A formal system or logical calculus is any well-defined system of abstract thought based on the model of mathematics. A formal system need not be mathematical as such; for example, Spinoza's Ethics imitates the form of Euclid's Elements. Spinoza employed Euclidiean elements such as "axioms" or "primitive truths", rules of inferences, etc., so that a calculus can be built using these.
Some theorists[which?] use the term formalism as a rough synonym for formal system, but the term is also used to refer to a particular style[which?] of notation, for example, Paul Dirac's bra–ket notation.
The system thus consists of valid formulas built up through finite combinations of the primitive symbols—combinations that are formed from the axioms in accordance with the stated rules.
More formally, this can be expressed as the following:
- A finite set of symbols, that can be used for constructing formulas (i.e. finite strings of symbols).
- 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.
- A set of axioms or axiom schemata: each axiom must be a wff.
- A set of inference rules.
Inference and entailment
The 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]
- the syntax of a language is what the language looks like (more formally: the set of possible expressions that are valid utterances in the language) studied in formal language theory
- the semantics of a language are what the utterances of the language mean (which is formalized in various ways, depending on the type of language in question)
In computer science and linguistics a formal grammar is a precise description of a formal language: a set of strings. The two main categories of formal grammar are that of generative grammars, which are sets of rules for how strings in a language can be generated, and that of analytic grammars (or reductive grammar, which are sets of rules for how a string can be analyzed to determine whether it is a member of the language. In short, an analytic grammar describes how to recognize when strings are members in the set, whereas a generative grammar describes how to write only those strings in the set.
A logical system or, for short, logic, is a formal system together with a form of semantics,[disputed ] usually in the form of model-theoretic interpretation, which assigns truth values to sentences of the formal language, that is, formulae that contain no free variables. A logic is sound if all sentences that can be derived are true in the interpretation, and complete if, conversely, all true sentences can be derived.
Such deductive systems 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.
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 syntactic 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.
Panini is credited with the creation of the world's first formal system well before[when?] the innovations of Gottlob Frege and the subsequent development of mathematical logic in 19th century Europe.
The QED manifesto represented a subsequent unsuccessful effort at formalization of known mathematics.
Examples of formal systems include:
The following systems are variations of formal systems[clarification needed].
Formal proofs are sequences of well-formed formulas (or wff for short). For a wff to qualify as part of a proof, it 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.
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.
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 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.
This article is in a list format that may be better presented using prose. (January 2017)
- Encyclopædia Britannica, Formal system definition, 2007.
- 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. 
- "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.
- Hunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic, University of California Pres, 1971
- Raymond M. Smullyan, 1961. Theory of Formal Systems: Annals of Mathematics Studies, Princeton University Press (April 1, 1961) 156 pages ISBN 0-691-08047-X
- Stephen Cole Kleene, 1967. Mathematical Logic Reprinted by Dover, 2002. ISBN 0-486-42533-9
- Douglas Hofstadter, 1979. Gödel, Escher, Bach: An Eternal Golden Braid ISBN 978-0-465-02656-2. 777 pages.
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