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| '''Proof theory''' is a branch of [[mathematical logic]] that represents [[Mathematical proof|proof]]s as formal [[mathematical object]]s, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined [[data structures]] such as plain lists, boxed lists, or trees, which are constructed according to the [[axiom]]s and [[rule of inference|rules of inference]] of the logical system. As such, proof theory is [[syntax (logic)|syntactic]] in nature, in contrast to [[model theory]], which is [[Formal semantics (logic)|semantic]] in nature. Together with [[model theory]], [[axiomatic set theory]], and [[recursion theory]], proof theory is one of the so-called ''four pillars'' of the [[foundations of mathematics]].<ref name=wang>E.g., Wang (1981), pp. 3–4, and Barwise (1978).</ref>
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| Proof theory is important in [[philosophical logic]], where the primary interest is in the idea of a [[proof-theoretic semantics]], an idea which depends upon technical ideas in [[structural proof theory]] to be feasible.
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| ==History==
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| Although the formalisation of logic was much advanced by the work of such figures as [[Gottlob Frege]], [[Giuseppe Peano]], [[Bertrand Russell]], and [[Richard Dedekind]], the story of modern proof theory is often seen as being established by [[David Hilbert]], who initiated what is called [[Hilbert's program]] in the [[foundations of mathematics]]. [[Kurt Gödel]]'s seminal work on proof theory first advanced, then refuted this program: his [[Gödel's completeness theorem|completeness theorem]] initially seemed to bode well for Hilbert's aim of reducing all mathematics to a finitist formal system; then his [[Gödel's incompleteness theorem|incompleteness theorems]] showed that this is unattainable. All of this work was carried out with the proof calculi called the [[Hilbert system]]s.
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| In parallel, the foundations of [[structural proof theory]] were being founded. [[Jan Łukasiewicz]] suggested in 1926 that one could improve on [[Hilbert system]]s as a basis for the axiomatic presentation of logic if one allowed the drawing of conclusions from assumptions in the inference rules of the logic. In response to this [[Stanisław Jaśkowski]] (1929) and [[Gerhard Gentzen]] (1934) independently provided such systems, called calculi of [[natural deduction]], with Gentzen's approach introducing the idea of symmetry between the grounds for asserting propositions, expressed in [[introduction rule]]s, and the consequences of accepting propositions in the [[elimination rule]]s, an idea that has proved very important in proof theory.<ref>Prawitz (1965).</ref> Gentzen (1934) further introduced the idea of the [[sequent calculus]], a calculus advanced in a similar spirit that better expressed the duality of the logical connectives,<ref>Girard, Lafont, and Taylor (1988).</ref> and went on to make fundamental advances in the formalisation of intuitionistic logic, and provide the first [[combinatorial proof]] of the consistency of [[Peano arithmetic]]. Together, the presentation of natural deduction and the sequent calculus introduced the fundamental idea of [[analytic proof]] to proof theory.
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| ==Formal and informal proof==<!-- This section is linked from [[Black–Scholes]] and [[Mathematical proof]]-->
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| {{Main|Formal proof}}
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| The ''informal'' proofs of everyday mathematical practice are unlike the ''formal'' proofs of proof theory. They are rather like high-level sketches that would allow an expert to reconstruct a formal proof at least in principle, given enough time and patience. For most mathematicians, writing a fully formal proof is too pedantic and long-winded to be in common use.
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| Formal proofs are constructed with the help of computers in [[interactive theorem proving]].
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| Significantly, these proofs can be checked automatically, also by computer. (Checking formal proofs is usually simple, whereas ''finding'' proofs ([[automated theorem proving]]) is generally hard.) An informal proof in the mathematics literature, by contrast, requires weeks of [[peer review]] to be checked, and may still contain errors.
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| ==Kinds of proof calculi==
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| The three most well-known styles of [[proof calculi]] are:
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| *The [[Hilbert system|Hilbert calculi]]
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| *The [[natural deduction calculus|natural deduction calculi]]
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| *The [[sequent calculus|sequent calculi]]
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| Each of these can give a complete and axiomatic formalization of [[propositional logic|propositional]] or [[predicate logic]] of either the [[classical logic|classical]] or [[intuitionistic logic|intuitionistic]] flavour, almost any [[modal logic]], and many [[substructural logic]]s, such as [[relevance logic]] or
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| [[linear logic]]. Indeed it is unusual to find a logic that resists being represented in one of these calculi.
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| ==Consistency proofs==
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| {{Main|Consistency proof}}
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| As previously mentioned, the spur for the mathematical investigation of proofs in formal theories was [[Hilbert's program]]. The central idea of this program was that if we could give finitary proofs of consistency for all the sophisticated formal theories needed by mathematicians, then we could ground these theories by means of a metamathematical argument, which shows that all of their purely universal assertions (more technically their provable [[arithmetical hierarchy|<math>\Pi^0_1</math> sentences]]) are finitarily true; once so grounded we do not care about the non-finitary meaning of their existential theorems, regarding these as pseudo-meaningful stipulations of the existence of ideal entities.
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| The failure of the program was induced by [[Kurt Gödel]]'s [[Gödel's incompleteness theorems|incompleteness theorems]], which showed that any [[ω-consistent theory]] that is sufficiently strong to express certain simple arithmetic truths, cannot prove its own consistency, which on Gödel's formulation is a <math>\Pi^0_1</math> sentence.
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| Much investigation has been carried out on this topic since, which has in particular led to:
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| *Refinement of Gödel's result, particularly [[J. Barkley Rosser]]'s refinement, weakening the above requirement of ω-consistency to simple consistency;
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| *Axiomatisation of the core of Gödel's result in terms of a modal language, [[provability logic]];
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| *Transfinite iteration of theories, due to [[Alan Turing]] and [[Solomon Feferman]];
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| *The recent discovery of [[self-verifying theories]], systems strong enough to talk about themselves, but too weak to carry out the diagonal argument that is the key to Gödel's unprovability argument.
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| See also [[Mathematical logic]]
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| ==Structural proof theory==
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| {{Main|Structural proof theory}}
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| Structural proof theory is the subdiscipline of proof theory that studies proof calculi that support a notion of [[analytic proof]]. The notion of analytic proof was introduced by Gentzen for the sequent calculus; there the analytic proofs are those that are [[cut-elimination theorem|cut-free]]. His natural deduction calculus also supports a notion of analytic proof, as shown by [[Dag Prawitz]]. The definition is slightly more complex: we say the analytic proofs are the [[Natural deduction#Consistency.2C completeness.2C and normal forms|normal forms]], which are related to the notion of normal form in term rewriting. More exotic proof calculi such as [[Jean-Yves Girard]]'s [[proof net]]s also support a notion of analytic proof.
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| Structural proof theory is connected to [[type theory]] by means of the [[Curry-Howard correspondence]], which observes a structural analogy between the process of normalisation in the natural deduction calculus and beta reduction in the [[typed lambda calculus]]. This provides the foundation for the [[intuitionistic type theory]] developed by [[Per Martin-Löf]], and is often extended to a three way correspondence, the third leg of which are the [[cartesian closed category|cartesian closed categories]].
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| ==Proof-theoretic semantics==
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| {{Main|proof-theoretic semantics|logical harmony}}
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| In [[linguistics]], [[type-logical grammar]], [[categorial grammar]] and [[Montague grammar]] apply formalisms based on structural proof theory to give a formal [[natural language semantics]].
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| ==Tableau systems==
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| {{Main|Method of analytic tableaux}}
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| Analytic tableaux apply the central idea of analytic proof from structural proof theory to provide decision procedures and semi-decision procedures for a wide range of logics.
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| ==Ordinal analysis==
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| {{Main|Ordinal analysis}}
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| Ordinal analysis is a powerful technique for providing combinatorial consistency proofs for theories formalising arithmetic and analysis.
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| ==Logics from proof analysis==
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| {{Main|Substructural logic}}
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| Several important logics have come from insights into logical structure arising in structural proof theory.
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| ==See also==
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| {{Portal|Logic}}
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| *[[Intermediate logic]]
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| *[[Model theory]]
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| *[[Proof (truth)]]
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| *[[Proof techniques]]
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| ==Notes==
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| <references/>
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| ==References==
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| *J. Avigad, E.H. Reck (2001). [http://www.andrew.cmu.edu/user/avigad/Papers/infinite.pdf “Clarifying the nature of the infinite”: the development of metamathematics and proof theory]. Carnegie-Mellon Technical Report CMU-PHIL-120.
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| *J. Barwise (ed., 1978). Handbook of Mathematical Logic. North-Holland.
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| *[[A. S. Troelstra]], H. Schwichtenberg (1996). ''Basic Proof Theory''. In series ''Cambridge Tracts in Theoretical Computer Science'', Cambridge University Press, ISBN 0-521-77911-1.
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| *G. Gentzen (1935/1969). Investigations into logical deduction. In M. E. Szabo, editor, ''Collected Papers of Gerhard Gentzen''. North-Holland. Translated by Szabo from "Untersuchungen über das logische Schliessen", Mathematisches Zeitschrift 39: 176-210, 405-431.
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| * {{springer|title=Proof theory|id=p/p075430}}
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| *Luis Moreno & [[Bharath Sriraman]] (2005).''Structural Stability and Dynamic Geometry: Some Ideas on Situated Proof. International Reviews on Mathematical Education. Vol. 37, no.3, pp. 130–139'' [http://www.springerlink.com/content/n602313107541846/?p=74ab8879ce75445da488d5744cbc3818&pi=0]
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| *J. von Plato (2008). [http://plato.stanford.edu/entries/proof-theory-development/ The Development of Proof Theory]. [[Stanford Encyclopedia of Philosophy]].
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| *{{cite book
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| | last = Wang | first = Hao | authorlink = Hao Wang (academic)
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| | title = Popular Lectures on Mathematical Logic
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| | publisher = [[Van Nostrand|Van Nostrand Reinhold Company]]
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| | year = 1981
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| | isbn = 0-442-23109-1 }}
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| {{Logic}}
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| {{DEFAULTSORT:Proof Theory}}
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| [[Category:Mathematical logic|*P]]
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| [[Category:Proof theory| ]]
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| [[Category:Metalogic]]
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The advertising newsletter notes that with Jordan Kurland returnoninvestment and analytic information are an easy task to collect. Sometimes we've to depend on authorities promoting items to translate the data, but when you want to invest in online marketing you don't need to find out the mechanics of how data is published. You just have to recognize the statistics and review the info over a regular basis, the content notes.
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