Formal system
tertiary sources.(December 2024) ) |
A formal system is an
In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in mathematics.[2]
The term formalism is sometimes a rough synonym for formal system, but it also refers to a given style of
Concepts

A formal system has the following:[3][4][5]
- Formal language, which is a set of well-formed formulas, which are strings of symbols from an alphabet, formed by a formal grammar (consisting of production rules or formation rules).
- Deductive system, deductive apparatus, or proof system, which has rules of inference that take axioms and infers theorems, both of which are part of the formal language.
A formal system is said to be
Formal language
Part of Formal languages |
A formal language is a language that is defined by a formal system. Like languages in linguistics, formal languages generally have two aspects:
- the syntax is what the language looks like (more formally: the set of possible expressions that are valid utterances in the language)
- the semantics are what the utterances of the language mean (which is formalized in various ways, depending on the type of language in question)
Usually only the
), which are sets of rules for how a string can be analyzed to determine whether it is a member of the language.Deductive system
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A deductive system, also called a deductive apparatus,
Such deductive systems preserve
In order to sustain its deductive integrity, a deductive apparatus must be definable without reference to any
The logical consequence (or entailment) of the system by its logical foundation is what distinguishes a formal system from others which may have some basis in an abstract model. Often the formal system will be the basis for or even identified with a larger theory or field (e.g. Euclidean geometry) consistent with the usage in modern mathematics such as model theory.[clarification needed]
An example of a deductive system would be the rules of inference and axioms regarding equality used in first order logic.
The two main types of deductive systems are proof systems and formal semantics.[8][9]
Proof system
Formal proofs are sequences of well-formed formulas (or WFF for short) that might either be an axiom or be the product of applying an inference rule on previous WFFs in the proof sequence. The last WFF in the sequence is recognized as a theorem.
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 WFFs for which there is a proof. Thus all axioms are considered theorems. 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.
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 a 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. 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.
Formal semantics of logical system
A logical system is a deductive system (most commonly
A logical system is:
- Sound, if each well-formed formula that can be inferred from the axioms is satisfied by every model of the logical system.
- Semantically complete, if each well-formed formula that is satisfied by every model of the logical system can be inferred from the axioms.
An example of a logical system is
History
Early logic systems includes Indian logic of Pāṇini, syllogistic logic of Aristotle, propositional logic of Stoicism, and Chinese logic of Gongsun Long (c. 325–250 BCE) . In more recent times, contributors include George Boole, Augustus De Morgan, and Gottlob Frege. Mathematical logic was developed in 19th century Europe.
See also
- List of formal systems
- Formal method– Mathematical program specifications
- Formal science – Study of abstract structures described by formal systems
- Logic translation – Translation of a text into a logical system
- Rewriting system– Replacing subterm in a formula with another term
- Substitution instance– Concept in logic
- Theory (mathematical logic) – Set of sentences in a formal language
References
- ^ a b Hunter 1996, p. 7.
- ^ a b Zach, Richard (31 July 2003). "Hilbert's Program". Hilbert's Program, Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University.
- ^ "Formal System". PlanetMath.
- ^ Rapaport, William J. (25 March 2010). "Syntax & Semantics of Formal Systems". University of Buffalo.
- ^ "Formal System". PrfWiki.
- ^ "Reductive grammar". Dictionary of Scientific and Technical Terms (6th ed.). McGraw-Hill.
Reductive grammar: (computer science) A set of syntactic rules for the analysis of strings to determine whether the strings exist in a language.
- ^ Rulifson, Johns F. (April 1968). "A Tree Meta for the XDS 940" (PDF). Augmentation Research Center. Retrieved 30 November 2024.
There are two classes of formal-language definition compiler-writing schemes. The productive grammar approach is the most common. A productive grammar consists primarrly of a set of rules that describe a method of generating all possible strings of the language. The reductive or analytical grammar technique states a set of rules that describe a method of analyzing any string of characters and deciding whether that string is in the language.
- ^ a b "Deductive Apparatus". PrfWiki. Retrieved 30 November 2024.
- ^ van Fraassen, Bas C. (2016) [1971]. Formal Semantics and Logic (PDF). Nousoul Digital Publishers. p. 12.
Metalogic can in turn be roughly divided into two parts: proof theory and formal semantics... The division is not exact; many questions have been dealt with from both points of view, and some proof-theoretic methods and results are indispensable in semantics.
- ISBN 9780198532132.
Sources
- )
Further reading
- ISBN 978-0-465-02656-2. 777 pages.
- ISBN 0-486-42533-9
- ISBN 0-691-08047-X
External links
Media related to Formal systems at Wikimedia Commons
- Encyclopædia Britannica, Formal system definition, 2007.
- Daniel Richardson, Formal systems, logic and semantics
- Formal System at PlanetMath.
- Encyclopedia of Mathematics, Formal system
- Peter Suber, Formal Systems and Machines: An Isomorphism Archived 2011-05-24 at the Wayback Machine, 1997.
- Ray Taol, Formal Systems
- What is a Formal System?: Some quotes from John Haugeland's `Artificial Intelligence: The Very Idea' (1985), pp. 48–64.