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In [[mathematical logic]], '''structural proof theory''' is the subdiscipline of [[proof theory]] that studies [[proof calculi]] that support a notion of [[analytic proof]]. | |||
==Analytic proof== | |||
{{Main|analytic proof}} | |||
The notion of analytic proof was introduced into proof theory by [[Gerhard Gentzen]] for the [[sequent calculus]]; the analytic proofs are those that are [[cut-elimination theorem|cut-free]]. His [[natural deduction calculus]] also supports a notion of analytic proof, as was 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 (abstract rewriting)|normal form]] in [[term rewriting]]. | |||
==Structures and connectives== | |||
The term ''structure'' in structural proof theory comes from a technical notion introduced in the sequent calculus: the sequent calculus represents the judgement made at any stage of an inference using special, extra-logical operators which we call structural operators: in <math>A_1, \dots, A_m \vdash B_1, \dots, B_n</math>, the commas to the left of the [[Turnstile (symbol)|turnstile]] are operators normally interpreted as conjunctions, those to the right as disjunctions, whilst the turnstile symbol itself is interpreted as an implication. However, it is important to note that there is a fundamental difference in behaviour between these operators and the [[logical connective]]s they are interpreted by in the sequent calculus: the structural operators are used in every rule of the calculus, and are not considered when asking whether the subformula property applies. Furthermore, the logical rules go one way only: logical structure is introduced by logical rules, and cannot be eliminated once created, while structural operators can be introduced and eliminated in the course of a derivation. | |||
The idea of looking at the syntactic features of sequents as special, non-logical operators is not old, and was forced by innovations in proof theory: when the structural operators are as simple as in Getzen's original sequent calculus there is little need to analyse them, but proof calculi of [[deep inference]] such as display logic support structural operators as complex as the logical connectives, and demand sophisticated treatment. | |||
==Cut-elimination in the sequent calculus== | |||
{{Main|Cut-elimination}} | |||
{{Expand section|date=December 2009}} | |||
==Natural deduction and the formulae-as-types correspondence== | |||
{{Main|Natural deduction}} | |||
{{Expand section|date=December 2009}} | |||
==Logical duality and harmony== | |||
{{Main|Logical harmony}} | |||
{{Expand section|date=December 2009}} | |||
==Display logic== | |||
{{Expand section|date=December 2009}} | |||
==Calculus of structures== | |||
{{Main|Calculus of structures}} | |||
{{Expand section|date=December 2009}} | |||
== References == | |||
* {{cite book|author1=Sara Negri|author2=Jan Von Plato|title=Structural proof theory|year=2001|publisher=Cambridge University Press|isbn=978-0-521-79307-0}} | |||
* {{cite book|author1=Anne Sjerp Troelstra|author2=Helmut Schwichtenberg|title=Basic proof theory|year=2000|publisher=Cambridge University Press|isbn=978-0-521-77911-1|edition=2nd}} | |||
{{DEFAULTSORT:Structural Proof Theory}} | |||
[[Category:Proof theory]] | |||
Revision as of 21:30, 6 January 2014
In mathematical logic, structural proof theory is the subdiscipline of proof theory that studies proof calculi that support a notion of analytic proof.
Analytic proof
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The notion of analytic proof was introduced into proof theory by Gerhard Gentzen for the sequent calculus; the analytic proofs are those that are cut-free. His natural deduction calculus also supports a notion of analytic proof, as was shown by Dag Prawitz; the definition is slightly more complex — we say the analytic proofs are the normal forms, which are related to the notion of normal form in term rewriting.
Structures and connectives
The term structure in structural proof theory comes from a technical notion introduced in the sequent calculus: the sequent calculus represents the judgement made at any stage of an inference using special, extra-logical operators which we call structural operators: in , the commas to the left of the turnstile are operators normally interpreted as conjunctions, those to the right as disjunctions, whilst the turnstile symbol itself is interpreted as an implication. However, it is important to note that there is a fundamental difference in behaviour between these operators and the logical connectives they are interpreted by in the sequent calculus: the structural operators are used in every rule of the calculus, and are not considered when asking whether the subformula property applies. Furthermore, the logical rules go one way only: logical structure is introduced by logical rules, and cannot be eliminated once created, while structural operators can be introduced and eliminated in the course of a derivation.
The idea of looking at the syntactic features of sequents as special, non-logical operators is not old, and was forced by innovations in proof theory: when the structural operators are as simple as in Getzen's original sequent calculus there is little need to analyse them, but proof calculi of deep inference such as display logic support structural operators as complex as the logical connectives, and demand sophisticated treatment.
Cut-elimination in the sequent calculus
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. Template:Expand section
Natural deduction and the formulae-as-types correspondence
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. Template:Expand section
Logical duality and harmony
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. Template:Expand section
Display logic
Calculus of structures
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. Template:Expand section
References
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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534