Wednesday, May 11, 2011

Why is Logic Logical?

For years I have puzzled over the validity of logic. Why does one idea compel another? What is the nature of that compulsion? For example, why is the “law” of the excluded middle true: A thing cannot be, and not be, simultaneously. A equals A, and A does not equal not-A. There is nothing “in the middle” between A and not-A. That’s what Aristotle said, and it’s been true ever since. But is it only true by convention, or does logic follow some natural laws, either laws of the world or laws of the mind?

In day-to-day experience, the middle is not excluded. There is the luxury car and there is the economy car, and plenty of choices in between. There is one dollar, and no dollars, and fifty cents in between. There are guilty and innocent, and shades in between. So why is it true that there is nothing in between A and not-A?

At first consideration it seems that the difference is that the law of the excluded middle is about existence. It says a thing cannot BE and not-BE simultaneously. That’s about what IS. By contrast, everyday examples are all about degrees of qualities that all exist. The economy car exists, and so does the luxury car, and all the ones in between. The qualities of price and value vary along some (abstract) dimension, but all of it exists.

But we cannot say that THIS particular car (not in the abstract, but this one right here) exists and doesn’t exist at the same time. Why not? Because that would be illogical. But why? That is the question.

Is it a matter of abstraction? In algebra, which is very abstract, we all agree that A cannot be equal to not-A. that is uncontroversial. But we refuse to say the same about a particular stone.

The difference seems to boil down to what exists and doesn’t exist. But how is that determined? How do we know what exists and doesn’t exist? Do flying elephants exist? Well, yes and no. It depends on what you mean by “exist.” They exist in animated movies and in the minds of millions of children, but not on game reserves in Africa.

So do we restrict the scope of the question to things that exist physically, not mentally? That would seem an arbitrary restriction. Anyway, it would make algebra and logic, and science, higher mathematics, and law, and much else, not susceptible to the law of the excluded middle, and by extension, not susceptible to logic and reason. The purpose of logic is to bring the order of reason to abstraction. So it can’t be right to exclude mental abstraction from logic.

Besides, even in the so-called physical world, there are counterexamples to the law of the excluded middle. Light exists as light waves and as photons, simultaneously. That seems to violate the rule, doesn’t it? Hawking radiation around a black hole exists and doesn’t exist at the same time. There aren’t too many examples like that however, and in general, we tend to quarantine the principles of relativity theory when we consider logic in general.

I think the answer lies not in abstraction itself, but in the human capacity for discrimination. When we are ignorant of a thing or a topic, we cannot perceive distinctions. Someone who does not know wine literally cannot distinguish between cabernet and merlot. A person who does not know philosophy cannot tell the difference between Kantian and Cartesian ideas. Someone who does not know airplanes cannot tell if they are about to board a Boeing or an Airbus. I remember once looking over a locksmith’s shoulder as he fixed a lock on my door. “Look at that!” he exclaimed in disgust when he took off the outer cowling to expose the insides of the lock. “The quality these days is just disgusting.” I saw nothing but a jumble of metal parts. I wasn’t disgusted because I didn’t know what I was supposed to be seeing. I failed to discriminate what he did.

After training or other experience however, it becomes possible to discriminate parts from wholes and parts from other parts. Then a person can discuss the merits of cabernet and merlot, or well-made from poorly-made lock mechanisms. It works the same in the world of abstract ideas. It takes instruction or experience to discriminate democracy from authoritarianism and A from not-A.

Simple sensory discriminations enable abstraction. A door lock is a door lock, but a well-made lock is an abstraction, it is a kind of lock, or a category of locks. Once the discrimination has been made and conceptualized, multiple instances of a like kind can be grouped into an abstract category.

Thus “dog” is a category of animals, but that abstraction was developed only after I became able to discriminate dogs from cats, and from other kinds of animals. In turn, that discrimination was explicitly taught by parents and teachers, who dwell obsessively on helping children discriminate categories of animals. Why that is considered important is a separate mystery. Finally, there must have been some sensory discrimination at the bottom, by which I learned to identify my dog, a particular, concrete, sensory dog, as a “dog” and discriminated it from myself. So the sequence of abstraction goes from a particular, sensory being that exists right now in my presence, to a category of all such animals, which are then further discriminated and contrasted with other animals, and so on up the chain of abstraction.

The sequence of discrimination, conceptualization, and categorization is so automatic that I suspect it is a faculty of the human mind. Teachers teach us how to discriminate and identify, and categorize dogs, cats, forms of government, and much else, but nobody teaches us how to discriminate in the first place. We just do it.

Other animals discriminate in a similar way. In classical conditioning, a type of learning, the dog learns to salivate when the bell rings. Why? We say the dog has “associated” the bell with forthcoming food. However the dog first had to discriminate the bell from the general background noise, and also the occurrence of food from other events, and also the fact that the bell sounds just before food appears. Those are all sensory discriminations that the dog learns fairly easily, without the benefit of language. As far as we know the dog does not conceptualize any of it, but does manage somehow to generalize a more-or-less abstract category about what we would call the conditioned stimulus, because if a buzzer is sounded instead of a bell, the dog salivates in the same way he would to the bell. He obviously has an abstract category of sorts.

I’m not aware of any animal species with a nervous system that is not susceptible to classical conditioning, so I would have to conclude that discrimination and abstraction are built into the architecture of animal neurology.

Does that answer the question of what compels one idea to follow another and why logic is logical? Partially, it does. But the rules of logic are themselves so abstract that it is difficult to believe they are neurological manifestations. Suppose a proposal says that if p exists, then q will always occur. But if we look and find that q did not occur, what is the only logical conclusion? It has to be that p does not exist. This rule is the absolute foundation of reasoning in science and statistics. What makes it valid?

According to the analysis given here, that rule of logic, called modus tollens, is valid because it is an abstraction of sensory, bodily experience that many humans have discriminated and agreed is universal. We have all observed that if the bulb inside the refrigerator is working, then when you open the door, there will be light. If you do not see light, the conclusion is that the bulb is not working. Enough people have had experience like this, so that as a community, we have agreed the relationships involved are worthy of becoming a “law,” the law of modus tollens. It’s logical because we all say it is, not because of neurology.

The implication of this finding is that reason compels one idea to follow from another because of generalization of discriminations that many people have similarly made and conceptualized and categorized. The validity of logic is a social construct, not a natural phenomenon.

So what are we to make of the situation where people do not agree? Different groups insist that their god and only their god exists. Is there any concrete sensory discrimination at the bottom of those abstractions? I would say, no, and virtually all scientists would agree with me. Are there neurological differences supporting the abstractions? No. The human nervous system and brain is 99.999% similar across individuals.

But are there discriminations among abstract ideas beneath the disagreements? Of course there are. Different groups have different ideas about history, justice, virtue, beauty, and many other abstract categories, and they assiduously teach these discriminations to their children. Higher abstractions are based on discriminations made among lower abstractions and it is around these higher abstractions that wars are fought. Fundamentally though, the mid-level abstractions upon which they are based do not rest upon sensory discriminations. The validity of logic in the abstract realms is socially constructed.

At the bottom we are all the same kind of animal and make the same kinds of sensory discriminations and the same kinds of basic abstractions. It is only our teachers that guide us to abstractions among the abstractions, and therefore to differences we will kill for. Anybody can discriminate a brown skin from a white skin, narrow eyes from round eyes, male from female, but what those differences mean must be taught to us. There is no universal sensory or neurological basis, and therefore no intrinsic rationality that justifies what our teachers make of those differences. Whether my god or your god is the true god, is essentially culturally constructed, and we would say, “not logical.”

Ideas compel other ideas then, not because there is some intrinsic validity to the rules of logic that make it so, but only for two reasons.

One, because concrete, sensory discriminations that anyone, even a dog, can make, seem universal, as in classical conditioning. Red is different from blue, and we all agree on that, regardless of culture. Therefore it is “logical” to insist that Red cannot be Blue and vice versa.

And Two, logic is logical because the teachers in a cultural tradition decide, based on contingent values (that is, arbitrarily), that some abstract ideas “should” compel other abstract ideas. That compulsion is valid inasmuch as everybody lives in a culture and nobody can live outside of culture, so nobody is immune from cultural values. So if “The Bible is the word of God,” it follows that the Biblical God is the “correct” God. That is cultural logic.

These two kinds of logic are both valid, but for different reasons.