{{Short description, Systematic method for producing proofs
In
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
, and in particular
proof theory
Proof theory is a major branchAccording to , proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. consists of four corresponding parts, with part D being about "Proof The ...
, a proof procedure for a given
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
is a systematic method for producing proofs in some
proof calculus
In mathematical logic, a proof calculus or a proof system is built to prove statements.
Overview
A proof system includes the components:
* Formal language: The set ''L'' of formulas admitted by the system, for example, propositional logic or fir ...
of (provable) statements.
Types of proof calculi used
There are several types of proof calculi. The most popular are
natural deduction
In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning. This contrasts with Hilbert-style systems, which instead use ...
,
sequent calculi
In mathematical logic, sequent calculus is a style of formal logical argumentation in which every line of a proof is a conditional tautology (called a sequent by Gerhard Gentzen) instead of an unconditional tautology. Each conditional tautology ...
(i.e., Gentzen-type systems), Hilbert systems, and semantic tableaux or trees. A given proof procedure will target a specific proof calculus, but can often be reformulated so as to produce proofs in other proof styles.
Completeness
A proof procedure for a logic is ''complete'' if it produces a proof for each provable statement. The
theorem
In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...
s of logical systems are typically
recursively enumerable
In computability theory, a set ''S'' of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable if:
*There is an algorithm such that the ...
, which implies the existence of a complete but usually extremely inefficient proof procedure; however, a proof procedure is only of interest if it is reasonably efficient.
Faced with an unprovable statement, a complete proof procedure may sometimes succeed in detecting and signalling its unprovability. In the general case, where provability is only a semidecidable property, this is not possible, and instead the procedure will diverge (not terminate).
See also
*
Automated theorem proving
Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving mathematical theorems by computer programs. Automated reasoning over mathematical proof was a majo ...
*
Proof complexity In logic and theoretical computer science, and specifically proof theory and computational complexity theory, proof complexity is the field aiming to understand and analyse the computational resources that are required to prove or refute statements. ...
*
Deductive system
A formal system is an abstract structure and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms.
In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in math ...
References
*
Willard Quine
Willard Van Orman Quine ( ; known to his friends as "Van"; June 25, 1908 – December 25, 2000) was an American philosopher and logician in the analytic tradition, recognized as "one of the most influential philosophers of the twentieth centur ...
1982 (1950). ''Methods of Logic''. Harvard Univ. Press.
Proof theory