## Foundations of Mathematics## Mathematical and Formal Logic |

## Who was responsible for expanding ideas of predicate calculus? |

The German philosopher and mathematician Friedrich Ludwig Gottlob Frege (1848–1925) presented a way to rearrange sentences to make their logic clearer and to show how the sentences relate in various ways in his 1879 treatise *Begriffsschrift* (German for “Concept Script”). Before Frege began his work, formal logic (in the form of propositional or sentential calculus; see above) used such words as “and”, “or”, and so on, but the method could not break the sentences down into smaller parts. For example, formal logic could not show how the sentence, “Cats are animals,” actually entails, “parts of cats are parts of animals.”

Frege added words such as “all,” “some,” and “none,” using variables and quantifiers to rearrange the sentences, therefore making them more precise in their meaning. He also developed two of the major qualifier symbols for predicate calculus, the upside-down A and the backward E .

Frege’s work began modern logical theory, even though his work was considered defective in several respects and was considered awkward to use. By the 1910s and 1920s, Frege’s system was modified and streamlined into today’s predicate calculus.