Formal system

Source: Wikipedia, the free encyclopedia.

A formal system is an

axioms by a set of inference rules.[1]

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

notation, for example, Paul Dirac's bra–ket notation
.

Concepts

This diagram shows the syntactic entities that may be constructed from formal languages. The symbols and strings of symbols may be broadly divided into nonsense and well-formed formulas. A formal language can be thought of as identical to the set of its well-formed formulas, which may be broadly divided into theorems and non-theorems.

A formal system has the following:[3][4][5]

A formal system is said to be

semidecidable sets
, respectively.

Formal language

Part of
Formal languages
Key concepts
Applications
Main articles: Formal language, Formal grammar, Syntax (logic), and Logical form

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

analytic grammars (or reductive grammar[6][7]
), which are sets of rules for how a string can be analyzed to determine whether it is a member of the language.

Deductive system

Main articles: Inference, Logical consequence, and Deductive reasoning

A deductive system, also called a deductive apparatus,

rules of inference that can be used to derive theorems of the system.[9]

Such deductive systems preserve

justification or belief
may be preserved instead.

In order to sustain its deductive integrity, a deductive apparatus must be definable without reference to any

intended interpretation of the language. The aim is to ensure that each line of a derivation is merely a logical consequence of the lines that precede it. There should be no element of any interpretation
of the language that gets involved with the deductive nature of the system.

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]

Proof system

Main articles:
Proof system and Formal proof

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

Main articles: Semantics of logic, Interpretation (logic), and Model theory

A logical system is a deductive system (most commonly

model
of the logical system.

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

nonnegative integers and gives the symbols their usual meaning.[10] There are also non-standard models of arithmetic
.

History

Main articles:
Formal logical systems

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.

foundational crisis of mathematics, that was eventually tempered by Gödel's incompleteness theorems.[2] The QED manifesto
represented a subsequent, as yet unsuccessful, effort at formalization of known mathematics.

See also
  • List of formal systems
  • Formal method
     – Mathematical program specifications
  • Formal science – Branch of science
  • 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

  1. ^ "Formal system | Logic, Symbols & Axioms | Britannica". www.britannica.com. Retrieved 2023-10-10.
  2. ^ a b Zach, Richard (31 July 2003). "Hilbert's Program". Hilbert's Program, Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University.
  3. ^ "formal system". planetmath.org. Retrieved 2023-10-10.
  4. ^ Rapaport, William J. (25 March 2010). "Syntax & Semantics of Formal Systems". University of Buffalo.
  5. ^ "Definition:Formal System - ProofWiki". proofwiki.org. Retrieved 2023-10-16.
  6. ^ Reductive grammar: (computer science) A set of syntactic rules for the analysis of strings to determine whether the strings exist in a language. "Sci-Tech Dictionary McGraw-Hill Dictionary of Scientific and Technical Terms" (6th ed.). McGraw-Hill.[unreliable source?] About the Author Compiled by The Editors of the McGraw-Hill Encyclopedia of Science & Technology (New York, NY) an in-house staff who represents the cutting-edge of skill, knowledge, and innovation in science publishing. [1]
  7. ^ "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." "The TREE-META Compiler-Compiler System: A Meta Compiler System for the Univac 1108 and General Electric 645, University of Utah Technical Report RADC-TR-69-83. C. Stephen Carr, David A. Luther, Sherian Erdmann" (PDF). Retrieved 5 January 2015.
  8. ^ a b "Definition:Deductive Apparatus - ProofWiki". proofwiki.org. Retrieved 2023-10-10.
  9. ^ Hunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California Press, 1971
  10. ISBN 9780198532132
    .

Further reading
Look up formalisation in Wiktionary, the free dictionary.
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