. Fuzzy logic is yin, not yang, although Professor Zadeh was not a woman.
A woman’s heart “feels things the eyes cannot see, and knows what the mind cannot understand,” as Robert Vallett wrote. Another famous proponent of intuition, flourishing in the early twentieth century, was the French-Jewish philosopher Henri Bergson. In “Creative Evolution,” Bergson renounced logic as an adequate measure of what can and cannot be; using creative intuition, élan vital, life exceeds our logic, overflows and surrounds it. Bertrand Russell countered that Professor Bergson “wants to turn us into bees with his notion of intuition.”
Still, Bertrand Russell in his “Problems of Philosophy” writes about universal truths–inborn or self evident truths, which all normal human beings recognize without needing proof, as opposed to sense data obtained from observations. For example, the statement “two plus two equals four” is a universal truth: once you have pointed out this statement to a rational human being, no further elucidation or explanation is necessary.
Plato, in the Socratic dialogue “Meno,” gives an impressive demonstration of universal truths. At a symposium, Socrates questions an illiterate slave boy serving wine and leads the boy to prove a non-trivial mathematical theorem step-by-step by using universal truths.
Complex statements, however, hard to prove, are broken down into simpler statements, on which we can get a better handle. Statements which are obviously true to a normal person are accepted as such without proof; they are called axioms, from the Greek axioma, an authoritative statement. Axioms are said to be self evident; for example the statement “a thing cannot be and be not at the same time,” or double negation: “He is not truthful,” means he is a liar. Modern symbolic logic, however, allows for axioms which are not self-evident. The consequences of starting with such axioms can be some very unusual theorems. For example, while Euclid’s geometry was based on self-evident axioms, modern non-Euclidean geometry is not, and leads to strange theorems, theorems which, however, better explain some scientific phenomena, such as relativity. In non-Euclidean geometry two parallel lines do meet, at infinity. Clearly, the choice of axioms is pragmatic: we choose those that lead to useful results. That’s logic.
Logic as we know it today is a Western invention. In the East people have used other kinds of logic for thousands of years. The writings of the great Chinese mystic, Lao Tzu are replete with paradoxical statements that defy Aristotle’s as well as Russell’s logic, for example, “Not-doing, nothing is left undone,” or “Heaviness is the root of lightness.” In Lao Tzu we also find the statement, “Though you lose the body, you do not die.” This last one is similar to Jesus’, “If you cling to your life, you will lose it.”
Whether you subscribe to Western or Eastern logic, where do you suppose the innate, universal truths, come from? What is the source of that knowledge which is not derived from sense data? The British empiricists, such as David Hume, John Locke, and Bishop George Berkeley, said that experience is the source of universal truths. The mind absorbs specific impressions from the senses and abstracts them subconsciously. These abstractions appear to come from nowhere at a later date when we have forgotten the specific impressions we have accumulated over time. Such an explanation seems feasible, especially if we take into account the accumulated experiences of mankind over the millennia. Some of the abstractions we use as self-evident truths may be abstractions woven into the genes we have inherited, similar to the complex instinctive behaviors of animals building nests or performing ritual courtships. Here is an area of logic which borders on intuition. The abstractions we call universal truths are wired into our brains to help us deal with the problems we face in human society.
It has been said that the structure of mathematics bears a close resemblance to the structure of the universe; that is why mathematics is such a powerful tool for explaining phenomena and predicting outcomes in nature. We use the self-evident rules of logic to prove theorems in mathematics. Not surprisingly, mathematics (and music), reflect nature and explain it, since nature is the source our experiences over the generations. Such is tribal or folk wisdom also.
In “Principia Mathematica,” Alfred North Whitehead and Bertrand Russell demonstrated that all mathematics can be deduced from logical principles. In the fifth century BC, the cult of Pythagoras of Samos, known for the theorem of geometry on right triangles, believed they could explain pretty much everything in the universe, including music, with numbers and mathematics. Pythagoras built a cult because he wandered into mysticism and intuition, far away from logic. Today scientists use mathematics as the basic medium for explaining phenomena in physics, chemistry, astronomy, and in all the natural sciences. Sciences that used to be called soft, because they did not use mathematics, are now becoming much harder. Economists, psychologists, and biologists are expressing their theories in mathematical form and using statistics and probability to organize their data.
Albert Einstein wrote: “How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?” Also, he wrote: “The most incomprehensible thing about the universe is that it is comprehensible.” Eugene Wigner in a 1960 paper wrote: “It is difficult to avoid the impression that a miracle confronts us here, quite comparable in its striking nature to the miracle that the human mind can string a thousand arguments together without getting itself into contradictions, or to the two miracles of the laws of nature and of the human mind’s capacity to divine them.” This statement is known as Wigner’s Puzzle: the unreasonable effectiveness of mathematics in the natural sciences.
Logical thought is what nature has trained humans to do; ordinary logic is the reasoning we are accustomed to use; it is linear, very helpful in solving routine or old problems in nature. When we face new problems from nature, we shift to non-linear or intuitive thinking to come up with novel solutions. Non-linear thinking is more creative, carried out mostly by the subconscious. Non-linear thinking does not dispense with the law of causality (we still observe cause and effect in what we study); effects are not proportional to causes, connections being more subtle and complex. For example, linear thinking says if I exercise, I strengthen my muscles; if I exercise twice as hard, my muscles will get twice as strong. We do not get this effect in fact; strength may double or be halved.
Intuitive or rational, all thinking uses concepts expressed in words or other means. All animals with nervous systems use some form of language: body language is common, such as that of a dog begging for love or food; pictorial words used by male peacocks and other birds; music in human compositions or that of the songbirds; aromas in flowers and animal glands; taste sensations in fruits provided by plants to enlist animals in plant reproduction with the spreading of seeds and natural fertilizer. To deal with logical statements we use a language, which used to be with natural words until the nineteenth century, but is now with mathematical symbols.
Specialists in the law, physics, music, painting, medicine, biology, or economics have developed over some centuries their unique language systems for communicating within themselves and with others to do their work efficiently. Using words incomprehensible to laymen, physicians, attorneys, specialists use technical words to express the principles of their particular discipline and provide useful services. And each discipline has its own logic, which we cannot easily follow; thus we go to these practitioners with trust, expecting favorable results from their dealings. If we have obtained references, usually our trust is rewarded, but sometimes we are taken. Using intuition, most women and some men, may sense the deception behind the inscrutable language of a specialist.
To develop a language we need words or symbols; symbols are both shorter and more precise. Words represent concepts: the products of classifying sense data and thoughts. In concepts we have levels of content such as species, genus, or higher level groupings that hold more things. A genus is like a file the holds referents or things. For example, the word chair is a grouping or classification of all sorts of chairs, such as lounge chair, dining chair, beach chair, and it is a species. Chair belongs in the larger grouping of the genus furniture. Differentia is that property of a species in a genus which distinguishes the species from other members of the genus. For example, chair is distinguished from table because we sit on a chair. Furniture is a species in the genus of house furnishings. We define words or symbols with other words or symbols, called elemental definitions or primitives which we accept on faith. We are also free to create new meaning for an old word or coin a word to define something new, producing a stipulative definition. Again with elemental and stipulative words, we enter the realm of intuition.
A word is the expression of a concept, a response in the mind to a collection of impressions from outside the self or inside. As such, it is an organizing function of a bunch of things that have a common attribute, which is important, essential, or useful, called a consistent principle. Or, the principle of classification is something which differentiates our grouping from other collections of things. For example, chairs are a pieces of furniture on which we sit; similarly, beds are pieces of furniture on sit we sleep.
Couches are pieces of furniture on which we recline. We occasionally sleep on couches, or sit on them without reclining, which gives couches a fuzzy definition, not readily acceptable in logic. In biology, we have the duckbill platypus and the Florida manatee, which have bedeviled taxonomists since they were discovered.
Taxonomy continues somehow, because of the need for convenient classifications. To find and use things we need to classify them: to have a place for each item and put each in its place, as we do in files or in a library. In logic, we must classify things into groups that are mutually exclusive and jointly exhaustive. Mutually exclusive means the groupings must not overlap. For example, chairs cannot include tables in their group. Jointly exhaustive means our groupings must include all objects in our universe of discourse. We cannot leave any object out of the discussion which should normally be included. These rules are needed to avoid ambiguities and confusion, to communicate and to carry on arguments about things conclusively. In the real world, however, some items overlap and defy classification; other things like the platypus don’t seem to belong to any logical category, called outliers in statistics, they are like mistakes God has made in creation or anomalies which present opportunities for discovery and new theories, as intuition would have it.
Now, to organize words into sentences or statements, we establish a grammar or syntax: rules for making sentences. Any statement can be judged to be true or false by using rules of logic, inferences or syllogisms. For example, the rule of modus ponens, which says: B follows logically from A; A is true; therefore, B is true. Here is another inference rule: A is true, B is true; therefore, the combination of A and B is true. Some rules are called syllogisms. For example, the disjunctive syllogism says: A, or B is true; A is not true; therefore, B is true.
Another way of showing the truth of a complex logical statement is to employ truth tables. In a truth table we show all possibilities given certain conditions exhaustively. As a simple example: when is the combination A and B together true? In the table we show headings of A, B, and A+B (logical +). Then we indicate the possible value of A, of B, and the outcome under A+B, as true or false. When A is false, B is false, A+B is false. When A is false, B is true, A+B is false. When A is true, B is false, A+B is false. Finally, when A is true, B is true, A +B is true. Conclusion: A + B together is true only if both A and B are true. Who can argue against a truth table? Nobody who agrees with the definitions, the axioms involved, and the rules of reasoning we are using. It is hard to argue against the proof of a statement based on an exhaustive table of all possibilities. With modern computers we can do exhaustive tables for complex cases with some ease. If you are human, truth tables and logical rules must strike you as evidently true and you are ready to accept them and use them in your thinking, openly or subconsciously.
We all use some logical rules subconsciously in our behavior, a part of our human heritage. Sometimes, we also act without logic subconsciously, also part of our nature. We would do better if we try to apply logical principles consciously, training ourselves in their use at all times, until their use becomes automatic and subconscious. We would then behave more effectively in our relations with others and with natural forces. Sherlock Holmes, the fictional detective, applied induction, deduction, inference and scientific methods to find the culprits in a crime; and so do modern-day CSI sleuths.
In court logic, we allow what is true or false only; something which is partly true is excluded. When you testify in court, you are asked to tell the truth, the whole truth, and nothing but the truth. Sir, do you love your wife? Yes, and no, your Honor; I love some things about her and hate other things. Sir, you have to answer yes, or no. As to the whole truth, it is impossible to tell all about anything. Telling nothing but the truth leaves out a whole world of context that every case has. Still, we abide by these fictions in court and in logic, because they are convenient and allow us to get on with the business at hand, solving problems that arise. Was Kevorkian, “Dr. Death,” guilty of murder in assisted suicides, or innocent? He was both, something not acceptable in our current law practices. Faced with such dilemmas, it is easier for authorities to put someone in prison or in the streets, rather than face a situation of uncomfortable complexity.
It is hard dealing with complexity and we can be lost in a labyrinth of words trying to prove whether something is true or false. Today logicians and mathematicians like to talk about rigor in their proofs. They point out that in the past their colleagues proved theorems and other statements with insufficient rigor, and those proofs were fallacious: proofs given contained logical errors, causing their statements to go into rigor mortis instead of the truth. Rigor means stiffness in Latin. What we mean by rigor in logic is: exactness, precision, freedom from error, and a high level of completeness and correctness using a formal language, leading to a nearly total confidence in the truths we show. A formal language is a symbolic language capable of high precision and exactness, without ambiguity and vagueness. But there can be no perfect rigor in anything; perfect rigor is an asymptote which we can approach but never reach in our logical thinking.
To get perfect rigor, to be absolutely sure of the truth of statements, is the holy grail of thinkers. René Descartes found certainty in the statement, “I think, therefore I exist.” He also believed the existence of God was another certainty on which he could count. For most of us, as we approach total rigor we find that it brings about sound results we can use. In the end, we accept the rules of logic and their rigorous use, the assumptions we call axioms and the primitive definitions, because they help us communicate with other thinkers and with nature. Logic to a pragmatist is whatever works.
Fundamentally, we set up definitions, axioms, and rules in order to communicate with each other and after some effort to come to an agreement and mutual conclusion. Then we can cooperate. If not, we fight; and so do religious fanatics who deny reason, relying too much on intuition and revelation. There is, however, an alternative to fighting. We can agree to start with different assumptions from what is generally accepted as obviously true and see where such a course takes us. If we find the outcome desirable and useful we can accept it; if not, we can reject it, and start all over with different or the old assumptions to get together in thought. Either way, fighting is not necessary if we have respect for each other’s opinions, relying on common sense or logic.
Give me sure-footed logic instead of intuition, when I have to make a serious decision. Logic is a powerful tool in thinking more effectively. It may be possible to think without words, syntax, or rules of reasoning; some people speak of something called energy, responding to vibrations in life. Modern science and technology have not made their solid advances relying on mind vibrations, but with the use of logical and mathematical symbols and the application of sound reasoning. Intuition is fine, if it is backed up with logic and research of the facts. If not so endowed, you can call it fantasy, wishful thinking, or daydreaming. To have fun and to be creative, dreaming can light the way to exploration and adventure. Yes, lead with your right brain, but follow through with your left. We achieve our greatest successes in science, art, politics, or business by combining logic with intuition, the yin and yang, quite as the union of male and female produces life. In our bodies, our right and left brains are connected with the corpus callosum. So in the mind, logic is coupled with intuition and emotion. Humor, a play of the intellect, produces a strong emotion, laughter; laughter stimulates the intellect. Similarly, logic can control emotions and intuition, and intuition inspires logic. Moreover, the mind affects the body, and vice versa. This is why cognitive therapy works in controlling obsessive compulsive behaviors, phobias, and depression, making the body and mind healthier, and more adaptable in the stresses of living.