Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics.
Although the layperson may think that mathematical logic is the logic of mathematics, the truth is rather that it more closely resembles the mathematics of logic. It comprises those parts of logic that can be modelled mathematically. Earlier appellations were symbolic logic (as opposed to philosophical logic); and metamathematics, which is now restricted as a term to some aspects of proof theory.
Mathematical logic was the name given by Peano to what is also known as symbolic logic. In essentials, it is still the logic of Aristotle, but from the point of view of notation it is written as a branch of abstract algebra.
Attempts to treat the operations of formal logic in a symbolic or algebraic way were made by some of the more philosophical mathematicians, such as Leibniz and Lambert; but their labors remained little known and isolated. It was George Boole and then Augustus De Morgan, in the middle of the nineteenth century, who presented a systematic mathematical (of course non-quantitative) way of regarding logic. The traditional, Aristotelian doctrine of logic was reformed and completed; and out of it developed an adequate instrument for investigating the fundamental concepts of mathematics. It would be misleading to say that the foundational controversies that were alive in the period 1900-1925 have all been settled; but philosophy of mathematics was greatly clarified by the 'new' logic.
While the traditional development of logic (see list of topics in logic) put heavy emphasis on forms of arguments, the attitude of current mathematical logic might be summed up as the combinatorial study of content. This covers both the syntactic (for example, sending a string from a formal language to a compiler program to write it as sequence of machine instructions), and the semantic (constructing specific models or whole sets of them, in model theory).
Topics in mathematical logic
The main areas of mathematical logic include model theory, proof theory and recursion theory. Axiomatic set theory is sometimes considered too. There are many overlaps with computer science, since many early pioneers in computer science, such as Alan Turing, were mathematicians and logicians.
The Curry-Howard isomorphism between proofs and programs relates to proof theory; intuitionistic logic and linear logic are significant here. Calculi such as the lambda calculus and combinatory logic are nowadays studied mainly as idealized programming languages.
Some fundamental results
Some important results are:
- Putative proofs of universal validity of first-order formulas can be checked for validity, algorithmically. In technical language, the set of proofs is primitive recursive. Essentially, this is Gödel's completeness theorem, although that theorem is usually stated in a way that does not make it obvious that it has anything to do with algorithms.
- The set of valid first-order formulas is not computable, i.e., there is no algorithm for checking for universal validity. There is, however, an algorithm that behaves as follows: Given a first-order formula as its input, the algorithm eventually halts if the formula is universally valid, and runs forever otherwise. If the algorithm has been running for a trillion years, the answer remains unknown. In other words, this set is "recursively enumerable", or, as it is sometimes more suggestively put, "semi-decidable".
- The set of all universally valid second-order formulas is not even recursively enumerable. This is a consequence of Gödel's incompleteness theorem.
- Cut-elimination in sequent calculus.
- A. S. Troelstra, H. Schwichtenberg. Basic Proof Theory (Cambridge Tracts in Theoretical Computer Science). Cambridge University Press. ISBN 0521779111
- An exposition aimed at mathematicians is Boolos (not to be confused with George Boole) and Jeffrey's book Computability and Logic (ISBN 0521007585).
- Mathematical Logic around the world
- Polyvalued logic
- Computability logic A new direction in mathematical logic, turning it from a theory of truth into a theory of computability.
- Computability logic
- Game semantics
- Provability logic
- Interpretability logic
- Sequent calculus
- Intuitionistic logic
- Predicate logic