Piero Scaruffi(Copyright © 2013 Piero Scaruffi | Legal restrictions )
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.
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