Everything about Formal Proof totally explained
» see also
Mathematical proof,
Proof theory, and
Axiomatic system.
A
formal proof or
derivation is a finite sequence of
sentences (called
well-formed formulas in the case of a
formal language) each of which is an
axiom or follows from the preceding sentences in the sequence by a
rule of inference. The last sentence in the sequence is a
theorem of a
formal system. The notion of theorem isn't in general
effective, therefore there may be no method by which we can always find a proof of a given sentence or determine that none exists. The concept of
deduction is a
generalization of the concept of proof.
The theorem is a
syntactic consequence of all the wffs preceding it in the proof. For a wff to qualify as part of a proof, it must be the result of applying a rule of the
deductive apparatus of some formal system to the previous wffs in the proof sequence.
Formal proofs often are constructed with the help of computers in
interactive theorem proving. Significantly, these proofs can be checked automatically, also by computer. Checking formal proofs is usually trivial, whereas
finding proofs (
automated theorem proving) is typically quite hard.
Background
Formal language
A
formal language is an organized
set of
symbols the essential feature of which is that it can
be precisely defined in terms of just the shapes and locations of those symbols. Such a language can be defined, then, without any
reference to any
meanings of any of its expressions; it can exist before any
formal interpretation is assigned to it -- that is, before it has any meaning. Formal proofs are expressed in some formal language
Formal grammar
A
formal grammar (also called
formation rules) is a precise description of a the
well-formed formulas of a formal language. It is synonymous with the
set of
strings over the
alphabet of the formal language which constitute well formed formulas. However, it doesn't describe their
semantics (for example what they mean).
Formal systems
A
formal system (also called a
logical calculus, or a
logical system) consists of a formal language together with a
deductive apparatus (also called a
deductive system). The deductive apparatus may consist of a set of
transformation rules (also called
inference rules) or a set of
axioms, or have both. A formal system is used to
derive one expression from one or more other expressions.
Formal interpretations
An
interpretation of a formal system is the assignment of meanings to the symbols, and truth-values to the sentences of a formal system. The study of formal interpretations is called
formal semantics.
Giving an interpretation is synonymous with
constructing a model.
Further Information
Get more info on 'Formal Proof'.
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