On the historical development of logic

From Aristotle to the first artificial intelligence (“AI”) program.

Logic is a language for reasoning. The word “logic” comes from the greek term “logikos”, which means “word”, but has deeper resonances of intelligibility or ordered thought. Logic is formally defined as the science which addresses the validity of inference and demonstration. The deeper understanding of logic, however, is more nuanced and reveals its enduring appeal to philosophers and its influence on computer science, and ultimately the field of artificial intelligence.

The origins of logic can be traced back to the works of Aristotle who lived during the fourth century BC. His thoughts on logic are spread across six works collectively known as the “Organon”.

Aristotle wanted to establish rules which would allow Greek citizens to distinguish arguments which are formally valid and correct from those which are invalid and therefore wrong. Two fundamental laws of Aristotelian logic are the law of noncontradiction and the law of the excluded middle.

The law of noncontradiction states that no proposition can be both true and false and the law of the excluded middle says that a proposition must be either true or false.

Aristotle designed his formal logic to be able to determine the validity of an argument regardless of the matter being discussed. The engine of Aristotle’s formal logic was the syllogism — a form of argument with two premises and a conclusion such as “All men are mortal, Socrates is a man, therefore Socrates is mortal”

Aristotle’s logic, though foundational, was not by modern standards a robust system, and over time it began to exhibit serious formal defects and an over reliance on the deductive syllogism

The Stoics contribution to logic went virtually unnoticed for centuries until medieval philosophers such as Albert of Saxony and Pohl of Venice concentrated on shoring up Aristotelian logic and making incremental contributions of their own.

However, the advance of science demanded new systems of thought and the English philosopher Francis Bacon formulated a system that challenged the orthodoxy of Aristotle’s Organon.

Originally Bacon was presenting himself as writing an updated Organon on a new logic for interrogating the world. He argued that the old logic was useful for presenting what it is that you already knew but it was not helpful for actually discovering new things. What Bacon was suggesting involved beginning by going out into the world and collecting facts, organizing them into tables and being able to draw theoretical conclusions from what it is that we organized into these tables.

What Bacon wanted to do was give us a rational way of organising experience and interrogating nature so that with this combination of reason and experience we could make progress.

In the 19th century philosophers began actively applying the structures of mathematics to logic. This mathematical turn marked the beginning of what we would recognize today as modern logic, and it really exploded after 1854 when George Boole, a British mathematician who was a very important contributor to the theory of differential equations and the creator of the calculus of finite differences, created what is called the “Algebra Of Classes”.

Boole was a self-taught mathematician from a working-class background and expanded on the earlier work of the German philosopher Gottfried Leibniz to create the first algebra of logic.

He used algebraic symbols to express hypothetical propositions.His arguments would be taken up and improved by the mathematician Gottlob Frege giving rise to modern propositional logic.

The decisive breakthrough really came in the 19th century and although other people were seeing how you could improve on Aristotle, it was really Frege who made the decisive break and introduced the modern era of logic.

Gottlob Frege a was an unknown, somewhat obscure scholar in his own day, and a mathematician who took it upon himself to demonstrate that the laws of arithmetic were ultimately reducible to logic — a position known as “logicism” — and along the way he made the greatest single contribution to the study of logic in the history of the discipline.

He was the first to identify the distinction between the logical axioms and rules which are necessary to make a deduction. He took the first steps towards making logic axiomatic. He created the propositional function and he refined Aristotle’s concept of quantification.

Frege’s monumental work “The foundations of arithmetic” profoundly influenced the English philosopher Bertrand Russell who wrote Principia Mathematica with his mentor Alfred North Whitehead.

Next to Aristotle’s Organon, Principia Mathematica is undoubtedly the most influential book on logic ever written. In it Russell took up the development of logicism that Frege had set in motion.

German mathematician David Hilbert was an avid student of the Principia Mathematica and in the early 1920s he formulated a proposal for formalizing mathematics in axiomatic form.

This proposal came to be known as Hilbert’s program and brought logic into the realm of language.

Hilbert’s program was to exert a defining influence on the czech mathematician and logician Kurt Godel who argued that all logical systems are essentially incomplete.

Godel proved that in a formal axiomatic system you simply can’t have both consistency and completeness. Either you can’t prove everything within a particular system or you couldn’t guarantee that there could not be any contradictions.

As logic evolved in the 20th century one area where it had tremendous impact is in the development of computer science.

In 1847, George Boole, who was discussed earlier, developed a theory of binary logic and arithmetic known as Boolean algebra. Boolean algebra is a binary algebra,in which a proposition can have the value true or false only. Boolean logic provided a formal language that allowed mathematicians to explicitly describe the logic proposed by Aristotle in a precise and unambiguous way.

The information processing in contemporary computers employs the binary principles of boolean algebra. People who were designing computers in the 1940’s, especially Alan Turing, were interested in the principles of artificial intelligence.

A Turing machine is the abstract representation of how a computer works and is named for the the same Alan Turing. The Turing machine is an invention which tries to formalize the idea of a computer. It is the idea of a mechanical procedure which will eventuate in a in a result.

Alan Turing is considered to be one of the fathers of artificial intelligence. He wrote a paper describing what is now known as the “Turing Test”. The test involves a person asking questions via terminal to both a person and a intelligent machine. He states that if the person could not tell the machine apart from the person, the machine was somewhat intelligent.

In 1955, Newell, Simon and Shaw from Carnegie Institute of Technology developed “Logic Theorist”, considered by many to be the first artificial intelligence program. The program was a decision tree system for finding proofs for mathematical theorems. The impact that the Logic Theorist made on both the public and the field of artificial intelligence has made it a crucial stepping stone in developing the artificial intelligence field.

To conclude, one can see that we can almost draw a direct line from the original teachings and questions of Aristotle to the development of machines and artificial intelligence programs. What has been very interesting to come to understand is the clear link through history of each of the main contributors to the field and how their knowledge compounded due to the work of the previous. We have seen remarkable advances in the field of AI since the first program was published but knowing that solid logic underlies the entire field gives more comfort as we face a more uncertain future which will largely be shaped by artificial intelligence.