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mathematical logic

n. (context logic English) A subfield of logic and mathematics consisting of both the mathematical study of logic and the application of this study to other areas of mathematics, exemplified by questions on the expressive power of formal logics and the deductive power of formal proof systems.

mathematical logic

n. any logical system that abstracts the form of statements away from their content in order to establish abstract criteria of consistency and validity [syn: symbolic logic, formal logic]

Mathematical logic

Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.

Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory. These areas share basic results on logic, particularly first-order logic, and definability. In computer science (particularly in the ACM Classification) mathematical logic encompasses additional topics not detailed in this article; see Logic in computer science for those.

Since its inception, mathematical logic has both contributed to, and has been motivated by, the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and analysis. In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory. Contemporary work in the foundations of mathematics often focuses on establishing which parts of mathematics can be formalized in particular formal systems (as in reverse mathematics) rather than trying to find theories in which all of mathematics can be developed.

Usage examples of "mathematical logic".

But, in developing it, its protagonists have moved away from how biological brains and psychological minds might work and instead concentrated on solving problems embedded in the silicon of computer chips and in mathematical logic - an approach which may produce bigger and better machines, but has become entirely indifferent to their relationship with the biological systems they were once attempting to model.

It seems to be the rule for top people to come to mathematical logic only after considerable work in other areas.

I suppose then, youll be applying mathematical logic to our problems on the Council?

For on a planet as technologically advanced as this, the mathematical logic of the computer would have long since taken over the menial administration from the highly sensitive Perseans.

They rest, not upon the unchanging verities of mathematical logic, but upon observable facts of the real world.

He had read of the pure, mathematical logic that formed the essence of his god, and understood it to be the ultimate goal he’.

If there is to be parallelism, it is easy to prove by mathematical logic that the causation in physical and psychical matters must be of the same sort, and it is impossible that mnemic causation should exist in psychology but not in physics.

If one white life was worth a dozen blacks, then obviously there was no mathematical logic in arresting a black for Mary Phagan’.

If one white life was worth a dozen blacks, then obviously there was no mathematical logic in arresting a black for Mary Phagan’s murder.

Had the guards been able to see and think and react, they would have discovered that the clouds were not clouds at all, but two dense, pulsing masses with a mathematical logic about them.

With their penchant for linguistic analysis, mathematical logic, and scientific empiricism, they have aligned philosophy with the mystique of science, have begun to transform the philosopher's library or mountain retreat into something nearer to a laboratory, and, as William Earle said, would come to work in white coats if they thought they could get away with it.

It seemed intuitively right that the fellow with the fake hair and the others should have been willing to sacrifice their lives to save one unrelated comrade, even though it defied mathematical logic.