
Just as the ancient Greeks were the originators of the tradition of Western logical thought, they were the first people to study the processes of logical thought in any depth. Their work is summed up in Aristotle\'s treatises on logic (4th century Â BCE). They were hampered by a lack of any systematic notation for the processes of logic, and so relied on the use of language to explain what they were doing. This approach soon becomes extremely complex and unwieldy, but Greek logic was the pinnacle of the art for over 2,000 years. Indeed, many of the greatest medieval minds failed even to understand the work of Aristotle, let alone to build on it.
Further great advances of logic only began in the late 19th century. Most mathematical reasoning up to that time was informal, not really based on even the principles of logic that were known. The work of Gottlob Frege (1848  1925), though widely ignored until championed by , Bertrand Russell (1872  1970), brought about a revolution in the whole field of mathematics. Like many great advances in mathematics, Frege\'s was the result of an improvement in notation; he invented symbolic logic, the use of symbols to represent ideas. Frege\'s symbolic logic (and its cousin, Boolean logic) meant that mathematical ideas could be precisely written down for the first time, without being dependent on the inconsistencies and vaguenesses of language. (For example, notational vagueness in Newton\'s time makes it sometimes difficult today to have a precise idea of what he really meant in his various mathematical works.)
Once the significance of Frege\'s ideas had been grasped, mathematicians were not slow to apply them in as many areas as possible. The greatest exponent of this was Frege\'s popularizer, Bertrand Russell. Today, the use of symbols in mathematics means that what is being said is crystal clear (at least to those initiated in the subject), with (if used properly) no possibility of ambiguity; it has, however, made the subject among the most incomprehensible of all human knowledge to the layperson.
Today, that logic itself has limitations is more clearly understood than it was in the first flush of excitement at Frege\'s work; in some cases, the best that can be hoped for is consistency (that is, absence of contradictions) rather than truth. SMcL
See also GÃ¶del\'s incompleteness theorem.Further reading I.M. Copi, Introduction to Logic. 
