Logic and the mind

This article discusses the relationship between the formal logic and the mind.

For a long time people believed that intelligence is equivalent to conceptual understanding and reasoning. A part of this belief was that the mind works according to logic. Over the course of the two millennia since Aristotle, many people have identified the power of intelligence with logic. Founders of artificial intelligence in the 1950s and 60s believed that by relying on rules of logic they would soon develop computers with intelligence far exceeding the human mind.

Aristotle invented logic. However, Aristotle did not appear to think that the mind works logically; he used logic as a supreme way of argument, not as a theory of the mind. This can be deduced from Aristotelian writings, for example, in “Rhetoric for Alexander” Aristotle lists dozens of topics on which Alexander had to speak publicly. For each topic, Aristotle identified two opposite positions (e.g. make peace or declare war; use torture or don’t for extracting the truth, etc.). For each of the opposite positions, Aristotle gives logical arguments, to argue either way. Therefore, for Aristotle, logic is a tool to express previously made decisions, not the mechanism of the mind. Logic can only provide deductions from first principles, but cannot indicate what the first principles should be.

To explain the mind, Aristotle developed a theory of Forms. But during the following centuries the subtleties of Aristotelian thoughts were not always understood. With the advent of science, the idea that intelligence is equivalent to logic was gaining grounds. In the 19th century mathematicians turned their attention to logic. George Boole noted what he thought was not completed in Aristotle’s theory. The foundation of logic, since Aristotle, was a law of excluded middle (or excluded third): every statement is either true or false, any middle alternative is excluded. But Aristotle also emphasized that logical statements should not be formulated too precisely (say, a measure of wheat should not be defined with an accuracy of a single grain), that language implies the adequate accuracy, and everyone has his mind to decide what is reasonable.
Boole thought that the contradiction between exactness of the law of excluded middle and vagueness of language should be corrected. A new branch of mathematics, formal logic was born. Prominent mathematicians contributed to the development of formal logic, including George Boole, Gottlob Frege, Georg Cantor, Bertrand Russell, David Hilbert, and Kurt Gödel. Logicians ‘threw away’ uncertainty of language and founded formal mathematical logic based on the law of excluded middle. Most of physicists today agree that exactness of mathematics is an inseparable part of physics, but formal logicians went beyond this. Hilbert developed an approach named formalism, which rejected the intuition as a part of scientific investigation and thought to define scientific objects formally in terms of axioms or rules. Hilbert was sure that his logical theory also described mechanisms of the mind: “The fundamental idea of my proof theory is none other than to describe the activity of our understanding, to make a protocol of the rules according to which our thinking actually proceeds.” In the 1900 he formulated famous Entscheidungsproblem: to define a set of logical rules sufficient to prove all past and future mathematical theorems. This entailed formalization of scientific creativity and the entire human thinking.

In 1902 Bertrand Russell exposed an inconsistency of formal procedures by introducing a set R as follows: R is a set of all sets which are not members of themselves. Is R a member of R? If it is not, then it should belong to R according to the definition, but if R is a member of R, this contradicts the definition. Thus, either way we get a contradiction. This became known as the Russell's paradox. Its joking formulation is as follows: A barber shaves everybody who does not shave himself. Does the barber shave himself? Either answer to this question (yes or no) leads to a contradiction. This barber, like Russell’s set can be logically defined, but cannot exist. For the next 25 years mathematicians where trying to develop a self-consistent mathematical logic, free from the paradoxes of this type. But, in 1931, Gödel has proved that it is not possible, formal logic was inconsistent, self-contradictory.

Belief in logic has deep psychological roots related to functioning of human mind. A major part of any perception and cognition process is not accessible to consciousness directly. We are conscious about the ‘final states’ of these processes, which are perceived by our minds as ‘concepts’ approximately obeying formal logic. For this reason prominent mathematicians believed in logic. Even after the Gödelian proof, founders of artificial intelligence still insisted that logic is sufficient to explain working of the mind.

New directions in artificial intelligence, especially computational intelligence, explore alternatives to formal logic as foundations of the mind. For example, Dynamic logic holds logic not as a fundamental mechanism of the mind, but the result of mind’s operations. Dynamic logic gives a mathematical explanation of how logic appears from illogical states of the mind.

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