formal system

In logic, a formal language together with a deductive apparatus by which some well-formed formulas can be derived from others. Each formal system has a formal language composed of primitive symbols that figure in certain rules of formation (statements concerning the expressions allowable in the system) and a set of theorems developed by inference from a set of axioms. In an axiomatic system, the primitive symbols are undefined and all other symbols are defined in terms of them. In Euclidean geometry, for example, such concepts as “point,” “line,” and “lies on” are usually posited as primitive terms. From the primitive symbols, certain formulas are defined as well formed, some of which are listed as axioms; and rules are stated for inferring one formula as a conclusion from one or more other formulas taken as premises. A theorem within such a system is a formula capable of proof through a finite sequence of well-formed formulas, each of which either is an axiom or is validly inferred from earlier formulas.

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In philosophy, reasoning that fails to establish its conclusion because of deficiencies in form or wording. Formal fallacies are types of deductive argument that instantiate an invalid inference pattern (see deduction; validity); an example is “affirming the consequent: If A then B; B; therefore, A.” Informal fallacies are types of inductive argument the premises of which fail to establish the conclusion because of their content. There are many kinds of informal fallacy; examples include argumentum ad hominem (“argument against the man”), which consists of attacking the arguer instead of his argument; the fallacy of false cause, which consists of arguing from the premise that one event precedes another to the conclusion that the first event is the cause of the second; the fallacy of composition, which consists of arguing from the premise that a part of a thing has a certain property to the conclusion that the thing itself has that property; and the fallacy of equivocation, which consists of arguing from a premise in which a term is used in one sense to a conclusion in which the term is used in another sense.

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The term formal has a number of uses, including:

General

• relating to formality
• opposite of informal

Social

• Formal occasion
• Formal attire worn on such occasions
• Formals are particular meals at some British universities
• In Australian or British English, a formal is the equivalent of the American prom dance.
• In American English, a "formal" is organized by a college or university organization, typically but not always a fraternity or sorority.
• Informal occasion (included as Informal redirect here); see Dress code (Western) for the distinctions between Formal, Semi-formal, and Informal
• Informal attire (not to be confused with Casual -- see Dress code (Western))

Philosophical

• relating to form, i.e. appearance rather than essence.
• relating to Formalism, i.e. emphasis on form over content or meaning.
• formal logic logical argument based only on the form and not on the meaning.
• Formal cause, Aristotle's intrinsic, determining cause.

Mathematics

• formal power series, a generalization of power series without requiring convergence, used in combinatorics;
• formal calculation, a calculation which is systematic, but without a rigorous justification;
• formal set theory as opposed to naive set theory;
• formal derivative, an operation on elements of a polynomial ring which mimics the form of the derivative from calculus.

Logic and Language

• formal system, an abstract means of generating inferences in a formal language;
• formal language, comprising the symbolic "words" or "sentences" of a formal system;
• formal grammar, a grammar describing a formal language;
• formal proof, a fully rigorous proof as is possible only in a formal system.

Computer science

• formal methods in computer science, including:
• formal specification describes what a system should do, not how it should do it
• formal verification proves correctness of a system

See also the Language section of Mathematics, above

Linguistic

• Formal equivalence word-for-word translation (especially of the Bible)

See also the Language section of Mathematics, above

Chemical

• A unit of concentration
• short or prefix form for formalin
• short or prefix form for formaldehyde
• synonym for dimethoxymethane

See also

• Informal economy