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A formal proof is a sequence of statements, verbal or mathematical, that is used to demonstrate the logical necessity of a given conclusion. Such proofs require rigorous and precise use of language, as linguistic ambiguity can easily render a proof meaningless. In many cases, in order to avoid this problem, one may present a formal proof symbolically or mathematically in order to avoid the confusion introduced by language as much as is possible. Such strictly formal proofs generally start with one or more well-established or theoretical premises. These premises are followed by axioms or statements that follow logically from the premises' preceding statements and terminate in a final conclusion or proven theorem that, like the preceding statements, is a logically necessary result of the initial premises and axioms.
In contrast to a formal proof, most arguments in day-to-day life rely on common language and are not generally logically rigorous. They may, for instance, not originate from a well-constructed set of premises, or they may rely on rhetorical appeals — as to emotion or authority — which have no place in a formal proof. While a formal proof is valuable because of its ability to demonstrate the correctness of a statement based on a set of premises, it is important to remember that formal proofs have little or no use when discussing anything that cannot be conclusively demonstrated in the realm of logic. Also, they only apply within the context of the original premises and do not, therefore, demonstrate universal truths.
Most formal proofs are based in a "formal language" composed either of a subset of normal language or in symbols. A mathematical formal proof, for instance, is expressed using the symbols used in mathematics and does not rely at all on verbal language. In many cases, words are substituted for symbols so that even a non-mathematical formal proof can be understood in the form of simple symbolic logic without the use of potentially-ambiguous words.
Many different fields, usually in academia, make use of formal proofs. The most obvious example is mathematics, a field that is largely based in the use of proofs. Similarly, computer science relies on the use of strict, formal logical progressions in order to ensure absolutely precise instructions are given to computers. Philosophy, especially analytic philosophy, also relies on the use of formal proofs to demonstrate the correctness of various philosophical claims within the context of various previously established or theoretical premises.
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