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A logical symbol is a fundamental
concept Concepts are defined as abstract ideas. They are understood to be the fundamental building blocks of the concept behind principles, thoughts and beliefs. They play an important role in all aspects of cognition. As such, concepts are studied by s ...
in
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...
, tokens of which may be marks or a configuration of marks which form a particular pattern. Although the term "symbol" in common use refers at some times 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 consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules. The alphabet of a formal language consists of sym ...
s studied in
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...
, 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 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 any interpretation of them. A symbol or string 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. A formal language can ...
if it is consistent with the formation rules of the language. In a
formal system A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A for ...
a symbol may be used as a token in formal operations. The set of formal symbols in a
formal language In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules. The alphabet of a formal language consists of sym ...
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 may be a variable (member from a
universe of discourse In the formal sciences, the domain of discourse, also called the universe of discourse, universal set, or simply universe, is the set of entities over which certain variables of interest in some formal treatment may range. Overview The doma ...
), a constant, a function (mapping to another member of universe) or a
predicate Predicate or predication may refer to: * Predicate (grammar), in linguistics * Predication (philosophy) * several closely related uses in mathematics and formal logic: ** Predicate (mathematical logic) **Propositional function ** Finitary relation ...
(mapping to T/F). Formal symbols are usually thought of as purely syntactic structures, composed into larger structures using a
formal grammar In formal language theory, a grammar (when the context is not given, often called a formal grammar for clarity) describes how to form strings from a language's alphabet that are valid according to the language's syntax. A grammar does not describe ...
, 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 public intellectual: a linguist, philosopher, cognitive scientist, historian, social critic, and political activist. Sometimes called "the father of modern linguistics", Chomsky ...
(it was this work that resulted in the Chomsky hierarchy in formal languages). The
generative grammar Generative grammar, or generativism , is a linguistic theory that regards linguistics as the study of a hypothesised innate grammatical structure. It is a biological or biologistic modification of earlier structuralist theories of linguisti ...
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 ...
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 __notoc__ 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 ...
. 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 con ...
, by philosophers like
Stevan Harnad Stevan Robert Harnad (Hernád István Róbert, Hesslein István, born June 2, 1945, Budapest) is a Hungarian-born cognitive scientist based in Montreal, Canada. Education Harnad was born in Budapest, Hungary. He did his undergraduate work at McG ...
, 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 conc ...
and Ronald Langacker.