Symbol (formal)
Encyclopedia
For other uses see Symbol (disambiguation)
In logic, symbols build literal utility to illustrate ideas. A symbol is an abstraction, token
s 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
s studied in mathematics
and logic
, the term "symbol" refers to the idea, and the marks are considered to be a token
instance of the symbol.
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
if it is consistent with the formation rule
s of the language.
In a formal system
a symbol may be used as a token in formal operations. The set of formal symbols in a formal language
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 an universe of discourse), 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
structures, composed into larger structures using a formal grammar
, though sometimes they may be associated with an interpretation or model (a formal semantics), that define it in terms of other formal symbols.
Symbol
s such as ∧ or ¬ or
are not formal symbols, in that their semantics
is fixed - they are logical constants.
Formal symbols on the other hand are purely syntactic entities with no necessary association. However, in formal semantics, one attempts to construct models or interpretations based on higher-order logic
s like lambda calculus
that provide an interpretation for the symbol in terms of what sets variables may belong to (first-order semantics, e.g. Montague grammar
), or in terms of possible worlds where a statement may be true (modal logic
semantics, e.g. Kripke semantics
. However, these interpretations are themselves defined in terms of other formal (and therefore syntactic) symbols, and are not grounded
in entities outside the formal system; hence they can be challenged as a case of circular definition
.
(it was this work that resulted in the Chomsky hierarchy
in formal languages). The generative grammar
model looked upon syntax as autonomous from semantics. Building on these models, the logician Richard Montague
proposed that semantics could also be constructed on top of the formal structure:
This is the philosophical premise underlying Montague grammar.
However, this attempt to equate linguistic symbols with formal symbols has been challenged widely, particularly in the tradition of cognitive linguistics
, by philosophers like Stevan Harnad
, and linguists like George Lakoff
and Ronald Langacker
.
Symbol (disambiguation)
Symbols are objects, characters, or other representations of ideas, concepts, objects, or abstractions.Symbol may also refer to:-Computer science:* Symbol , the smallest amount of data transmitted at a time in digital communications...
In logic, symbols build literal utility to illustrate ideas. A symbol is an abstraction, token
Type-token distinction
In disciplines such as philosophy and knowledge representation, the type-token distinction is a distinction that separates an abstract concept from the objects which are particular instances of the concept...
s 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
Formal language
A formal language is a set of words—that is, finite strings of letters, symbols, or tokens that are defined in the language. The set from which these letters are taken is the alphabet over which the language is defined. A formal language is often defined by means of a formal grammar...
s studied in mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
and logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
, the term "symbol" refers to the idea, and the marks are considered to be a token
Type-token distinction
In disciplines such as philosophy and knowledge representation, the type-token distinction is a distinction that separates an abstract concept from the objects which are particular instances of the concept...
instance of the symbol.
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
Interpretation (logic)
An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation...
of them.
A symbol or string
String (computer science)
In formal languages, which are used in mathematical logic and theoretical computer science, a string is a finite sequence of symbols that are chosen from a set or alphabet....
of symbols may comprise a well-formed formula
Well-formed formula
In mathematical logic, a well-formed formula, shortly wff, often simply formula, is a word which is part of a formal language...
if it is consistent with the formation rule
Formation rule
In mathematical logic, formation rules are rules for describing which strings of symbols formed from the alphabet of a formal language are syntactically valid within the language. These rules only address the location and manipulation of the strings of the language. It does not describe anything...
s of the language.
In a formal system
Formal system
In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive an expression from one or more other premises that are antecedently supposed or derived . The axioms and rules may be called a deductive apparatus...
a symbol may be used as a token in formal operations. The set of formal symbols in a formal language
Formal language
A formal language is a set of words—that is, finite strings of letters, symbols, or tokens that are defined in the language. The set from which these letters are taken is the alphabet over which the language is defined. A formal language is often defined by means of a formal grammar...
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
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
may be a variable (member from an universe of discourse), 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
Syntax (logic)
In logic, syntax is anything having to do with formal languages or formal systems without regard to any interpretation or meaning given to them...
structures, composed into larger structures using a formal grammar
Formal grammar
A formal grammar is a set of formation rules for strings in a formal language. The rules describe how to form strings from the language's alphabet that are valid according to the language's syntax...
, though sometimes they may be associated with an interpretation or model (a formal semantics), that define it in terms of other formal symbols.
Symbol
Symbol
A symbol is something which represents an idea, a physical entity or a process but is distinct from it. The purpose of a symbol is to communicate meaning. For example, a red octagon may be a symbol for "STOP". On a map, a picture of a tent might represent a campsite. Numerals are symbols for...
s such as ∧ or ¬ or
are not formal symbols, in that their semanticsSemantics
Semantics is the study of meaning. It focuses on the relation between signifiers, such as words, phrases, signs and symbols, and what they stand for, their denotata....
is fixed - they are logical constants.
Formal Symbols versus traditional symbols
Traditional symbols are signs that stand for or represent some thing else, e.g. a portrait of a person, the resistance symbol in a circuit diagram, a phrase like "the horse" that refers to an animal, etc..Formal symbols on the other hand are purely syntactic entities with no necessary association. However, in formal semantics, one attempts to construct models or interpretations based on higher-order logic
Higher-order logic
In mathematics and logic, a higher-order logic is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and a stronger semantics...
s like lambda calculus
Lambda calculus
In mathematical logic and computer science, lambda calculus, also written as λ-calculus, is a formal system for function definition, function application and recursion. The portion of lambda calculus relevant to computation is now called the untyped lambda calculus...
that provide an interpretation for the symbol in terms of what sets variables may belong to (first-order semantics, e.g. Montague grammar
Montague grammar
Montague grammar is an approach to natural language semantics, named after American logician Richard Montague. The Montague grammar is based on formal logic, especially higher order predicate logic and lambda calculus, and makes use of the notions of intensional logic, via Kripke models...
), or in terms of possible worlds where a statement may be true (modal logic
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
semantics, e.g. Kripke semantics
Kripke semantics
Kripke semantics is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke. It was first made for modal logics, and later adapted to intuitionistic logic and other non-classical systems...
. However, these interpretations are themselves defined in terms of other formal (and therefore syntactic) symbols, and are not grounded
Symbol grounding
The Symbol Grounding Problem is related to the problem of how words get their meanings, and hence to the problem of what meaning itself really is. The problem of meaning is in turn related to the problem of consciousness, or how it is that mental states are meaningful...
in entities outside the formal system; hence they can be challenged as a case of circular definition
Circular definition
A circular definition is one that uses the term being defined as a part of the definition or assumes a prior understanding of the term being defined. Either the audience must already know the meaning of the key term, or the definition is deficient in including the term to be defined in the...
.
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 ChomskyNoam Chomsky
Avram Noam Chomsky is an American linguist, philosopher, cognitive scientist, and activist. He is an Institute Professor and Professor in the Department of Linguistics & Philosophy at MIT, where he has worked for over 50 years. Chomsky has been described as the "father of modern linguistics" and...
(it was this work that resulted in the Chomsky hierarchy
Chomsky hierarchy
Within the field of computer science, specifically in the area of formal languages, the Chomsky hierarchy is a containment hierarchy of classes of formal grammars....
in formal languages). The generative grammar
Generative grammar
In theoretical linguistics, generative grammar refers to a particular approach to the study of syntax. A generative grammar of a language attempts to give a set of rules that will correctly predict which combinations of words will form grammatical sentences...
model looked upon syntax as autonomous from semantics. Building on these models, the logician Richard Montague
Richard Montague
Richard Merett Montague was an American mathematician and philosopher.-Career:At the University of California, Berkeley, Montague earned an B.A. in Philosophy in 1950, an M.A. in Mathematics in 1953, and a Ph.D. in Philosophy 1957, the latter under the direction of the mathematician and logician...
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."
This is the philosophical premise underlying Montague grammar.
However, this attempt to equate linguistic symbols with formal symbols has been challenged widely, particularly in the tradition of cognitive linguistics
Cognitive linguistics
In linguistics, cognitive linguistics refers to the branch of linguistics that interprets language in terms of the concepts, sometimes universal, sometimes specific to a particular tongue, which underlie its forms...
, by philosophers like Stevan Harnad
Stevan Harnad
Stevan Harnad is a cognitive scientist.- Career :Harnad was born in Budapest, Hungary. He did his undergraduate work at McGill University and his graduate work at Princeton University's Department of Psychology...
, and linguists like George Lakoff
George Lakoff
George P. Lakoff is an American cognitive linguist and professor of linguistics at the University of California, Berkeley, where he has taught since 1972...
and Ronald Langacker
Ronald Langacker
Ronald Wayne Langacker is an American linguist and professor emeritus at the University of California, San Diego. He is best known as one of the founders of the cognitive linguistics movement and the creator of Cognitive Grammar....
.