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Hilbert systemFrom Wikipedia, the free encyclopediaJump to: navigation , search

In mathematical physics , Hilbert system is an infrequently used term for a physicalsystem described by a C*-algebra .

This article needs attention from an expert on the subject . See the talk page fordetails. WikiProject Mathematics or the Mathematics Portal may be able to helprecruit an expert. (March 2011)

In logic , especially mathematical logic , a Hilbert system , sometimes called Hilbert calculus or Hilbert – Ackermann system , is a type of system of formal deduction attributed toGottlob Frege [1] and David Hilbert . These deductive systems are most often studied for first-order logic , but are of interest for other logics as well.

Most variants of Hilbert systems take a characteristic tack in the way they balance a trade-off between logical axioms and rules of inference .[1] Hilbert systems can be characterised by thechoice of a large number of schemes of logical axioms and a small set of rules of inference . The most commonly studied Hilbert systems have either just one rule of inference — modusponens , for propositional logics — or two — with generalisation , to handle predicate logics , as well — and several infinite axiom schemes. Hilbert systems for propositional modallogics , sometimes called Hilbert-Lewis systems , are generally axiomatised with twoadditional rules, the necessitation rule and the uniform substitution rule.

A characteristic feature of the many variants of Hilbert systems is that the context is notchanged in any of their rules of inference, while both natural deduction and sequent calculus contain some context-changing rules. Thus, if we are interested only in the derivability of tautologies , no hypothetical judgments, then we can formalize the Hilbert system in such away that its rules of inference contain only judgments of a rather simple form. The samecannot be done with the other two deductions systems: as context is changed in some of theirrules of inferences, they cannot be formalized so that hypothetical judgments could beavoided — not even if we want to use them just for proving derivability of tautologies.

Systems of natural deduction take the opposite task, including many deduction rules but veryfew or no axiom schemes.

Contents[hide ]

1 Formal deductions o 1.1 Logical axioms

2 Conservative extensions o 2.1 Existential quantification

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o 2.2 Conjunction and Disjunction 3 Metatheorems 4 Alternative axiomatizations 5 Further connections 6 Notes

7 References 8 External links

[edit ] Formal deductions

In a Hilbert-style deduction system, a formal deduction is a finite sequence of formulas inwhich each formula is either an axiom or is obtained from previous formulas by a rule of inference. These formal deductions are meant to mirror natural-language proofs, althoughthey are far more detailed.

Suppose Γ is a set of formulas, considered as hypotheses . For example Γ could be a set of axioms for group theory or set theory . The notation means that there is a deductionthat ends with φ using as axioms only logical axioms and elements of Γ. Thus, informally,

means that φ is provable assuming all the formulas in Γ.

Hilbert-style deduction systems are characterized by the use of numerous schemes of logicalaxioms . An axiom scheme is an infinite set of axioms obtained by substituting all formulas

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of some form into a specific pattern. The set of logical axioms includes not only thoseaxioms generated from this pattern, but also any generalization of one of those axioms. Ageneralization of a formula is obtained by prefixing zero or more universal quantifiers on theformula; thus

is a generalization of .

[edit ] Logical axioms

There are several variant axiomatisations of predicate logic, since for any logic there isfreedom in choosing axioms and rules that characterise that logic. We describe here aHilbert system with nine axioms and just the rule modus ponens, which we call the one-rule axiomatisation and which describes classical equational logic. We deal with aminimal language for this logic, where formulas use only the connectives and and

only the quantifier . Later we show how the system can be extended to includeadditional logical connectives, such as and , without enlarging the class of deducibleformulas.

The first four logical axiom schemes allow (together with modus ponens) for themanipulation of logical connectives.

P1.P2.P3.

P4.

The axiom P1 is redundant, as it follows from P3, P2 and modus ponens.These axioms describe classical propositional logic ; without axiom P4we get (minimal) intuitionistic logic . Full intuitionistic logic is achievedby adding instead the axiom P4i for ex falso quodlibet , which is anaxiom of classical propositional logic.

P4i.

Note that these are axiom schemes, which represent infinitely manyspecific instances of axioms. For example, P1 might represent theparticular axiom instance , or it might represent

: the φ is a place where any formula can beplaced. A variable such as this that ranges over formulae is called a'schematic variable'.

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With a second rule of uniform substitution (US), we can change eachof these axiom schemes into a single axiom, replacing eachschematic variable by some propositional variable that isn'tmentioned in any axiom to get what we call the substitutionalaxiomatisation. Both formalisations have variables, but where the

one-rule axiomatisation has schematic variables that are outside thelogic's language, the substitutional axiomatisation uses propositionalvariables that do the same work by expressing the idea of a variableranging over formulae with a rule that uses substitution.

US. Let φ( p) be a formula with one or more instances of the propositional variable p ,and let ψ be another formula. Then from φ( p), infer φ(ψ) .

The next three logical axiom schemes provide ways to add,manipulate, and remove universal quantifiers.

Q5. where t may be substituted for x inQ6.Q7. where x is a free variable of .

These three additional rules extend the propositionalsystem to axiomatise classical predicate logic . Likewise, these three rules extend system forintuitionstic propositional logic (with P1-3 and P4i)to intuitionistic predicate logic .

Universal quantification is often given an alternativeaxiomatisation using an extra rule of generalisation(see the section on Metatheorems), in which case therules Q5 and Q6 are redundant.

The final axiom schemes are required to work withformulas involving the equality symbol.

I8. x = x for every variable x.

I9.

[edit ] Conservativeextensions

It is common to include in a Hilbert-stylededuction system only axioms forimplication and negation. Given theseaxioms, it is possible to form conservative

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extensions of the deduction theorem thatpermit the use of additional connectives.These extensions are called conservative

because if a formula φ involving newconnectives is rewritten as a logically

equivalent formula θ invol ving onlynegation, implication, and universalquantification, then φ is derivable in theextended system if and only if θ is derivablein the original system. When fully extended,a Hilbert-style system will resemble moreclosely a system of natural deduction .

[edit ] Existential quantification

Introduction

Elimination

where x is not a free variable of ψ.

[edit ] Conjunction andDisjunction

Conjunction introduction andelimination

introduction:elimination left:elimination right:

Disjunctionintroduction andelimination

introduction left:introduction right:

elimination:

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[edit ]Metatheorems

BecauseHilbert-stylesystems havevery fewdeductionrules, it iscommon toprovemetatheorems that showthatadditionaldeductionrules add nodeductivepower, in thesense that adeductionusing thenewdeductionrules can be

convertedinto adeductionusing onlythe originaldeductionrules.

Somecommonmetatheorem

s of this formare:

Thedeductiontheorem:

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if andonly if

.

if andonly if

and

. Contrapo

sition: If

then

. Generali

zation: If

and x does notoccurfree inanyformula

of Γ then

.

[edit ]Alternativeaxiomatizations

The axiom 3above iscredited toŁukasiewicz .[2] Theoriginalsystem by

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Frege hadaxioms P2and P3 butfour otheraxioms

instead of axiom P4(see Frege'spropositionalcalculus ).Russell andWhitehead alsosuggested asystem withfive

propositionalaxioms.

[edit ]Furtherconnections

Axioms P1,

P2 and P3,with thedeductionrule modusponens(formalisingintuitionisticpropositionallogic ),correspondtocombinatorylogic basecombinatorsI , K and S with theapplicationoperator.Proofs in the

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Hilbertsystem thencorrespondtocombinator

terms incombinatorylogic. Seealso Curry-Howardcorrespondence.

[edit ]Notes

1. ^ a b

Máté&Ruzsa1997:129

2. ^ A.Tarski,Logic,seman

tics,metamathematics,Oxford,1956

[edit ]References

Curry,HaskellB.;RobertFeys

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(1958).Combinatory

LogicVol. I . 1.

Amsterdam:NorthHolland.

Monk, J.Donald(1976).

Mathematical

Logic .Graduate

Texts inMathematics.Berlin,NewYork:Springer-Verlag . ISBN 978-0-387-90170-1 .

Ruzsa,Imre;Máté,András(1997)(inHungarian).

Bevezeté s amodernlogikába .Budapest: OsirisKiadó.

Tarski,Alfred(1990)(in

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Hungarian).

Bizonyítás ésigazság .

Budapest:Gondolat. It is aHungariantranslation of AlfredTarski 'sselected

papersonsemantictheory of truth .

DavidHilbert(1927)"Thefoundations of mathematics",translated byStephanBauer-Menglerberg andDagfinnFøllesdal(pp. 464

– 479).in:o van

Heijenoort,Jean(196

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7,3rdprinting1976

).FromFrege toGödel: ASour ce

Book in

Mathemat ical

Logic,1879 – 1931.CambrdigeMA:HarvardUniversityPress.ISBN 0-674-32449-8(pbk.).

Hilbert's 1927, Based on an earlier 1925 "foundations" lecture (pp. 367 – 392), presents his17 axioms -- axioms of implication #1-4, axioms about & and V #5-10, axioms of

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negation #11- 12, his logical ε -axiom #13, axioms of equality #14-15, and axioms of number #16-17 -- along with the other necessary elements of his Formalist "proof theory"-- e.g. induction axioms, recursion axioms, etc; he also offers up a spirited defenseagainst L.E.J. Brouwer's Intuitionism. Also see Hermann Weyl's (1927) comments andrebuttal (pp. 480 – 484), Paul Bernay's (1927) appendix to Hilbert's lecture (pp. 485 – 489)

and Luitzen Egbertus Jan Brouwer's (1927) response (pp. 490 – 495)

Kleene,StephenCole(1952,10thimpr

essionwith1971corrections).

Introduction of

Metamathemat ics .AmsterdamNY:NorthHollandPublishingCompany.ISBN 07204

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21039.

See in particular Chapter IV Formal System (pp. 69 – 85) wherein Kleene presentssubchapters §16 Formal symbols, §17 Formation rules, §18 Free and bound variables

(including substitution), §19 Transformation rules (e.g. modus ponens) -- and from thesehe presents 21 "postulates" -- 18 axioms and 3 "immediate-consequence" relationsdivided as follows: Postulates for the propostional calculus #1-8, Additional postulatesfor the predicate calculus #9-12, and Additional postulates for number theory #13-21.

[edit ]Exter

nallinks

Farmer,W.M."Propositionallogic" (pdf).http:/

/imps.mcmaster.ca/ cours

es/SE-2F03-05/slides/ 02-prop-

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logic.pdf .Itdescribes

(amongothers) apartof theHilbert-stylededu

ctionsystem(restrictedtopropositionalcalculus ).

Retrievedfrom"http://en.wikipedia.org/ wiki/ Hilbert_system" Viewpageratings

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RatethispageWhat'sthis?

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ctive

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Well-written

Iamhighlyknowledgeableabout thistopic

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I do not see this as David Hilbert/Blaise Pascal. I appreciate limitation.

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