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**These are excerpts and elaborations from my book "The Nature of Consciousness"**

The history of Logic starts
with the Greeks. Pythagoras’s' theorem stands as a paradigm that would influence all of western
science: a relationship between physical quantities that is both abstract and
eternal. It talks about a triangle, a purely abstract figure which can be applied
to many practical cases, and it states a fact that is always true, regardless
of the weather, the season, the millennium. Euclides built the first system of Logic
when he wrote his "Elements" (around 350 BC). From just five axioms (there
is a straight line between two points, a straight line can be extended to
infinite, there is a circle with any given center and radius, all right angles
are equal, two parallel lines never meet), he could deduct a wealth of theorems
by applying the same inference rules over and over again. Then, of course, Aristotle wrote his "Organon"
and showed that we employ more than one “syllogism” (more than one kind of
reasoning). Although he listed several kinds of logical thinking, only three
were widely known and eventually became the foundations of Logic. The law of
the excluded middle states that an object cannot have both a property and the
opposite property (i cannot be both rich and poor). “Modus ponens” states that:
if all B's are C's and all A's are B's, then all A's are C's. “Modus tollens”
states that: if all B's are C's and no A's are C's then no A's are B's. After centuries of Roman
indifference and of medieval neglect, Logic resumed its course. Studies on
logic, from the "Dialectica" of the French philosopher Pierre Abelard (1100 AD) to the
"Introductiones Logicam" of the English philosopher William of
Shyreswood (1200 AD), had actually been
studies on language. Logic was truly reborn with the "Summa Totius
Logicae" of another Englishman, William Ockham (1300 AD), who discussed how
people reason and learn. Three centuries later Francis Bacon’s "Novum Organum" (1620) and Rene' Descartes’ "Discours de la Methode" (1937) hailed the analytic method
over the dialectic method and therefore started the age of modern Science. The
German mathematician Gottfried Leibniz emphasized the fact that
reasoning requires symbols in his "De Arte Combinatoria" (1676) and
co-discovered calculus with Isaac Newton. In 1761 the Swiss mathematician Leonhard Euler showed how to do symbolic logic
with diagrams. The British philosopher John Stuart Mill tried to apply logic outside of
science in his "System of Logic" (1843). Non-numerical algebra was
formalized by the British mathematician Augustus De Morgan in "The Foundations of
Algebra" (1844). Another Englishman, George
Boole, was so fascinated by the progress of symbolic logic that in "The
Laws Of Thought" (1854) he claimed that logic could be applied to thought
in general: instead of solving mathematical problems such as equations, one
would be able to derive a logical argument. Boole's ideas evolved into
"propositional logic" and then "predicate logic", which
fascinated philosopher-mathematicians such as Gottlob Frege in Germany, Giuseppe Peano in Italy, Charles Sanders Peirce in the United States, and
Bertrand Russell in Britain. Thought became more and more formalized. Frege's "Foundations of Arithmetic" (1884) and "Sense and
meaning" (1892), Peano's "Arithmetices Principia Nova Methodo Exposita" (1889), and
Russell's "Principia Mathematica" (1903) moved philosophy towards an
"axiomatization" of thought. Back to the beginning of the chapter "Machine Intelligence" | Back to the index of all chapters |