I believe that Classical AI uses deductive thought processes. For example, given as a set of constraints, deduce a conclusion.
What are some examples of successfully applying Classical AI to real-world problems?
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The term classical AI refers to the concept of intelligence that was broadly accepted after the Dartmouth Conference and basically refers to a kind of intelligence that is strongly symbolic and oriented to logic and language processing. One basic point is the duality body vs. mind. It's in this period that the mind starts to be compared with computer software.
Deep Blue, whose aim in life was to be the master of chess, ruling over the (not-so) intelligent mankind
Expert systems, which are computer programs that strongly rely on the type of constraints and conclusions that you refer to, in order to accomplish feats of apparent intelligence
Fuzzy logic, which is an extension of multivalued logic, but with continuous values instead of discrete ones
Note that in all cases the hardware (once compared with the body) does not play any role: Intelligence is abstract and independent from the material world.