Symbol (formal)

Token in a mathematical or logical formula

This diagram shows the syntactic entities that may be constructed from formal languages. The symbols and strings of symbols may be broadly divided into nonsense and well-formed formulas. A formal language can be thought of as identical to the set of its well-formed formulas. The set of well-formed formulas may be broadly divided into theorems and non-theorems.

A logical symbol is a fundamental concept in logic, tokens of which may be marks or a configuration of marks which form a particular pattern.[citation needed] 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 languages 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.[dubious - discuss] In logic, symbols build literal utility to illustrate ideas.

## Overview

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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 rules 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")[1][page needed]

A formal symbol as used in first-order logic may be a variable (member from a 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).

## Words modeled as formal symbols

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

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."[2][page needed]

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.

## References

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1. ^ John Hopcroft, Rajeev Motwani and Jeffrey Ullman, Introduction to Automata Theory, Languages, and Computation, 2000
2. ^ Richard Montague, Universal Grammar, 1970

## See also

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* List of mathematical symbols
* List of logic symbols
* Terminal and nonterminal symbols

* v
* t
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Mathematical logic

General|

* Axiom
* list
* Cardinality
* First-order logic
* Formal proof
* Formal semantics
* Foundations of mathematics
* Information theory
* Lemma
* Logical consequence
* Model
* Theorem
* Theory
* Type theory

Theorems (list)
and paradoxes|

* Gödel's completeness and incompleteness theorems
* Tarski's undefinability
* Banach–Tarski paradox
* Cantor's theorem, paradox and diagonal argument
* Compactness
* Halting problem
* Lindström's
* Löwenheim–Skolem
* Russell's paradox

Logics|

| Traditional|

* Classical logic
* Logical truth
* Tautology
* Proposition
* Inference
* Logical equivalence
* Consistency
* Equiconsistency
* Argument
* Soundness
* Validity
* Syllogism
* Square of opposition
* Venn diagram

|

Propositional|

* Boolean algebra
* Boolean functions
* Logical connectives
* Propositional calculus
* Propositional formula
* Truth tables
* Many-valued logic
* 3
* finite
* ∞

Predicate|

* First-order
* list
* Second-order
* Monadic
* Higher-order
* Fixed-point
* Free
* Quantifiers
* Predicate
* Monadic predicate calculus

Set theory|

* Set
* hereditary
* Class
* (Ur-)Element
* Ordinal number
* Extensionality
* Forcing
* Relation
* equivalence
* partition
* Set operations:
* intersection
* union
* complement
* Cartesian product
* power set
* identities

Types of sets|

* Countable
* Uncountable
* Empty
* Inhabited
* Singleton
* Finite
* Infinite
* Transitive
* Ultrafilter
* Recursive
* Fuzzy
* Universal
* Universe
* constructible
* Grothendieck
* Von Neumann

Maps and cardinality|

* Function/Map
* domain
* codomain
* image
* In/Sur/Bi-jection
* Schröder–Bernstein theorem
* Isomorphism
* Gödel numbering
* Enumeration
* Large cardinal
* inaccessible
* Aleph number
* Operation
* binary

Set theories|

* Zermelo–Fraenkel
* axiom of choice
* continuum hypothesis
* General
* Kripke–Platek
* Morse–Kelley
* Naive
* New Foundations
* Tarski–Grothendieck
* Von Neumann–Bernays–Gödel
* Ackermann
* Constructive

Formal systems (list),
language and syntax|

* Alphabet
* Arity
* Automata
* Axiom schema
* Expression
* ground
* Extension
* by definition
* conservative
* Relation
* Formation rule
* Grammar
* Formula
* atomic
* closed
* ground
* open
* Free/bound variable
* Language
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* Logical connective
* ¬
* ∨
* ∧
* →
* ↔
* =
* Predicate
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* variable
* propositional variable
* Proof
* Quantifier
* ∃
* !
* ∀
* rank
* Sentence
* atomic
* spectrum
* Signature
* String
* Substitution
* Symbol
* function
* logical/constant
* non-logical
* variable
* Term
* Theory
* list

Example axiomatic
systems (list)|

* of arithmetic:
* Peano
* second-order
* elementary function
* primitive recursive
* Robinson
* Skolem
* of the real numbers
* Tarski's axiomatization
* of Boolean algebras
* canonical
* minimal axioms
* of geometry:
* Euclidean:
* Elements
* Hilbert's
* Tarski's
* non-Euclidean

* Principia Mathematica

Proof theory|

* Formal proof
* Natural deduction
* Logical consequence
* Rule of inference
* Sequent calculus
* Theorem
* Systems
* axiomatic
* deductive
* Hilbert
* list
* Complete theory
* Independence (from ZFC)
* Proof of impossibility
* Ordinal analysis
* Reverse mathematics
* Self-verifying theories

Model theory|

* Interpretation
* function
* of models
* Model
* equivalence
* finite
* saturated
* spectrum
* submodel
* Non-standard model
* of arithmetic
* Diagram
* elementary
* Categorical theory
* Model complete theory
* Satisfiability
* Semantics of logic
* Strength
* Theories of truth
* semantic
* Tarski's
* Kripke's
* T-schema
* Transfer principle
* Truth predicate
* Truth value
* Type
* Ultraproduct
* Validity

Computability theory|

* Church encoding
* Church–Turing thesis
* Computably enumerable
* Computable function
* Computable set
* Decision problem
* decidable
* undecidable
* P
* NP
* P versus NP problem
* Kolmogorov complexity
* Lambda calculus
* Primitive recursive function
* Recursion
* Recursive set
* Turing machine
* Type theory

Related|

* Abstract logic
* Algebraic logic
* Automated theorem proving
* Category theory
* Concrete/Abstract category
* Category of sets
* History of logic
* History of mathematical logic
* timeline
* Logicism
* Mathematical object
* Philosophy of mathematics
* Supertask

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