Mathematical logic got its first great boost forward in the 19th Century, though at the time there was little practical use to it. The great mathematicians on the continent were working trying to understand magnetism and its implications to physics, but a hearty band of British oddballs decided to revolutionize the study of logic.
George Boole is considered the originator of modern mathematical logic, so much so that the field is called Boolean Algebra. He wrote his important treatise The Laws of Thought in the 1850s.
Augustus de Morgan is another important pioneer. The ways to distribute a not sign ~ through parentheses are called de Morgan's laws. (v stands for the or operator and the ^ is the and operator.)
~(p v q) = ~p ^ ~q
~(p ^ q) = ~p v ~q
He was one of the first professors at University College London, the first major school in Great Britain that accepted students who were not Church of England, which meant that Catholics, Jews, protestants of denominations other that Church of England and those who professed no faith whatsoever could get a first class education in Great Britain due to deMorgan and a few others dedicated to education and religious freedom.
Charles Babbage was another British logician, and he wanted to take mathematical logic to the next step. He designed the world's first mechanical computers, the Difference Engine and the Analytical Engine, both of which were designed to run on steam. Problems arose when trying to build the machines and neither was ever completed.
The literary genre known as steampunk, a derivative of cyberpunk, is based on the "what if" world of Babbage actually succeeding at making a computer.
This did not stop Countess Ada Lovelace from designing programs for Babbage's machines. Besides the acclaim for her work in logic, she was the only legitimate daughter of the famed British poet, libertine and addict Lord Byron, who was as dedicated to illogic as his daughter was devoted to its opposite. The Countess Lovelace was educated by de Morgan and is given credit as the first computer programmer. The language Ada is named after her.
The best known of the British eccentrics fascinated with logic in the 19th Century was Charles Dodgson a.k.a. Lewis Carroll. While still famous for Alice in Wonderland, he was also a mathematician, clergyman and photographer, and enjoyed putting together logic puzzles based on the ideas of syllogism using silly but logical statements. For example:
(a) No ducks waltz.
(b) No officers ever decline to waltz.
(c) All my poultry are ducks.
Therefore (d) None of my poultry are officers.
Makes sense to me. Sort of.