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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