Formal proofs use precise language or symbols to logically demonstrate a conclusion from established premises. They are used in mathematics, computer science, and philosophy, but have limited use in everyday arguments.
A formal proof is a sequence of statements, verbal or mathematical, 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, to avoid this problem, a formal proof can be presented symbolically or mathematically to avoid as much confusion as possible by the language. Such strictly formal tests generally begin with one or more established or theoretical premises. These premises are followed by axioms or statements that follow logically from the previous statements of the premises and end in a final conclusion or proved theorem which, like the previous statements, is a logically necessary result of the initial premises and axioms.
In contrast to a formal proof, most arguments in everyday life are based on common language and are generally not logically rigorous. They may, for example, not come from a well-constructed set of premises, or they may rely on rhetorical appeals—as to emotion or authority—that have no place in formal proof. While formal proof is valuable for its ability to prove the correctness of a claim based on a number of premises, it is important to remember that formal proofs have little or no use when arguing about something that cannot be proven conclusively. in the realm of logic. Furthermore, they apply only in the context of the original premises and, therefore, do not demonstrate universal truths.
Most formal proofs are based on a “formal language” composed of a subset of normal language or symbols. A formal mathematical proof, for example, is expressed using the symbols used in mathematics and does not rely on verbal language at all. In many cases, words are replaced by symbols so that even a formal non-mathematical proof can be understood as simple symbolic logic without the use of potentially ambiguous words.
Many different fields, usually in academia, make use of formal testing. The most obvious example is mathematics, a field that relies heavily on the use of proofs. Similarly, computer science relies on the use of strict and formal logical progressions to ensure that computers are given absolutely precise instructions. Philosophy, especially analytic philosophy, also relies on the use of formal proofs to prove the correctness of various philosophical claims in the context of various previously established or theoretical premises.
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