Protasiakos Kai Katigorimatikos Logismos

Dom tou Majmatoc

IProtasiak Logik (Propositional Logic).

IKathgorhmatik Logik (Predicate Logic), h opoa enaignwst kai wc Prwtobjmia Logik (First-Order Logic).

Autc oi shmeiseic enai basismnec stic shmeiseic tou IanHodkinson ap to Kolgio Imperial, tou Panepisthmou touLondnou.

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Bibla

IAp th Logik sto Logik Programmatism kai thn Prolog.G. Mhtakdhc. Ekdseic Kardamtsa.

IStoiqea Majhmatikc Logikc. A. Tzoubrac.

Ekdseic Zth.

ILogik: H dom tou Epiqeirmatoc. D. Portdhc, S.

Ylloc kai D. Anapolitnoc. Ekdseic Neflh.

IDiakrit Majhmatik kai Majhmatik Logik, Tmoc G':

Majhmatik Logik. K. Dhmhtrakpouloc.

Ellhnik Anoikt Panepistmio.

I Logic in Computer Science: Modelling and Reasoning aboutSystems. M. Huth kai M. Ryan. Cambridge University Press.

I Artificial Intelligence: A Modern Approach. S. Russel kai P.Norvig.Kuklofore ellhnik kdosh me ttlo Teqnht Nohmosnh -

Ma Sgqronh Prosggish, ekdseic Kleidrijmoc.

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1. Logik: Eisagwg

ITo kommti thc Filosofac pou pragmateetai morfc

sullogismo kai skyhc, eidik epagwgc kai episthmonikc

mejdouc. Epiplon, Logik enai h susthmatik qrsh

sumbolikn teqnikn kai majhmatikn mejdwn gia ton

upologism morfn gkurou epagwgiko epiqeirmatoc.

ISe aut to mjhma ja asqolhjome me th qrsh thc

Logikc gia thn perigraf kai anptuxh prodiagrafn

(specification) logismiko, kai ton sumperasm (reasoning)sqetik me tic prodiagrafc.

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Logik: Eisagwg

IH Logik enai o logismc thc Plhroforikc: mamajhmatik bsh gia thn antimetpish plhroforac kai

gia ton sumperasm sqetik me th sumperifor

logismiko.

IParqei ma kal exskhsh gia to swst sumperasm kai

thn akrib perigraf.

IMerikc ap tic polplokec efarmogc Plhroforikc

qreizontai kai qrhsimopoion Logik. Pq, h Prolog enaima glssa programmatismo Logikc. H glssa bsewn

dedomnwn SQL enai basismnh se idec Logikc. Oiprodiagrafc eufun praktrwn logismiko (intelligentsoftware agents) ekfrzontai me th qrsh thc Logikc.

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Logik: Eisagwg

H Logik brskei efarmog (sunjwc apotele to upbajro)se arketc epistmec:

IFusik: Kbantologikc Logikc (quantum logics) qounepinohje gia thn jemelwsh thc kbantomhqanikc (quantummechanics).

IYuqologa: qrsh Logikn gia thn melth thc anjrpinhc

skyhc.

IGlwssologa: qrsh Logikc gia thn jemelwsh Logikn

Grammatikn (logical grammar) me skop thn anlush thcqrshc thc glssac ap touc anjrpouc. Pc, gia

pardeigma, qrhsimopoiontai ta onmata, oi antwnumec,

oi posodektec (loi-meriko-kpoioi klp), oi qrnoi

(upersuntlikoc -aristoc, klp).

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Logik: Eisagwg

'Ena ssthma Logikc perilambnei:

ISuntaktik (Syntax): ma tupik glssa (pwc maglssa programmatismo) h opoa qrhsimopoietai gia na

ekfrsei nnoiec, gia thn anaparstash thc gnshc.

IShmasiologa (Semantics). H shmasiologa tousustmatoc exetzei ton trpo me ton opoo oi ekfrseic

pou epitrpontai ap to suntaktik thc glssac

sqetzontai me to pragmatik touc perieqmeno.

ISnolo kannwn sumperasmatologac (Proof theory). Oikannec sumperasmatologac epitrpoun thn exagwg

logikn sumperasmtwn ap th gnsh tou sustmatoc.

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2. Protasiak Logik

Paraktw enai na mroc progrmmatoc:

if count>0 and not found thendecrement count; look for next entry;

end if

ITo prgramma perilambnei basikc (atomikc)protseic, pwc count>0, found, oi opoec enai alhjec yeudec ktw ap orismnec sunjkec.

IMporon na qrhsimopoihjon sndesmoi pwc and, or,not gia na anaptuqjon pio polplokec protseic ap ticatomikc protseic.

Ih telik polplokh prtash ja enai alhjc yeudc.

H protasiak logik den enai pol ekfrastik. H

kathgorhmatik logik enai pio ekfrastik.

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Protasiak Logik

I'Ola ta edh Logikc baszontai se kpoio bajm sthn

Protasiak Logik.

IJloume na antimetwpzoume if-tests opoiasdpotepoluplokthtac kai na meletsoume ta genik

qarakthristik touc.

IDen jloume apl na apotimome if-testsJloume na gnwrzoume an do if-tests qoun to dio nhma,pte to na sunepgetai to llo, an na if-test mporepot na enai alhjc ( yeudc).

IH Protasiak Logik perilambnei ma shmantik omda

upologistikn problhmtwn, kai sunepc enai antikemeno

melthc twn algorjmwn kai thc jewrac poluplokthtac.

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2.1 Suntaktik

Prta, prpei na dsoume na akrib orism sta edh twn

if-tests ta opoa ja antimetwpsoume. Autc o orismc ja macdsei thn tupik glssa (formal language) thc ProtasiakcLogikc. To alfbhto thc glssac thc Protasiakc Logikc

qei 3 sustatik.

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Prto sustatik: 'Atoma

De mac apasqole poia atomik gegonta (pq, count>0,found) ja antimetwpsoume. Arke na mporon na proun maalhjotim (dhlad na enai alhj yeud).

Oi protasiakc metablhtc (propositional variables) toma(propositional atoms) grfontai sunjwc wc p, p, p0, p1, p2, . . . wc q, r , . . .

Enai san tic metablhtc x , y , z sta Majhmatik.

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Detero sustatik: Logiko sndesmoi

'Eqoume touc paraktw sundsmouc (connectives):

Iszeuxh (kai): grfetai wc

Irnhsh (den): grfetai wc

Idizeuxh (): grfetai wc

Isunepagwg (an-tte): grfetai wc

Iisodunama (an kai mno an): grfetai wc

I truth kai falsity: grfontai wc > kai

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Paradegmata

ITo test count>0 and not found grfetai wc(count > 0) found, swc p q

ITa smbola gia szeuxh, dizeuxh, sunepagwg kai

isodunama parnoun do paramtrouc kai grfontai wc:

p q, p q, p q, p qIH rnhsh parnei ma parmetro kai grfetai wc:p, q

ITa >, den qoun paramtrouc.Enai logikc stajerc (pwc to pi).

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Trto sustatik: Shmea Stxhc

Qreiazmaste parenjseic gia na aposafhnsoume ta if-tests.

Pq, sthn arijmhtik to 1 2 + 3 den enai xekjaro, up thnnnoia ti mporome na to diabsoume wc (1 2) + 3 1 (2 + 3). H diafor pazei rlo.

Omowc, to p q r mpore na diabaste wc:I (p q) rI p (q r)

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Protseic

'Olec oi akoloujec sumblwn pou ekfrzoun toma,

sundsmouc kai parenjseic onomzontai ekfrseic.

Aut pou qoume onomsei if-tests sth Logik lgontaiprotseic (formulas) kalosqhmatismnec (well-formed)ekfrseic.

Orismc 2.1 (Prtash)

IKje tomo (p, q, r klp) enai prtash.

ITa > kai enai protseic.

IAn to A enai prtash tte enai kai to (A).

IAn ta A,B enai protseic tte enai kai ta(A B), (A B), (A B) kai (A B).

IOi mnec ekfrseic thc glssac pou enai protseic, enai

autc pou kataskeuzontai me touc parapnw kannec.

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Paradegmata Protsewn

Ta paraktw enai protseic:

I pI (p)I ((p) >)I (((p) >))I ((p) (((p) >)))Qreiazmaste kpoiouc kannec gia na meisoume ton arijm

parenjsewn.

Austhr milntac, ta apotelsmata ja enai suntomografec

protsewn. All jewrontai protseic.

Mporome pnta na paraleyoume tic exwterikc parenjseic:

pq, ta p kai ((p) >) enai saf.

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Kannec Apaloifc Parenjsewn

Gia na paraleyoume perissterec parenjseic, orzoume thn

paraktw proteraithta sundsmwn:

(isqurteroc) ,,,, (pio adnamoc)

Enai pwc sthn arijmhtik pou to enai isqurtero ap to+ kai sunepc to 2+34 diabzetai wc 2+(34) kai qi wc(2+3)4. Omowc:I p q r diabzetai wc p (q r) kai qi wc (p q) rI p q diabzetai wc (p) q kai qi wc (p q)I p q r diabzetai wc (p (q)) r kai qi wc

p ((q r)) p ((q) r))Genik mwc protenetai h qrsh twn parenjsewn tan bohjei

thn anagnwsimthta, parlo pou oi parenjseic mporon na

apaloifjon. Pq, to p q r s t enai pol dskolona katanohje.

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Kannec Apaloifc Parenjsewn

Ti gnetai me protseic pwc p q r ; H proteraithta twnsundsmwn den mac bohjei se aut thn perptwsh. H

parapnw prtash prpei na diabaste wc p (q r) wc(p q) r ;

Se ttoiec periptseic prpei opwsdpote na qrhsimopoiome

parenjseic.

Ap thn llh, oi protseic p q r kai p q r enai safecpwc enai.

'Opwc ja dome h logik shmasa enai h dia kai stic do

periptseic pwc kai na qrhsimopoisoume tic parenjseic. Oi

protseic (p q) r kai p (q r) enai diaforetikc. 'Allapwc ja dome qoun thn dia alhjotim se kje perptwsh:

enai logik isodnamec.'Ara de mac apasqole pc ja aposafhnsoume thn prtash

p q r . (Sunjwc enai (p q) r .)18 / 232

Dndra

'Eqoume dexei pc na diabzoume me safneia ma prtash

(formula).

Se kje prtash antistoiqe na dndro (formation tree), nadigramma dhlad pou deqnei pc kataskeuzetai h prtash

aut ap toma me th bojeia twn sundsmwn.

Pq, h prtash p q (p q r) qei to paraktw dndro:

p

pq

q r

-

V

/\

->

->

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Krioc Sndesmoc

Prosxte ti o sndesmoc sth rza (koruf) tou dndrou enai

. Autc enai o krioc sndesmoc (principal connective) thcprtashc p q (p q r). Aut h prtash qei thnlogik morf A B .Kje mh atomik prtash qei na krio sndesmo, o opooc

kajorzei th logik morf thc prtashc.

I p q r qei krio sndesmo . H logik thc morf enaiA B .

I (p q) qei krio sndesmo . H logik thc morfenai A.

I p q r qei krio sndesmo (pijantata to detero). Hlogik thc morf enai A B .

I p q r qei krio sndesmo . H logik thc morf enaiA B .Poia enai ta dndra autn twn protsewn;

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Upo-protseic

Oi upo-protseic (subformulas) mac prtashc A enai oiprotseic pou qrhsimopoiontai gia thn anptuxh thc A, pwcaut orzetai ap ton Orism 2.1.

Oi upo-protseic antistoiqon stouc kmbouc, sta

upo-dndra, tou dndrou thc A.

Oi upo-protseic thc p q (p q r) enai:p q (p q r)p q p q rp q p q r

p q r

Prosoq: uprqoun duo diaforetikc upo-protseic p.

Ta p q kai p q den enai upo-protseic, enaiupo-akoloujec (sub-strings) sumblwn.21 / 232

Suntomografec 1in Ai ,

ni=1 Ai ,

ni=1

Ai ,

1inAi

Ta parapnw enai suntomografec gia thn prtash

A1 A2 An. 1in Ai ,

ni=1 Ai ,

ni=1

Ai ,

1inAi

Ta parapnw enai suntomografec gia thn prtash

A1 A2 An.

Enai pwc sthn 'Algebra, pou to

ni=1 ai enai suntomografatou a1 + a2 + + an.22 / 232

Orologa

Orismc 2.2

IMa prtash thc morfc >,, p, gia na tomo p,onomzetai atomik prtash (atomic formula).

IStoiqeidhc tpoc (literal) enai kje atomik prtash hrnhsh thc.

IProgrammatikc tpoc (clause) enai ma dizeuxh enc perissotrwn stoiqeiwdn tpwn.

Pq, oi protseic p,r ,,> enai lec stoiqeideic tpoi.

Paradegmata programmatikn tpwn:

p, p, p q r , p p p > q.

O kenc programmatikc tpoc, o opooc den periqei kanna

stoiqeidh tpo, antimetwpzetai pwc h prtash (stobiblo o kenc programmatikc tpoc sumbolzetai me ).23 / 232

2.2 Shmasiologa

Gnwrzoume pc na diabzoume kai na grfoume protseic

(formulas). Poio enai mwc to nhma touc; Me lla lgia,poia enai h shmasiologa touc;

Oi logiko sndesmoi (,,,,) qoun kpoia antistoiqame thn Ellhnik glssa.

All ta Ellhnik enai ma fusik glssa, gemth me

asfeiec kai eidikc periptseic. H metfrash metax

Ellhnikn kai Logikc den enai pnta ekolh.

Qreiazmaste na akrib trpo me ton opoo na dnoume nhma

stic protseic thc Logikc.

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Katstash, Aponom aljeiac

Ma katstash (situation) enai kti to opoo orzei an kjetomo enai alhjc yeudc.

Aponom aljeiac onomzoume kje sunrthsh F : Q 7 {,}pou Q enai to snolo twn atmwn thc glssac. Dhlad maaponom aljeiac dnei alhjotimc sta toma thc glssac.

Gia ta if-tests ma katstash enai na shmeo sthn ektleshtou progrmmatoc. Oi trqousec timc twn metablhtn tou

progrmmatoc kajorzoun an kje atomik kfrash enc if-test(pq, x > 0, x=y klp) enai alhjc yeudc.

Gia thn kfrash if (it rains) then (it is cloudy) ma katstashenai o kairc.

Gia ta toma p, q, . . . ma katstash diatupnei poia apaut enai alhj kai poia enai yeud.

Uprqei pnw ap ma katstash. Profanc, se ma

diaforetik katstash oi alhjotimc ja enai diaforetikc.

Gnwrzontac ma katstash mporome na upologsoume to

nhma kje prtashc, dhlad an h prtash enai alhjc

yeudc se aut thn katstash. Phganoume ap aplc

protseic se pio snjetec, pwc ja dome eujc amswc.

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Katstash, Aponom aljeiac

Uprqei pnw ap ma katstash. Profanc, se ma

diaforetik katstash oi alhjotimc ja enai diaforetikc.

Gnwrzontac ma katstash mporome na upologsoume to

nhma kje prtashc, dhlad an h prtash enai alhjc

yeudc se aut thn katstash. Phganoume ap aplc

protseic se pio snjetec, pwc ja dome eujc amswc.

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Ermhnea

Orismc 2.3 ( Ermhnea (Evaluation) )

H alhjotim mac prtashc se ma dedomnh katstash

orzetai wc exc.

IH katstash mac plhrofore gia tic alhjotimc twn

atmwn.

IH prtash > enai alhjc kai h enai yeudc.Ac upojsoume ti A,B enai protseic, kai ti gnwrzoume ticalhjotimc touc sth dedomnh katstash. Tte se aut thn

katstash:

IH prtash A enai alhjc an h A enai yeudc, kai yeudcan h A enai alhjc.

IH prtash AB enai alhjc an h A kai h B enai alhjec.Se opoiadpote llh perptwsh h A B enai yeudc.

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Ermhnea

Orismc 2.3 (Ermhnea) (sunqeia)

IH prtash A B enai alhjc an h A enai alhjc h Benai alhjc kai h A kai h B enai alhjec. Seopoiadpote llh perptwsh h A B enai yeudc.(Periektik dizeuxh.)

IH prtash A B enai alhjc an h A enai yeudc h Benai alhjc. Allic, an h A enai alhjc kai h B enaiyeudc tte h A B enai yeudc.

IH prtash A B enai alhjc an h A kai h B qoun thndia alhjotim (enai kai oi do alhjec kai oi do

yeudec).

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Pnakec Aljeiac gia touc sundsmouc

Mporome na ekfrsoume autoc touc kannec

qrhsimopointac pnakec aljeiac gia touc sundsmouc.

Grfoume 1 gia thn tim alhjc kai 0 gia thn tim yeudc.Aut den qoun sqsh me ta >, ta opoa enai protseic kaiqi alhjotimc.

A A > 1 0 1 0

0 1 1 0

A B A B A B A B A B1 1 1 1 1 1

1 0 0 1 0 0

0 1 0 1 1 0

0 0 0 0 1 1

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Paradegmata

Ac upojsoume ti h prtash p enai alhjc kai h q enaiyeudc se ma sugkekrimnh katstash. Tte se aut thn

katstash h prtash:

I p enai yeudc.I q enai alhjc.I p q enai yeudc.I p q enai alhjc.I p q enai yeudc.I q p enai alhjc.I p q enai yeudc kai h p q enai alhjc.

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Paradegmata

Ac upojsoume tra ti h prtash p enai yeudc kai h q enaialhjc. Se aut thn kainorgia katstash h prtash:

I p enai tra alhjc.I q enai tra yeudc.I p q enai akma yeudc.I p q enai akma alhjc.I p q enai tra alhjc.I q p enai tra yeudc.I p q enai akma yeudc kai h p q enai akmaalhjc.

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Paradegmata

Ac upojsoume pli ti h prtash p enai alhjc kai h q enaiyeudc se ma sugkekrimnh katstash. Poia enai h alhjotim

twn paraktw protsewn:

I p qI q qI p qI p q qI (p >)I (q (p q))

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2.3 Sqsh tou Protasiako Logismo me th Fusik

Glssa

H tupopohsh thc fusikc glssac, dhlad h metfrash twn

protsewn thc fusikc glssac sthn Protasiak Logik den

enai pnta ekolh. Enai mwc aparathth gia orismnec

efarmogc.

Merik stoiqea pou prpei na prosxoume sthn tupopohsh

thc fusikc glssac:

IApomonnoume tic atomikc protseic.

IProsqoume an kpoiec ap tic atomikc protseic enai

tautshmec.

IAntikajistome diakritc atomikc protseic me diakrit

toma (p, q, r , . . . ).

IEntopzoume louc touc logikoc sundsmouc (aut apaite

thn apdosh twn sundsmwn thc fusikc glssac se

antstoiqouc thc tupikc).

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Tupopohsh thc fusikc glssac

IEntopzoume tic mikrterec (snjetec) protseic oi opoec

apotelon th snjeth prtash, kai tic diakrnoume me

parenjseic.

IEntopzoume ton krio logik sndesmo pou sundei tic

snjetec protseic metax touc, kai proqwrome ap' autn

stouc epimrouc sundsmouc.

IApoddoume thn prtash qrhsimopointac ta sqetik

logik smbola kai parenjseic.

Paradegmata tupopohshc thc fusikc glssac uprqoun sto

biblo Logik: H Dom tou Epiqeirmatoc.

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Pardeigma: An den psei o plhjwrismc kai anboun taepitkia, tte to dhmsio qroc ja psei an kai mno an

katafgoume se exwterik daneism.

Atomikc protseic:

p: o plhjwrismc pfteiq: ta epitkia anebanounr : to dhmsio qroc pfteis: katafegoume se exwterik daneism

Logiko sndesmoi: an . . . tte, den, kai, an kai mno an

Prth prosggish logikc morfc:

an (den p kai q), tte (r an kai mno an s)Krioc sndesmoc: an . . . tteLogik morf: (p q) (r s)35 / 232

Problmata sthn tupopohsh thc fusikc glssac

To all mpore na shmanei kai.ja bgw xw all brqei metafrzetai seja bgw xw brqei.

ektc an genik shmanei ja bgw xw ektc an brqei metafrzetai seja bgw xw ja brqei.Epshc mpore na ekfraste wc (ja brqei) ja bgw xw.

To suqn shmanei kaimporec na preic kotpoulo yri sunjwc shmaneimporec na preic kotpoulo mporec na preic yri

o Ginnhc kai h Mara enai eutuqismnoi maz prpei nametafraste wc ma atomik prtash, qi wc

o Ginnhc enai eutuqismnoc h Mara enai eutuqismnh36 / 232

Ap thn tupik glssa sth fusik

Arqik h metfrash thc tupikc glssac sth fusik fanetai

ekolh. Pq:

I p q metafrzetai se p kai q.I brqei ja bgw xw metafrzetai se an brqei tteden ja bgw xw.

All uprqoun kai problmata. Oi polplokec protseic thc

Protasiakc Logikc enai dskolo na apodojon sth fusik

glssa.

O sndesmoc enai dskolo na apodoje. Sth fusik glssaqrhsimopoiome to an . . . tte me diaforetikoc trpouc, kaiqi pnta prosektik.

H prtash Emai o Ppac emai jeoc enai alhjc(giat;). All ja lgame pot ti h prtash thc fusikcglssac An emai o Ppac tte emai jeoc enai alhjc;

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3. Epiqeirmata, egkurthta

Gnwrzoume pc na diabzoume, na grfoume kai na

ermhneoume (evaluate) protseic thc Protasiakc Logikc.Gnwrzoume epshc merik prgmata sqetik me th metfrash

metax thc fusikc glssac kai thc Protasiakc Logikc.

H Logik qrhsimopoietai kai gia epiqeirhmatologa. Pq,

paraktw qoume to exc epiqerhma:

IO Swkrthc enai ndrac.

IOi ndrec enai jnhto.

I'Ara o Swkrthc enai jnhtc.

Enai aut na gkuro epiqerhma;

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3.1 'Egkura epiqeirmata

To prohgomeno epiqerhma enai gkuro an gia kje

katstash sthn opoa o Swkrthc enai ndrac kai oi ndrec

enai jnhto, o Swkrthc enai ndrac.

'Opwc qoume anafrei, sthn Protasiak Logik ma

katstash orzei thn alhjotim kje atmou.

Orismc 3.1 ( 'Egkuro Epiqerhma (Valid Argument) )

Dojousn twn protsewn A1,A2, . . . ,An,B , na epiqerhma

A1, . . . ,An, ra B

enai gkuro an: h prtash B enai alhjc se kje katstashsthn opoa lec oi protseic A1, . . . ,An enai alhjec. 'An isqeiaut tte grfoume A1, . . . ,An |= B .To smbolo |= diabzetai wc logik sunpeia (logicalentailment/implication).39 / 232

Paradegmata epiqeirhmtwn

Ac upojsoume ti A,B enai tuqaec protseic. To epiqerhma:

I A, ra A enai gkuro giat se kje katstash, an h Aenai alhjc tte h A enai alhjc. A |= A.

I A B , ra A enai gkuro. A B |= A.I A, ra A B den enai gkuro: uprqoun katastseicpou h A enai alhjc kai h A B enai yeudc. SunepcA 2 A B .

I A,A B , ra B enai gkuro. To noma auto touepiqeirmatoc enai modus ponens.

I A B,B , ra A enai epshc gkuro. To nomaauto tou epiqeirmatoc enai modus tollens.

I A B,B , ra A den enai gkuro. A B,B 2 A.

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3.2 'Egkurec, ikanopoisimec, isodnamec protseic

Treic basikc idec sqetzontai me ta gkura epiqeirmata.

Orismc 3.2 ( 'Egkurh Prtash (Valid Formula) )

Ma prtash enai logik gkurh logik alhjc an enai

alhjc se kje katstash. Grfoume |= A tan ma prtashA enai gkurh.

Oi gkurec protseic onomzontai kai tautologec (tautologies).

Orismc 3.3 ( Ikanopoisimh Prtash (Satisfiable Formula) )

Ma prtash enai ikanopoisimh epalhjesimh an enai

alhjc se toulqiston ma katstash.

Orismc 3.4 ( Isodnamec Protseic (Equivalent Formulas) )

Duo protseic A,B enai logik isodnamec an enai alhjecstic diec akribc katastseic. Sumbolik A B .

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Paradegmata

Prtash 'Egkurh Ikanopoisimh

> nai nai qi qip qi nai

p p qi qip p nai nai

Isodnamh; p > p qp p nai qi qi

p p qi nai qip q qi qi nai

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3.3 Sqseic metax twn tessrwn ennoin

Ta gkura epiqeirmata kai oi gkurec, ikanopoisimec kai

isodnamec protseic orzontai metax touc. Pq:

epiqerhma egkurthta ikanopoih- isodunama

simthta

A |= B A B gkurh A B (A B) >mh ikanopoisimh

> |= A A gkurh A A >mh ikanopoisimh

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4. Elgqontac thn egkurthta twn protsewn

'Ara qreizetai na antimetwpsoume mno gkurec protseic.

Pc katalabanoume an ma prtash enai gkurh; Elgqoumeti h prtash enai alhjc se kje katstash.

Aut mpore na epiteuqje gia protseic thc Protasiakc

Logikc, all upologistik enai na dskolo prblhma.

Gia thn Kathgorhmatik Logik, thn opoa ja dome sta

epmena majmata, enai pol pio dskolo prblhma, giat oi

katastseic enai pollc kai pio polplokec. All uprqoun

kpoioi trpoi na dexoume ti orismnec protseic thc

Kathgorhmatikc Logikc enai gkurec.

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Trpoi na elegqje h egkurthta twn protsewn

Uprqoun diforoi trpoi na upologsoume an ma prtash

thc Protasiakc Logikc enai gkurh qi:

IPnakec aljeiac. Elgqoume an se lec tic pijanc

sqetikc katastseic h prtash enai alhjc.

I'Amesh majhmatik epiqeirhmatologa (directmathematical argument).

IIsodunamec. 'Eqoume ma lsta ap qrsima zeugria

isodnamwn protsewn. Qrhsimopoiome autc tic

isodunamec gia na metatryoume thn arqik prtash sthn

prtash >, h opoa enai gkurh.IDifora sustmata apodexewn (proof systems), pwc pq,kannec fusikc sumperasmatologac (natural deduction),sustmata Hilbert, shmasiologiko pnakec (semantictableaux).

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4.1 Pnakec aljeiac

Ac dexoume ti h prtash (p q) (p q) enai gkurh.Grfoume 1 gia thn tim alhjc kai 0 gia thn tim yeudc.

'Eqoume do toma, pou to kajna parnei ma ap tic do

alhjotimc. Oi alhjotimc llwn atmwn den mac apasqolon.

'Ara qoume tsseric sqetikc katastseic. Ermhneoume lec

tic upo-protseic thc prtashc mac se kje katstash:

p q p q p p q (p q) (p q)1 1 1 0 1 1

1 0 0 0 0 1

0 1 1 1 1 1

0 0 1 1 1 1

Blpoume ti h prtash (p q) (p q) enai alhjc (qeitim 1) se lec tic tsseric katastseic. 'Ara enai gkurh.

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Isodunama kai ikanopoihsimthta

O dioc pnakac deqnei ti oi protseic p q kai p q enaiisodnamec: se kje ma ap tic tsseric katastseic qoun

thn dia alhjotim.

Ti elgqoume gia na dexoume ikanopoihsimthta;

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Meionektmata kai pleonektmata twn pinkwn aljeiac

Enai kourastik kai enai ekolo na gnei ljoc.

All douleoun pnta, toulqiston gia thn Protasiak

Logik pou mno peperasmnec katastseic sqetzontai me

ma prtash. (Aut den isqei gia thn Kathgorhmatik Logik.

Sthn perptwsh aut prpei na brome llouc trpouc.)

Oi pnakec aljeiac mporon epiplon na qrhsimopoihjon gia

na dexoume ikanopoihsimthta kai isodunama.

Deqnoun pso dskolo enai na apodeiqje h egkurthta,

ikanopoihsimthta, klp. Ma prtash me n toma qreizetai2n grammc ston pnaka aljeiac thc. Prosjtontac na tomodiplasizei tic grammc tou pnaka.

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4.2 'Amesh majhmatik epiqeirhmatologa (Directargument)

Ac dexoume ti h prtash p p q enai gkurh.

Ac proume ma tuqaa katstash, h opoa dnei alhjotimc

sta toma p, q. Ja dexoume ti h prtash p p q enaialhjc se aut thn katstash.

'Ara prpei na dexoume ti AN to p enai alhjc se aut thnkatstash TOTE enai alhjc kai h prtash p q.

An to p enai alhjc, tte kai na ap ta p, q enai alhjc, rah prtash p q enai alhjc.

'Askhsh: dexte ti h prtash A (A B) B enai gkurh.

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'Amesh majhmatik epiqeirhmatologa

Ac dexoume ti h prtash (A B) C enai logik isodnamhme thn prtash A (B C ). (Omowc, ja mporosame nadexoume ti h prtash (AB)C A (B C ) enai gkurh.)

Ac proume ma opoiadpote katstash.

H (A B) C enai alhjc se aut thn katstash an kai mnoan h A B kai h C enai alhjec (smfwna me ton orism thcshmasiologac tou ).

To parapnw isqei an kai mno an h A kai h B enai alhjec,kai h C enai epshc alhjc, dhlad enai lec alhjec.

Aut isqei an kai mno an h A enai alhjc, kai epshc h B kaih C enai alhjec.

Aut isqei an kai mno an h A kai h B C enai alhjec.50 / 232

'Amesh majhmatik epiqeirhmatologa

Aut isqei an kai mno an h A (B C ) enai alhjc.

'Ara oi protseic (A B) C kai A (B C ) qoun thn diaalhjotim se aut thn katstash. H katstash tan tuqaa,

ra oi protseic enai logik isodnamec.

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'Amesh Majhmatik Epiqeirhmatologa

Ac dexoume ti h prtash ((p q) p) p, h opoa enaignwst wc nmoc tou Peirce, enai gkurh.

Ac proume ma tuqaa katstash. An to p enai alhjc seaut thn katstash, tte h prtash ((p q) p) p enaialhjc, giat kje prtash thc morfc A B enai alhjctan h B enai alhjc.

An to p den enai alhjc tte ja enai yeudc se aut thnkatstash.

Tte h prtash p q enai alhjc, giat kje prtash thcmorfc A B enai alhjc tan h A enai yeudc.

Tte h prtash (p q) p enai yeudc giat kje prtashthc morfc A B enai yeudc tan h A enai alhjc kai h Benai yeudc.

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'Amesh Majhmatik Epiqeirhmatologa

Tte h prtash ((p q) p) p enai alhjc giat kjeprtash thc morfc A B enai alhjc tan h A enai yeudc.

Aut h morf epiqeirhmatologac onomzetai

epiqeirhmatologa me periptseic (argument by cases): to penai alhjc to p enai yeudc. Den uprqoun llecperiptseic. Autc enai o nmoc thc tou trtou apoklesewc

(law of excluded middle).

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4.3 Isodunamec

Oi isodunamec mac bohjon sto na aplopoisoume ma

prtash na metatryoume ma prtash se ma llh, pnta

diathrntac th logik isodunama.

'Ara an mporsoume na metatryoume ma prtash sthn

prtash > tte xroume ti h arqik prtash enai gkurh.

Ja anafroume ma lsta isodunamin oi opoec enai qrsimec

gi' aut ton skop.

Oi isodunamec enai pollc, all lec ekfrzoun basikc

logikc arqc pou ja prpei na gnwrzete. Na elgxete ti oi

protseic enai ntwc isodnamec qrhsimopointac pnakec

aljeiac mesh epiqeirhmatologa.

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Isodunamec pou perilambnoun to sndesmo

Stic paraktw isodunamec ta A,B,C ekfrzoun tuqaecprotseic. Gia suntoma pollc forc ja grfoume

isodnamec ant gia logik isodnamec.

1. H prtash A B enai logik isodnamh me thn B A(antimetajetikthta tou ).2. H prtash A A enai logik isodnamh me thn A(autopjeia tou ).3. H prtash A > enai logik isodnamh me thn A.4. Oi protseic A kai A A enai isodnamec me thn .5. H prtash (A B) C enai isodnamh me thn A (B C )(prosetairistikthta tou ).

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6. H prtash A B enai isodnamh me thn B A(antimetajetikthta tou ).7. H prtash AA enai isodnamh me thn A (autopjeia tou).8. Oi protseic A > kai A A enai isodnamec me thn >.9. H prtash A enai isodnamh me thn A.10. H prtash (A B) C enai isodnamh me thn A (B C )(prosetairistikthta tou ).

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Isodunamec pou perilambnoun touc sundsmouc ,kai

11. H prtash > enai isodnamh me thn .12. H prtash enai isodnamh me thn >.13. H prtash A enai isodnamh me thn A.14. H prtash A A enai isodnamh me thn >.15. H prtash > A enai isodnamh me thn A.16. H prtash A > enai isodnamh me thn >.17. H prtash A enai isodnamh me thn >.18. H prtash A enai isodnamh me thn A.19. H prtash A B enai isodnamh me thn A B kai thn(A B).20. H prtash (A B) enai isodnamh me thn A B .

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21. H prtash A B enai isodnamh me tic:I (A B) (B A),I (A B) (A B),I A B.22. H prtash (A B) enai isodnamh me tic:

I A B,I A B,I (A B) (A B).

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Nmoi De Morgan kai epimeristikthtac twn ,

Nmoi De Morgan:

23. H prtash (A B) enai isodnamh me thn A B .24. H prtash (A B) enai isodnamh me thn A B .

Nmoi epimeristikthtac twn ,:25. H prtash A (B C ) enai isodnamh me thn

(A B) (A C ).26. H prtash A (B C ) enai isodnamh me thn

(A B) (A C ).27. Oi protseic A (A B) kai A (A B) enai isodnamecme thn A.

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Apodeiknontac egkurthta me isodunamec

Prte opoiadpote upo-prtash mac prtashc A kaiantikatastste thn me ma isodnamh prtash.

Epanalbete to prohgomeno bma. To apotlesma ja enai

ma prtash logik isodnamh me thn A.An to apotlesma enai h prtash > tte h A enai gkurh.Pq, ac dexoume ti h prtash p p q enai gkurh.IAut h prtash enai isodnamh me thn p (p q) (apthn isodunama 19).

IH teleutaa enai isodnamh me thn (p p) q (ap thnisodunama 10).

IH teleutaa enai isodnamh me thn > q (ap thnisodunama 8).

IH teleutaa enai isodnamh me thn > (ap thn 8 pli), hopoa enai gkurh.

Ftsame sthn prtash >, h opoa enai gkurh. 'Ara hprtash p p q enai gkurh.60 / 232

Pardeigma

Ac dexoume ti h prtash A B enai isodnamh me thnB A.IH prtash B A enai isodnamh me thn (B) A(ap thn isodunama 19).

IH teleutaa enai isodnamh me thn A B (ap thnisodunama 6).



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Pardeigma

Ac dexoume ti h prtash (p q) (p q) enai logikisodnamh me thn p.

IH prtash (p q) (p q) enai isodnamh me thnp (q q) (anstrofh qrsh thc isodunamac 26).

IH teleutaa enai isodnamh me thn p (ap thnisodunama 4).

IH teleutaa enai isodnamh me thn p (ap thn isodunama9).

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Kanonikc morfc

Oi isodunamec mac epitrpoun na metatryoume ma prtash

se ma isodnam thc se kanonik morf (normal form).Uprqoun do koinc kanonikc morfc:

Orismc 4.1 (Diazeuktik Kanonik Morf, Suzeuktik

Kanonik Morf)

IMa prtash enai se diazeuktik kanonik morf (DKM)

(disjunctive normal form (DNF)) an enai ma dizeuxhsuzexewn stoiqeiwdn tpwn, kai den mpore peraitrw na

aplopoihje allzontac morf.

Paradegmata:

p q r(p q) r (p q r)Anti-pardeigma: (p p) (q > q)

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Kanonikc morfc

Orismc 4.1 (sunqeia)

IMa prtash enai se suzeuktik kanonik morf (SKM)

(conjunctive normal form (CNF)) an enai ma szeuxhdiazexewn stoiqeiwdn tpwn (dhlad ma szeuxh

programmatikn tpwn), kai den mpore peraitrw na

aplopoihje allzontac morf.

Pardeigma:

(p q) (q r) (p q)

O Orismc 2.2 orzei touc stoiqeideic tpouc kai touc

programmatikoc tpouc.

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Metatrpontac ma prtash se kanonik morf

1. Apalofoume touc sundsmouc ,:Antikajistome lec tic upo-protseic A B me A B .Antikajistome lec tic upo-protseic A B me(A B) (A B).2. Prowjome tic arnseic msa stic parenjseic mqri ta

toma qrhsimopointac touc nmouc De Morgan. Bgzoumetic diplc arnseic (antikajistome thn A me thn A).3. Qrhsimopoiome touc kannec epimeristikthtac gia na

katalxoume sthn epijumht kanonik morf.

4. Aplopoiome: antikajistome tic upo-protseic p p me, kai tic p p me >. Antikajistome tic > p me >, tic> p me p, tic p me p, kai tic p me . H isodunama27 enai epshc suqn qrsimh. Epanalambnoume mqri h

prtash na mhn enai peraitrw aplopoisimh.

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Pardeigma: h p q (p r) se DKM

p q (p r) [arqik prtash](p q) (p r) [apaloif ](p q) ((p r) (p r)) [apaloif ,]p q (p r) (p r) [De Morgan, apaloif ]p (p r) (p r) q [isodunama 6]p (p r) q [isodunama 27]H teleutaa prtash enai se DKM. Mporome na thn

aplopoisoume peraitrw an jloume proswrin na fgoume

ap thn DKM:

[(p p) (p r)] q [isodunama 26-epimeristikthta][> (p r)] q [isodunama 8]p r q [isodunama 3]H shmantik aplopohsh thc arqikc prtashc deqnei to

pleonkthma thc metatropc mac prtashc se kanonik morf.

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5. Sustmata apodexewn: fusik sumperasmatologa

'Ena ssthma apodexewn mac bohjei na deqnoume ti ma

prtash enai gkurh qrhsimopointac mno suntaktikoc

kannec den qrhsimopoiome to nhma twn protsewn. Ja

mporosame na gryoume na prgramma logismiko to opoo

efarmzei autoc touc kannec.

H automatopoihmnh sumperasmatologa (automated reasoning)enai ma anaptussmenh ereunhtik perioq.

Uprqoun arket sustmata apodexewn. Sta epmena

majmata ja asqolhjome me to ssthma kannwn fusikc

sumperasmatologac (natural deduction (ND)).

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Ti enai to ssthma ND;IMa tupopohsh thc meshc epiqeirhmatologac (directargument).

IXekinntac swc ap kpoia dojnta gegonta kpoiec

protseic A1, . . . ,An qrhsimopoiome touc kannec tousustmatoc gia sumperasmatologa proc ma prtash B .An epitqoume, grfoume A1, . . . ,An ` B .

IKat th sumperasmatologa pargontai endimesec

protseic. Autc apotelon thn apdeixh thc B ap ticA1, . . . ,An. Kje bma thc apdeixhc enai na gkuroepiqerhma.

IUprqoun do kannec gia kje sndesmo: nac gia na

eisgoume to sndesmo (dhlad na eisgoume ma prtash

thc opoac enai o krioc sndesmoc), kai nac gia na

qrhsimopoisoume to sndesmo (na qrhsimopoisoume ma

prtash thc opoac enai o krioc sndesmoc). Oi kannec

enai basismnoi sthn shmasiologa twn sundsmwn, thn

opoa edame sta prohgomena majmata.

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5.1 Kannec ND gia touc sundsmouc ,,Uprqoun do kannec gia kje sndesmo. Oi kannec

baszontai sto nhma twn sundsmwn.

Kannec gia to sndesmo :I(eisagwg , -introduction, I ). Gia na eisgoume maprtash thc morfc A B , prpei na qoume dh eisgeitic protseic A kai B .

1 A to qoume apodexei aut . . ..

.

. (lla sumpersmata)

2 B kai aut . . .3 A B I (1,2)H arjmhsh twn grammn enai aparathth gia thn

katanhsh thc apdeixhc.

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Kannec ND gia to sndesmo

I(apaloif , -elimination, E ). An qoume katafrei nagryoume A B , tte mporome na gryoume A kai/ B .1 A B to qoume apodexei aut2 A E (1)3 B E (1)

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Upojseic

Stic apodexeic ND prpei suqn na knoume upojseic gia naapodexoume aut pou jloume.

Ma upjesh enai apl ma prtash, all qrhsimopoietai me

na eidik trpo. Upojtoume ti (qoume ma katstash sthn

opoa) h prtash enai alhjc. 'Epeita mpore na

proqwrsoume sthn apdeix mac kajc gnwrzoume

perisstera gia thn katstash.

Pq, o Srlok Qolmc upojtei gia mia stigm ti autc pou

kleye to logo th nqta tan gnwstoc. Tte enai sgouroc

ti ta skuli tou stblou ja gbgizan kai sunepc ja eqan

xupnsei oi flakec tou stblou.

All oi flakec den xpnhsan kat th dirkeia thc nqtac. O

Qolmc sumperanei ti ta skuli den gbgisan th nqta, ra

autc pou kleye to logo den tan gnwstoc.

H upjesh tou tan ljoc parl' aut mporose na knei

aut thn upjesh, kai tan qrsimo pou thn kane.

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Kannec ND gia to sndesmo I(eisagwg , -introduction, I ). Gia na eisgoumema prtash thc morfc A B , upojtoume ti A kaiapodeiknoume ti B .Kat th dirkeia thc apdeixhc, mporome na

qrhsimopoisoume thn A kai otidpote llo qoume dhapodexei. All den mporome na qrhsimopoisoume thn A otidpote llo ap thn apdeixh thc B ap thn Aargtera (giat aut h apdeixh tan basismnh se ma

epiplon upjesh). Sunepc, apomonnoume thn apdeixh

thc B ap thn A se na kout:

1 A upjesh(h apdeixh)

2 B to apodexame!

3 A B I (1, 2)Kama ap tic protseic pou enai msa sto kout den

mpore na qrhsimopoihje argtera.

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Sto ssthma ND ta kouti qrhsimopoiontai tan knoumeepiplon upojseic. H prth gramm msa sto kout prpei

pnta na onomzetai upjesh me ma exaresh, thn opoaja dome argtera.

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I(apaloif , -elimination, E ). An qoume katafreina gryoume A kai A B , me opoiadpote seir, ttemporome na gryoume B (Modus Ponens).

1 A B to apodexame aut . . ..

.

.

2 A kai aut . . .3 B E (1, 2)

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I(eisagwg , -introduction, I ). Gia na apodexoumema prtash thc morfc A B , apodeiknoume thn A, (anprotimme) thn B .

1 A to apodexame aut . . .2 A B I (1)H B mpore na enai opoiadpote prtash!

1 B to apodexame aut . . .2 A B I (1)H A mpore na enai opoiadpote prtash!

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Kannec ND gia to sndesmo I(apaloif , -elimination, E ). Gia na apodexoumekti ap ma prtash thc morfc A B , prpei na toapodexoume upojtontac thn A, KAI na to apodexoumeupojtontac thn B .Aut enai epiqeirhmatologa me periptseic.

1 A B to qoume apodexei aut2 A upjesh 5 B upjesh

3

.

.

. prth apdeixh 6

.

.

. deterh apdeixh

4 C to apodexame 7 C to apodexame xan

8 C E (1, 2, 4, 5, 7)Oi upojseic A,B den mporon na qrhsimopoihjonargtera, sunepc topojetontai se (diplan) kouti.

Kama ap tic protseic pou enai msa sta kouti den

mpore na qrhsimopoihje argtera.

H C mpore na enai ma opoiadpote prtash, all kai tado kouti prpei na teleinoun me thn dia prtash C .76 / 232

5.2 Paradegmata

Akma den qoume dexei touc kannec gia touc sundsmouc

,>,. Parl' aut mporome na dome kpoia paradegmata.Pardeigma 5.1

Ac apodexoume ti A A B me to ssthma ND (pou oi A,Benai tuqaec protseic).

Thn egkurthta autc thc prtashc thn apodexame me mesh

epiqeirhmatologa nwrtera sto mjhma. Sugkrnete tic do

apodexeic.

1 A upjesh2 A B I (1)3 A A B I (1, 2)

H apdeixh sth fusik glssa: Se kpoia katstashupojtoume ti h A enai alhjc. Tte kai h AB enai alhjc.'Ara h A A B enai alhjc se opoiadpote katstash.77 / 232

Oi kannec enai suntaktiko

Oi kannec tou sustmatoc ND enai basismnoi sthnshmasiologa twn sundsmwn. Parl' aut enai suntaktiko

kannec. Ma apdeixh me to ssthma ND enai suntaktik.

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Paradegmata

Pardeigma 5.2

Dojeshc thc prtashc A (B C ) apodexte ti AB C .1 A (B C ) dedomno2 A B upjesh3 A E (2)4 B C E (1,3)5 B E (2)6 C E (4, 5)7 A B C I (2, 6)Sth fusik glssa: Jewrome ti h prtash A (B C )enai alhjc (se kpoia katstash). Gia na dexoume ti h

A B C enai alhjc, upojtoume ti h A B enai epshcalhjc (se aut thn katstash), kai deqnoume ti h C enaialhjc. Efson qoume A B tte qoume kai A. Epomnwc,qrhsimopointac thn A (B C ) blpoume ti h B C enaialhjc. H A B mac dnei th B . Sunepc qoume kai thn C .79 / 232

Sumperanoume:

Se kje katstash sthn opoa h prtash A (B C ) enaialhjc, h A B C enai epshc alhjc.

Ap ed kai sto exc suqn ja paralepoume th frsh enaialhjc.

80 / 232

To smbolo `Apodexame thn A B C ap th dojesa A (B C ).Sunepc mporome na gryoume:

A (B C ) ` A B COrismc 5.3

'Estw A1, . . . ,An,B tuqaec protseic.

A1, . . . ,An ` B

shmanei ti h uprqei ma apdeixh (fusikc

sumperasmatologac (ND)) thc B , xekinntac ap ticprotseic A1, . . . ,An (dedomna).

I ` B (dhlad tan n=0) shmanei ti mporome naapodexoume thn B qwrc dedomna. Me lla lgia h Benai jerhma (thc fusikc sumperasmatologac).

I A1, . . . ,An ` B diabzetai wc h B enai apodeximh ap ticA1, . . . ,An kai ` B diabzetai wc h B enai jerhma.81 / 232

To smbolo `

IDen prpei na sugqoume to ` me to .To ` enai suntaktik kai perilambnei apodexeic.To enai shmasiologik kai perilambnei katastseic.

82 / 232

Paradegmata

Pardeigma 5.4

A B C ` A (B C ).

1 A B C dedomno2 A upjesh3 B upjesh4 A B I (2,3)5 C E (1,4)6 B C I (3, 5)7 A (B C ) I (2, 6)

Sth fusik glssa: Jewrome ti h prtash AB C enaialhjc. Gia na dexoume ti h A (B C ) enai alhjc,upojtoume A kai deqnoume ti B C . Gia na dexoume B Cupojtoume B kai deqnoume ti C . All tra qoume A kai B ,ra kai AB . Gnwrzoume ti AB C . Sunepc qoume C .83 / 232

5.3 Kannec ND gia to sndesmo

O sndesmoc qei treic kannec. Oi do prtoiantimetwpzoun thn A wc thn A .I(eisagwg , -introduction, I ). Gia na apodexoumema prtash thc morfc A, upojtoume A kaiapodeiknoume .Wc sunjwc, den mporome na qrhsimopoisoume thn Aargtera. Sunepc klenoume thn apdeixh tou ap thnA se na kout:

1 A upjesh

2

.

.

. lla sumpersmata

3 to apodexame!4 A I (1, 3)

84 / 232


I(apaloif , -elimination, E ). Ap thn A kai thn Asumperanoume .1 A to apodexame . . .2

.

.

. lla sumpersmata

3 A to apodexame kai aut4 E (1, 3)

I(apaloif , -elimination, ). Ap thn Asumperanoume A.

1 A to apodexame . . .2 A (1)To pardeigma 5.8 knei qrsh tou kanna .

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5.4 Paradegmata

Pardeigma 5.5 (O Qolmc kai to logo)

I s: autc pou kleye to logo th nqta tan gnwstocI b: o skloc gbgise th nqtaI w : oi flakec tou stblou xpnhsan

1 s b dedomno2 b w dedomno3 w dedomno4 s upjesh5 b E (4, 1)6 w E (5, 2)7 E (3, 6)8 s I (4, 7)

86 / 232

Paradegmata

Pardeigma 5.6

Ac apodexoume ti A ` A.

1 A dedomno

2 A upjesh3 E (1, 2)4 A I (2, 3)

87 / 232

Paradegmata

Pardeigma 5.7

Ac apodexoume ti (A B) ` A.

1 (A B) dedomno2 A upjesh3 A B I (1)4 E (1, 3)5 A I (2, 4)

Sth fusik glssa: Dojeshc thc prtashc (A B), aneqame A tte ja eqame A B , dhlad ja eqame antfash.'Ara qoume A. Aut enai ma akma kfansh tou nmou thctou trtou apoklesewc (law of excluded middle).

88 / 232

Otidpote enai apodeximo ap thn antfash

Pardeigma 5.8

Ac apodexoume ti ` A gia kje prtash A.

1 dedomno2 A upjesh3 (1)4 A I (2, 3)5 A (4)

Prosxte th qrsh tou

sth gramm 3 gia na

dikaiologsoume ma prtash h opoa enai dh diajsimh.

Otidpote enai apodeximo ap thn antfash.

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Otidpote enai apodeximo ap thn antfash

Uprqoun do trpoi na to katanosoume:

1. Apl efarmzoume touc kannec ND.

2. Jewrsame wc dedomno ma katstash sthn opoa h enai alhjc kai apodexame ti kai h A enai alhjc. Allden uprqei katstash pou h enai alhjc. Sunepcden uprqei lgoc na mhn deqtome opoiadpote sunpeia

autc thc jerhshc.

Epeid den uprqei katstash sthn opoa h enai alhjc,mporome na apodexoume thn mno ktw ap antifatikcupojseic se na kout miac apdeixhc. Suqn knoume ttoiec

upojseic. Pq, se epiqeirhmatologa me periptseic (qrsh

kanna E ). Se autc tic periptseic mporome nasumpernoume opoiadpote prtash (kajc apodexame thn

). Sunepc enai qrsimo na deqnoume ti . Dete topardeigma 5.11 paraktw.

90 / 232

Apodeiknontac thn A APardeigma 5.9

Ac apodexoume ti ` A A (pol qrsimh apdeixh).1 (A A) upjesh2 A upjesh3 A A I (2)4 E (1,3)5 A I (2,4)6 A A I (5)7 E (1, 6)8 (A A) I (1, 7)9 A A (8)Sth fusik glssa: Upojtoume ti (A A). An A tteA A, dhlad ja eqame antfash. 'Ara A (gramm 5). Ttemwc qoume A A to opoo enai antfash. Ara h arqikupjesh enai yeudc, dhlad enai (A A). SunepcA A.91 / 232

AdministratorRectangle


5.5 Kannec gia to sndesmo >

I(eisagwg >, >-introduction, >I ). Mporome naeisgoume to sndesmo > opoudpote. Aut bbaia denenai pol qrsimo.

I(apaloif >, >-elimination, >E ). Den mporome naapodexoume tpota kainorgio ap >.

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5.6 Kannec gia to sndesmo I(eisagwg , -introduction, I ). Gia na apodexoume prpei na apodexoume A kai A (gia opoiadpoteprtash A).

1 A to apodexame . . .2

.

.

. lla sumpersmata

3 A to apodexame kai aut4 I (1, 3)Autc o kannac enai o dioc me ton E . Uprqoun dhladdo onmata gia ton dio kanna.

I(apaloif , -elimination, E ). Mporome naapodexoume opoiadpote prtash ap ! (Dete topardeigma 5.8).

1 to apodexame . . .2 A E (1)

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5.7 Lmmata

'Ena lmma enai kti pou qoume dh apodexei kai mac

bohjei na apodexoume aut pou jloume.

Sto pardeigma 5.9 apodexame ` A A. Aut h apdeixhenai pol qrsimh. Diaire to epiqerhma se do periptseic, Akai A. Aut dieukolnei ma apdeixh giat gnwrzoumeperisstera gia kje perptwsh.

Stic apodexeic ND ja mporete na eisgete thn prtashAA wc lmma (dhlad ja grfete lmma sth dexi stlhthc apdeixhc), qwrc na qreizetai na thn apodexete.

Enai to monadik lmma pou ja mporete na qrhsimopoisete

(qwrc na to apodexete).

Prpei mwc esec na dialxete poia prtash A jaqrhsimopoisete.

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Paradegmata

Pardeigma 5.10

Ac apodexoume ti ` ((A B) A) A (nmoc tou Peirce).

1 (A B) A upjesh2 A A lmma3 A upjesh 5 A upjesh6 A upjesh7 I (5,6)8 B E (7)9 A B I (6,8)4 A

(3) 10 A E (1,9)11 A E (2, 3, 4, 5, 10)12 ((A B) A) A I (1, 11)

95 / 232





5.8 Kannec gia to sndesmo Antimetwpzoume mia prtash thc morfc A B wc(A B) (B A), kai isodnama wc (A B) (A B).I(eisagwg , -introduction, I ). Gia na apodexoumeA B apodeiknoume A B kai B A. Enallaktik,gia na apodexoume A B apodeiknoume A kai B , apodeiknoume A kai B (poia perptwsh protimme).Prth enallaktik. . .

1 B A to apodexame . . .. . .2 A B kai aut . . .3 A B I (1,2)Deterh enallaktik. . .

1 B to apodexame . . .. . .2 A kai aut . . .3 A B I (1,2)96 / 232

5.8 Kannec gia to sndesmo

Trth enallaktik. . .

1 A to apodexame . . .. . .2 B kai aut . . .3 A B I (1,2)

97 / 232

Kannec gia to sndesmo

I(apaloif , -elimination, E ). Ap thn A B kaithn A, qoume thn B .Ap thn A B kai thn B , qoume thn A.

1 A B to apodexame . . .. . .2 A kai aut . . .3 B E (1,2)

1 A B to apodexame . . .. . .2 B kai aut . . .3 A E (1,2)

98 / 232

5.9 Anakptontec kannec (Derived Rules)

Oi basiko kannec tou sustmatoc ND (oi opooi qounparousiaste wc tra) mporon na sunduaston gia

paraqjon noi kannec. Oi anakptontec auto kannec DEN

enai aparathtoi se mia apdeixh, all knoun thn apdeixh

pio sunoptik.

'Enac anakptwn kannac pou ja mporete na qrhsimopoiete

onomzetai apdeixh ap antfash (Proof by Contradiction(PC)):

Gia na apodexoume A, upojtoume A kai apodeiknoume .

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Anakptontec kannec

O kannac PC enai o sunduasmc twn kannwn I kai :

1 A upjesh2

.

.

. lla sumpersmata

3 to apodexame!4 A I (1, 3)5 A (4)

H parapnw apdeixh grfetai wc exc me th qrsh tou

kanna PC:

1 A upjesh2

.

.

. lla sumpersmata

3 to apodexame!4 A PC (1,3)

H qrsh tou kanna PC meinei thn apdeixh kat ma gramm.100 / 232

5.10 Sumboulc gia apodexeic ND

1. Mjaite touc kannec. Lste askseic.

2. Gryte thn apdeixh me mesh majhmatik

epiqeirhmatologa. 'Epeita tupopoiste thn apdeixh me

touc kannec ND.

3. An duskoleeste na apodexete ma prtash A, tte taparaktw mpore na bohjsoun:

I pi A pi ( pi pi ,proof by contradiction (PC) ).

I pi B B pi B. pi pi pipi. pi 5.10 .

101 / 232

Paradegmata

Pardeigma 5.11

A B, C A, (B C ) ` C .Ac upojsoume ti qoume C . Tte ja qoume kai A. Allqoume wc dedomno ti A B , ra ja qoume B . 'Eqoumedhlad B kai C , to opoo dhmiourge antfash, kajc qoumewc dedomno (B C ).

'Ara prpei na qoume C .

H parapnw epiqeirhmatologa enai ekolo na metafraste se

apdeixh ND.

102 / 232

Pardeigma 5.11

1 A B dedomno2 C A dedomno3 (B C ) dedomno4 C upjesh5 A E (2,4)6 A upjesh 9 B upjesh7 I (5,6)8 B E (7) 10 B (9)11 B E (1, 6, 8, 9, 10)12 B C I (11, 4)13 I (3, 12)14 C PC (4, 13)

103 / 232




Kouti: ti prpei na jummaste

Kje kout msa se ma apdeixh prpei na plhro tic

paraktw propojseic:

1. 'Ena kout pnta arqzei me ma upjesh (h monadik

exaresh afor ton kanna I sthn kathgorhmatiklogik.)

2. Ma upjesh mpore na lbei qra mno sthn prth

gramm enc koutio.

3. Msa se na kout mporome na qrhsimopoisoume

opoiadpote prtash ap prohgomenec grammc thc

apdeixhc, ektc ap tic protseic pou brskontai se

kouti pou qoun klesei prin ap to trqon shmeo thc

apdeixhc.

104 / 232

Kouti: ti prpei na jummaste

4. Oi monadiko trpoi na exgoume plhrofora ap na

kout enai me th qrsh twn kannwn I , E , I , kai PC(epiplon me thn qrsh twn kannwn E kai I sthnkathgorhmatik logik). H prth gramm met ap kje

kout prpei na dikaiologhje me th qrsh enc ap autoc

touc kannec.

5. Kama ap tic protseic pou enai msa se na kout den

mpore na qrhsimopoihje xw ap to kout, ektc an

exaqje smfwna me touc kannec pou anafrontai sthn

propjesh 4.

An na kout den plhro ma ap tic parapnw propojseic

tte h apdeixh enai ljoc.

105 / 232

Qrsh koutin

'Estw ti jloume na apodexoume ti A ` A.

1 A dedomno2 A (1) h kalterh apdeixh!

1 A dedomno2 A upjesh3 I (2,1)4 A I (1,2)

swst all qi xupno

106 / 232

Ljoc qrsh koutin

1 A dedomno2 A upjesh3 I (2,1)4 A E (3)

ljoc!

1 A dedomno2 A upjesh3 I (2,1)4 A E (3)5 A (4)

ljoc!

107 / 232

5.11 ` kai O basikc mac stqoc enai na exetzoume th logik sunpeia

( ).Jumhjete ti A1, . . . ,An B enai alhjc an h B enai alhjcse kje katstash pou oi A1, . . . ,An enai lec alhjec.H ` den enai qrsimh an den mac bohjei na dexoume .Orismc 5.12

'Ena jerhma enai ma prtash h opoa mpore na apodeiqje

ap na dojn ssthma apodexewn.

Mac endiafrei to ssthma kannwn fusikc

sumperasmatologac. 'Ara na jerhma enai kje prtash Apou ` A.Orismc 5.13

'Ena ssthma apodexewn enai orj an kje jerhma enai

gkuro, kai plrec an kje gkurh prtash enai jerhma.

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Orjthta kai Plhrthta tou sustmatoc NDMpore na deiqte ti to ssthma kannwn fusikc

sumperasmatologac (ND) enai orj kai plrec.

Jerhma 5.14 (orjthta sustmatoc ND)

'Estw ti oi A1, . . . ,An,B enai tuqaec protseic. AnA1, . . . ,An ` B , tte A1, . . . ,An B .Gia n=0:Kje apodeximh prtash enai gkurh.To ssthma ND den knei pot ljh.

Jerhma 5.15 (plhrthta sustmatoc ND)

'Estw ti oi A1, . . . ,An,B enai tuqaec protseic. AnA1, . . . ,An B , tte A1, . . . ,An ` B .Gia n=0:Kje gkurh prtash enai apodeximh.To ssthma ND mpore na apodexei lec tic gkurecprotseic.109 / 232

Orjthta kai Plhrthta tou sustmatoc ND

'Ara mporome na qrhsimopoiome to ssthma ND gia naelgqoume thn egkurthta twn protsewn.

110 / 232

Sunpeia

Orismc 5.16 (sunpeia)

Mia prtash A enai sunepc an 0 A.'Ena snolo protsewn {A1, . . . ,An} enai sunepc an0 1in Ai .Smfwna me ta jewrmata orjthtac kai plhrthtac

(Jewrmata 5.14 kai 5.15) qoume:

Jerhma 5.17

Mia prtash enai sunepc an kai mno an enai ikanopoisimh.

H nnoia thc ikanopoihsimthtac dnetai ap ton Orism 3.3.

111 / 232

PandoraPandora: www.doc.ic.ac.uk/pandoraNew Pandora: www.doc.ic.ac.uk/pandora/newpandoraAut enai do ekdseic enc progrmmatoc logismiko me to

opoo mporete na knete apodexeic qrhsimopointac touc

kannec ND.

'Eqei anaptuqje ap foithtc tou Kolegou Imperial. Oifoithtc tou majmatoc Majhmatikc Logikc to qoun brei

pol qrsimo sto na majanoun na anaptssoun apodexeic me

to ssthma ND.

Diabste to help tou progrmmatoc Pandora kai exaskhjeteme ta tutorials tou. Lste tic askseic sto Pandora stoNew Pandora kai elgxte an oi lseic sac enai swstc.

Shmewsh: To prgramma Pandora onomzei to lmma ` A AEM (ap to law of Excluded Middle).112 / 232

6. Pnakec Beth

Oi pnakec Beth apl pnakec (tableaux) apotelon nassthma apodexewn. Qrhsimopointac suntaktikoc kannec

mporome na exetzoume an ma prtash enai gkurh.

O pnakac miac snjethc prtashc kataskeuzetai me bsh

touc pnakec twn protsewn pou emfanzontai sth snjeth

prtash.

113 / 232

Atomiko pnakec

Orzoume atomikoc pnakec gia touc sundsmouc thc

Protasiakc Logikc.

A /\ B

B

A

-(A /\ B)

-B-A

O isqurismc ti ma prtash A B enai alhjc apaite thn Aalhj kai thn B alhj (gramm).O isqurismc ti ma prtash (A B) enai alhjc apaitethn A yeud thn B yeud (diakldwsh).

114 / 232

Atomiko pnakec

A V B

BA

A -> B

B-A

-(A V B)

-B

-A

-(A -> B)

-B

A

Mia diakldwsh stouc atomikoc pnakec dhlnei dizeuxh en

ma gramm dhlnei szeuxh.

115 / 232

Atomiko pnakec

A B

-AA

B -B

-(A B)

-AA

-B B

- -A

A

-p p

116 / 232

Pardeigma

Gia na kataskeusoume ton pnaka miac snjethc prtashc Kxekinme grfontac K K sthn koruf tou pnaka. Metanaptssoume ton pnaka thc K smfwna me touc atomikocpnakec.

Pardeigma 6.1

Na kataskeuaste o pnakac thc (A A) (B (C D)).

(A /\ -A) V (B V (C /\ D))

B V (C /\ D)A /\ -A

-A

A C /\ DB

D

C

117 / 232

Pardeigma

Autc enai nac teleiwmnoc pnakac me treic kldouc. Oi

kldoi arqzoun ap thn koruf. O aristerc kldoc enai

antifatikc, periqei tic antifatikc protseic A kai A.Sumbolzoume thn antifatikthta enc kldou

upogrammzontac thn teleutaa tou prtash.

Oi lloi do kldoi den enai antifatiko.

Msw tou pnaka thc (A A) (B (C D)) blpoume ti hupjesh ti aut h prtash enai alhjc enai ktw ap

orismnec sunjkec swst, pq tan h B enai alhjc oi C kaiD enai alhjec.

Mporete na xeqwrsete touc atomikoc pnakec ston parapnw

pnaka;

118 / 232

Orismo

Orismc 6.2

'Enac kmboc enc pnaka lgetai qrhsimopoihmnoc an

emfanzetai san koruf atomiko pnaka. Allic lgetai

aqrhsimopohtoc.

Orismc 6.3

'Enac kldoc enc pnaka lgetai antifatikc an oi A kai Aenai kmboi tou kldou (gia kpoia tuqaa prtash A).

Orismc 6.4

'Enac pnakac lgetai teleiwmnoc an kannac mh antifatikc

kldoc den qei aqrhsimopohtouc kmbouc. Allic lgetai

hmitelc.

Orismc 6.5

'Enac pnakac lgetai antifatikc an loi oi kldoi tou enai

antifatiko.

119 / 232

Pardeigma

Pardeigma 6.6

Na kataskeuaste o pnakac tou nmou tou Peirce((A B) A) A.

-(((A -> B ) -> A ) -> A )

(A -> B ) -> A

A

-A

-(A -> B)

A

-B120 / 232

Pardeigma

Diaisjhtik: An nac teleiwmnoc pnakac me koruf K brejeantifatikc, aut shmanei ti dokimsame louc touc dunatoc

trpouc pou ja mporosan na knoun thn prtash K yeudkai apotqame. 'Ara h K enai alhjc se lec tic katastseic.

121 / 232

Pardeigma

((A -> B ) -> A ) -> A

-((A -> B ) -> A) A

A -> B

-A

-A B

122 / 232

To smbolo `BOrismc 6.7

Ma Beth-apdeixh miac prtashc K enai nac teleiwmnocantifatikc pnakac me koruf K . H prtash K lgetaiBeth-apodeximh an qei Beth-apdeixh.

To ti ma prtash enai Beth-apodeximh to sumbolzoume me

`B KH prtash K enai Beth-apodeximh ap tic protseicA1, . . . ,An an uprqei teleiwmnoc antifatikc pnakac mekoruf K kai epmeno kmbo A1 An. Sumbolik

A1, . . . ,An `B K

Orismc 6.8

Ma Beth-diyeush miac prtashc K enai nac teleiwmnocantifatikc pnakac me koruf K . H prtash K lgetaiBeth-diayesimh an uprqei ma Beth-diyeush thc.123 / 232

Sustmata apodexewn

Ja qrhsimopoiome ta smbola `ND kai `B gia na xeqwrzoumetic apodexeic tou sustmatoc ND ap tic Beth apodexeic.

124 / 232

Pardeigma

Pardeigma 6.9

Na apodexete ti `B ((p q) (p q)).

- -((p V q) (-p /\ -q))

(p V q) (-p /\ -q)

p V q

-p /\ -q

-(p V q)

-(-p /\ -q)

qp

-p

-q

-p

-q

-p

-q

- -p - -q

qp125 / 232

Pardeigma

Pardeigma 6.10

Na apodexete ti M K , M K `B K .

-K

(M V K) /\ (M -> K)

M V K

M

M -> K

K

-M K

126 / 232

Orjthta kai Plhrthta twn Beth-apodexewn

Jerhma 6.11 (orjthta Beth-apodexewn)

'Estw ti oi A1, . . . ,An,K enai tuqaec protseic. AnA1, . . . ,An `B K , tte A1, . . . ,An K .Gia n=0:Kje Beth-apodeximh prtash enai gkurh.To ssthma Beth-apodexewn den knei pot ljh.

Jerhma 6.12 (plhrthta Beth-apodexewn)

'Estw ti oi A1, . . . ,An,K enai tuqaec protseic. AnA1, . . . ,An K , tte A1, . . . ,An `B K .Gia n=0:Kje gkurh prtash enai Beth-apodeximh.To ssthma Beth-apodexewn mpore na apodexei lec ticgkurec protseic.

127 / 232

Orjthta kai Plhrthta twn Beth-apodexewn

'Ara mporome na qrhsimopoiome touc pnakec Beth gia naelgqoume thn egkurthta twn protsewn.

128 / 232

Sunpeia

Mia prtash A enai sunepc an 0B A.'Ena snolo protsewn {A1, . . . ,An} enai sunepc an0B

1in Ai .

Smfwna me ta jewrmata orjthtac kai plhrthtac twn

Beth-apodexewn ma prtash enai sunepc an kai mno anenai ikanopoisimh.

H nnoia thc ikanopoihsimthtac dnetai ap ton Orism 3.3.

129 / 232

Pardeigma

Pardeigma 6.13

Na apodexete ti p (p q).Epeid to ssthma Beth-apodexewn enai orj kai plrec,arke na dexoume ti `B p (p q).

-(p -> (p V q))

p

-(p V q)

-q

-p

130 / 232

Pardeigma

Pardeigma 6.14

Na apodexete ti oi protseic p q kai q p enai isodnamec.Arke na dexoume ti `B (p q) (q p). (Giat;).

-((p /\ q) (q /\ p))

p /\ q

-(q /\ p)

-(p /\ q)

q /\ p

p

q

-q -p

q

p

-p -q

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Sumboulc gia Beth-apodexeic

IMjaite touc atomikoc pnakec. Lste askseic.

IDen uprqei ousidhc lgoc pou na kajorzei th seir

anptuxhc twn protsewn, all mejodologik h anptuxh

prta twn suzexewn (grammc) kai peita twn diazexewn

(diakladseic) odhge kat kanna se aplosterouc

pnakec.

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Pardeigma

Pardeigma 6.15

Na exetsete an M K , M K `B M.

-M

(M V K) /\ (M -> K)

M V K

M

M -> K

K

-M K

O parapnw pnakac enai teleiwmnoc all mh antifatikc.

'Ara M K , M K 0B M.133 / 232

Pardeigma

Pardeigma 6.16

Na brejon oi katastseic stic opoec h prtash

(p q) (p q) enai yeudc.

-((p -> q) (p V q))

p -> q

-(p V q)

-(p -> q)

p V q

q-p

-p

-q

-p

-q

p

-q

p q

Ap touc mh antifatikoc kldouc enc teleiwmnou pnaka me

koruf A, parnoume tic katastseic pou h A enai yeudc.134 / 232

Kathgorhmatik Logik

H Kathgorhmatik Logik ( Prwtobjmia Logik) enai ma

epktash thc Protasiakc Logikc.

Sta epmena majmata:

Ija parousisoume to suntaktik kai th shmasiologa thc

Kathgorhmatikc Logikc,

Ija dome th sqsh thc Kathgorhmatikc Logikc me th

fusik glssa,

Ija dsoume touc orismoc tou gkurou epiqeirmatoc kai

thc gkurhc prtashc sthn Kathgorhmatik Logik, kai

Ija dome trpouc gia na apodeiknoume thn egkurthta

sthn Kathgorhmatik Logik:

I pi.I .I pi.

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H angkh gia ma pio ekfrastik glssa

H Protasiak Logik enai ma plosia glssa, all enai

perioristik sthn diatpwsh idiottwn kai sqsewn.

Pq: An o Girgoc enai njrwpoc tte o Girgoc enai jnhtcgrfetai wc A B .Ti gnetai me th Mara, ton Ksta, klp;

Epiplon den mpore na elegqje me thn Protasiak Logik h

egkurthta twn epiqeirhmtwn pwc:

I'Oloi oi gtoi enai templhdec (p).

IO Mtsoc enai gtoc (q).

I'Ara o Mtsoc enai templhc (r).

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7. Perlhyh Kathgorhmatikc Logikc

7.1 Nec Atomikc Protseic

Mqri stigmc, jewrosame frseic pwc o upologistc enaiSun kai o Fthc agrase staflia wc atomikc, dqwceswterik dom.

Tra ja dome thn dom touc.

Jewrome thn mrka Sun wc ma idithta (property, attribute)thn opoa nac upologistc (kai lla prgmata) mpore na

qei na mhn qei. 'Ara eisgoume:

I'Ena smbolo kathgormatoc (relation symbol, predicatesymbol) Sun. 'Eqei 1 parmetro ra lme ti o bajmctou (arity) enai 1.

I'Ena smbolo kathgormatoc . 'Eqei 2 paramtrouc ra lme ti o bajmc tou enai 2.

IStajerc (constants), gia na onomzoume antikemena. Pq,, , -002, .

Sunepc Sun() kai (, ) enai do necatomikc protseic.

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7.2 Posodektec

Arqik mpore na fanetai ti to

(, )

den enai diaforetik ap aut pou grfame sthn Protasiak

Logik:

o Fwthc agorase stafulia

All h Kathgorhmatik Logik qei mhqanismoc gia na

metablei tic paramtrouc tou kathgormatoc .

Sunepc mporome na ekfrsoume tic idithtec thc sqshc

.

Oi mhqanismo auto onomzontai posodektec.

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Ti enai oi posodektec;

'Enac posodekthc ekfrzei ma posthta (ap prgmata ta

opoa qoun ma idithta).

Paradegmata:

I'Oloi oi foithtc diabzoun pol.

IKpoioi foithtc koimontai.

IOi perissteroi kajhghtc enai trelo.

IOkt stic dka gtec to protimon.

IKannac den enai pio xupnoc ap esna.

IToulqiston xi foithtc den koimontai.

IUprqoun peiroi prtoi arijmo.

IUprqoun pio poll PC ap Mac.

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Posodektec sthn Kathgorhmatik Logik

'Eqoume do posodektec:

I ( (A) ): gia kje (kajolikc posodekthc).I ( (E ) ): uprqei (uparxiakc posodekthc).Kpoioi lloi posodektec mporon na ekfraston me touc

parapnw. (Kje nac ap touc parapnw posodektec mpore

na ekfraste msw tou llou.) All den uprqoun posodektec

sthn Prwtobjmia Logik gia na ekfrsoun sqseic pwc

uprqoun peiroikai perisstero ap.

Edame ekfrseic pwc , . Aut enai stajerc pwcto pi.

Gia na ekfrsoume protseic pwc loi oi upologistc enaiSun qreiazmaste metablhtc oi opoec mporon na prountimc ap to snolo lwn twn upologistn.

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7.3 Metablhtc

Ja qrhsimopoisoume metablhtc gia na ekfrsoume

posodeixa. 'Eqoume na snolo V metablhtn: pqx , y , z , u, v ,w , x0, x1, x2, . . .Merikc forc ja grfoume x y gia na pome opoiadpotemetablht.Pra ap tic protseic pwc Sun(), ja grfoume kaiprotseic pwc Sun(x).

IGia na ekfrsoume thn prtash ta pnta enai Sun jagrfoume x Sun(x).Aut diabzetai wc: Gia kje x , to x enai Sun.

IGia na ekfrsoume thn prtash kti enai Sun jagrfoume x Sun(x).Aut diabzetai wc: Uprqei x to opoo enai Sun.

O Fthc agrase na Sungrfetai wcx (Sun(x) (, x))

Uprqei x , to opoo enai Sun kai o Fthc to agrase.141 / 232

8. Suntaktik Kathgorhmatikc Logikc

'Opwc kname sthn Protasiak Logik, tsi kai tra ja dome

prta to suntaktik kai met th shmasiologa.

8.1 Glssa

Orismc 8.1

Ma glssa enai na snolo ap stajerc kai smbola

kathgorhmtwn sugkekrimnou bajmo.

Sunjwc sumbolzoume ma glssa wc L. Grfoume c , d , . . .gia stajerc kai P,Q,R,S , . . . gia smbola kathgorhmtwn.

Argtera ja exetsoume kai ta smbola sunartsewn.

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Ma apl glssa

To snolo twn sumblwn mac glssac L exarttai ap autpou jloume na ekfrsoume.

Ja qrhsimopoisoume wc pardeigma ma glssa L h opoaperilambnei:

Itic stajerc , , , , , ,-002, kai c

Ita smbola kathgorhmtwn Sun, pi, (bajmc 1)

Ito smbolo kathgormatoc (bajmc 2)

Shmewsh: H L perilambnei smbola (suntaktik). Gia nadome to nhma (shmasiologa) twn sumblwn ja prpei na

dome poia morf qei ma katstash sthn Kathgorhmatik

Logik.

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8.2 'Oroi (Terms)

Gia na grfoume protseic sthn Kathgorhmatik Logik

qreiazmaste rouc (terms), me touc opoouc onomzoumeantikemena. Oi roi den enai protseic, den qoun alhjotimc,

Orismc 8.2 ('Oroc)

Dojeshc mac glssac L:

1. Kje stajer thc L enai nac L-roc.

2. Kje metablht enai nac L-roc.

3. Tpota llo den enai L-roc.

'Enac basikc roc (ground term) enai nac roc o opooc denperilambnei metablhtc.

Paradegmata rwn:

I , (basiko roi).I x , y , x56 (mh basiko roi).

Argtera ja exetsoume kai ta smbola sunartsewn.

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8.3 Tpoi Kathgorhmatikc Logikc

Orismc 8.3 ( Tpoc (formula) )

'Estw ma glssa L.

1. An R enai na smbolo kathgormatoc bajmo n thc L,kai t1, . . . , tn enai L-roi, tte to R(t1, . . . , tn) enai nacatomikc L-tpoc.

2. An t, t enai L-roi tte t=t enai nac atomikc L-tpoc.3. Oi ,> enai atomiko L-tpoi.4. An A,B enai L-tpoi, tte kai oi (A), (A B), (A B),

(A B), (A B) enai L-tpoi.5. An h A enai nac L-tpoc kai x ma metablht, tte oi

(x A) kai (x A) enai L-tpoi.6. Tpota llo den enai L-tpoc.

H proteraithta twn sundsmwn enai pwc sthn Protasiak

Logik. Oi posodektec x kai x enai tso isquro so kai h.145 / 232

Paradegmata tpwn

1. (, x)O Fthc agrase to x.

2. x (, x)O Fthc agrase kti.

3. x ((x) pi(x))Kje kajhghtc enai njrwpoc.

4. x ((, x) Sun(x))'O,ti agrase o Tsoc enai Sun.

Anaptssoume to dndro (formation tree) enc tpou kaigrfoume touc upo-tpouc tou (subformulas) pwc sthnProtasiak Logik.

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Paradegmata tpwn

5. x ((, x) (, x))H Sofa agrase ti agrase kai o Tsoc.

6. x (, x) x (, x)An o Tsoc agrase ta pnta, tte kai h Sofaagrase ta pnta.

7. xy (x , y)Ta pnta agrasan kti.

8. yx (x , y)Kti agorsthke ap ta pnta.

9. xy (x , y)Kti agrase ta pnta.

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9. Shmasiologa Kathgorhmatikc Logikc

'Opwc sthn Protasiak Logik, tsi kai tra prpei na

diatupsoume

Iti enai ma katstash sthn Kathgorhmatik Logik

Ipc na ermhneoume touc tpouc thc Kathgorhmatikc

Logikc se ma dedomnh katstash

9.1 Domc (katastseic sthn Kathgorhmatik Logik)

Orismc 9.1

'Estw ma glssa L. Ma L-dom (h opoa merikc forconomzetai montlo) M:

Iqei na mh-ken snolo ap antikemena ta opoa h Mgnwrzei. To snolo aut onomzetai to pedo timn(domain) smpan (universe) thc M, kai grfetai wcdom(M).

Idiatupnei ti shmanoun ta smbola thc L se sqsh me taparapnw antikemena.

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Domc

H apdosh (interpretation) mac stajerc sth M enai naantikemeno sto dom(M).

H apdosh (interpretation) enc sumblou kathgormatoc sthM enai ma sqsh sto dom(M).

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Pardeigma domc

Gia thn glssa L pou qrhsimopoiome wc pardeigma, maL-dom prpei na diatupnei:Ipoia antikemena enai sto smpan thc

Ipoia ap ta antikemena thc enai o Tsoc, h Sofa, klp

Ipoia antikemena enai Sun, kajhghtc, njrwpoi

Ipoia antikemena agrasan lla antikemena

Paraktw uprqei na digramma miac sugkekrimnhc L-domc,thn opoa onomzoume M.Uprqoun 12 antikemena sto smpan thc M.'Eqoume blei etiktec (pq Fthc) se merik antikemena giana dexoume to nhma twn stajern thc L (pq ).Oi apodseic (nnoiec) twn kathgorhmtwn Sun, pianaparstantai wc perioqc sto digramma.

H apdosh (nnoia) tou kathgormatoc anaparstatai me marec koukkdec.

H apdosh (nnoia) tou kathgormatoc anaparstataime kateujunmenec grammc metax antikeimnwn.

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H dom M

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Tsoc Tasoc

Den prpei na uprqei sgqush metax tou antikeimnou methn etikta Tsoc, tou smpantoc thc M, me to smbolo(stajer) thc glssac L.

(Qrhsimopoi diaforetikc grammatoseirc gia na diakrnw

metax twn sumblwn mac glssac kai twn antikeimnwn tou

smpantoc mac domc thc glssac.)

Aut enai diaforetik prgmata. To smbolo enaisuntaktik, en to enai shmasiologik. Gia th dom M, enai na noma gia to antikemeno me thn etikta Tsoc.

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Tsoc Tasoc

Gia na apofgoume thn sgqush ja uiojetsoume thn

paraktw smbash:

Smbash 9.2

'Estw M ma L-dom kai c ma stajer thc L. Grfoume cM

gia thn apdosh thc c sth M. To antikemeno tou smpantocthc M enai aut pou onomzei h c sth M.

'Ara M=to antikemeno me etikta Tsoc.

Se ma llh dom, to smbolo mpore na onomzei (nashmanei) kti llo.

H nnoia mac stajerc c enai to antikemeno cM to opoo qei

apodoje sto c ap th M. Ma stajer (kai kje smbolo thcL) qei tsec nnoiec sec kai o arijmc twn L-domn.

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Ta uploipa smbola

H glssa L pou qrhsimopoiome wc pardeigma qei stajerckai kathgormata prtou kai deutrou bajmo.

Gia na anaparastsoume grafik ma dom M autc thcglssac:

Izwgrafsame na snolo ap antikemena (to smpan thc

M)

Idhlsame poia antikemena thc M onomzontai ap poiecstajerc thc L

Idhlsame poia antikemena ikanopoion ta kathgormata

prtou bajmo (Sun, pi, ) smfwna me thM

Izwgrafsame kateujunmenec grammc metax twn

antikeimnwn pou ikanopoion to kathgrhma deutrou

bajmo () smfwna me th M. H katejunsh thcgrammc qei shmasa.

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Ta uploipa smbola

An uprqan perisstera ap na kathgormata deutrou

bajmo sthn L tte ja tan aparathto na bloume etiktecstic kateujunmenec grammc.

Genik, den uprqei ekoloc trpoc na anaparastsoume

grafik apodseic kathgorhmtwn trtou megalterou

bajmo.

Kathgormata bajmo 0 enai ta dia me ta toma thc

Protasiakc Logikc.

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9.2 Alhjotimc se ma dom (mh tupik perigraf)

Pte enai nac tpoc qwrc posodektec alhjc se ma dom;

IO tpoc Sun() enai alhjc sth M, giat M enaina antikemeno to opoo h M lei ti enai Sun.Aut to grfoume M Sun().To diabzoume wc h M lei ti Sun().Prosoq: Aut enai ma diaforetik qrsh tou apaut tou Orismo 3.1.

To smbolo qrhsimopoietai gia do diaforetikocskopoc.

IOmowc, o tpoc (, ) enai alhjc sth M.Sumbolik M (, ).

IO tpoc (, ) enai yeudc sth M, giat hM den lei ti h stajer onomzei na antikemeno to opoo ton eaut tou.Sumbolik M 2 (, ).

IOmowc, M 2 Sun() (, ).156 / 232

Ma llh dom

Paraktw enai ma llh L-dom, h M .

Uprqoun mno 10 antikemena sto smpan thc M .157 / 232

Sqetik me th M

I M 2 (, ).I M =.I M pi() Sun().I M (, ) (, c).

Poia h alhjotim twn paraktw tpwn;

I (, ) pi().I (c, ) Sun() pi().

158 / 232

Ermhneontac tpouc me posodektec (mh tupik

perigraf)

Pte enai nac tpoc me posodektec alhjc se ma dom;

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Ermhneontac posodektec

Pc exetzoume an o tpoc x (x , ) enai alhjcsth M;Sumbolik, M x (x , );Sth fusik glssa, uposthrzei h M ti kti agrase tonErm;

Gia na enai alhjc o parapnw tpoc, prpei na uprqei na

antikemeno x sto smpan thc M tsi ste

M (x , ).

Me lla lgia, h M prpei na uposthrzei ti

(x , ), pou =M .Uprqei ttoio antikemeno: blpoume sto digramma ti to xmpore na enai, pq, M .

Sunepc, M x (x , ).160 / 232

Pardeigma:

M x(agorase(Tasoc, x) agorase(Sofia, x));

Gia kje antikemeno x sto smpan thc M, enai o tpoc(, x) (, x) alhjc sth M;

Sth M uprqoun 12 pijan x . Prpei na elgxoume an o tpoc(, x) (, x) enai alhjc sth M giakje na ap aut.

O tpoc (, x) (, x) ja enaialhjc sth M gia kje na antikemeno x ttoio ste o tpoc(, x) enai yeudc sth M. (Giat;) 'Ara qreizetaina elgxoume ta x gia ta opoa o tpoc (, x)enai alhjc.

161 / 232

Pardeigma:

M x(agorase(Tasoc, x) agorase(Sofia, x));

O tpoc (, x) enai alhjc mno gia na x , gia toM .

Gia to antikemeno =M , o tpoc (, ) enaiepshc alhjc sthn M. 'Ara o tpoc(, ) (, ) enai alhjc sth M.

'Ara o tpoc (, x) (, x) enaialhjc sth M gia kje antikemeno x thc M. Sunepc,M x((, x) (, x)).

162 / 232

'Askhsh: Poia enai alhj sth M ;

x(Sun(x) (, x))x((x) pi(x))x(Sun(x) y (y , x))163 / 232

Ermhneontac tpouc (mh tupik perigraf)

Gia na snjeto tpo pwc x(Sun(x) y (y , x)):Brete pc metafrzetai kje upo-tpoc sth fusik glssa,

xekinntac ap touc atomikoc upo-tpouc (ta flla tou

dndrou (formation tree) ) ftnontac mqri thn arqik prtash(rza tou dndrou).

Autc enai suqn nac kalc trpoc na ermhneoume tpouc.

Pq, o parapnw tpoc anafrei ti uprqei na x to opooenai Sun kai kti to agrase (enai dhlad h katlhxh mackateujunmenhc grammc). 'Ara elgqoume tic grammc.

164 / 232

9.3 Alhjotimc se ma dom (tupik perigraf)

Edame pc na ermhneoume orismnouc tpouc se ma dom

koitntac to digramma thc domc.

Qreiazmaste mwc na pio tupik/majhmatik trpo gia na

ermhneoume louc touc tpouc thc Kathgorhmatikc Logikc

se domc.

Sth Protasiak Logik, upologzame thn alhjotim mac

prtashc se ma katstash upologzontac tic alhjotimc twn

upo-protsewn thc, xekinntac ap tic atomikc upo-protseic

(ta flla tou dndrou thc (formation tree) ) ftnontac mqri thrza tou dndrou.

Sth Kathgorhmatik Logik ta prgmata den enai tso

apl...

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'Ena prblhma

O tpoc x((, x) Sun(x)) enai alhjc sth domthc seldac 163. To dndro tou enai:

Mporome na ermhnesoume autn ton tpo xekinntac ap ta

flla, katalgontac sth rza;

Enai o tpoc (, x) alhjc sth M;Enai o tpoc Sun(x) alhjc sth M;

Den enai loi oi tpoi thc Kathgorhmatikc Logikc alhjec

yeudec se ma dom!

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Elejerec kai desmeumnec metablhtc

Prpei na elgxoume pc emfanzontai oi metablhtc stouc

tpouc.

Orismc 9.3

'Estw A nac tpoc.

1. H emfnish mac metablhtc x se na atomik upo-tpotou A lgetai desmeumnh (bound) an brsketai ktw apna posodekth x x sto dndro (formation tree) tou A.2. Allic h emfnish thc metablhtc lgetai elejerh (free).

3. Oi elejerec metablhtc enc tpou A enai oi metablhtcoi opoec qoun elejerec emfanseic ston A.

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Pardeigma

x(R(x , y) R(y , z) z(S(x , z) R(z , y)))

Oi elejerec metablhtc tou tpou enai oi y , z .Shmewsh: h z qei elejerec kai desmeumnec emfanseic.168 / 232

Protseic

Orismc 9.4

Ma prtash (sentence) enai nac tpoc qwrc elejerecmetablhtc.

Paradegmata:

IO tpoc x((, x) Sun(x)) enai prtash.

IOi upo-tpoi tou:

I (, x) Sun(x)I (, x)I Sun(x)

den enai protseic.

Poiec enai protseic;

I (, )I (, x)I x=xI x(y(y=x) x=y)I xy(x=y z(R(x , z) R(y , z)))169 / 232

Prto prblhma: elejerec metablhtc

Oi protseic enai alhjec yeudec se ma dom.

All oi tpoi pou den enai protseic den enai alhjec

yeudec se ma dom!

'Enac tpoc me elejerec metablhtc den enai alhjc yeudc

se ma dom M, giat oi elejerec metablhtc tou den qounnhma sth M. Enai san na rwtme enai to x=7 alhjc;Den mporome na ermhnesoume na tpo A thcKathgorhmatikc Logikc se ma dom pwc ermhneame

protseic thc Protasiakc Logikc se katastseic, giat h

dom den enai ma oloklhrwmnh katstash: den dnei nnoiecstic elejerec metablhtc tou A.

'Ara prpei na dsoume timc stic elejerec metablhtc prin

ermhnesoume na tpo.

Aut prpei na to knoume akma kai tan oi timc autc den

ephrezoun thn ermhnea tou tpou (pq, tan o tpoc enai

x=x).170 / 232

Ekqrhsh timn stic metablhtc (assignment)Ma ermhneutik sunrthsh metablhtn (ESM) dnei timc stic

elejerec metablhtc.

Ma ESM enai gia tic metablhtc ,ti ma dom gia tic

stajerc.

Orismc 9.5

'Estw M ma dom. Ma ermhneutik sunrthsh metablhtn(ESM) h sto plasio thc M antistoiqe se kje metablht naantikemeno ap to smpan thc M:

h : V dom(M)(V enai to snolo twn metablhtn kai dom(M) enai to snolotwn antikeimnwn (smpan) thc M.)Gia ma ESM h kai ma metablht x , grfoume h(x) gia toantikemeno to opoo qei antistoiqhje sth x ap thn h.

171 / 232

Ekqrhsh timn stic metablhtc (assignment)

Dojeshc mac L-domc M kai mac ESM h sto plasio thc M,qoume ma oloklhrwmnh katstash. Sunepc, mporome naermhnesoume:

Iopoiodpote L-ro. H tim tou ja enai na antikemeno stosmpan thc M.

Iopoiodpote atomik ( mh) tpo mac glssac L. H timtou ja enai alhjc yeudc.

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Ermhneontac rouc

Ma dom kai ma ESM dnoun ma oloklhrwmnh katstash.Mac dnoun tic alhjotimc lwn twn atomikn tpwn.

Ja dexoume prta pc ermhneoume rouc kai met pc

ermhneoume tpouc.

Orismc 9.6

'Estw ma glssa L, M ma L-dom, kai h ma ESM stoplasio thc M.Tte gia kje L-ro t, h tim tou t sth M smfwna me thn henai to antikemeno thc M to opoo qei antistoiqhje ston tap:

ITh M, an to t enai ma stajer. Se aut thn perptwsh htim tou t enai tM .

IThn h, an to t enai ma metablht. Se aut thn perptwshh tim tou t enai h(t).

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Ermhneontac rouc: pardeigma

(1) H tim tou rou sth M smfwna me thn h enai toantikemeno me etikta Tsoc. (Ap ed kai sto exc ja toanaparist wc Tsoc M , all qi wc .)(2) H tim thc x sth M smfwna me thn h enai Ermc.

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Shmasiologa tpwn qwrc posodektec

Mporome tra na ermhnesoume opoiodpote tpo qwrc

posodektec.

'Estw ma L-dom M kai ma ESM h. Grfoume M, h A an otpoc A enai alhjc sth M smfwna me thn h. Allicgrfoume M, h 2 A.

Orismc 9.7

1. 'Estw ti R enai na smbolo kathgormatoc bajmo nmac glssac L, kai t1, . . . , tn enai L-roi. Ac upojsoumeti h tim tou ti sth M smfwna me thn h enai ai , giakje i=1, . . . , n (dete ton Orism 9.6).Tte M, h R(t1, . . . , tn) an h M uposthrzei ti h ditaxh(a1, . . . , an) qei th sqsh R . Allic, M, h 2 R(t1, . . . , tn).2. An t, t enai roi, tte M, h t=t an ta t kai t qoun thndia tim sth M smfwna me thn h. An den qoun thn diatim tte M, h 2 t=t .

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Shmasiologa tpwn qwrc posodektec


3. M, h > kai M, h 2 .4. M, h A B an M, h A kai M, h B . Allic,

M, h 2 A B .5. Omowc orzoume thn ermhnea twn tpwn A, A B ,

A B kai A B .

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Ermhneontac tpouc qwrc posodektec: pardeigma

I M, h pi(z)I M, h x=I M, h 2 (, v) z=177 / 232

Detero prblhma: desmeumnec metablhtc

Gnwrzoume pc na dnoume timc se elejerec metablhtc: me

ma ESM. Me aut ton trpo mporsame na ermhnesoume

louc touc tpouc qwrc posodektec.

All uprqoun tpoi pou qoun desmeumnec metablhtc. Oi

timc autn twn metablhtn den dnontai, kai den prpei na

dnontai, ap thn katstash, kajc oi metablhtc autc

elgqontai ap posodektec.

Pc upologzoume tic timc desmeumnwn metablhtn;Do trpoi:

1. Elgqoume tic pijanc ESM:

I pi pi pi pi.

I pi pi.

2. Qrhsimopoiome paiqndia (Hintikka games), ta opoa macbohjon na katalboume tic epidrseic twn posodeiktn,

kai eidiktera twn emfwleumnwn (nested) posodeiktn.178 / 232

Shmasiologa tpwn me posodektec


Ac upojsoume ti gnwrzoume pc na ermhnesoume na tpo Asth M smfwna me opoiadpote ESM. 'Estw x maopoiadpote metablht, kai h ma opoiadpote ESM. Tte:

6. M, h xA an uprqei kpoia ESM g h opoa sumfwneme thn h gia lec tic metablhtc ektc, pijanc, ap th x ,kai isqei M, g A. An den uprqei ttoia ESM tteM, h 2 xA.7. M, h xA an M, g A gia kje ESM g h opoa sumfwneme thn h gia lec tic metablhtc ektc, pijanc, ap th x .Allic, M, h 2 xA.

H g sumfwne me thn h gia lec tic metablhtc ektc, pijanc,ap th x shmanei ti g(y)=h(y) gia lec tic metablhtc yektc thc x . (Aut den apokleei thn perptwsh g(x)=h(x).)

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Ermhneontac tpouc me posodektec: pardeigma

I M, h x pi(x)giat, pq M, g pi(x) pou g(x)=Fthc.

I M, h 2 x pi(x)giat, pq M, g 2 pi(x) pou g(x)='Arhc.

I M, h x((, x) pi(x))(Elgxte sugkekrimna th g me g(x)=Ermc kai th g meg(x)='Arhc.)180 / 232

Smbash gia elejerec metablhtc

H paraktw smbash enai qrsimh gia na grfoume kai

ermhneoume tpouc.

Sta bibla suqn blpoume ekfrseic thc morfc:

'Estw A(x1, . . . , xn) nac tpoc.

H parapnw kfrash dhlnei ti oi elejerec metablhtc tou

A enai metax twn x1, . . . , xn.Shmewsh: Oi x1, . . . , xn prpei na enai lec diaforetikc.Epiplon, den enai aparathto na qoun lec oi x1, . . . , xnelejerec emfanseic ston A.Pardeigma: 'Estw o tpoc C :

x(R(x , y) yS(y , z))Ja mporosame na gryoume ton tpo wc:

I C (y , z)I C (x , z , v , y)I C (an den qrhsimopoisoume aut th smbash)all qi wc C (x).181 / 232

Smbash gia Ermhneutikc Sunartseic Metablhtn

Gia kje tpo A, an isqei M, h A qi, exarttai ap thnh(x) mno gia tic metablhtc x oi opoec qoun elejerecemfanseic ston A.'Ara gia na tpo A(x1, . . . , xn), an h(x1)=a1, . . . , h(xn)=an, tteenai OK na gryoume M A(a1, . . . , an) ant gia M, h A.IAc upojsoume ti mac qoun dsei na tpo C (y , z) ttoioste

x(R(x , y) yS(y , z))An pq h(y)=a, h(z)=b, tte mporome na gryoume

M C (a, b), M x(R(x , a) yS(y , b))ant gia M, h C . Shmewsh: mno oi elejerec emfanseictou y ston C qoun antikatastaje ap to a. Oidesmeumnec emfanseic tou y paramnoun wc qoun.

IGia ma prtash S , an isqei M, h S qi, den exarttaikajlou ap thn h. 'Ara mporome na gryoume M S .182 / 232

Pc na ermhneoume tpouc;

Edame ton orism thc aljeiac se ma dom (Alfred Tarski,dekaeta 1950).

Poioc mwc enai o kalteroc trpoc na elgqoume an isqei

M A;ISe merikc periptseic mporome na metafrzoume ton Asth fusik glssa kai na elgqoume an enai alhjc sth

M (dete selda 164).

IQrhsimopoiome ton Orism 9.7 kai elgqoume lec tic

pijanc ESM. Pq, gia ton tpo x((x) Sun(x))elgqoume, gia la ta x , an to x enai kajhghtc tteenai kai Sun.

IQrhsimopoiome ta paiqndia Hintikka, ta opoa macbohjon na katalboume tic epidrseic twn posodeiktn.

Me ta paiqndia Hintikka den ja asqolhjome se aut tomjhma.

183 / 232

10. Sqsh tou Kathgorhmatiko Logismo me th fusik

glssa

10.1 Ap thn tupik glssa sth fusik

H metfrash twn protsewn thc Kathgorhmatikc Logikc sth

fusik glssa den enai pol duskolterh ap th metfrash

twn protsewn thc Protasiakc Logikc sth fusik glssa.

Oi metablhtc prpei apaloifjon: sth fusik glssa den tic

qrhsimopoiome.

x((x) (x=) (x , ))Gia kje x , an to x enai nac kajhghtc kai to x den enai oFthc tte to x agrase ton Trtwn.Kje kajhghtc ektc ap ton Fth agrase ton Trtwn.(Mpore kai o Fthc na ton agrase.)

184 / 232

Ap thn tupik glssa sth fusik

xyz((x , y) (x , z) (y=z))Uprqoun x , y , z ttoia ste to x agrase to y , to xagrase to z , kai to y den enai to z .Kti agrase toulqiston do diaforetik prgmata.

x(yz((x , y) (x , z) (y=z)) x=)Gia kje x , an to x agrase do diaforetik prgmata, tteto x enai o Tsoc.Otidpote agrase do diaforetik prgmata enai o Tsoc.Prosoq: h parapnw prtash den lei ti o Tsoc agrase

do diaforetik prgmata, all ti kannac lloc den

agrase do diaforetik prgmata.

185 / 232

10.2 Tupopohsh thc fusikc glssac

Ekfrste tic upo-nnoiec sth logik. 'Epeita sunjste tic

upo-nnoiec se ma logik prtash.

IUpo-nnoia: to x agorsthke/ to x qeiagorast:y (y , x).

IOtidpote agorsthke den enai njrwpoc:

x(y (y , x) pi(x)).Prosoq: h metfrash se

xy((y , x) pi(x)) den enai swst.IKje Sun agorsthke: x(Sun(x) y (y , x)).

IKpoio Sun qei na agorast:x(Sun(x) y (y , x)).

I'Oloi oi agorastc enai njrwpoi kai kajhghtc:

x(y (x , y) o x enai agorastc

pi(x) (x)).

IKannac kajhghtc den agrase na Sun:x((x) y((x , y) (Sun(y)) o x agrase na Sun

).

186 / 232


Suqn qreizetai na pome:

I 'Oloi oi kajhghtc enai njrwpoi:x((x) pi(x)).OQI x((x) pi(x)).OQI x (x) x pi(x).

I 'Oloi oi kajhghtc enai njrwpoi Sun:x((x) pi(x) Sun(x)).

I Tpota den enai Sun kai kajhghtc:x(Sun(x) (x)), x(Sun(x) (x)).Sunepc o tpoc x(A B) enai pol suqnc.Epshc oi x(A B), x(A B), x(A B), x(A B)qreizontai suqn: lne ti ta pnta/kti enai A kai/ B .

O tpoc x(A B), eidik tan h x qei elejerh emfnishston A, enai spnioc.An gryete autn ton tpo elgxte an qete knei ljoc.

187 / 232


IUprqei toulqiston na Sun: x Sun(x).

IUprqoun toulqiston do Sun:xy(Sun(x) Sun(y) x 6=y), xy(Sun(y) y 6=x).

IUprqoun toulqiston tra Sun:xyz(Sun(x) Sun(y) Sun(z) x 6=y y 6=z x 6=z), xyz(Sun(z) z 6=x z 6=y).

IDen uprqei Sun: x Sun(x).

IUprqei to pol na Sun:1. xy(Sun(x) Sun(y) x 6=y)

(pi Sun)2. xy(Sun(x) Sun(y) x=y).3. xy(Sun(y) x=y).

IUprqei akribc na Sun:1. pi Sunpi pi

Sun.2. xy(Sun(y) x=y).188 / 232

11. Smbola sunartsewn

Sthn arijmhtik qrhsimopoiome sunartseic pwc pq +, , ,x .Sthn Kathgorhmatik Logik mporome na ekfrsoume

sunartseic.

'Ena smbolo sunrthshc enai san na smbolo kathgormatoc

ma stajer, all apoddetai se ma dom wc ma sunrthsh.

Kje smbolo sunrthshc qei na sugkekrimno bajm (arity).

Sumbolzoume sunjwc ta smbola sunartsewn me f , g .Ap ed kai sto exc uiojetome thn paraktw epktash tou

Orismo 8.1:

Orismc 11.1

Ma glssa enai na snolo ap stajerc, kai smbola

kathgorhmtwn kai sunartsewn sugkekrimnou bajmo.

Prosoq: den prpei na uprqei sgqush metax mac ESM hsto plasio mac domc M, me ma sunrthsh f mac glssac L.

189 / 232

'Oroi me smbola sunartsewn

Tra mporome epektenoume ton Orism 8.2:

Orismc 11.2 ('Oroc)

Dojeshc mac glssac L:

1. Kje stajer thc L enai nac L-roc.

2. Kje metablht enai nac L-roc.

3. An f enai na smbolo sunrthshc bajmo n thc L, kait1, . . . , tn enai L-roi, tte o f (t1, . . . , tn) enai nac L-roc.

4. Tpota llo den enai L-roc.

Pardeigma:

'Estw ti h L qei ma stajer c, na smbolo sunrthshc fprtou bajmo, kai na smbolo sunrthshc g deutroubajmo. Tte ta paraktw enai L-roi:I cI f (c)I g(x , x) (wc sunjwc h x enai metablht)I g(f (c), g(x , x))190 / 232

Shmasiologa sumblwn sunartsewn

Prpei epiplon na epektenoume ton Orism 9.1: an h L qeismbola sunartsewn, ma L-dom prpei na diatupnei tonhma touc.

Gia kje smbolo sunrthshc f bajmo n thc L, ma L-dom Mprpei na lei poio antikemeno (ap to smpan thc)

antistoiqe h f se kje ditaxh (a1, . . . , an), pou a1, . . . , anenai antikemena tou smpantoc thc M.Sumbolzoume wc f M(a1, . . . , an) to antikemeno tou smpantocthc M pou h f antistoiqe sth ditaxh (a1, . . . , an).'Ena smbolo sunrthshc bajmo 0 jewretai ma stajer.

Pardeigma

Sthn arijmhtik, h M mpore na uposthrzei ti oi sunartseic+ kai ekfrzoun thn prsjesh kai ton pollaplasiasmarijmn: to 5 qei antistoiqhje me to 2+3, to 8 me to 24, klp.Den enai aparathto h M na to uposthrzei aut. Mpore nauposthrzei enallaktik ti h sunrthsh ekfrzei thnprsjesh.

191 / 232

Ermhneontac rouc me smbola sunartsewn

Tra mporome na epektenoume ton Orism 9.6:

Orismc 11.3

H tim enc L-rou t se ma L-dom M smfwna me ma ESM hsto plasio thc M orzetai wc exc:

IAn o t enai stajer tte h tim tou enai to antikemenotM to opoo qei antistoiqhje ston t ap th M.

IAn o t enai metablht tte h tim tou enai to antikemenoh(t) to opoo qei antistoiqhje ston t ap thn h.

IAn o t enai sunrthsh thc morfc f (t1, . . . , tn), kai oitimc twn rwn t1, . . . , tn sth M smfwna me thn h enaiantstoiqa a1, . . . , an, tte h tim tou t enai to antikemenof M(a1, . . . , an).

Sunepc h tim enc rou sth M smfwna me thn h enai pntana antikemeno tou smpantoc thc M. Me lla lgia nacroc den qei alhjotim.

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Shmasiologa tpwn me smbola sunartsewn

O orismc 9.7 den qreizetai allag, pra ap to ti

qrhsimopoietai plon me ton Orism 11.3 (kai qi me ton

Orism 9.6).

193 / 232

Arijmhtiko roi

'Estw ma glssa L gia arijmhtik kai gia progrmmatalogismiko pou qrhsimopoion arijmoc, h opoa perilambnei:

Itic stajerc 0, 1, 2, . . . (upogrammzw aut ta smbola giana ta xeqwrsw ap touc fusikoc arijmoc 0, 1, 2, . . . ),

Ita smbola sunartsewn deutrou bajmo +,,, kai

Ita smbola kathgorhmtwn deutrou bajmo ,.Apoddoume ta parapnw smbola se ma dom me smpan to

snolo {0, 1, . . . }, pou to 0 apoddetai sto 0, to 1 sto 1, klp,to +(x , y) ekfrzei x+y , to

Sunoyzontac

Ma L-dom M perilambnei:

Ina mh-ken snolo, to dom(M)

Igia kje stajer c L, na stoiqeo cM dom(M)

Igia kje smbolo sunrthshc f L bajmo n, masunrthsh f M : dom(M)n dom(M)

Documents

Protasiakos Kai Katigorimatikos Logismos