Formal system

In formal logic, a formal system (also called a logical system,^[1] a logistic system,^[1] a logical calculus,^[2] or simply a logic^[1]) consists of a formal language together with a deductive system (also called a deductive apparatus) which consists of a set of inference rules and/or axioms. Logic is the study of the principles of valid demonstration and Inference. A formal language is a set of words, ie finite strings of letters, or symbols. A deductive system (also called a deductive apparatus of a Formal system) consists of the Axioms (or Axiom schemata and Rules of inference In Logic, a rule of inference (also called a transformation rule) is a function from sets of formulae to formulae In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject A formal system is used to derive one expression from one or more other expressions antecedently expressed in the system. see also Mathematical proof, Proof theory, and Axiomatic system. These expressions are called axioms, in the case of those previously supposed to be true, or theorems, in the case of those derived. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements A formal system may be formulated and studied for its intrinsic properties, or it may be intended as a description (i. e. a model) of external phenomena. Scientific modelling is the process of generating abstract, conceptual, Graphical and or mathematical models.

1 Overview
2 Related subjects
3 References
4 Further reading
5 See also
6 External links

Overview

Each formal system has a formal language, which is composed by primitive symbols. A formal language is a set of words, ie finite strings of letters, or symbols. The musical instrument is spelled Cymbal. A symbol is something --- such as an object, Picture, written word a sound a piece These symbols act on certain rules of formation and are developed by inference from a set of axioms. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject The system thus consists of any number of formulas built up through finite combinations of the primitive symbols—combinations that are formed from the axioms in accordance with the stated rules. ^[3]

Formal systems in mathematics consist of the following elements:

A finite set of symbols (i. e. the alphabet), that can be used for constructing formulas (i. In Computer science, an alphabet is a usually finite set of characters or digits e. finite strings of symbols).
A grammar, which tells how well-formed formulas (abbreviated wff) are constructed out of the symbols in the alphabet. Grammar is the field of Linguistics that covers the Rules governing the use of any given natural language. In Mathematical logic, a well-formed formula (often abbreviated WFF, pronounced "wiff" or "wuff" is a Symbol or string of symbols (a It is usually required that there be a decision procedure for deciding whether a formula is well formed or not.
A set of axioms or axiom schemata: each axiom must be a wff. In Mathematical logic, an axiom schema generalizes the notion of Axiom.
A set of inference rules. In Logic, a rule of inference (also called a transformation rule) is a function from sets of formulae to formulae

A formal system is said to be recursive (i. In Computability theory, a set of Natural numbers is called recursive, computable or decidable if there is an Algorithm e. effective) if the set of axioms and the set of inference rules are decidable sets or semidecidable sets, according to context. In Computability theory, a set of Natural numbers is called recursive, computable or decidable if there is an Algorithm In Computability theory, traditionally called Recursion theory, a set S of Natural numbers is called recursively enumerable, computably

Some theorists use the term formalism as a rough synonym for formal system, but the term is also used to refer to a particular style of notation, for example, Paul Dirac's bra-ket notation. Bra-ket notation is a standard notation for describing Quantum states in the theory of Quantum mechanics composed of angle brackets (chevrons and Vertical

Related subjects

Formal language

Main article: Formal language

A formal language is a set A of strings (finite sequences) on a fixed alphabet α. A formal language is a set of words, ie finite strings of letters, or symbols.

Formal grammar

Main article: Formal grammar

In computer science and linguistics a formal grammar is a precise description of a formal language: a set of strings. In Formal semantics, Computer science and Linguistics, a formal grammar (also called formation rules) is a precise description of a Formal Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their Linguistics is the scientific study of Language, encompassing a number of sub-fields A formal language is a set of words, ie finite strings of letters, or symbols. In Computer programming and some branches of Mathematics, a string is an ordered Sequence of Symbols. The two main categories of formal grammar are that of generative grammars, which are sets of rules for how strings in a language can be generated, and that of analytic grammars, which are sets of rules for how a string can be analyzed to determine whether it is a member of the language. In Theoretical linguistics, generative grammar refers to a particular approach to the study of Syntax. In Formal semantics, Computer science and Linguistics, a formal grammar (also called formation rules) is a precise description of a Formal In short, an analytic grammar describes how to recognize when strings are members in the set, whereas a generative grammar describes how to write only those strings in the set.

Formal proofs

Main article: Formal proof

Formal proofs are sequences of wffs. see also Mathematical proof, Proof theory, and Axiomatic system. For a wff to qualify as part of a proof, it might either be an axiom or be the product of applying an inference rule on previous wffs in the proof sequence. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject The last wff in the sequence is recognized as a theorem. In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements

The point of view that generating formal proofs is all there is to mathematics is often called formalism. The philosophy of mathematics is the branch of Philosophy that studies the philosophical assumptions foundations and implications of Mathematics. David Hilbert founded metamathematics as a discipline for discussing formal systems. David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most In general metamathematics or meta-mathematics is a scientific reflection and Knowledge about mathematics seen as an entity/ object in Human Any language that one uses to talk about a formal system is called a metalanguage. In Logic and Linguistics, a metalanguage is a Language used to make statements about statements in another language which is called the Object The metalanguage may be nothing more than ordinary natural language, or it may be partially formalized itself, but it is generally less completely formalized than the formal language component of the formal system under examination, which is then called the object language, that is, the object of the discussion in question. Object language has meaning in contexts of computer programming and operation and in linguistics and logic

Once a formal system is given, one can define the set of theorems which can be proved inside the formal system. This set consists of all wffs of which there is a proof for. Thus all axioms are considered theorems. Unlike the grammar for wffs, there is no guarantee that there will be a decision procedure for deciding whether a given wff is a theorem or not. In Logic, the term decidable refers to the existence of an Effective method for determining membership in a set of formulas The notion of theorem just defined should not be confused with theorems about the formal system, which, in order to avoid confusion, are usually called metatheorems. In Mathematical logic, a metatheorem is a Statement about Theorems or about some axiomatic Theory.

Formal interpretations

Main articles: Formal semantics, Formal interpretation, and Interpretation (logic)

A formal interpretation of a formal system is the assignment of meanings, to the symbols, and truth-values to the sentences of the formal system. See also Formal semantics of programming languages. Formal semantics is the study of the Semantics, or Interpretations A formal interpretation or model is the assignment of Meanings to the Symbols and Truth-values to the Sentences of a Formal In Logic an interpretation gives meaning to an artificial or Formal language or to a sentence of such a language by assigning a denotation (extension The study of formal interpretations is called Formal semantics. See also Formal semantics of programming languages. Formal semantics is the study of the Semantics, or Interpretations Giving an interpretation is synonymous with constructing a model. In Universal algebra and in Model theory, a structure consists of an underlying set along with a collection of Finitary functions and relations

An interpreted formal system is a formal language for which both syntactical rules for deduction, and semantical rules of interpretation are given. A deductive system (also called a deductive apparatus of a Formal system) consists of the Axioms (or Axiom schemata and Rules of inference A formal interpretation or model is the assignment of Meanings to the Symbols and Truth-values to the Sentences of a Formal An interpreted formal system can be expressed as the ordered quadruple <α, $\mathcal{I}$ , $\mathcal{D}$ d, $\mathcal{D}$ >. Where, in the case of extensional metalanguages, $\mathcal{D}$ is the relation of value assignment for the sentences of the language and in the case of intensional metalanguages, it is relation of designation, i. Not to be confused with the homophone Intention; or the related concept of Intentionality. e. , the relation between an expression and its intension; and where $\mathcal{D}$ d is the relation of direct derivability. see also Mathematical proof, Proof theory, and Axiomatic system. This relation is understood in a comprehensive sense such that the primitive sentences of the formal system are taken as directly derivable from the empty set of sentences. Here axioms are stated, some similar to those stated for a formal system, and some like those for an interpreted formal language. Usually, we wish for $\mathcal{D}$ d to be truth-preserving (that is, any sentence which is directly derivable from true sentences is itself true), however other modalities can also preserved in such a system. A modal logic is any system of formal logic that attempts to deal with modalities. We can formulate an axiom for these purposes with use of the term "true". For any $\mathcal{I}$ _i_₁,. . . , $\mathcal{I}$ _i_{_n}, $\mathcal{I}$ _j, p₁,. . . ,p_n,q if $\mathcal{D}$ d( $\mathcal{I}$ _j,{ $\mathcal{I}$ _i_₁,. . . , $\mathcal{I}$ _i_{_n}}), $\mathcal{D}$ ( $\mathcal{I}$ _i_₁,p₁) and . . . and $\mathcal{D}$ ( $\mathcal{I}$ _i_{_n},p_n) and p₁ and . . . and p_n, q.

For interpreted formal systems there are also alternative, more explicit definitions which include ds, or both ds and D, analogous to those given for interpreted formal languages.

References

^ ^a ^b ^c Audi, Robert (Editor). Robert Audi (born November 1941 is a philosopher whose major work has focused on epistemology Ethics —especially on Ethical intuitionism, and the theory of action The Cambridge Dictionary of Philosophy. The Cambridge Dictionary of Philosophy is a Dictionary of philosophical terms published by Cambridge University Press and edited by Robert Second edition, Cambridge University Press, 1999. Cambridge University Press (known colloquially as CUP is a Publisher given a Royal Charter by Henry VIII in 1534 ISBN 978-0521631365 (hardcover) and ISBN 978-0521637220 (paperback).
^ Rudolf Carnap. Rudolf Carnap ( May 18, 1891 &ndash September 14, 1970) was an influential German -born philosopher who was active in Introduction to Symbolic Logic and its Applications,Dover, 1958. p. 101.
^ Encyclopædia Britannica, Formal system definition, 2007.

External links

Encyclopædia Britannica, Formal system definition, 2007. In Mathematics, an axiomatic system is any set of Axioms from which some or all axioms can be used in conjunction to logically derive Theorems In Mathematical logic, a proof calculus corresponds to a family of Formal systems that use a common style of formal inference for its Inference rules. Formal ethics is a Formal logical system for describing and evaluating the form as opposed to the content of ethical principles In Mathematical logic and Computer science, lambda calculus, also written as λ-calculus, is a Formal system designed to investigate function This is a technical mathematical article about the area of mathematical logic variously known as "propositional calculus" or "propositional logic" In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject A formal language is a set of words, ie finite strings of letters, or symbols. In Computer science and Software engineering, formal methods are particular kind of Mathematically -based techniques for the specification, development A formal science is a theoretical study that is concerned with theoretical Formal systems, for instance Logic, Mathematics, Systems theory and In Mathematical logic, Gödel's incompleteness theorems, proved by Kurt Gödel in 1931 are two Theorems stating inherent limitations of all but the most The notion of institution has been created by Joseph Goguen and Rod Burstall in the late 1970'sin order to deal with the "population explosion among the In Propositional logic, a substitution instance of a Propositional formula is a second formula obtained by replacing symbols of the original formula by other formulas
Christer Blomqvist, a introduction to formal systems, webpage 1997.
What is a Formal System?: Some quotes from John Haugeland's `Artificial Intelligence: The Very Idea' (1985), pp. 48-64.
Heinrich Herre Formal Language and systems, 1997.
Peter Suber, Formal Systems and Machines: An Isomorphism, 1997.

Dictionary

-noun

(logic) The grouping of a formal language and a set of inference rules and/or axioms.

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |

Contents