A logical symbol is a fundamental

concept A concept is an abstract idea that serves as a foundation for more concrete principles, thoughts, and beliefs. Concepts play an important role in all aspects of cognition. As such, concepts are studied within such disciplines as linguistics, ...

logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...

, tokens of which may be marks or a configuration of marks which form a particular pattern. Although the term ''symbol'' 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 the

formal language In logic, mathematics, computer science, and linguistics, a formal language is a set of strings whose symbols are taken from a set called "alphabet". The alphabet of a formal language consists of symbols that concatenate into strings (also c ...

s studied in

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

and

, the term ''symbol'' refers to the idea, and the marks are considered to be a token instance of the symbol. In logic, symbols build literal utility to illustrate ideas.

Overview

Symbols of a formal language need not be symbols ''of'' anything. For instance there are

logical constant In logic, a logical constant or constant symbol of a language \mathcal is a symbol that has the same semantic value under every interpretation of \mathcal. Two important types of logical constants are logical connectives and quantifiers. The e ...

s 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 any interpretation of them. A symbol or

string String or strings may refer to: *String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects Arts, entertainment, and media Films * ''Strings'' (1991 film), a Canadian anim ...

of symbols may comprise a

well-formed formula In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language. The abbreviation wf ...

if it is consistent with the

formation rule In mathematical logic, formation rules are rules for describing well-formed words over the alphabet of a formal language. These rules only address the location and manipulation of the strings of the language. It does not describe anything else a ...

s of the language. In a

formal system A formal system is an abstract structure and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms. In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in ma ...

a symbol may be used as a token in formal operations. The set of formal symbols in a

is referred to as an alphabet (hence each symbol may be referred to as a "letter") A formal symbol as used in

first-order logic First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over ...

may be a variable (member from a

universe of discourse In the formal sciences, the domain of discourse or universe of discourse (borrowing from the mathematical concept of ''universe'') is the set of entities over which certain variables of interest in some formal treatment may range. It is also ...

), a constant, a function (mapping to another member of universe) or a predicate (mapping to T/F). Formal symbols are usually thought of as purely

syntactic In linguistics, syntax ( ) is the study of how words and morphemes combine to form larger units such as phrases and sentences. Central concerns of syntax include word order, grammatical relations, hierarchical sentence structure (constituency ...

structures, composed into larger structures using a

formal grammar A formal grammar is a set of Terminal and nonterminal symbols, symbols and the Production (computer science), production rules for rewriting some of them into every possible string of a formal language over an Alphabet (formal languages), alphabe ...

, though sometimes they may be associated with an interpretation or model (a formal semantics).

Can words be modeled as formal symbols?

The move to view units in natural language (e.g. English) as formal symbols was initiated by

Noam Chomsky Avram Noam Chomsky (born December 7, 1928) is an American professor and public intellectual known for his work in linguistics, political activism, and social criticism. Sometimes called "the father of modern linguistics", Chomsky is also a ...

(it was this work that resulted in the

Chomsky hierarchy The Chomsky hierarchy in the fields of formal language theory, computer science, and linguistics, is a containment hierarchy of classes of formal grammars. A formal grammar describes how to form strings from a formal language's alphabet that are v ...

in formal languages). The

generative grammar Generative grammar is a research tradition in linguistics that aims to explain the cognitive basis of language by formulating and testing explicit models of humans' subconscious grammatical knowledge. Generative linguists, or generativists (), ...

model looked upon syntax as autonomous from semantics. Building on these models, the logician

Richard Montague Richard Merritt Montague (September 20, 1930 – March 7, 1971) was an American mathematician and philosopher who made contributions to mathematical logic and the philosophy of language. He is known for proposing Montague grammar to formalize th ...

proposed that semantics could also be constructed on top of the formal structure: :There is in my opinion no important theoretical difference between natural languages and the artificial languages of logicians; indeed, I consider it possible to comprehend the syntax and semantics of both kinds of language within a single natural and mathematically precise theory. On this point I differ from a number of philosophers, but agree, I believe, with Chomsky and his associates."Richard Montague, ''Universal Grammar'', 1970 This is the philosophical premise underlying

Montague grammar Montague grammar is an approach to natural language semantics, named after American logician Richard Montague. The Montague grammar is based on mathematical logic, especially higher-order predicate logic and lambda calculus, and makes use of th ...

. However, this attempt to equate linguistic symbols with formal symbols has been challenged widely, particularly in the tradition of

cognitive linguistics Cognitive linguistics is an interdisciplinary branch of linguistics, combining knowledge and research from cognitive science, cognitive psychology, neuropsychology and linguistics. Models and theoretical accounts of cognitive linguistics are cons ...

, by philosophers like

Stevan Harnad Stevan Robert Harnad (Hernád István Róbert, Hesslein István, born 1945) is a Canadian cognitive scientist based in Montreal. Early life and education Harnad was born in Budapest, Hungary. He did his undergraduate work at McGill University an ...

, and linguists like

George Lakoff George Philip Lakoff ( ; born May 24, 1941) is an American cognitive linguist and philosopher, best known for his thesis that people's lives are significantly influenced by the conceptual metaphors they use to explain complex phenomena. The ...

and Ronald Langacker.

Overview

Can words be modeled as formal symbols?

References

See also