Formal system

In logic and mathematics, together with the allied branches of computer science, information theory, and statistics, a formal system is an idealized and abstract language or formal grammar used for modeling purposes. Formalization is the act of creating a formal system, in an attempt to capture the essential features of a real-world or conceptual system in formal language. A model is a structure that can be used to provide an interpretation of the abstract or symbolic language that is embodied in the formal system.

Characteristics of Formal System

Formal systems have the virtue that absolutely certain proofs—or absolutely certain knowledge—can be found within them, and nowhere else. If the formal system is itself consistent, then it is possible to prove, within the given system, that a given conclusion follows from given premises. This means that, within that system, if the given premises are true, then the conclusion cannot be false. Thus we have absolutely certain knowledge that this conclusion is true if the premises are true. This is, of course, a highly restricted notion of certain (absolute) knowledge, but it is the only place or domain in which genuine or unassailable or unquestionable absolute knowledge exists, despite all claims to the contrary.

In mathematics, formal proofs are the product of formal systems, consisting of axioms and rules of deduction. Theorems are then recognized as the possible 'last lines' of formal proofs. The point of view that this picture encompasses in mathematics has been called formalism. That term has been used both approvingly and pejoratively. On the other hand, David Hilbert founded metamathematics as a discipline designed for discussing formal systems; it is not assumed that the metalanguage in which proofs are studied is itself less informal than the usual habits of mathematicians suggest. To contrast with the metalanguage, or language in which the formal system is itself stated, the language described by a formal grammar is often called an object language (i.e., the object of discussion—this distinction may have been introduced by Carnap).

It has become common to speak of a formalism, more-or-less synonymously with a formal system within standard mathematics invented for a particular purpose. This may not be much more than a notation, such as Dirac's bra-ket notation.

Mathematical formal systems consist of the following:

A finite set of symbols which can be used for constructing formulae.
A grammar, i.e. a way of constructing well-formed formulae out of the symbols, such that it is possible to find a decision procedure for deciding whether a formula is a well-formed formula (wff) or not.
A set of axioms or axiom schemata: each axiom has to be a wff.
A set of inference rules.
A set of theorems. This set includes all the axioms, plus all wffs which can be derived from previously-derived theorems by means of rules of inference. Unlike the grammar for wffs, there is no guarantee that there will be a decision procedure for deciding whether a given wff is a theorem or not.

References
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The Journal of Symbolic Logic has been a repository of literature on formal systems.
Addison, John, Leon Henkin, and Alfred Tarski, eds. Proceedings of the International Symposium of the Theory of Models, Berkeley, 1963. Amsterdam, 1965.
Boole, George. The Mathematical Analysis of Logic, Being an Essay Toward a Calculus of Deductive Reasoning. Oxford: Basil Blackwell, 1998. Reproduction of the 1847 ed. in Cambridge, England. ISBN 1855065835
Broy, Manfred, Stephan Merz, and Katharina Spies, eds. Formal Systems Specification: The RPC-Memory Specification Case Study. Berlin & New York: Springer, 1996. ISBN 3540619844
Church, Alonzo. Introduction to Mathematical Logic. Vol. I. Princeton, NJ: Princeton University Press, 1956.
Frege, Gottlob. Die Grundlagen der Arithmetik, eine logisch-mathemataische Untersuchung über den Begriff der Zahl. Breslau 1884, reprinted 1934. Trans. as The Foundations of Arithmetic: A Logico-Mathematical Enquiry Into the Concept of Number. English trans. by J.L. Austin, Northwestern University Press, 2nd Rev edition, 1980. ISBN 0810106051
Hilbert, David, and Wilhelm Ackermann. Grundzüge der theoretischen Logik. 3rd ed. Berlin 1949. 2nd ed. trans. by Lewis M. Hammond as Principles of Mathematical Logic. American Mathematical Society, 1999. ISBN 0821820249
Quine, W.V.O. Mathematical Logic. Rev. ed. Harvard University Press, 2003. ISBN 0674554515
Quine, W.V.O. Philosophy of Logic: 2nd Edition. Harvard University Press, 2006. ISBN 0674665635
Russell, Bertrand, and Alfred North Whitehead. Principia Mathematica. 3 vols. Cambridge: Cambridge University Press, 1910-1913.
Tarski, Alfred. A Decision Method for Elementary Algebra and Geometry. Berkeley: University of California Press, 1951.

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