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Artificial Intelligence, as a field of research, was basically created by a man named John McCarthy, who came up with the name and convened the first academic workshop on the subject (in Dartmouth, in 1956). John McCarthy (who is also known as the creator of Lisp) was the student of Alonzo Church, who gave his name to the Church-Turing Thesis and is remembered for his description of the Lambda calculus, a model of computation equivalent to universal Turing machines. These are all logicians, btw, mathematicians steeped in the work of the founders of the field of mathematical logic as it was developed mainly in the 1920's by Frege, Hillbert, Gödel, Russel and Whitehead, and others (of course there was much work done before the 1920's, e.g. by Boole, Leibnitz... Aristotle... but it only really got properly systematised in the 1920's).

So it makes sense that the field of AI was closely tied with logic, for a long time: because its founder was a logician.

There were different strains of research in subjects and with goals adjacent to and overlapping with AI, as John McCarthy laid them down. For example, statistical pattern recognition or cybernetics (the latter intuition I owe to a comment by Carl Hewitt here, on HN). And of course we should not forget that the first "artificial neuron", described by Pitts and McCulloch in 1938 (long before everything else of the sort) was a propositional logic circuit, because at the time the models of human cognition were based on the propositional calculus.

The problem with McCarthy's programme, of logic-based AI, is that logic is full of incompleteness and NP-hardness results, and the more expressive power you demand from a logic language, the more of those endless pits of undecidability and despair you fall down. So work in the field advanced veerrryyy slooowwllyyy and of course much of it was the usual kind of incremental silly-walk stuff I point out in the comment above. To make matters worse, while significant academic results drip-dripped slowly and real-world applications were few and far between, you had prominent researchers overpromising and launching themselves into wild flights of fancy with abandon based on very early actual results (certainly not McCarthy! and I don't want to point fingers but cough Marvin Minsky cough).

Then this wretched dog by the name of Sir James Lighthill :spits: was commissioned by the British Science Research Council, the holders of the Spigot of the Tap of Funding, to write a report on the progress of AI reserach and the report was full of misunderstandings and misrepresentations and the Spigot was turned off and logic-based AI research died. Not just logic-based AI- Lighthill's :spits: report is the reason the UK doesn't have a robotics sector to speak of today. Then, while all this was happening in the US and Europe, the Japanese caught the bug of logic-based AI and they decided to spend a bunch of money to corner the market for computers like they had with electrical devices and automobiles, and they instituted something called the Fifth Generation Computer project (special-purpose computer architectures to run logic programming languages). That flopped and it took with it one of the few branches of classical AI that had actually delivered, automated theorem proving.

The first link I posed in the comment above is to a televised debate between Lighthill on the one side and on the other side John McCarthy, Donald Michie (dean of AI in the UK) and er, some other guy from cognitive science in the US, I always forget his name and I'm probably doing him a disservice. You may want to watch that if you are curious about what happened. Pencil pushers without an inkling of understanding killed logic-based AI research is what happened. That was a bit like the comet that killed the dinosaurs and gave the mammals a chance. Meaning those adjacent to AI research directions, like probabilistic reasoning, connectionism and pattern recognition found an opening and they took their chance and dominated research since then. I am well aware of the connotations of my own metaphor. What can I say? Dinosaurs are cool. Ask any five-year old to explain why.