Formal system

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In logic, a formal system (usually synonymous with a logical system,^[1] a logistic system,^[1] a logical calculus,^[2] or simply a logic^[1], although a logical system can be expressed nonformally, and formal systems can be constructed so as to be nonlogical) consists of a formal language together with a deductive system (also called a deductive apparatus) which consists of a set of inference rules and/or axioms.

A formal system is used to derive one expression from one or more other expressions antecedently expressed in the system. These expressions are called axioms, in the case of those previously supposed to be true, or theorems, in the case of those derived. Formal systems are useful because their theorems can be interpreted as expressing logical truths or because their transformation rules can be interpreted as logically valid rules of inference. A formal system may be formulated and studied for its intrinsic properties, or it may be intended as a description (i.e. a model) of external phenomena.

Overview

Each formal system can be considered to contain a "formal language, which is composed by primitive symbols. These symbols act on certain rules of formation and are developed by inference from a set of axioms. The system thus consists of any number of formulas built up through finite combinations of the primitive symbols—combinations that are formed from the axioms in accordance with the stated rules".^[3]

Formal systems in mathematics consist of the following elements:

1. A finite set of symbols (i.e. the alphabet), that can be used for constructing formulas (i.e. finite strings of symbols).
2. A grammar, which tells how well-formed formulas (abbreviated wff) are constructed out of the symbols in the alphabet. It is usually required that there be a decision procedure for deciding whether a formula is well formed or not.
3. A set of axioms or axiom schemata: each axiom must be a wff.
4. A set of inference rules.

A formal system is said to be recursive (i.e. effective) if the set of axioms and the set of inference rules are decidable sets or semidecidable sets, according to context.

Some theorists use the term formalism as a rough synonym for formal system, but the term is also used to refer to a particular style of notation, for example, Paul Dirac's bra-ket notation.

Related subjects

Formal language

A formal language is a set A of strings (finite sequences) on a fixed alphabet α. While a formal language can be thought of as identical to its set of well-formed formula, a formal system cannot be thought of as identical with its set of theorems. Two different formal systems may produce the same set of theorems.

Formation rules

Formation rules are rules for describing which strings of symbols formed from the alphabet of a formal language are syntactically valid within the language. These rules only address the location and manipulation of the strings of the language. It does not describe anything else about a language, such as its semantics (i.e. what the strings mean). (See also formal grammar).

Deductive system

A deductive system (also called a deductive apparatus of a formal system) consists of the axioms (or axiom schemata) and rules of inference that can be used to derive the theorems of the system.^[4]

A deductive system is intended to preserve deductive qualities in the formulas that are expressed in the system. Usually the quality we are concerned with is truth as opposed to falsehood. However, other modalities, such as justification or belief may be preserved instead.

Formal proofs

Formal proofs are sequences of strings. For a string to qualify as part of a proof, it might either be an axiom or be the product of applying an inference rule on previous strings in the proof sequence. The last string in the sequence is recognized as a theorem.

The point of view that generating formal proofs is all there is to mathematics is often called formalism. David Hilbert founded metamathematics as a discipline for discussing formal systems. Any language that one uses to talk about a formal system is called a metalanguage. The metalanguage may be nothing more than ordinary natural language, or it may be partially formalized itself, but it is generally less completely formalized than the formal language component of the formal system under examination, which is then called the object language, that is, the object of the discussion in question.

Once a formal system is given, one can define the set of theorems which can be proved inside the formal system. This set consists of all strings for which there is a proof. Thus all axioms are considered theorems. Unlike the grammar for strings, there is no guarantee that there will be a decision procedure for deciding whether a given string is a theorem or not. The notion of theorem just defined should not be confused with theorems about the formal system, which, in order to avoid confusion, are usually called metatheorems.

Interpretations of formal systems

An interpretation of a formal system is the assignment of meanings to the symbols, and truth-values to the sentences of the formal system. The study of formal interpretations is called Formal semantics. Giving an interpretation is synonymous with constructing a model.

An interpreted formal system is a formal language for which both syntactical rules for deduction, and semantical rules of interpretation are given.

See also

Examples of formal systems Axiomatic system Proof calculus Formal ethics Lambda calculus Propositional calculus

Logic portal Systems science portal Other related topics Axiom Formal Formal language Formal method Formal science Gödel's incompleteness theorems Institution substitution instance

References

1. ^ ^{a b c} Audi, Robert (Editor). The Cambridge Dictionary of Philosophy. Second edition, Cambridge University Press, 1999. ISBN 978-0521631365 (hardcover) and ISBN 978-0521637220 (paperback).
2. ^ Rudolf Carnap. Introduction to Symbolic Logic and its Applications,Dover, 1958. p. 101.
3. ^ Encyclopædia Britannica, Formal system definition, 2007.
4. ^ Hunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California Pres, 1971

Further reading

• Dov M. Gabbay (ed.), What is a logical system?, Oxford University Press, 1994, ISBN 0198538596
• Raymond M. Smullyan, Theory of Formal Systems: Annals of Mathematics Studies, Princeton University Press (April 1, 1961) 156 pages ISBN 069108047X
• S. C. Kleene, 1967. Mathematical Logic Reprinted by Dover, 2002. ISBN 0486425339

External links

• What is a Formal System?: Some quotes from John Haugeland's `Artificial Intelligence: The Very Idea' (1985), pp. 48-64.
• Heinrich Herre Formal Language and systems, 1997.
• Peter Suber, Course in Logical Systems, 1997.