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{{Short description, Fundamental theory of logical analysis In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, an analytic proof is a proof of a theorem in
analysis Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
that only makes use of methods from analysis, and that does not predominantly make use of algebraic or geometrical methods. The term was first used by
Bernard Bolzano Bernard Bolzano (, ; ; ; born Bernardus Placidus Johann Nepomuk Bolzano; 5 October 1781 – 18 December 1848) was a Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction, also known for his liberal ...
, who first provided a non-analytic proof of his
intermediate value theorem In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval. This has two imp ...
and then, several years later provided a proof of the theorem that was free from intuitions concerning lines crossing each other at a point, and so he felt happy calling it analytic (Bolzano 1817). Bolzano's philosophical work encouraged a more abstract reading of when a demonstration could be regarded as analytic, where a proof is analytic if it does not go beyond its subject matter (Sebastik 2007). In
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 ...
, an analytic proof has come to mean a proof whose structure is simple in a special way, due to conditions on the kind of inferences that ensure none of them go beyond what is contained in the assumptions and what is demonstrated.


Structural proof theory

In proof theory, the notion of analytic proof provides the fundamental concept that brings out the similarities between a number of essentially distinct proof calculi, so defining the subfield of structural proof theory. There is no uncontroversial general definition of analytic proof, but for several proof calculi there is an accepted notion. For example: * In
Gerhard Gentzen Gerhard Karl Erich Gentzen (24 November 1909 – 4 August 1945) was a German mathematician and logician. He made major contributions to the foundations of mathematics, proof theory, especially on natural deduction and sequent calculus. He died ...
's
natural deduction calculus 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 a ...
the analytic proofs are those in normal form; that is, no formula occurrence is both the principal premise of an elimination rule and the conclusion of an introduction rule; * In Gentzen's
sequent calculus 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 tautolog ...
the analytic proofs are those that do not use the cut rule. However, it is possible to extend the
inference rule Rules of inference are ways of deriving conclusions from premises. They are integral parts of formal logic, serving as norms of the logical structure of valid arguments. If an argument with true premises follows a rule of inference then the co ...
s of both calculi so that there are proofs that satisfy the condition but are not analytic. For example, a particularly tricky example of this is the ''analytic cut rule'', used widely in the tableau method, which is a special case of the cut rule where the cut formula is a
subformula In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbol (formal), symbols from a given alphabet (computer science), alphabet that is part ...
of side formulae of the cut rule: a proof that contains an analytic cut is by virtue of that rule not analytic. Furthermore, proof calculi that are not analogous to Gentzen's calculi have other notions of analytic proof. For example, the calculus of structures organises its inference rules into pairs, called the up fragment and the down fragment, and an analytic proof is one that only contains the down fragment.


See also

* Proof-theoretic semantics


References

*
Bernard Bolzano Bernard Bolzano (, ; ; ; born Bernardus Placidus Johann Nepomuk Bolzano; 5 October 1781 – 18 December 1848) was a Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction, also known for his liberal ...
(1817). Purely analytic proof of the theorem that between any two values which give results of opposite sign, there lies at least one real root of the equation. In ''Abhandlungen der koniglichen bohmischen Gesellschaft der Wissenschaften'' Vol. V, pp.225-48. * Frank Pfenning (1984). Analytic and Non-analytic Proofs. In ''Proc. 7th International Conference on Automated Deduction''. * Jan Šebestik (2007)
Bolzano's Logic
Entry in the ''
Stanford Encyclopedia of Philosophy The ''Stanford Encyclopedia of Philosophy'' (''SEP'') is a freely available online philosophy resource published and maintained by Stanford University, encompassing both an online encyclopedia of philosophy and peer-reviewed original publication ...
''. Proof theory Methods of proof