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Aformal system (ordeductive system) is anabstract structure andformalization of anaxiomatic system used fordeducing, usingrules of inference,theorems fromaxioms.[1]
In 1921,David Hilbert proposed to use formal systems as the foundation of knowledge inmathematics.[2]However, in 1931Kurt Gödel proved that anyconsistent formal system sufficiently powerful to express basic arithmetic cannot prove its owncompleteness. This effectively showed thatHilbert's program was impossible as stated.
The termformalism is sometimes a rough synonym forformal system, but it also refers to a given style ofnotation, for example,Paul Dirac'sbra–ket notation.
A formal system has the following components, as a minimum:[3][4][5]
A formal system is said to berecursive (i.e. effective) or recursively enumerable if the set of axioms and the set of inference rules aredecidable sets orsemidecidable sets, respectively.
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Aformal language is a language that uses a set of strings whose symbols are taken from a specific alphabet, and operations used to form sentences from them. Like languages inlinguistics, formal languages generally have two aspects:
Usually only thesyntax of a formal language is considered via the notion of aformal grammar. The two main categories of formal grammar are that ofgenerative grammars, which are sets of rules for how strings in a language can be written, and that ofanalytic grammars (or reductive grammar[6][unreliable source?][7]), which are sets of rules for how a string can be analyzed to determine whether it is a member of the language.
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Adeductive system, also called adeductive apparatus,[8] consists of theaxioms (oraxiom schemata) andrules of inference that can be used toderivetheorems of the system.[1]
In order to sustain its deductive integrity, adeductive apparatus must be definable without reference to anyintended interpretation of the language. The aim is to ensure that each line of aderivation is merely alogical consequence of the lines that precede it. There should be no element of anyinterpretation of the language that gets involved with the deductive nature of the system.
Thelogical 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 largertheory or field (e.g.Euclidean geometry) consistent with the usage in modern mathematics such asmodel theory.[clarification needed]
An example of a deductive system would be the rules of inference andaxioms regarding equality used infirst order logic.
The two main types of deductive systems are proof systems and formal semantics.[8][9]
Formal proofs are sequences ofwell-formed formulas (or WFF for short) that might either be anaxiom or be the product of applying an inference rule on previous WFFs in the proof sequence.
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 adecision 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 calledformalism.David Hilbert foundedmetamathematics as a discipline for discussing formal systems. Any language that one uses to talk about a formal system is called ametalanguage. 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 theobject language, that is, the object of the discussion in question. The notion oftheorem just defined should not be confused withtheorems about the formal system, which, in order to avoid confusion, are usually calledmetatheorems.
Alogical system is a deductive system (most commonlyfirst order logic) together with additionalnon-logical axioms. According tomodel theory, a logical system may be giveninterpretations which describe whether a givenstructure - the mapping of formulas to a particular meaning - satisfies a well-formed formula. A structure that satisfies all the axioms of the formal system is known as amodel of the logical system.
A logical system is:
An example of a logical system isPeano arithmetic. The standard model of arithmetic sets thedomain of discourse to be thenonnegative integers and gives the symbols their usual meaning.[10] There are alsonon-standard models of arithmetic.
Early logic systems includes Indian logic ofPāṇini, syllogistic logic of Aristotle, propositional logic of Stoicism, and Chinese logic ofGongsun Long (c. 325–250 BCE).[citation needed] In more recent times, contributors includeGeorge Boole,Augustus De Morgan, andGottlob Frege.Mathematical logic was developed in 19th centuryEurope.
David Hilbert instigated aformalist movement calledHilbert’s program as a proposed solution to thefoundational crisis of mathematics, that was eventually tempered byGödel's incompleteness theorems.[2] TheQED manifesto represented a subsequent, as yet unsuccessful, effort at formalization of known mathematics.
Reductive grammar: (computer science) A set of syntactic rules for the analysis of strings to determine whether the strings exist in a language.
There are two classes of formal-language definition compiler-writing schemes. The productivegrammar 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 oranalytical 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.
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.
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