Alogical symbol is a fundamentalconcept inlogic,tokens of which may be marks or a configuration of marks which form a particular pattern.[citation needed] Although the termsymbol in common use sometimes refers to the idea being symbolized, and at other times to the marks on a piece of paper or chalkboard which are being used to express that idea; in theformal languages studied inmathematics andlogic, the termsymbol refers to the idea, and the marks are considered to be atoken instance of the symbol.[dubious –discuss] In logic, symbols build literal utility to illustrate ideas.
Symbols of a formal language need not be symbolsof anything. For instance there arelogical constants which do not refer to any idea, but rather serve as a form of punctuation in the language (e.g. parentheses). Symbols of a formal language must be capable of being specified without any reference to anyinterpretation of them.
A symbol orstring of symbols may comprise awell-formed formula if it is consistent with theformation rules of the language.
In aformal system a symbol may be used as a token in formal operations. The set of formal symbols in aformal language is referred to as an alphabet (hence each symbol may be referred to as a "letter")[1][page needed]
A formal symbol as used infirst-order logic may be a variable (member from auniverse of discourse), a constant, a function (mapping to another member of universe) or apredicate (mapping to T/F).
Formal symbols are usually thought of as purelysyntactic structures, composed into larger structures using aformal grammar, though sometimes they may be associated with an interpretation or model (aformal semantics).
The move to view units in natural language (e.g. English) as formal symbols was initiated byNoam Chomsky (it was this work that resulted in theChomsky hierarchy in formal languages). Thegenerative grammar model looked upon syntax as autonomous from semantics. Building on these models, the logicianRichard Montague proposed that semantics could also be constructed on top of the formal structure:
This is the philosophical premise underlyingMontague grammar.
However, this attempt to equate linguistic symbols with formal symbols has been challenged widely, particularly in the tradition ofcognitive linguistics, by philosophers likeStevan Harnad, and linguists likeGeorge Lakoff andRonald Langacker.
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