Symbolic logic is the basis for electronic logic circuits and hence digital computing.
The German mathematician and philosopher Gottfried Wilhelm von Liebniz laid the foundations of symbolic logic as a partnership between mathematics and Classical Logic. It was Aristotle the Greek philosopher that laid down the three basic principles of classical logic and so it is often referred to as Aristotelian logic.
The three principles are:
1. The principle of identity.
A thing is itself: A is A
2. The principle of the excluded middle.
A proposition is either true or false: either A or not A
3. The principle of contradiction.
No proposition can be true and false: A cannot be both A and not A
This classical logic was built upon by Liebniz who was trying to find a language where errors in thinking would be equivalent to arithmetical errors.
Later the English mathematician and logician George Boole (1815-1864) devised a system for logical reasoning now commonly referred to as Boolean algebra. As with many things in mathematics his system had to wait for a century before it found an application. In 1938 an American postgraduate student Claude Shannon applied Boolean algebra to the design of telephone switching networks. From this point on Boolean algebra became the basis for all electronic network design and analysis of binary systems.
Bertrand Russell and his former teacher Alfred North Whitehead in their three volume Principia Mathematica tried to derive mathematics from self-evident logical principles. They did not completely reach their goal, but it was an important work in the development of logic and mathematics.
In 1900 the German Mathematician David Hilbert suggested 23 Important Mathematical Problems to be targeted during the 20th century. Kurt Gödel and his Incompleteness Theorem solved one of Hilbert's Problems in 1931. Gödel stated that any system adequate to describe arithmetic must contain statements that can neither be proved nor disproved.