dom tou maj matoc · dom tou maj matoc i protasiak logik ( propositional logic). i kathgorhmatik...

Dom tou Maj matoc

I Protasiak Logik (Propositional Logic).

I Kathgorhmatik Logik (Predicate Logic), h opoÐa eÐnaignwst kai wc Prwtob�jmia Logik (First-Order Logic).

Autèc oi shmei¸seic eÐnai basismènec stic shmei¸seic tou IanHodkinson apì to Kolègio Imperial, tou PanepisthmÐou touLondÐnou.

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BiblÐa

I Apì th Logik sto Logikì Programmatismì kai thn Prolog.G. MhtakÐdhc. Ekdìseic KardamÐtsa.

I StoiqeÐa Majhmatik c Logik c. A. Tzoub�rac.Ekdìseic Z th.

I Logik : H dom tou Epiqeir matoc. D. PortÐdhc, S.YÔlloc kai D. Anapolit�noc. Ekdìseic Nefèlh.

I Diakrit� Majhmatik� kai Majhmatik Logik , Tìmoc G':

Majhmatik Logik . K. Dhmhtrakìpouloc.Ellhnikì Anoiktì Panepist mio.

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.KukloforeÐ ellhnik èkdosh me tÐtlo Teqnht NohmosÔnh -

MÐa SÔgqronh Prosèggish, ekdìseic Kleid�rijmoc.

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1. Logik : Eisagwg

I To komm�ti thc FilosofÐac pou pragmateÔetai morfècsullogismoÔ kai skèyhc, eidik� epagwgèc kai episthmonikècmejìdouc. Epiplèon, Logik eÐnai h susthmatik qr shsumbolik¸n teqnik¸n kai majhmatik¸n mejìdwn gia tonupologismì morf¸n ègkurou epagwgikoÔ epiqeir matoc.

I Se autì to m�jhma ja asqolhjoÔme me th qr sh thcLogik c gia thn perigraf kai an�ptuxh prodiagraf¸n(specification) logismikoÔ, kai ton sumperasmì (reasoning)sqetik� me tic prodiagrafèc.

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Logik : Eisagwg

I H Logik eÐnai o “logismìc thc Plhroforik c”: mÐamajhmatik b�sh gia thn antimet¸pish plhroforÐac kaigia ton sumperasmì sqetik� me th sumperifor�logismikoÔ.

I Parèqei mÐa kal ex�skhsh gia to swstì sumperasmì kaithn akrib perigraf .

I Merikèc apì tic polÔplokec efarmogèc Plhroforik cqrei�zontai kai qrhsimopoioÔn Logik . Pq, h Prolog eÐnaimÐa gl¸ssa programmatismoÔ Logik c. H gl¸ssa b�sewndedomènwn SQL eÐnai basismènh se idèec Logik c. Oiprodiagrafèc eufu¸n praktìrwn logismikoÔ (intelligentsoftware agents) ekfr�zontai me th qr sh thc Logik c.

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Logik : Eisagwg

H Logik brÐskei “efarmog ” (sun jwc apoteleÐ to upìbajro)se arketèc epist mec:

I Fusik : Kbantologikèc Logikèc (quantum logics) èqounepinohjeÐ gia thn jemelÐwsh thc kbantomhqanik c (quantummechanics).

I YuqologÐa: qr sh Logik¸n gia thn melèth thc anjr¸pinhcskèyhc.

I GlwssologÐa: qr sh Logik c gia thn jemelÐwsh Logik¸nGrammatik¸n (logical grammar) me skopì thn an�lush thcqr shc thc gl¸ssac apì touc anjr¸pouc. P¸c, giapar�deigma, qrhsimopoioÔntai ta onìmata, oi antwnumÐec,oi posodeÐktec (ìloi-merikoÐ-k�poioi klp), oi qrìnoi(upersuntèlikoc -aìristoc, klp).

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Logik : Eisagwg

'Ena sÔsthma Logik c perilamb�nei:

I Suntaktikì (Syntax): mÐa tupik gl¸ssa (ìpwc mÐagl¸ssa programmatismoÔ) h opoÐa qrhsimopoieÐtai gia naekfr�sei ènnoiec, gia thn anapar�stash thc gn¸shc.

I ShmasiologÐa (Semantics). H shmasiologÐa tousust matoc exet�zei ton trìpo me ton opoÐo oi ekfr�seicpou epitrèpontai apì to suntaktikì thc gl¸ssacsqetÐzontai me to pragmatikì touc perieqìmeno.

I SÔnolo kanìnwn sumperasmatologÐac (Proof theory). Oikanìnec sumperasmatologÐac epitrèpoun thn exagwg logik¸n sumperasm�twn apì th gn¸sh tou sust matoc.

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2. Protasiak Logik

Parak�tw eÐnai èna mèroc progr�mmatoc:if count>0 and not found then

decrement count; look for next entry;end if

I To prìgramma perilamb�nei basikèc (“atomikèc”)prot�seic, ìpwc count>0, found, oi opoÐec eÐnai alhjeÐc yeudeÐc k�tw apì orismènec sunj kec.

I MporoÔn na qrhsimopoihjoÔn sÔndesmoi ìpwc and, or,not gia na anaptuqjoÔn pio polÔplokec prot�seic apì ticatomikèc prot�seic.

I h telik polÔplokh prìtash ja eÐnai alhj c yeud c.

H protasiak logik den eÐnai polÔ ekfrastik . Hkathgorhmatik logik eÐnai pio ekfrastik .

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Protasiak Logik

I 'Ola ta eÐdh Logik c basÐzontai se k�poio bajmì sthnProtasiak Logik .

I Jèloume na antimetwpÐzoume if-tests opoiasd potepoluplokìthtac kai na melet soume ta genik�qarakthristik� touc.

I Den jèloume apl� na apotimoÔme if-testsJèloume na gnwrÐzoume an dÔo if-tests èqoun to Ðdio nìhma,pìte to èna sunep�getai to �llo, an èna if-test mporeÐpotè na eÐnai alhjèc ( yeudèc).

I H Protasiak Logik perilamb�nei mÐa shmantik om�daupologistik¸n problhm�twn, kai sunep¸c eÐnai antikeÐmenomelèthc twn algorÐjmwn kai thc jewrÐac poluplokìthtac.

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2.1 Suntaktikì

Pr¸ta, prèpei na d¸soume èna akrib orismì sta eÐdh twnif-tests ta opoÐa ja antimetwpÐsoume. Autìc o orismìc ja macd¸sei thn tupik gl¸ssa (formal language) thc Protasiak cLogik c. To alf�bhto thc gl¸ssac thc Protasiak c Logik cèqei 3 sustatik�.

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Pr¸to sustatikì: 'Atoma

De mac apasqoleÐ poia atomik� gegonìta (pq, count>0,found) ja antimetwpÐsoume. ArkeÐ na mporoÔn na p�roun mÐaalhjotim (dhlad na eÐnai alhj yeud ).

Oi protasiakèc metablhtèc (propositional variables) �toma

(propositional atoms) gr�fontai sun jwc wc p, p′, p0, p1, p2, . . . wc q, r , . . .

EÐnai san tic metablhtèc x , y , z sta Majhmatik�.

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DeÔtero sustatikì: LogikoÐ sÔndesmoi

'Eqoume touc parak�tw sundèsmouc (connectives):

I sÔzeuxh (kai): gr�fetai wc ∧I �rnhsh (den): gr�fetai wc ¬I di�zeuxh ( ): gr�fetai wc ∨I sunepagwg (an-tìte): gr�fetai wc →I isodunamÐa (an kai mìno an): gr�fetai wc ↔I truth kai falsity: gr�fontai wc > kai ⊥

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ParadeÐgmata

I To test count>0 and not found gr�fetai wc(count > 0) ∧ ¬found, Ðswc p ∧ ¬q

I Ta sÔmbola gia sÔzeuxh, di�zeuxh, sunepagwg kaiisodunamÐa paÐrnoun dÔo paramètrouc kai gr�fontai wc:p ∧ q, p ∨ q, p → q, p ↔ q

I H �rnhsh ¬ paÐrnei mÐa par�metro kai gr�fetai wc:¬p, ¬q

I Ta >,⊥ den èqoun paramètrouc.EÐnai logikèc stajerèc (ìpwc to π).

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TrÐto sustatikì: ShmeÐa StÐxhc

Qreiazìmaste parenjèseic gia na aposafhnÐsoume ta if-tests.

Pq, sthn arijmhtik to 1− 2 + 3 den eÐnai xek�jaro, upì thnènnoia ìti mporoÔme na to diab�soume wc (1− 2) + 3 1− (2 + 3). H diafor� paÐzei rìlo.

OmoÐwc, to p ∧ q ∨ r mporeÐ na diabasteÐ wc:

I (p ∧ q) ∨ r

I p ∧ (q ∨ r)

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Prot�seic'Olec oi akoloujÐec sumbìlwn pou ekfr�zoun �toma,sundèsmouc kai parenjèseic onom�zontai ekfr�seic.

Aut� pou èqoume onom�sei if-tests sth Logik lègontaiprot�seic (formulas) kalosqhmatismènec (well-formed)ekfr�seic.

Orismìc 2.1 (Prìtash)

I K�je �tomo (p, q, r klp) eÐnai prìtash.

I Ta > kai ⊥ eÐnai prot�seic.

I An to A eÐnai prìtash tìte eÐnai kai to (¬A).

I An ta A,B eÐnai prot�seic tìte eÐnai kai ta

(A ∧ B), (A ∨ B), (A→ B) kai (A↔ B).

I Oi mìnec ekfr�seic thc gl¸ssac pou eÐnai prot�seic, eÐnai

autèc pou kataskeu�zontai me touc parap�nw kanìnec.

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ParadeÐgmata Prot�sewn

Ta parak�tw eÐnai prot�seic:

I p

I (¬p)

I ((¬p) ∧ >)

I (¬((¬p) ∧ >))

I ((¬p)→ (¬((¬p) ∧ >)))

Qreiazìmaste k�poiouc kanìnec gia na mei¸soume ton arijmìparenjèsewn.Austhr� mil¸ntac, ta apotelèsmata ja eÐnai suntomografÐecprot�sewn. All� jewroÔntai prot�seic.MporoÔme p�nta na paraleÐyoume tic exwterikèc parenjèseic:pq, ta ¬p kai ¬((¬p) ∧ >) eÐnai saf .

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Kanìnec Apaloif c ParenjèsewnGia na paraleÐyoume perissìterec parenjèseic, orÐzoume thnparak�tw proteraiìthta sundèsmwn:

(isqurìteroc) ¬,∧,∨,→,↔ (pio adÔnamoc)

EÐnai ìpwc sthn arijmhtik ìpou to × eÐnai isqurìtero apì to+ kai sunep¸c to 2+3×4 diab�zetai wc 2+(3×4) kai ìqi wc(2+3)×4. OmoÐwc:

I p ∨ q ∧ r diab�zetai wc p ∨ (q ∧ r) kai ìqi wc (p ∨ q) ∧ r

I ¬p ∧ q diab�zetai wc (¬p) ∧ q kai ìqi wc ¬(p ∧ q)

I p ∧ ¬q → r diab�zetai wc (p ∧ (¬q))→ r kai ìqi wcp ∧ (¬(q → r)) p ∧ ((¬q)→ r))

Genik� ìmwc proteÐnetai h qr sh twn parenjèsewn ìtan bohj�eithn anagnwsimìthta, parìlo pou oi parenjèseic mporoÔn naapaloifjoÔn. Pq, to p → ¬q ∨ ¬¬r ∧ s → t eÐnai polÔ dÔskolona katanohjeÐ.

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Kanìnec Apaloif c ParenjèsewnTi gÐnetai me prot�seic ìpwc p → q → r ; H proteraiìthta twnsundèsmwn den mac bohj�ei se aut thn perÐptwsh. Hparap�nw prìtash prèpei na diabasteÐ wc p → (q → r) wc(p → q)→ r ;

Se tètoiec peript¸seic prèpei opwsd pote na qrhsimopoioÔmeparenjèseic.

Apì thn �llh, oi prot�seic p ∧ q ∧ r kai p ∨ q ∨ r eÐnai safeÐcìpwc eÐnai.'Opwc ja doÔme h logik shmasÐa eÐnai h Ðdia kai stic dÔopeript¸seic ìpwc kai na qrhsimopoi soume tic parenjèseic. Oiprot�seic (p ∧ q) ∧ r kai p ∧ (q ∧ r) eÐnai diaforetikèc. 'Allaìpwc ja doÔme èqoun thn Ðdia alhjotim se k�je perÐptwsh:eÐnai “logik� isodÔnamec”.'Ara de mac apasqoleÐ p¸c ja aposafhnÐsoume thn prìtashp ∧ q ∧ r . (Sun jwc eÐnai (p ∧ q) ∧ r .)

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Dèndra'Eqoume deÐxei p¸c na diab�zoume me saf neia mÐa prìtash(formula).

Se k�je prìtash antistoiqeÐ èna dèndro (formation tree), ènadi�gramma dhlad pou deÐqnei p¸c kataskeu�zetai h prìtashaut apì �toma me th bo jeia twn sundèsmwn.Pq, h prìtash ¬p ∨ q → (p → q ∧ r) èqei to parak�tw dèndro:

p

pq

q r

-

V

/\

->

->

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KÔrioc SÔndesmoc

Prosèxte ìti o sÔndesmoc sth rÐza (koruf ) tou dèndrou eÐnai→. Autìc eÐnai o kÔrioc sÔndesmoc (principal connective) thcprìtashc ¬p ∨ q → (p → q ∧ r). Aut h prìtash èqei thnlogik morf A→ B .K�je mh atomik prìtash èqei èna kÔrio sÔndesmo, o opoÐockajorÐzei th logik morf thc prìtashc.

I p ∧ q → r èqei kÔrio sÔndesmo →. H logik thc morf eÐnaiA→ B .

I ¬(p → ¬q) èqei kÔrio sÔndesmo ¬. H logik thc morf eÐnai ¬A.

I p ∧ q ∧ r èqei kÔrio sÔndesmo ∧ (pijanìtata to deÔtero). Hlogik thc morf eÐnai A ∧ B .

I p ∨ q ∧ r èqei kÔrio sÔndesmo ∨. H logik thc morf eÐnaiA ∨ B .

Poia eÐnai ta dèndra aut¸n twn prot�sewn;

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Upo-prot�seicOi upo-prot�seic (subformulas) mÐac prìtashc A eÐnai oiprot�seic pou qrhsimopoioÔntai gia thn an�ptuxh thc A, ìpwcaut orÐzetai apì ton Orismì 2.1.

Oi upo-prot�seic antistoiqoÔn stouc kìmbouc, staupo-dèndra, tou dèndrou thc A.

Oi upo-prot�seic thc ¬p ∨ q → (p → q ∧ r) eÐnai:¬p ∨ q → (p → q ∧ r)¬p ∨ q p → q ∧ r¬p q p q ∧ r

p q r

Prosoq : up�rqoun duo diaforetikèc upo-prot�seic p.

Ta p ∨ q kai p → q den eÐnai upo-prot�seic, eÐnaiupo-akoloujÐec (sub-strings) sumbìlwn.

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SuntomografÐec ∧1≤i≤n Ai ,

∧ni=1 Ai ,

n∧i=1

Ai ,∧

1≤i≤n

Ai

Ta parap�nw eÐnai suntomografÐec gia thn prìtashA1 ∧ A2 ∧ · · · ∧ An. ∨

1≤i≤n Ai ,∨n

i=1 Ai ,

n∨i=1

Ai ,∨

1≤i≤n

Ai

Ta parap�nw eÐnai suntomografÐec gia thn prìtashA1 ∨ A2 ∨ · · · ∨ An.

EÐnai ìpwc sthn 'Algebra, ìpou to∑n

i=1 ai eÐnai suntomografÐatou a1 + a2 + · · ·+ an.

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OrologÐa

Orismìc 2.2

I MÐa prìtash thc morf c >,⊥, p, gia èna �tomo p,onom�zetai atomik prìtash (atomic formula).

I Stoiqei¸dhc tÔpoc (literal) eÐnai k�je atomik prìtash h

�rnhsh thc.

I Programmatikìc tÔpoc (clause) eÐnai mÐa di�zeuxh enìc

perissotèrwn stoiqeiwd¸n tÔpwn.

Pq, oi prot�seic p,¬r ,¬⊥,> eÐnai ìlec stoiqei¸deic tÔpoi.

ParadeÐgmata programmatik¸n tÔpwn:p, ¬p, p ∨ ¬q ∨ r , p ∨ p ∨ ¬p ∨ ¬⊥ ∨ > ∨ ¬q.

O kenìc programmatikìc tÔpoc, o opoÐoc den perièqei kanènastoiqei¸dh tÔpo, antimetwpÐzetai ìpwc h prìtash ⊥ (stobiblÐo o kenìc programmatikìc tÔpoc sumbolÐzetai me �).

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2.2 ShmasiologÐa

GnwrÐzoume p¸c na diab�zoume kai na gr�foume prot�seic(formulas). Poio eÐnai ìmwc to nìhma touc; Me �lla lìgia,poia eÐnai h shmasiologÐa touc;

Oi logikoÐ sÔndesmoi (∧,¬,∨,→,↔) èqoun k�poia antistoiqÐame thn Ellhnik gl¸ssa.

All� ta Ellhnik� eÐnai mÐa fusik gl¸ssa, gem�th meas�feiec kai eidikèc peript¸seic. H met�frash metaxÔEllhnik¸n kai Logik c den eÐnai p�nta eÔkolh.

Qreiazìmaste èna akrib trìpo me ton opoÐo na dÐnoume nìhmastic prot�seic thc Logik c.

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Kat�stash, Aponom al jeiacMÐa kat�stash (situation) eÐnai k�ti to opoÐo orÐzei an k�je�tomo eÐnai alhjèc yeudèc.

Aponom al jeiac onom�zoume k�je sun�rthsh F : Q 7→ {α,ψ}ìpou Q eÐnai to sÔnolo twn atìmwn thc gl¸ssac. Dhlad mÐaaponom al jeiac dÐnei alhjotimèc sta �toma thc gl¸ssac.

Gia ta if-tests mÐa kat�stash eÐnai èna shmeÐo sthn ektèleshtou progr�mmatoc. Oi trèqousec timèc twn metablht¸n touprogr�mmatoc kajorÐzoun an k�je atomik èkfrash enìc if-test(pq, x > 0, x=y klp) eÐnai alhj c yeud c.

Gia thn èkfrash “if (it rains) then (it is cloudy)” mÐa kat�stasheÐnai o kairìc.

Gia ta �toma p, q, . . . mÐa kat�stash diatup¸nei poia apìaut� eÐnai alhj kai poia eÐnai yeud .

Up�rqei p�nw apì mÐa kat�stash. Profan¸c, se mÐadiaforetik kat�stash oi alhjotimèc ja eÐnai diaforetikèc.GnwrÐzontac mÐa kat�stash mporoÔme na upologÐsoume tonìhma k�je prìtashc, dhlad an h prìtash eÐnai alhj c yeud c se aut thn kat�stash. PhgaÐnoume apì aplècprot�seic se pio sÔnjetec, ìpwc ja doÔme eujÔc amèswc.

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Kat�stash, Aponom al jeiac

Up�rqei p�nw apì mÐa kat�stash. Profan¸c, se mÐadiaforetik kat�stash oi alhjotimèc ja eÐnai diaforetikèc.

GnwrÐzontac mÐa kat�stash mporoÔme na upologÐsoume tonìhma k�je prìtashc, dhlad an h prìtash eÐnai alhj c yeud c se aut thn kat�stash. PhgaÐnoume apì aplècprot�seic se pio sÔnjetec, ìpwc ja doÔme eujÔc amèswc.

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ErmhneÐa

Orismìc 2.3 ( ErmhneÐa (Evaluation) )

H alhjotim mÐac prìtashc se mÐa dedomènh kat�stash

orÐzetai wc ex c.

I H kat�stash mac plhroforeÐ gia tic alhjotimèc twn

atìmwn.

I H prìtash > eÐnai alhj c kai h ⊥ eÐnai yeud c.

Ac upojèsoume ìti A,B eÐnai prot�seic, kai ìti gnwrÐzoume tic

alhjotimèc touc sth dedomènh kat�stash. Tìte se aut thn

kat�stash:

I H prìtash ¬A eÐnai alhj c an h A eÐnai yeud c, kai yeud c

an h A eÐnai alhj c.

I H prìtash A∧B eÐnai alhj c an h A kai h B eÐnai alhjeÐc.

Se opoiad pote �llh perÐptwsh h A ∧ B eÐnai yeud c.

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ErmhneÐa

Orismìc 2.3 (ErmhneÐa) (sunèqeia)

I H prìtash A ∨ B eÐnai alhj c an h A eÐnai alhj c h BeÐnai alhj c kai h A kai h B eÐnai alhjeÐc. Se

opoiad pote �llh perÐptwsh h A ∨ B eÐnai yeud c.

(Periektik di�zeuxh.)

I H prìtash A→ B eÐnai alhj c an h A eÐnai yeud c h BeÐnai alhj c. Alli¸c, an h A eÐnai alhj c kai h B eÐnai

yeud c tìte h A→ B eÐnai yeud c.

I H prìtash A↔ B eÐnai alhj c an h A kai h B èqoun thn

Ðdia alhjotim (eÐnai kai oi dÔo alhjeÐc kai oi dÔo

yeudeÐc).

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PÐnakec Al jeiac gia touc sundèsmoucMporoÔme na ekfr�soume autoÔc touc kanìnecqrhsimopoi¸ntac pÐnakec al jeiac gia touc sundèsmouc.

Gr�foume 1 gia thn tim “alhj c” kai 0 gia thn tim “yeud c”.Aut� den èqoun sqèsh me ta >,⊥ ta opoÐa eÐnai prot�seic kaiìqi alhjotimèc.

A ¬A > ⊥1 0 1 00 1 1 0

A B A ∧ B A ∨ B A→ B A↔ B

1 1 1 1 1 11 0 0 1 0 00 1 0 1 1 00 0 0 0 1 1

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ParadeÐgmata

Ac upojèsoume ìti h prìtash p eÐnai alhj c kai h q eÐnaiyeud c se mÐa sugkekrimènh kat�stash. Tìte se aut thnkat�stash h prìtash:

I ¬p eÐnai yeud c.

I ¬q eÐnai alhj c.

I p ∧ q eÐnai yeud c.

I p ∨ q eÐnai alhj c.

I p → q eÐnai yeud c.

I q → p eÐnai alhj c.

I p ↔ q eÐnai yeud c kai h ¬p ↔ q eÐnai alhj c.

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ParadeÐgmata

Ac upojèsoume t¸ra ìti h prìtash p eÐnai yeud c kai h q eÐnaialhj c. Se aut thn kainoÔrgia kat�stash h prìtash:

I ¬p eÐnai t¸ra alhj c.

I ¬q eÐnai t¸ra yeud c.

I p ∧ q eÐnai akìma yeud c.

I p ∨ q eÐnai akìma alhj c.

I p → q eÐnai t¸ra alhj c.

I q → p eÐnai t¸ra yeud c.

I p ↔ q eÐnai akìma yeud c kai h ¬p ↔ q eÐnai akìmaalhj c.

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ParadeÐgmata

Ac upojèsoume p�li ìti h prìtash p eÐnai alhj c kai h q eÐnaiyeud c se mÐa sugkekrimènh kat�stash. Poia eÐnai h alhjotim twn parak�tw prot�sewn:

I p → ¬q

I q → q

I ¬p ∨ ¬q

I p ∧ q → q

I ¬(p ↔ >)

I ¬(q → ¬(p ∨ q))

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2.3 Sqèsh tou ProtasiakoÔ LogismoÔ me th Fusik Gl¸ssa

H tupopoÐhsh thc fusik c gl¸ssac, dhlad h met�frash twnprot�sewn thc fusik c gl¸ssac sthn Protasiak Logik deneÐnai p�nta eÔkolh. EÐnai ìmwc aparaÐthth gia orismènecefarmogèc.Merik� stoiqeÐa pou prèpei na prosèxoume sthn tupopoÐhshthc fusik c gl¸ssac:

I Apomon¸noume tic atomikèc prot�seic.

I Prosèqoume an k�poiec apì tic atomikèc prot�seic eÐnaitautìshmec.

I AntikajistoÔme diakritèc atomikèc prot�seic me diakrit��toma (p, q, r , . . . ).

I EntopÐzoume ìlouc touc logikoÔc sundèsmouc (autì apaiteÐthn apìdosh twn sundèsmwn thc fusik c gl¸ssac seantÐstoiqouc thc tupik c).

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TupopoÐhsh thc fusik c gl¸ssac

I EntopÐzoume tic mikrìterec (sÔnjetec) prot�seic oi opoÐecapoteloÔn th sÔnjeth prìtash, kai tic diakrÐnoume meparenjèseic.

I EntopÐzoume ton kÔrio logikì sÔndesmo pou sundèei ticsÔnjetec prot�seic metaxÔ touc, kai proqwroÔme ap' autìnstouc epimèrouc sundèsmouc.

I ApodÐdoume thn prìtash qrhsimopoi¸ntac ta sqetik�logik� sÔmbola kai parenjèseic.

ParadeÐgmata tupopoÐhshc thc fusik c gl¸ssac up�rqoun stobiblÐo “Logik : H Dom tou Epiqeir matoc”.

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Par�deigma: “An den pèsei o plhjwrismìc kai anèboun taepitìkia, tìte to dhmìsio qrèoc ja pèsei an kai mìno ankatafÔgoume se exwterikì daneismì”.

Atomikèc prot�seic:p: o plhjwrismìc pèfteiq: ta epitìkia anebaÐnounr : to dhmìsio qrèoc pèfteis: katafeÔgoume se exwterikì daneismì

LogikoÐ sÔndesmoi: an . . . tìte, den, kai, an kai mìno an

Pr¸th prosèggish logik c morf c:an (den p kai q), tìte (r an kai mìno an s)KÔrioc sÔndesmoc: an . . . tìteLogik morf : (¬p ∧ q)→ (r ↔ s)

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Probl mata sthn tupopoÐhsh thc fusik c gl¸ssac

To “all�” mporeÐ na shmaÐnei “kai”.“ja bgw èxw all� brèqei” metafr�zetai se“ja bgw èxw ∧ brèqei”.

“ektìc an” genik� shmaÐnei “ ”“ja bgw èxw ektìc an brèqei” metafr�zetai se“ja bgw èxw ∨ ja brèqei”.EpÐshc mporeÐ na ekfrasteÐ wc “¬ (ja brèqei) → ja bgw èxw”.

To “ ” suqn� shmaÐnei “kai”“mporeÐc na p�reic kotìpoulo y�ri” sun jwc shmaÐnei“mporeÐc na p�reic kotìpoulo ∧ mporeÐc na p�reic y�ri”

“o Gi�nnhc kai h MarÐa eÐnai eutuqismènoi mazÐ” prèpei nametafrasteÐ wc mÐa atomik prìtash, ìqi wc“o Gi�nnhc eÐnai eutuqismènoc ∧ h MarÐa eÐnai eutuqismènh”

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Apì thn tupik gl¸ssa sth fusik

Arqik� h met�frash thc tupik c gl¸ssac sth fusik faÐnetaieÔkolh. Pq:

I “p ∧ q” metafr�zetai se “p kai q”.

I “brèqei → ¬ ja bgw èxw” metafr�zetai se “an brèqei tìteden ja bgw èxw”.

All� up�rqoun kai probl mata. Oi polÔplokec prot�seic thcProtasiak c Logik c eÐnai dÔskolo na apodojoÔn sth fusik gl¸ssa.O sÔndesmoc → eÐnai dÔskolo na apodojeÐ. Sth fusik gl¸ssaqrhsimopoioÔme to “an . . . tìte” me diaforetikoÔc trìpouc, kaiìqi p�nta prosektik�.H prìtash “EÐmai o P�pac → eÐmai �jeoc” eÐnai alhj c(giatÐ;). All� ja lègame potè ìti h prìtash thc fusik cgl¸ssac “An eÐmai o P�pac tìte eÐmai �jeoc” eÐnai alhj c;

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3. Epiqeir mata, egkurìthta

GnwrÐzoume p¸c na diab�zoume, na gr�foume kai naermhneÔoume (evaluate) prot�seic thc Protasiak c Logik c.GnwrÐzoume epÐshc merik� pr�gmata sqetik� me th met�frashmetaxÔ thc fusik c gl¸ssac kai thc Protasiak c Logik c.

H Logik qrhsimopoieÐtai kai gia epiqeirhmatologÐa. Pq,parak�tw èqoume to ex c epiqeÐrhma:

I O Swkr�thc eÐnai �ndrac.

I Oi �ndrec eÐnai jnhtoÐ.

I 'Ara o Swkr�thc eÐnai jnhtìc.

EÐnai autì èna ègkuro epiqeÐrhma;

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3.1 'Egkura epiqeir mataTo prohgoÔmeno epiqeÐrhma eÐnai ègkuro an gia k�jekat�stash sthn opoÐa o Swkr�thc eÐnai �ndrac kai oi �ndreceÐnai jnhtoÐ, o Swkr�thc eÐnai �ndrac.

'Opwc èqoume anafèrei, sthn Protasiak Logik mÐakat�stash orÐzei thn alhjotim k�je atìmou.

Orismìc 3.1 ( 'Egkuro EpiqeÐrhma (Valid Argument) )

Dojous¸n twn prot�sewn A1,A2, . . . ,An,B , èna epiqeÐrhma

A1, . . . ,An, �ra B

eÐnai ègkuro an: h prìtash B eÐnai alhj c se k�je kat�stash

sthn opoÐa ìlec oi prot�seic A1, . . . ,An eÐnai alhjeÐc. 'An isqÔei

autì tìte gr�foume A1, . . . ,An |= B .

To sÔmbolo “|=” diab�zetai wc “logik sunèpeia” (logicalentailment/implication).

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ParadeÐgmata epiqeirhm�twn

Ac upojèsoume ìti A,B eÐnai tuqaÐec prot�seic. To epiqeÐrhma:

I “A, �ra A” eÐnai ègkuro giatÐ se k�je kat�stash, an h AeÐnai alhj c tìte h A eÐnai alhj c. A |= A.

I “A ∧ B , �ra A” eÐnai ègkuro. A ∧ B |= A.

I “A, �ra A ∧ B” den eÐnai ègkuro: up�rqoun katast�seicìpou h A eÐnai alhj c kai h A ∧ B eÐnai yeud c. Sunep¸cA 2 A ∧ B .

I “A,A→ B , �ra B” eÐnai ègkuro. To ìnoma autoÔ touepiqeir matoc eÐnai modus ponens.

I “A→ B,¬B , �ra ¬A” eÐnai epÐshc ègkuro. To ìnomaautoÔ tou epiqeir matoc eÐnai modus tollens.

I “A→ B,B , �ra A” den eÐnai ègkuro. A→ B,B 2 A.

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3.2 'Egkurec, ikanopoi simec, isodÔnamec prot�seic

Treic basikèc idèec sqetÐzontai me ta ègkura epiqeir mata.

Orismìc 3.2 ( 'Egkurh Prìtash (Valid Formula) )

MÐa prìtash eÐnai logik� ègkurh logik� alhj c an eÐnai

alhj c se k�je kat�stash. Gr�foume |= A ìtan mÐa prìtash

A eÐnai ègkurh.

Oi ègkurec prot�seic onom�zontai kai tautologÐec (tautologies).

Orismìc 3.3 ( Ikanopoi simh Prìtash (Satisfiable Formula) )

MÐa prìtash eÐnai ikanopoi simh epalhjeÔsimh an eÐnai

alhj c se toul�qiston mÐa kat�stash.

Orismìc 3.4 ( IsodÔnamec Prot�seic (Equivalent Formulas) )

Duo prot�seic A,B eÐnai logik� isodÔnamec an eÐnai alhjeÐc

stic Ðdiec akrib¸c katast�seic. Sumbolik� A ≡ B .

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ParadeÐgmata

Prìtash 'Egkurh Ikanopoi simh

> nai nai⊥ ìqi ìqip ìqi nai

p ∧ ¬p ìqi ìqip → p nai nai

IsodÔnamh; p > p → q

p ∧ p nai ìqi ìqip ∨ ¬p ìqi nai ìqi¬p ∨ q ìqi ìqi nai

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3.3 Sqèseic metaxÔ twn tess�rwn ennoi¸n

Ta ègkura epiqeir mata kai oi ègkurec, ikanopoi simec kaiisodÔnamec prot�seic orÐzontai metaxÔ touc. Pq:

epiqeÐrhma egkurìthta ikanopoih- isodunamÐasimìthta

A |= B A→ B ègkurh A ∧ ¬B (A→ B) ≡ >mh ikanopoi simh

> |= A A ègkurh ¬A A ≡ >mh ikanopoi simh

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4. Elègqontac thn egkurìthta twn prot�sewn

'Ara qrei�zetai na antimetwpÐsoume mìno ègkurec prot�seic.

P¸c katalabaÐnoume an mÐa prìtash eÐnai ègkurh; Elègqoumeìti h prìtash eÐnai alhj c se k�je kat�stash.

Autì mporeÐ na epiteuqjeÐ gia prot�seic thc Protasiak cLogik c, all� upologistik� eÐnai èna dÔskolo prìblhma.

Gia thn Kathgorhmatik Logik , thn opoÐa ja doÔme staepìmena maj mata, eÐnai polÔ pio dÔskolo prìblhma, giatÐ oikatast�seic eÐnai pollèc kai pio polÔplokec. All� up�rqounk�poioi trìpoi na deÐxoume ìti orismènec prot�seic thcKathgorhmatik c Logik c eÐnai ègkurec.

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Trìpoi na elegqjeÐ h egkurìthta twn prot�sewn

Up�rqoun di�foroi trìpoi na upologÐsoume an mÐa prìtashthc Protasiak c Logik c eÐnai ègkurh ìqi:

I PÐnakec al jeiac. Elègqoume an se ìlec tic pijanècsqetikèc katast�seic h prìtash eÐnai alhj c.

I 'Amesh “majhmatik ” epiqeirhmatologÐa (direct‘mathematical’ argument).

I IsodunamÐec. 'Eqoume mÐa lÐsta apì qr sima zeug�riaisodÔnamwn prot�sewn. QrhsimopoioÔme autèc ticisodunamÐec gia na metatrèyoume thn arqik prìtash sthnprìtash >, h opoÐa eÐnai ègkurh.

I Di�fora sust mata apodeÐxewn (proof systems), ìpwc pq,kanìnec fusik c sumperasmatologÐac (natural deduction),sust mata Hilbert, shmasiologikoÐ pÐnakec (semantictableaux).

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4.1 PÐnakec al jeiac

Ac deÐxoume ìti h prìtash (p → q)↔ (¬p ∨ q) eÐnai ègkurh.Gr�foume 1 gia thn tim alhj c kai 0 gia thn tim yeud c.

'Eqoume dÔo �toma, ìpou to kajèna paÐrnei mÐa apì tic dÔoalhjotimèc. Oi alhjotimèc �llwn atìmwn den mac apasqoloÔn.'Ara èqoume tèsseric sqetikèc katast�seic. ErmhneÔoume ìlectic upo-prot�seic thc prìtashc mac se k�je kat�stash:

p q p → q ¬p ¬p ∨ q (p → q)↔ (¬p ∨ q)

1 1 1 0 1 11 0 0 0 0 10 1 1 1 1 10 0 1 1 1 1

Blèpoume ìti h prìtash (p → q)↔ (¬p ∨ q) eÐnai alhj c (èqeitim 1) se ìlec tic tèsseric katast�seic. 'Ara eÐnai ègkurh.

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IsodunamÐa kai ikanopoihsimìthta

O Ðdioc pÐnakac deÐqnei ìti oi prot�seic p → q kai ¬p ∨ q eÐnaiisodÔnamec: se k�je mÐa apì tic tèsseric katast�seic èqounthn Ðdia alhjotim .

Ti elègqoume gia na deÐxoume ikanopoihsimìthta;

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Meionekt mata kai pleonekt mata twn pin�kwn al jeiac

EÐnai kourastik� kai eÐnai eÔkolo na gÐnei l�joc.

All� douleÔoun p�nta, toul�qiston gia thn Protasiak Logik ìpou mìno peperasmènec katast�seic sqetÐzontai memÐa prìtash. (Autì den isqÔei gia thn Kathgorhmatik Logik .Sthn perÐptwsh aut prèpei na broÔme �llouc trìpouc.)

Oi pÐnakec al jeiac mporoÔn epiplèon na qrhsimopoihjoÔn giana deÐxoume ikanopoihsimìthta kai isodunamÐa.

DeÐqnoun pìso dÔskolo eÐnai na apodeiqjeÐ h egkurìthta,ikanopoihsimìthta, klp. MÐa prìtash me n �toma qrei�zetai2n grammèc ston pÐnaka al jeiac thc. Prosjètontac èna �tomodiplasi�zei tic grammèc tou pÐnaka.

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4.2 'Amesh majhmatik epiqeirhmatologÐa (Directargument)

Ac deÐxoume ìti h prìtash p → p ∨ q eÐnai ègkurh.

Ac p�roume mÐa tuqaÐa kat�stash, h opoÐa dÐnei alhjotimècsta �toma p, q. Ja deÐxoume ìti h prìtash p → p ∨ q eÐnaialhj c se aut thn kat�stash.

'Ara prèpei na deÐxoume ìti AN to p eÐnai alhjèc se aut thnkat�stash TOTE eÐnai alhj c kai h prìtash p ∨ q.

An to p eÐnai alhjèc, tìte kai èna apì ta p, q eÐnai alhjèc, �rah prìtash p ∨ q eÐnai alhj c.

'Askhsh: deÐxte ìti h prìtash A ∧ (A→ B)→ B eÐnai ègkurh.

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'Amesh majhmatik epiqeirhmatologÐaAc deÐxoume ìti h prìtash (A ∧ B) ∧ C eÐnai logik� isodÔnamhme thn prìtash A ∧ (B ∧ C ). (OmoÐwc, ja mporoÔsame nadeÐxoume ìti h prìtash (A∧B)∧C ↔ A∧ (B ∧C ) eÐnai ègkurh.)

Ac p�roume mÐa opoiad pote kat�stash.

H (A ∧ B) ∧ C eÐnai alhj c se aut thn kat�stash an kai mìnoan h A ∧ B kai h C eÐnai alhjeÐc (sÔmfwna me ton orismì thcshmasiologÐac tou ∧).

To parap�nw isqÔei an kai mìno an h A kai h B eÐnai alhjeÐc,kai h C eÐnai epÐshc alhj c, dhlad eÐnai ìlec alhjeÐc.

Autì isqÔei an kai mìno an h A eÐnai alhj c, kai epÐshc h B kaih C eÐnai alhjeÐc.

Autì isqÔei an kai mìno an h A kai h B ∧ C eÐnai alhjeÐc.

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'Amesh majhmatik epiqeirhmatologÐa

Autì isqÔei an kai mìno an h A ∧ (B ∧ C ) eÐnai alhj c.

'Ara oi prot�seic (A ∧ B) ∧ C kai A ∧ (B ∧ C ) èqoun thn Ðdiaalhjotim se aut thn kat�stash. H kat�stash tan tuqaÐa,�ra oi prot�seic eÐnai logik� isodÔnamec.

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'Amesh Majhmatik EpiqeirhmatologÐaAc deÐxoume ìti h prìtash ((p → q)→ p)→ p, h opoÐa eÐnaignwst wc nìmoc tou Peirce, eÐnai ègkurh.

Ac p�roume mÐa tuqaÐa kat�stash. An to p eÐnai alhjèc seaut thn kat�stash, tìte h prìtash ((p → q)→ p)→ p eÐnaialhj c, giatÐ k�je prìtash thc morf c A→ B eÐnai alhj cìtan h B eÐnai alhj c.

An to p den eÐnai alhjèc tìte ja eÐnai yeudèc se aut thnkat�stash.

Tìte h prìtash p → q eÐnai alhj c, giatÐ k�je prìtash thcmorf c A→ B eÐnai alhj c ìtan h A eÐnai yeud c.

Tìte h prìtash (p → q)→ p eÐnai yeud c giatÐ k�je prìtashthc morf c A→ B eÐnai yeud c ìtan h A eÐnai alhj c kai h BeÐnai yeud c.

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'Amesh Majhmatik EpiqeirhmatologÐa

Tìte h prìtash ((p → q)→ p)→ p eÐnai alhj c giatÐ k�jeprìtash thc morf c A→ B eÐnai alhj c ìtan h A eÐnai yeud c.

Aut h morf epiqeirhmatologÐac onom�zetaiepiqeirhmatologÐa me peript¸seic (argument by cases): to peÐnai alhjèc to p eÐnai yeudèc. Den up�rqoun �llecpeript¸seic. Autìc eÐnai o nìmoc thc tou trÐtou apokleÐsewc

(law of excluded middle).

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4.3 IsodunamÐec

Oi isodunamÐec mac bohjoÔn sto na aplopoi soume mÐaprìtash na metatrèyoume mÐa prìtash se mÐa �llh, p�ntadiathr¸ntac th logik isodunamÐa.

'Ara an mporèsoume na metatrèyoume mÐa prìtash sthnprìtash > tìte xèroume ìti h arqik prìtash eÐnai ègkurh.

Ja anafèroume mÐa lÐsta isodunami¸n oi opoÐec eÐnai qr simecgi' autì ton skopì.

Oi isodunamÐec eÐnai pollèc, all� ìlec ekfr�zoun basikèclogikèc arqèc pou ja prèpei na gnwrÐzete. Na elègxete ìti oiprot�seic eÐnai ìntwc isodÔnamec qrhsimopoi¸ntac pÐnakecal jeiac �mesh epiqeirhmatologÐa.

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IsodunamÐec pou perilamb�noun to sÔndesmo ∧

Stic parak�tw isodunamÐec ta A,B,C ekfr�zoun tuqaÐecprot�seic. Gia suntomÐa pollèc forèc ja gr�foume“isodÔnamec” antÐ gia “logik� isodÔnamec”.

1. H prìtash A ∧ B eÐnai logik� isodÔnamh me thn B ∧ A(antimetajetikìthta tou ∧).

2. H prìtash A ∧ A eÐnai logik� isodÔnamh me thn A(autop�jeia tou ∧).

3. H prìtash A ∧ > eÐnai logik� isodÔnamh me thn A.

4. Oi prot�seic ⊥ ∧ A kai ¬A ∧ A eÐnai isodÔnamec me thn ⊥.5. H prìtash (A ∧ B) ∧ C eÐnai isodÔnamh me thn A ∧ (B ∧ C )

(prosetairistikìthta tou ∧).

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IsodunamÐec pou perilamb�noun to sÔndesmo ∨

6. H prìtash A ∨ B eÐnai isodÔnamh me thn B ∨ A(antimetajetikìthta tou ∨).

7. H prìtash A∨A eÐnai isodÔnamh me thn A (autop�jeia tou∨).

8. Oi prot�seic A ∨ > kai A ∨ ¬A eÐnai isodÔnamec me thn >.9. H prìtash ⊥ ∨ A eÐnai isodÔnamh me thn A.

10. H prìtash (A ∨ B) ∨ C eÐnai isodÔnamh me thn A ∨ (B ∨ C )(prosetairistikìthta tou ∨).

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IsodunamÐec pou perilamb�noun touc sundèsmouc ¬,kai →

11. H prìtash ¬> eÐnai isodÔnamh me thn ⊥.12. H prìtash ¬⊥ eÐnai isodÔnamh me thn >.13. H prìtash ¬¬A eÐnai isodÔnamh me thn A.

14. H prìtash A→ A eÐnai isodÔnamh me thn >.15. H prìtash > → A eÐnai isodÔnamh me thn A.

16. H prìtash A→ > eÐnai isodÔnamh me thn >.17. H prìtash ⊥ → A eÐnai isodÔnamh me thn >.18. H prìtash A→ ⊥ eÐnai isodÔnamh me thn ¬A.

19. H prìtash A→ B eÐnai isodÔnamh me thn ¬A ∨ B kai thn¬(A ∧ ¬B).

20. H prìtash ¬(A→ B) eÐnai isodÔnamh me thn A ∧ ¬B .

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IsodunamÐec pou perilamb�noun to sÔndesmo ↔

21. H prìtash A↔ B eÐnai isodÔnamh me tic:I (A→ B) ∧ (B → A),I (A ∧ B) ∨ (¬A ∧ ¬B),I ¬A↔ ¬B.

22. H prìtash ¬(A↔ B) eÐnai isodÔnamh me tic:I A↔ ¬B,I ¬A↔ B,I (A ∧ ¬B) ∨ (¬A ∧ B).

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Nìmoi De Morgan kai epimeristikìthtac twn ∧,∨

Nìmoi De Morgan:

23. H prìtash ¬(A ∧ B) eÐnai isodÔnamh me thn ¬A ∨ ¬B .

24. H prìtash ¬(A ∨ B) eÐnai isodÔnamh me thn ¬A ∧ ¬B .

Nìmoi epimeristikìthtac twn ∧,∨:25. H prìtash A ∧ (B ∨ C ) eÐnai isodÔnamh me thn

(A ∧ B) ∨ (A ∧ C ).

26. H prìtash A ∨ (B ∧ C ) eÐnai isodÔnamh me thn(A ∨ B) ∧ (A ∨ C ).

27. Oi prot�seic A ∧ (A ∨ B) kai A ∨ (A ∧ B) eÐnai isodÔnamecme thn A.

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ApodeiknÔontac egkurìthta me isodunamÐecP�rte opoiad pote upo-prìtash mÐac prìtashc A kaiantikatast ste thn me mÐa isodÔnamh prìtash.Epanal�bete to prohgoÔmeno b ma. To apotèlesma ja eÐnaimÐa prìtash logik� isodÔnamh me thn A.An to apotèlesma eÐnai h prìtash > tìte h A eÐnai ègkurh.Pq, ac deÐxoume ìti h prìtash p → p ∨ q eÐnai ègkurh.

I Aut h prìtash eÐnai isodÔnamh me thn ¬p ∨ (p ∨ q) (apìthn isodunamÐa 19).

I H teleutaÐa eÐnai isodÔnamh me thn (¬p ∨ p) ∨ q (apì thnisodunamÐa 10).

I H teleutaÐa eÐnai isodÔnamh me thn > ∨ q (apì thnisodunamÐa 8).

I H teleutaÐa eÐnai isodÔnamh me thn > (apì thn 8 p�li), hopoÐa eÐnai ègkurh.

Ft�same sthn prìtash >, h opoÐa eÐnai ègkurh. 'Ara hprìtash p → p ∨ q eÐnai ègkurh.

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Par�deigma

Ac deÐxoume ìti h prìtash A→ B eÐnai isodÔnamh me thn¬B → ¬A.

I H prìtash ¬B → ¬A eÐnai isodÔnamh me thn ¬(¬B) ∨ ¬A(apì thn isodunamÐa 19).

I H teleutaÐa eÐnai isodÔnamh me thn ¬A ∨ ¬¬B (apì thnisodunamÐa 6).

I H teleutaÐa eÐnai isodÔnamh me thn ¬A ∨ B (apì thnisodunamÐa 13).

I H teleutaÐa eÐnai isodÔnamh me thn A→ B (apì thnisodunamÐa 19).

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Par�deigma

Ac deÐxoume ìti h prìtash (p ∨ ¬q) ∧ (p ∨ q) eÐnai logik�isodÔnamh me thn p.

I H prìtash (p ∨ ¬q) ∧ (p ∨ q) eÐnai isodÔnamh me thnp ∨ (¬q ∧ q) (“an�strofh” qr sh thc isodunamÐac 26).

I H teleutaÐa eÐnai isodÔnamh me thn p ∨ ⊥ (apì thnisodunamÐa 4).

I H teleutaÐa eÐnai isodÔnamh me thn p (apì thn isodunamÐa9).

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Kanonikèc morfèc

Oi isodunamÐec mac epitrèpoun na metatrèyoume mÐa prìtashse mÐa isodÔnam thc se kanonik morf (normal form).Up�rqoun dÔo koinèc kanonikèc morfèc:

Orismìc 4.1 (Diazeuktik Kanonik Morf , Suzeuktik Kanonik Morf )

I MÐa prìtash eÐnai se diazeuktik kanonik morf (DKM)(disjunctive normal form (DNF)) an eÐnai mÐa di�zeuxh

suzeÔxewn stoiqeiwd¸n tÔpwn, kai den mporeÐ peraitèrw na

aplopoihjeÐ all�zontac morf .

ParadeÐgmata:

p ∨ q ∨ ¬r(p ∧ ¬q) ∨ r ∨ (¬p ∧ q ∧ ¬r)Anti-par�deigma: (p ∧ p) ∨ (q ∧ > ∧ ¬q)

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Kanonikèc morfèc

Orismìc 4.1 (sunèqeia)

I MÐa prìtash eÐnai se suzeuktik kanonik morf (SKM)(conjunctive normal form (CNF)) an eÐnai mÐa sÔzeuxh

diazeÔxewn stoiqeiwd¸n tÔpwn (dhlad mÐa sÔzeuxh

programmatik¸n tÔpwn), kai den mporeÐ peraitèrw na

aplopoihjeÐ all�zontac morf .

Par�deigma:(p ∨ ¬q) ∧ (q ∨ r) ∧ (¬p ∨ q)

O Orismìc 2.2 orÐzei touc stoiqei¸deic tÔpouc kai toucprogrammatikoÔc tÔpouc.

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Metatrèpontac mÐa prìtash se kanonik morf

1. ApaloÐfoume touc sundèsmouc →,↔:AntikajistoÔme ìlec tic upo-prot�seic A→ B me ¬A ∨ B .AntikajistoÔme ìlec tic upo-prot�seic A↔ B me(A ∧ B) ∨ (¬A ∧ ¬B).

2. ProwjoÔme tic arn seic mèsa stic parenjèseic mèqri ta�toma qrhsimopoi¸ntac touc nìmouc De Morgan. Bg�zoumetic diplèc arn seic (antikajistoÔme thn ¬¬A me thn A).

3. QrhsimopoioÔme touc kanìnec epimeristikìthtac gia nakatal xoume sthn epijumht kanonik morf .

4. AplopoioÔme: antikajistoÔme tic upo-prot�seic p ∧ ¬p me⊥, kai tic p ∨ ¬p me >. AntikajistoÔme tic > ∨ p me >, tic>∧ p me p, tic ⊥∨ p me p, kai tic ⊥∧ p me ⊥. H isodunamÐa27 eÐnai epÐshc suqn� qr simh. Epanalamb�noume mèqri hprìtash na mhn eÐnai peraitèrw aplopoi simh.

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Par�deigma: h p ∧ q → ¬(p ↔ ¬r) se DKM

p ∧ q → ¬(p ↔ ¬r) [arqik prìtash]¬(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 [isodunamÐa 6]¬p ∨ (p ∧ r) ∨ ¬q [isodunamÐa 27]

H teleutaÐa prìtash eÐnai se DKM. MporoÔme na thnaplopoi soume peraitèrw an jèloume proswrin� na fÔgoumeapì thn DKM:

[(¬p ∨ p) ∧ (¬p ∨ r)] ∨ ¬q [isodunamÐa 26-epimeristikìthta][> ∧ (¬p ∨ r)] ∨ ¬q [isodunamÐa 8]¬p ∨ r ∨ ¬q [isodunamÐa 3]

H shmantik aplopoÐhsh thc arqik c prìtashc deÐqnei topleonèkthma thc metatrop c mÐac prìtashc se kanonik morf .

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5. Sust mata apodeÐxewn: fusik sumperasmatologÐa

'Ena sÔsthma apodeÐxewn mac bohj�ei na deÐqnoume ìti mÐaprìtash eÐnai ègkurh qrhsimopoi¸ntac mìno suntaktikoÔckanìnec � den qrhsimopoioÔme to nìhma twn prot�sewn. JamporoÔsame na gr�youme èna prìgramma logismikoÔ to opoÐoefarmìzei autoÔc touc kanìnec.

H automatopoihmènh sumperasmatologÐa (automated reasoning)eÐnai mÐa anaptussìmenh ereunhtik perioq .

Up�rqoun arket� sust mata apodeÐxewn. Sta epìmenamaj mata ja asqolhjoÔme me to sÔsthma kanìnwn fusik c

sumperasmatologÐac (natural deduction (ND)).

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Ti eÐnai to sÔsthma ND;I MÐa tupopoÐhsh thc �meshc epiqeirhmatologÐac (direct

argument).I Xekin¸ntac Ðswc apì k�poia dojènta gegonìta � k�poiec

prot�seic A1, . . . ,An � qrhsimopoioÔme touc kanìnec tousust matoc gia sumperasmatologÐa proc mÐa prìtash B .An epitÔqoume, gr�foume A1, . . . ,An ` B .

I Kat� th sumperasmatologÐa par�gontai endi�mesecprot�seic. Autèc apoteloÔn thn apìdeixh thc B apì ticA1, . . . ,An. K�je b ma thc apìdeixhc eÐnai èna ègkuroepiqeÐrhma.

I Up�rqoun dÔo kanìnec gia k�je sÔndesmo: ènac gia naeis�goume to sÔndesmo (dhlad na eis�goume mÐa prìtashthc opoÐac eÐnai o kÔrioc sÔndesmoc), kai ènac gia naqrhsimopoi soume to sÔndesmo (na qrhsimopoi soume mÐaprìtash thc opoÐac eÐnai o kÔrioc sÔndesmoc). Oi kanìneceÐnai basismènoi sthn shmasiologÐa twn sundèsmwn, thnopoÐa eÐdame sta prohgoÔmena maj mata.

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5.1 Kanìnec ND gia touc sundèsmouc ∧,→,∨

Up�rqoun dÔo kanìnec gia k�je sÔndesmo. Oi kanìnecbasÐzontai sto nìhma twn sundèsmwn.

Kanìnec gia to sÔndesmo ∧:I (eisagwg ∧, ∧-introduction, ∧I ). Gia na eis�goume mÐa

prìtash thc morf c A ∧ B , prèpei na èqoume dh eis�geitic prot�seic A kai B .

1 A to èqoume apodeÐxei autì . . .... (�lla sumper�smata)

2 B kai autì . . .3 A ∧ B ∧I (1,2)

H arÐjmhsh twn gramm¸n eÐnai aparaÐthth gia thnkatanìhsh thc apìdeixhc.

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Kanìnec ND gia to sÔndesmo ∧

I (apaloif ∧, ∧-elimination, ∧E ). An èqoume katafèrei nagr�youme A ∧ B , tìte mporoÔme na gr�youme A kai/ B .

1 A ∧ B to èqoume apodeÐxei autì2 A ∧E (1)3 B ∧E (1)

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UpojèseicStic apodeÐxeic ND prèpei suqn� na k�noume upojèseic gia naapodeÐxoume autì pou jèloume.MÐa upìjesh eÐnai apl� mÐa prìtash, all� qrhsimopoieÐtai meèna eidikì trìpo. Upojètoume ìti (èqoume mÐa kat�stash sthnopoÐa) h prìtash eÐnai alhj c. 'Epeita mporeÐ naproqwr soume sthn apìdeix mac kaj¸c gnwrÐzoumeperissìtera gia thn kat�stash.

Pq, o Sèrlok Qolmc upojètei gia mia stigm ìti autìc pouèkleye to �logo th nÔqta tan �gnwstoc. Tìte eÐnai sÐgourocìti ta skuli� tou st�blou ja g�bgizan kai sunep¸c ja eÐqanxupn sei oi fÔlakec tou st�blou.All� oi fÔlakec den xÔpnhsan kat� th di�rkeia thc nÔqtac. OQolmc sumperaÐnei ìti ta skuli� den g�bgisan th nÔqta, �raautìc pou èkleye to �logo den tan �gnwstoc.

H upìjesh tou tan l�joc � parìl' aut� mporoÔse na k�neiaut thn upìjesh, kai tan qr simo pou thn èkane. 71 / 232

Kanìnec ND gia to sÔndesmo →I (eisagwg →, →-introduction, → I ). Gia na eis�goume

mÐa prìtash thc morf c A→ B , upojètoume ìti A kaiapodeiknÔoume ìti B .Kat� th di�rkeia thc apìdeixhc, mporoÔme naqrhsimopoi soume thn A kai otid pote �llo èqoume dhapodeÐxei. All� den mporoÔme na qrhsimopoi soume thn A otid pote �llo apì thn apìdeixh thc B apì thn Aargìtera (giatÐ aut h apìdeixh tan basismènh se mÐaepiplèon upìjesh). Sunep¸c, apomon¸noume thn apìdeixhthc B apì thn A se èna koutÐ:

1 A upìjesh(h apìdeixh)

2 B to apodeÐxame!

3 A→ B → I (1, 2)

KamÐa apì tic prot�seic pou eÐnai mèsa sto koutÐ den

mporeÐ na qrhsimopoihjeÐ argìtera.

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Kanìnec ND gia to sÔndesmo →

Sto sÔsthma ND ta kouti� qrhsimopoioÔntai ìtan k�noumeepiplèon upojèseic. H pr¸th gramm mèsa sto koutÐ prèpeip�nta na onom�zetai “upìjesh” � me mÐa exaÐresh, thn opoÐaja doÔme argìtera.

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Kanìnec ND gia to sÔndesmo →

I (apaloif →, →-elimination, → E ). An èqoume katafèreina gr�youme A kai A→ B , me opoiad pote seir�, tìtemporoÔme na gr�youme B (Modus Ponens).

1 A→ B to apodeÐxame autì . . ....

2 A kai autì . . .3 B → E (1, 2)

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Kanìnec ND gia to sÔndesmo ∨

I (eisagwg ∨, ∨-introduction, ∨I ). Gia na apodeÐxoumemÐa prìtash thc morf c A ∨ B , apodeiknÔoume thn A, (anprotim�me) thn B .

1 A to apodeÐxame autì . . .2 A ∨ B ∨I (1)

H B mporeÐ na eÐnai opoiad pote prìtash!

1 B to apodeÐxame autì . . .2 A ∨ B ∨I (1)

H A mporeÐ na eÐnai opoiad pote prìtash!

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Kanìnec ND gia to sÔndesmo ∨I (apaloif ∨, ∨-elimination, ∨E ). Gia na apodeÐxoume

k�ti apì mÐa prìtash thc morf c A ∨ B , prèpei na toapodeÐxoume upojètontac thn A, KAI na to apodeÐxoumeupojètontac thn B .Aut eÐnai epiqeirhmatologÐa me peript¸seic.

1 A ∨ B to èqoume apodeÐxei autì

2 A upìjesh 5 B upìjesh

3... pr¸th apìdeixh 6

... deÔterh apìdeixh4 C to apodeÐxame 7 C to apodeÐxame xan�

8 C ∨E (1, 2, 4, 5, 7)

Oi upojèseic A,B den mporoÔn na qrhsimopoihjoÔnargìtera, sunep¸c topojetoÔntai se (diplan�) kouti�.KamÐa apì tic prot�seic pou eÐnai mèsa sta kouti� den

mporeÐ na qrhsimopoihjeÐ argìtera.

H C mporeÐ na eÐnai mÐa opoiad pote prìtash, all� kai tadÔo kouti� prèpei na telei¸noun me thn Ðdia prìtash C .

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5.2 ParadeÐgmata

Akìma den èqoume deÐxei touc kanìnec gia touc sundèsmouc¬,>,⊥. Parìl' aut� mporoÔme na doÔme k�poia paradeÐgmata.

Par�deigma 5.1

Ac apodeÐxoume ìti A→ A ∨ B me to sÔsthma ND (ìpou oi A,BeÐnai tuqaÐec prot�seic).

Thn egkurìthta aut c thc prìtashc thn apodeÐxame me �meshepiqeirhmatologÐa nwrÐtera sto m�jhma. SugkrÐnete tic dÔoapodeÐxeic.

1 A upìjesh2 A ∨ B ∨I (1)

3 A→ A ∨ B → I (1, 2)

H apìdeixh sth fusik gl¸ssa: “Se k�poia kat�stashupojètoume ìti h A eÐnai alhj c. Tìte kai h A∨B eÐnai alhj c.'Ara h A→ A ∨ B eÐnai alhj c se opoiad pote kat�stash”.

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Oi kanìnec eÐnai suntaktikoÐ

Oi kanìnec tou sust matoc ND eÐnai basismènoi sthnshmasiologÐa twn sundèsmwn. Parìl' aut� eÐnai suntaktikoÐkanìnec. MÐa apìdeixh me to sÔsthma ND eÐnai suntaktik .

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ParadeÐgmata

Par�deigma 5.2

DojeÐshc thc prìtashc A→ (B → C ) apodeÐxte ìti A∧B → C .

1 A→ (B → C ) dedomèno

2 A ∧ B upìjesh3 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 gl¸ssa: “JewroÔme ìti h prìtash A→ (B → C )eÐnai alhj c (se k�poia kat�stash). Gia na deÐxoume ìti hA ∧ B → C eÐnai alhj c, upojètoume ìti h A ∧ B eÐnai epÐshcalhj c (se aut thn kat�stash), kai deÐqnoume ìti h C eÐnaialhj c. Efìson èqoume A ∧ B tìte èqoume kai A. Epomènwc,qrhsimopoi¸ntac thn A→ (B → C ) blèpoume ìti h B → C eÐnaialhj c. H A ∧ B mac dÐnei th B . Sunep¸c èqoume kai thn C ”.

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SumperaÐnoume:

Se k�je kat�stash sthn opoÐa h prìtash A→ (B → C ) eÐnaialhj c, h A ∧ B → C eÐnai epÐshc alhj c.

Apì ed¸ kai sto ex c suqn� ja paraleÐpoume th fr�sh “eÐnaialhj c”.

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To sÔmbolo `ApodeÐxame thn A ∧ B → C apì th dojeÐsa A→ (B → C ).Sunep¸c mporoÔme na gr�youme:

A→ (B → C ) ` A ∧ B → C

Orismìc 5.3

'Estw A1, . . . ,An,B tuqaÐec prot�seic.

A1, . . . ,An ` B

shmaÐnei ìti h up�rqei mÐa apìdeixh (fusik c

sumperasmatologÐac (ND)) thc B , xekin¸ntac apì tic

prot�seic A1, . . . ,An (“dedomèna”).

I ` B (dhlad ìtan n=0) shmaÐnei ìti mporoÔme naapodeÐxoume thn B qwrÐc dedomèna. Me �lla lìgia h BeÐnai “je¸rhma” (thc fusik c sumperasmatologÐac).

I A1, . . . ,An ` B diab�zetai wc “h B eÐnai apodeÐximh apì ticA1, . . . ,An” kai ` B diab�zetai wc “h B eÐnai je¸rhma”.

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To sÔmbolo `

I Den prèpei na sugqèoume to ` me to �.To ` eÐnai suntaktikì kai perilamb�nei apodeÐxeic.To � eÐnai shmasiologikì kai perilamb�nei katast�seic.

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ParadeÐgmata

Par�deigma 5.4

A ∧ B → C ` A→ (B → C ).

1 A ∧ B → C dedomèno

2 A upìjesh3 B upìjesh4 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 gl¸ssa: “JewroÔme ìti h prìtash A∧B → C eÐnaialhj c. Gia na deÐxoume ìti h A→ (B → C ) eÐnai alhj c,upojètoume A kai deÐqnoume ìti B → C . Gia na deÐxoume B → Cupojètoume B kai deÐqnoume ìti C . All� t¸ra èqoume A kai B ,�ra kai A∧B . GnwrÐzoume ìti A∧B → C . Sunep¸c èqoume C ”.

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5.3 Kanìnec ND gia to sÔndesmo ¬

O sÔndesmoc ¬ èqei treic kanìnec. Oi dÔo pr¸toiantimetwpÐzoun thn ¬A wc thn A→ ⊥.

I (eisagwg ¬, ¬-introduction, ¬I ). Gia na apodeÐxoumemÐa prìtash thc morf c ¬A, upojètoume A kaiapodeiknÔoume ⊥.Wc sun jwc, den mporoÔme na qrhsimopoi soume thn Aargìtera. Sunep¸c kleÐnoume thn apìdeixh tou ⊥ apì thnA se èna koutÐ:

1 A upìjesh

2... �lla sumper�smata

3 ⊥ to apodeÐxame!

4 ¬A ¬I (1, 3)

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Kanìnec ND gia to sÔndesmo ¬

I (apaloif ¬, ¬-elimination, ¬E ). Apì thn A kai thn ¬AsumperaÐnoume ⊥.

1 ¬A to apodeÐxame . . .


3 A to apodeÐxame kai autì4 ⊥ ¬E (1, 3)

I (apaloif ¬¬, ¬¬-elimination, ¬¬). Apì thn ¬¬AsumperaÐnoume A.

1 ¬¬A to apodeÐxame . . .2 A ¬¬(1)

To par�deigma 5.8 k�nei qr sh tou kanìna ¬¬.

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5.4 ParadeÐgmata

Par�deigma 5.5 (O Qolmc kai to �logo)

I s: “autìc pou èkleye to �logo th nÔqta tan �gnwstoc”

I b: “o skÔloc g�bgise th nÔqta”

I w : “oi fÔlakec tou st�blou xÔpnhsan”

1 s → b dedomèno2 b → w dedomèno3 ¬w dedomèno

4 s upìjesh5 b → E (4, 1)6 w → E (5, 2)7 ⊥ ¬E (3, 6)

8 ¬s ¬I (4, 7)

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ParadeÐgmata

Par�deigma 5.6

Ac apodeÐxoume ìti A ` ¬¬A.

1 A dedomèno

2 ¬A upìjesh3 ⊥ ¬E (1, 2)

4 ¬¬A ¬I (2, 3)

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ParadeÐgmata

Par�deigma 5.7

Ac apodeÐxoume ìti ¬(A ∨ B) ` ¬A.

1 ¬(A ∨ B) dedomèno

2 A upìjesh3 A ∨ B ∨I (1)4 ⊥ ¬E (1, 3)

5 ¬A ¬I (2, 4)

Sth fusik gl¸ssa: “DojeÐshc thc prìtashc ¬(A ∨ B), aneÐqame A tìte ja eÐqame A ∨ B , dhlad ja eÐqame antÐfash.'Ara èqoume ¬A”. Aut eÐnai mÐa akìma èkfansh tou nìmou thctou trÐtou apokleÐsewc (law of excluded middle).

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Otid pote eÐnai apodeÐximo apì thn antÐfash

Par�deigma 5.8

Ac apodeÐxoume ìti ⊥ ` A gia k�je prìtash A.

1 ⊥ dedomèno

2 ¬A upìjesh3 ⊥

√(1)

4 ¬¬A ¬I (2, 3)5 A ¬¬(4)

Prosèxte th qr sh tou√

sth gramm 3 gia nadikaiolog soume mÐa prìtash h opoÐa eÐnai dh diajèsimh.

Otid pote eÐnai apodeÐximo apì thn antÐfash.

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Otid pote eÐnai apodeÐximo apì thn antÐfash

Up�rqoun dÔo trìpoi na to katano soume:

1. Apl� efarmìzoume touc kanìnec ND.

2. Jewr same wc dedomèno mÐa kat�stash sthn opoÐa h ⊥eÐnai alhj c kai apodeÐxame ìti kai h A eÐnai alhj c. All�den up�rqei kat�stash ìpou h ⊥ eÐnai alhj c. Sunep¸cden up�rqei lìgoc na mhn deqtoÔme opoiad pote sunèpeiaaut c thc je¸rhshc.

Epeid den up�rqei kat�stash sthn opoÐa h ⊥ eÐnai alhj c,mporoÔme na apodeÐxoume thn ⊥ mìno k�tw apì antifatikècupojèseic se èna koutÐ miac apìdeixhc. Suqn� k�noume tètoiecupojèseic. Pq, se epiqeirhmatologÐa me peript¸seic (qr shkanìna ∨E ). Se autèc tic peript¸seic mporoÔme nasumper�noume opoiad pote prìtash (kaj¸c apodeÐxame thn⊥). Sunep¸c eÐnai qr simo na deÐqnoume ìti ⊥. DeÐte topar�deigma 5.11 parak�tw.

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ApodeiknÔontac thn A ∨ ¬A

Par�deigma 5.9

Ac apodeÐxoume ìti ` A ∨ ¬A (polÔ qr simh apìdeixh).

1 ¬(A ∨ ¬A) upìjesh2 A upìjesh3 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 gl¸ssa: “Upojètoume ìti ¬(A ∨ ¬A). An A tìteA ∨ ¬A, dhlad ja eÐqame antÐfash. 'Ara ¬A (gramm 5). Tìteìmwc èqoume A ∨ ¬A to opoÐo eÐnai antÐfash. Ara h arqik upìjesh eÐnai yeud c, dhlad eÐnai ¬¬(A ∨ ¬A). Sunep¸cA ∨ ¬A”.

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Administrator

Rectangle

Administrator

Rectangle

5.5 Kanìnec gia to sÔndesmo >

I (eisagwg >, >-introduction, >I ). MporoÔme naeis�goume to sÔndesmo > opoud pote. Autì bèbaia deneÐnai polÔ qr simo.

I (apaloif >, >-elimination, >E ). Den mporoÔme naapodeÐxoume tÐpota kainoÔrgio apì >.

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5.6 Kanìnec gia to sÔndesmo ⊥I (eisagwg ⊥, ⊥-introduction, ⊥I ). Gia na apodeÐxoume ⊥

prèpei na apodeÐxoume A kai ¬A (gia opoiad poteprìtash A).

1 ¬A to apodeÐxame . . .


3 A to apodeÐxame kai autì4 ⊥ ⊥I (1, 3)

Autìc o kanìnac eÐnai o Ðdioc me ton ¬E . Up�rqoun dhlad dÔo onìmata gia ton Ðdio kanìna.

I (apaloif ⊥, ⊥-elimination, ⊥E ). MporoÔme naapodeÐxoume opoiad pote prìtash apì ⊥! (DeÐte topar�deigma 5.8).

1 ⊥ to apodeÐxame . . .2 A ⊥E (1)

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5.7 L mmata'Ena l mma eÐnai k�ti pou èqoume dh apodeÐxei kai macbohj�ei na apodeÐxoume autì pou jèloume.

Sto par�deigma 5.9 apodeÐxame ` A ∨ ¬A. Aut h apìdeixheÐnai polÔ qr simh. DiaireÐ to epiqeÐrhma se dÔo peript¸seic, Akai ¬A. Autì dieukolÔnei mÐa apìdeixh giatÐ gnwrÐzoumeperissìtera gia k�je perÐptwsh.

Stic apodeÐxeic ND ja mporeÐte na eis�gete thn prìtashA∨¬A wc l mma (dhlad ja gr�fete “l mma” sth dexi� st lhthc apìdeixhc), qwrÐc na qrei�zetai na thn apodeÐxete.

EÐnai to monadikì l mma pou ja mporeÐte na qrhsimopoi sete(qwrÐc na to apodeÐxete).

Prèpei ìmwc eseÐc na dialèxete poia prìtash A jaqrhsimopoi sete.

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ParadeÐgmata

Par�deigma 5.10

Ac apodeÐxoume ìti ` ((A→ B)→ A)→ A (nìmoc tou Peirce).

1 (A→ B)→ A upìjesh2 A ∨ ¬A l mma

3 A upìjesh 5 ¬A upìjesh6 A upìjesh7 ⊥ ⊥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)

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5.8 Kanìnec gia to sÔndesmo ↔AntimetwpÐzoume mia prìtash thc morf c A↔ B wc(A→ B) ∧ (B → A), kai isodÔnama wc (A ∧ B) ∨ (¬A ∧ ¬B).

I (eisagwg ↔, ↔-introduction, ↔ I ). Gia na apodeÐxoumeA↔ B apodeiknÔoume A→ B kai B → A. Enallaktik�,gia na apodeÐxoume A↔ B apodeiknÔoume A kai B , apodeiknÔoume ¬A kai ¬B (ìpoia perÐptwsh protim�me).Pr¸th enallaktik . . .

1 B → A to apodeÐxame . . .. . .

2 A→ B kai autì . . .3 A↔ B ↔ I (1,2)

DeÔterh enallaktik . . .

1 ¬B to apodeÐxame . . .. . .

2 ¬A kai autì . . .3 A↔ B ↔ I (1,2)

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5.8 Kanìnec gia to sÔndesmo ↔

TrÐth enallaktik . . .

1 A to apodeÐxame . . .. . .

2 B kai autì . . .3 A↔ B ↔ I (1,2)

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Kanìnec gia to sÔndesmo ↔

I (apaloif ↔, ↔-elimination, ↔ E ). Apì thn A↔ B kaithn A, èqoume thn B .Apì thn A↔ B kai thn B , èqoume thn A.

1 A↔ B to apodeÐxame . . .. . .

2 A kai autì . . .3 B ↔ E (1,2)

1 A↔ B to apodeÐxame . . .. . .

2 B kai autì . . .3 A ↔ E (1,2)

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5.9 AnakÔptontec kanìnec (Derived Rules)

Oi basikoÐ kanìnec tou sust matoc ND (oi opoÐoi èqounparousiasteÐ wc t¸ra) mporoÔn na sunduastoÔn giaparaqjoÔn nèoi kanìnec. Oi anakÔptontec autoÐ kanìnec DENeÐnai aparaÐthtoi se mia apìdeixh, all� k�noun thn apìdeixhpio sunoptik .

'Enac anakÔptwn kanìnac pou ja mporeÐte na qrhsimopoieÐteonom�zetai “apìdeixh apì antÐfash” (Proof by Contradiction(PC)):

Gia na apodeÐxoume A, upojètoume ¬A kai apodeiknÔoume ⊥.

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AnakÔptontec kanìnecO kanìnac PC eÐnai o sunduasmìc twn kanìnwn ¬I kai ¬¬:

1 ¬A upìjesh



4 ¬¬A ¬I (1, 3)5 A ¬¬(4)

H parap�nw apìdeixh gr�fetai wc ex c me th qr sh toukanìna PC:

1 ¬A upìjesh



4 A PC (1,3)

H qr sh tou kanìna PC mei¸nei thn apìdeixh kat� mÐa gramm .100 / 232

5.10 Sumboulèc gia apodeÐxeic ND

1. M�jaite touc kanìnec. LÔste ask seic.

2. Gr�yte thn apìdeixh me �mesh majhmatik epiqeirhmatologÐa. 'Epeita tupopoi ste thn apìdeixh metouc kanìnec ND.

3. An duskoleÔeste na apodeÐxete mÐa prìtash A, tìte taparak�tw mporeÐ na bohj soun:

I Υποθέστε ¬A και αποδείξτε ⊥ ( “απόδειξη από αντίφαση”,‘proof by contradiction’ (PC) ).

I Χρησιμοποιήστε το λήμμα B ∨ ¬B για μία κατάλληληπρόταση B. Αυτό έχει ως αποτέλεσμα τηνεπιχειρηματολογία με περιπτώσεις. Στο παράδειγμα 5.10

κάναμε χρήση αυτού του λήμματος.

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ParadeÐgmata

Par�deigma 5.11

A ∨ B, ¬C → ¬A, ¬(B ∧ ¬C ) ` C .

Ac upojèsoume ìti èqoume ¬C . Tìte ja èqoume kai ¬A. All�èqoume wc dedomèno ìti A B , �ra ja èqoume B . 'Eqoumedhlad B kai ¬C , to opoÐo dhmiourgeÐ antÐfash, kaj¸c èqoumewc dedomèno ¬(B ∧ ¬C ).

'Ara prèpei na èqoume C .

H parap�nw epiqeirhmatologÐa eÐnai eÔkolo na metafrasteÐ seapìdeixh ND.

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Par�deigma 5.11

1 A ∨ B dedomèno2 ¬C → ¬A dedomèno3 ¬(B ∧ ¬C ) dedomèno4 ¬C upìjesh5 ¬A → E (2,4)

6 A upìjesh 9 B upìjesh7 ⊥ ⊥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)

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Kouti�: ti prèpei na jumìmaste

K�je koutÐ mèsa se mÐa apìdeixh prèpei na plhroÐ ticparak�tw proôpojèseic:

1. 'Ena koutÐ p�nta arqÐzei me mÐa upìjesh (h monadik

exaÐresh afor� ton kanìna ∀I sthn kathgorhmatik logik .)

2. MÐa upìjesh mporeÐ na l�bei q¸ra mìno sthn pr¸thgramm enìc koutioÔ.

3. Mèsa se èna koutÐ mporoÔme na qrhsimopoi soumeopoiad pote prìtash apì prohgoÔmenec grammèc thcapìdeixhc, ektìc apì tic prot�seic pou brÐskontai sekouti� pou èqoun kleÐsei prin apì to trèqon shmeÐo thcapìdeixhc.

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Kouti�: ti prèpei na jumìmaste

4. Oi monadikoÐ trìpoi na ex�goume plhroforÐa apì ènakoutÐ eÐnai me th qr sh twn kanìnwn → I , ∨E , ¬I , kai PC(epiplèon me thn qr sh twn kanìnwn ∃E kai ∀I sthnkathgorhmatik logik ). H pr¸th gramm met� apì k�jekoutÐ prèpei na dikaiologhjeÐ me th qr sh enìc apì autoÔctouc kanìnec.

5. KamÐa apì tic prot�seic pou eÐnai mèsa se èna koutÐ denmporeÐ na qrhsimopoihjeÐ èxw apì to koutÐ, ektìc anexaqjeÐ sÔmfwna me touc kanìnec pou anafèrontai sthnproôpìjesh 4.

An èna koutÐ den plhroÐ mÐa apì tic parap�nw proôpojèseictìte h apìdeixh eÐnai l�joc.

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Qr sh kouti¸n

'Estw ìti jèloume na apodeÐxoume ìti ¬A ` ¬A.

1 ¬A dedomèno2 ¬A

√(1) h kalÔterh apìdeixh!

1 ¬A dedomèno

2 A upìjesh3 ⊥ ⊥I (2,1)

4 ¬A ¬I (1,2)

swstì all� ìqi èxupno

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L�joc qr sh kouti¸n

1 ¬A dedomèno

2 A upìjesh3 ⊥ ⊥I (2,1)

4 ¬A ⊥E (3)

l�joc!

1 ¬A dedomèno

2 A upìjesh3 ⊥ ⊥I (2,1)4 ¬A ⊥E (3)

5 ¬A√(4)

l�joc!

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5.11 ` kai �

O basikìc mac stìqoc eÐnai na exet�zoume th logik sunèpeia( � ).JumhjeÐte ìti A1, . . . ,An � B eÐnai alhj c an h B eÐnai alhj cse k�je kat�stash ìpou oi A1, . . . ,An eÐnai ìlec alhjeÐc.H ` den eÐnai qr simh an den mac bohj�ei na deÐxoume �.

Orismìc 5.12

'Ena je¸rhma eÐnai mÐa prìtash h opoÐa mporeÐ na apodeiqjeÐ

apì èna dojèn sÔsthma apodeÐxewn.

Mac endiafèrei to sÔsthma kanìnwn fusik csumperasmatologÐac. 'Ara èna je¸rhma eÐnai k�je prìtash Aìpou ` A.

Orismìc 5.13

'Ena sÔsthma apodeÐxewn eÐnai orjì an k�je je¸rhma eÐnai

ègkuro, kai pl rec an k�je ègkurh prìtash eÐnai je¸rhma.

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Orjìthta kai Plhrìthta tou sust matoc NDMporeÐ na deiqteÐ ìti to sÔsthma kanìnwn fusik csumperasmatologÐac (ND) eÐnai orjì kai pl rec.

Je¸rhma 5.14 (orjìthta sust matoc ND)

'Estw ìti oi A1, . . . ,An,B eÐnai tuqaÐec prot�seic. An

A1, . . . ,An ` B , tìte A1, . . . ,An � B .

Gia n=0:“K�je apodeÐximh prìtash eÐnai ègkurh”.“To sÔsthma ND den k�nei potè l�jh”.

Je¸rhma 5.15 (plhrìthta sust matoc ND)

'Estw ìti oi A1, . . . ,An,B eÐnai tuqaÐec prot�seic. An

A1, . . . ,An � B , tìte A1, . . . ,An ` B .

Gia n=0:“K�je ègkurh prìtash eÐnai apodeÐximh”.“To sÔsthma ND mporeÐ na apodeÐxei ìlec tic ègkurecprot�seic”.

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Orjìthta kai Plhrìthta tou sust matoc ND

'Ara mporoÔme na qrhsimopoioÔme to sÔsthma ND gia naelègqoume thn egkurìthta twn prot�sewn.

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Sunèpeia

Orismìc 5.16 (sunèpeia)

Mia prìtash A eÐnai sunep c an 0 ¬A.'Ena sÔnolo prot�sewn {A1, . . . ,An} eÐnai sunepèc an0 ¬

∧1≤i≤n Ai .

SÔmfwna me ta jewr mata orjìthtac kai plhrìthtac(Jewr mata 5.14 kai 5.15) èqoume:

Je¸rhma 5.17

Mia prìtash eÐnai sunep c an kai mìno an eÐnai ikanopoi simh.

H ènnoia thc ikanopoihsimìthtac dÐnetai apì ton Orismì 3.3.

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PandoraPandora: www.doc.ic.ac.uk/pandoraNew Pandora: www.doc.ic.ac.uk/pandora/newpandoraAut� eÐnai dÔo ekdìseic enìc progr�mmatoc logismikoÔ me toopoÐo mporeÐte na k�nete apodeÐxeic qrhsimopoi¸ntac touckanìnec ND.

'Eqei anaptuqjeÐ apì foithtèc tou KolegÐou Imperial. Oifoithtèc tou maj matoc Majhmatik c Logik c to èqoun breipolÔ qr simo sto na majaÐnoun na anaptÔssoun apodeÐxeic meto sÔsthma ND.

Diab�ste to “help” tou progr�mmatoc Pandora kai exaskhjeÐteme ta “tutorials” tou. LÔste tic ask seic sto Pandora stoNew Pandora kai elègxte an oi lÔseic sac eÐnai swstèc.

ShmeÐwsh: To prìgramma Pandora onom�zei to l mma ` A ∨ ¬AEM (apì to “law of Excluded Middle”).

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www.doc.ic.ac.uk/pandora

www.doc.ic.ac.uk/pandora/newpandora

6. PÐnakec Beth

Oi pÐnakec Beth apl� pÐnakec (tableaux) apoteloÔn ènasÔsthma apodeÐxewn. Qrhsimopoi¸ntac suntaktikoÔc kanìnecmporoÔme na exet�zoume an mÐa prìtash eÐnai ègkurh.

O pÐnakac miac sÔnjethc prìtashc kataskeu�zetai me b�shtouc pÐnakec twn prot�sewn pou emfanÐzontai sth sÔnjethprìtash.

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AtomikoÐ pÐnakec

OrÐzoume atomikoÔc pÐnakec gia touc sundèsmouc thcProtasiak c Logik c.

A /\ B

B

A

-(A /\ B)

-B-A

O isqurismìc ìti mÐa prìtash A ∧ B eÐnai alhj c apaiteÐ thn Aalhj kai thn B alhj (gramm ).O isqurismìc ìti mÐa prìtash ¬(A ∧ B) eÐnai alhj c apaiteÐthn A yeud thn B yeud (diakl�dwsh).

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AtomikoÐ pÐnakec

A V B

BA

A -> B

B-A

-(A V B)

-B

-A

-(A -> B)

-B

A

Mia diakl�dwsh stouc atomikoÔc pÐnakec dhl¸nei di�zeuxh en¸mÐa gramm dhl¸nei sÔzeuxh.

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AtomikoÐ pÐnakec

A <-> B

-AA

B -B

-(A <-> B)

-AA

-B B

- -A

A

-p p

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Par�deigmaGia na kataskeu�soume ton pÐnaka miac sÔnjethc prìtashc Kxekin�me gr�fontac K ¬K sthn koruf tou pÐnaka. Met�anaptÔssoume ton pÐnaka thc K sÔmfwna me touc atomikoÔcpÐnakec.

Par�deigma 6.1

Na kataskeuasteÐ o pÐnakac 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

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Par�deigma

Autìc eÐnai ènac teleiwmènoc pÐnakac me treic kl�douc. Oikl�doi arqÐzoun apì thn koruf . O aristerìc kl�doc eÐnaiantifatikìc, perièqei tic antifatikèc prot�seic A kai ¬A.SumbolÐzoume thn antifatikìthta enìc kl�douupogrammÐzontac thn teleutaÐa tou prìtash.Oi �lloi dÔo kl�doi den eÐnai antifatikoÐ.

Mèsw tou pÐnaka thc (A ∧ ¬A) ∨ (B ∨ (C ∧ D)) blèpoume ìti hupìjesh ìti aut h prìtash eÐnai alhj c eÐnai k�tw apìorismènec sunj kec swst , pq ìtan h B eÐnai alhj c oi C kaiD eÐnai alhjeÐc.

MporeÐte na xeqwrÐsete touc atomikoÔc pÐnakec ston parap�nwpÐnaka;

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OrismoÐ

Orismìc 6.2

'Enac kìmboc enìc pÐnaka lègetai qrhsimopoihmènoc anemfanÐzetai san koruf atomikoÔ pÐnaka. Alli¸c lègetai

aqrhsimopoÐhtoc.

Orismìc 6.3

'Enac kl�doc enìc pÐnaka lègetai antifatikìc an oi A kai ¬AeÐnai kìmboi tou kl�dou (gia k�poia tuqaÐa prìtash A).

Orismìc 6.4

'Enac pÐnakac lègetai teleiwmènoc an kanènac mh antifatikìc

kl�doc den èqei aqrhsimopoÐhtouc kìmbouc. Alli¸c lègetai

hmitel c.

Orismìc 6.5

'Enac pÐnakac lègetai antifatikìc an ìloi oi kl�doi tou eÐnai

antifatikoÐ.

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Par�deigma

Par�deigma 6.6

Na kataskeuasteÐ o pÐnakac tou nìmou tou Peirce((A→ B)→ A)→ A.

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

(A -> B ) -> A

A

-A

-(A -> B)

A

-B

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Par�deigma

Diaisjhtik�: An ènac teleiwmènoc pÐnakac me koruf ¬K brejeÐantifatikìc, autì shmaÐnei ìti dokim�same ìlouc touc dunatoÔctrìpouc pou ja mporoÔsan na k�noun thn prìtash K yeud kai apotÔqame. 'Ara h K eÐnai alhj c se ìlec tic katast�seic.

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Par�deigma

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

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

A -> B

-A

-A B

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To sÔmbolo `B

Orismìc 6.7

MÐa Beth-apìdeixh miac prìtashc K eÐnai ènac teleiwmènoc

antifatikìc pÐnakac me koruf ¬K . H prìtash K lègetai

Beth-apodeÐximh an èqei Beth-apìdeixh.

To ìti mÐa prìtash eÐnai Beth-apodeÐximh to sumbolÐzoume me

`B K

H prìtash K eÐnai Beth-apodeÐximh apì tic prot�seicA1, . . . ,An an up�rqei teleiwmènoc antifatikìc pÐnakac mekoruf ¬K kai epìmeno kìmbo A1 ∧ · · · ∧ An. Sumbolik�

A1, . . . ,An `B K

Orismìc 6.8

MÐa Beth-di�yeush miac prìtashc K eÐnai ènac teleiwmènoc

antifatikìc pÐnakac me koruf K . H prìtash K lègetai

Beth-diayeÔsimh an up�rqei mÐa Beth-di�yeush thc.123 / 232

Sust mata apodeÐxewn

Ja qrhsimopoioÔme ta sÔmbola `ND kai `B gia na xeqwrÐzoumetic apodeÐxeic tou sust matoc ND apì tic Beth apodeÐxeic.

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Par�deigma

Par�deigma 6.9

Na apodeÐxete ìti `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

Par�deigma

Par�deigma 6.10

Na apodeÐxete ìti M ∨ K , M → K `B K .

-K

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

M V K

M

M -> K

K

-M K

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Orjìthta kai Plhrìthta twn Beth-apodeÐxewn

Je¸rhma 6.11 (orjìthta Beth-apodeÐxewn)

'Estw ìti oi A1, . . . ,An,K eÐnai tuqaÐec prot�seic. An

A1, . . . ,An `B K , tìte A1, . . . ,An � K .

Gia n=0:“K�je Beth-apodeÐximh prìtash eÐnai ègkurh”.“To sÔsthma Beth-apodeÐxewn den k�nei potè l�jh”.

Je¸rhma 6.12 (plhrìthta Beth-apodeÐxewn)

'Estw ìti oi A1, . . . ,An,K eÐnai tuqaÐec prot�seic. An

A1, . . . ,An � K , tìte A1, . . . ,An `B K .

Gia n=0:“K�je ègkurh prìtash eÐnai Beth-apodeÐximh”.“To sÔsthma Beth-apodeÐxewn mporeÐ na apodeÐxei ìlec ticègkurec prot�seic”.

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Orjìthta kai Plhrìthta twn Beth-apodeÐxewn

'Ara mporoÔme na qrhsimopoioÔme touc pÐnakec Beth gia naelègqoume thn egkurìthta twn prot�sewn.

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Sunèpeia

Mia prìtash A eÐnai sunep c an 0B ¬A.'Ena sÔnolo prot�sewn {A1, . . . ,An} eÐnai sunepèc an0B ¬

∧1≤i≤n Ai .

SÔmfwna me ta jewr mata orjìthtac kai plhrìthtac twnBeth-apodeÐxewn mÐa prìtash eÐnai sunep c an kai mìno aneÐnai ikanopoi simh.

H ènnoia thc ikanopoihsimìthtac dÐnetai apì ton Orismì 3.3.

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Par�deigma

Par�deigma 6.13

Na apodeÐxete ìti � p → (p ∨ q).

Epeid to sÔsthma Beth-apodeÐxewn eÐnai orjì kai pl rec,arkeÐ na deÐxoume ìti `B p → (p ∨ q).

-(p -> (p V q))

p

-(p V q)

-q

-p

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Par�deigma

Par�deigma 6.14

Na apodeÐxete ìti oi prot�seic p ∧ q kai q ∧ p eÐnai isodÔnamec.

ArkeÐ na deÐxoume ìti `B (p ∧ q)↔ (q ∧ p). (GiatÐ;).

-((p /\ q) <-> (q /\ p))

p /\ q

-(q /\ p)

-(p /\ q)

q /\ p

p

q

-q -p

q

p

-p -q

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Sumboulèc gia Beth-apodeÐxeic

I M�jaite touc atomikoÔc pÐnakec. LÔste ask seic.

I Den up�rqei ousi¸dhc lìgoc pou na kajorÐzei th seir�an�ptuxhc twn prot�sewn, all� mejodologik� h an�ptuxhpr¸ta twn suzeÔxewn (grammèc) kai èpeita twn diazeÔxewn(diaklad¸seic) odhgeÐ kat� kanìna se aploÔsteroucpÐnakec.

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Par�deigma

Par�deigma 6.15

Na exet�sete an M ∨ K , M → K `B M.

-M

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

M V K

M

M -> K

K

-M K

O parap�nw pÐnakac eÐnai teleiwmènoc all� mh antifatikìc.'Ara M ∨ K , M → K 0B M.

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Par�deigma

Par�deigma 6.16

Na brejoÔn oi katast�seic stic opoÐec h prìtash(p → q)↔ (p ∨ q) eÐnai yeud c.

-((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

Apì touc mh antifatikoÔc kl�douc enìc teleiwmènou pÐnaka mekoruf ¬A, paÐrnoume tic katast�seic ìpou h A eÐnai yeud c. 134 / 232

Kathgorhmatik Logik

H Kathgorhmatik Logik ( Prwtob�jmia Logik ) eÐnai mÐaepèktash thc Protasiak c Logik c.Sta epìmena maj mata:

I ja parousi�soume to suntaktikì kai th shmasiologÐa thcKathgorhmatik c Logik c,

I ja doÔme th sqèsh thc Kathgorhmatik c Logik c me thfusik gl¸ssa,

I ja d¸soume touc orismoÔc tou ègkurou epiqeir matoc kaithc ègkurhc prìtashc sthn Kathgorhmatik Logik , kai

I ja doÔme trìpouc gia na apodeiknÔoume thn egkurìthtasthn Kathgorhmatik Logik :

I Αμεση επιχειρηματολογία.

I Ισοδυναμίες.

I Σύστημα κανόνων φυσικής συμπερασματολογίας.

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H an�gkh gia mÐa pio ekfrastik gl¸ssa

H Protasiak Logik eÐnai mÐa ploÔsia gl¸ssa, all� eÐnaiperioristik sthn diatÔpwsh idiot twn kai sqèsewn.Pq: “An o Gi¸rgoc eÐnai �njrwpoc tìte o Gi¸rgoc eÐnai jnhtìc”gr�fetai wc A→ B .Ti gÐnetai me th MarÐa, ton K¸sta, klp;

Epiplèon den mporeÐ na elegqjeÐ me thn Protasiak Logik hegkurìthta twn epiqeirhm�twn ìpwc:

I 'Oloi oi g�toi eÐnai tempèlhdec (p).

I O M tsoc eÐnai g�toc (q).

I 'Ara o M tsoc eÐnai tempèlhc (r).

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7. PerÐlhyh Kathgorhmatik c Logik c7.1 Nèec Atomikèc Prot�seicMèqri stigm c, jewroÔsame fr�seic ìpwc “o upologist c eÐnaiSun” kai “o F¸thc agìrase stafÔlia” wc atomikèc, dÐqwceswterik dom .T¸ra ja doÔme thn dom touc.JewroÔme thn m�rka Sun wc mÐa idiìthta (property, attribute)thn opoÐa ènac upologist c (kai �lla pr�gmata) mporeÐ naèqei na mhn èqei. 'Ara eis�goume:

I 'Ena sÔmbolo kathgor matoc (relation symbol, predicatesymbol) Sun. 'Eqei 1 par�metro � �ra lème ìti o bajmìc

tou (arity) eÐnai 1.I 'Ena sÔmbolo kathgor matoc αγορασε. 'Eqei 2 paramètrouc

� �ra lème ìti o bajmìc tou eÐnai 2.I Stajerèc (constants), gia na onom�zoume antikeÐmena. Pq,Ερμης, Φωτης, Αιθουσα-002, σταφυλια.

Sunep¸c Sun(Ερμης) kai αγορασε(Φωτης, σταφυλια) eÐnai dÔo nèecatomikèc prot�seic.

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7.2 PosodeÐktec

Arqik� mporeÐ na faÐnetai ìti to

αγορασε(Φωτης, σταφυλια)

den eÐnai diaforetikì apì autì pou gr�fame sthn Protasiak Logik :

o Fwthc agorase stafulia

All� h Kathgorhmatik Logik èqei mhqanismoÔc gia nametab�lei tic paramètrouc tou kathgor matoc αγορασε.

Sunep¸c mporoÔme na ekfr�soume tic idiìthtec thc sqèshcαγορασε.

Oi mhqanismoÐ autoÐ onom�zontai posodeÐktec.

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Ti eÐnai oi posodeÐktec;

'Enac posodeÐkthc ekfr�zei mÐa posìthta (apì pr�gmata taopoÐa èqoun mÐa idiìthta).

ParadeÐgmata:

I 'Oloi oi foithtèc diab�zoun polÔ.

I K�poioi foithtèc koimoÔntai.

I Oi perissìteroi kajhghtèc eÐnai treloÐ.

I Okt¸ stic dèka g�tec to protimoÔn.

I Kanènac den eÐnai pio èxupnoc apì esèna.

I Toul�qiston èxi foithtèc den koimoÔntai.

I Up�rqoun �peiroi pr¸toi arijmoÐ.

I Up�rqoun pio poll� PC apì Mac.

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PosodeÐktec sthn Kathgorhmatik Logik

'Eqoume dÔo posodeÐktec:

I ∀ ( (A) ): “gia k�je” (kajolikìc posodeÐkthc).

I ∃ ( (E ) ): “up�rqei” (uparxiakìc posodeÐkthc).

K�poioi �lloi posodeÐktec mporoÔn na ekfrastoÔn me toucparap�nw. (K�je ènac apì touc parap�nw posodeÐktec mporeÐna ekfrasteÐ mèsw tou �llou.) All� den up�rqoun posodeÐktecsthn Prwtob�jmia Logik gia na ekfr�soun sqèseic ìpwc“up�rqoun �peiroi”kai “perissìtero apì”.

EÐdame ekfr�seic ìpwc Ερμης, Φωτης. Aut� eÐnai stajerèc ìpwcto π.

Gia na ekfr�soume prot�seic ìpwc “ìloi oi upologistèc eÐnaiSun” qreiazìmaste metablhtèc oi opoÐec mporoÔn na p�rountimèc apì to sÔnolo ìlwn twn upologist¸n.

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7.3 MetablhtècJa qrhsimopoi soume metablhtèc gia na ekfr�soumeposodeixÐa. 'Eqoume èna sÔnolo V metablht¸n: pqx , y , z , u, v ,w , x0, x1, x2, . . .Merikèc forèc ja gr�foume x y gia na poÔme “opoiad potemetablht ”.Pèra apì tic prot�seic ìpwc Sun(Ερμης), ja gr�foume kaiprot�seic ìpwc Sun(x).

I Gia na ekfr�soume thn prìtash “ta p�nta eÐnai Sun” jagr�foume ∀x Sun(x).Autì diab�zetai wc: “Gia k�je x , to x eÐnai Sun”.

I Gia na ekfr�soume thn prìtash “k�ti eÐnai Sun” jagr�foume ∃x Sun(x).Autì diab�zetai wc: “Up�rqei x to opoÐo eÐnai Sun”.

“O F¸thc agìrase èna Sun”gr�fetai wc∃x (Sun(x) ∧ αγορασε(Φωτης, x))

“Up�rqei x , to opoÐo eÐnai Sun kai o F¸thc to agìrase”.141 / 232

8. Suntaktikì Kathgorhmatik c Logik c

'Opwc k�name sthn Protasiak Logik , ètsi kai t¸ra ja doÔmepr¸ta to suntaktikì kai met� th shmasiologÐa.

8.1 Gl¸ssa

Orismìc 8.1

MÐa gl¸ssa eÐnai èna sÔnolo apì stajerèc kai sÔmbola

kathgorhm�twn sugkekrimènou bajmoÔ.

Sun jwc sumbolÐzoume mÐa gl¸ssa wc L. Gr�foume c , d , . . .gia stajerèc kai P,Q,R,S , . . . gia sÔmbola kathgorhm�twn.

Argìtera ja exet�soume kai ta sÔmbola sunart sewn.

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MÐa apl gl¸ssa

To sÔnolo twn sumbìlwn mÐac gl¸ssac L exart�tai apì aut�pou jèloume na ekfr�soume.Ja qrhsimopoi soume wc par�deigma mÐa gl¸ssa L h opoÐaperilamb�nei:

I tic stajerèc Φωτης, Ερμης, Σοφια, Τασος, Τριτων, Αρης,Αιθουσα-002, kai c

I ta sÔmbola kathgorhm�twn Sun, ανθρωπος, καθηγητης(bajmìc 1)

I to sÔmbolo kathgor matoc αγορασε (bajmìc 2)

ShmeÐwsh: H L perilamb�nei sÔmbola (suntaktikì). Gia nadoÔme to nìhma (shmasiologÐa) twn sumbìlwn ja prèpei nadoÔme poia morf èqei mÐa kat�stash sthn Kathgorhmatik Logik .

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8.2 'Oroi (Terms)

Gia na gr�foume prot�seic sthn Kathgorhmatik Logik qreiazìmaste ìrouc (terms), me touc opoÐouc onom�zoumeantikeÐmena. Oi ìroi den eÐnai prot�seic, den èqoun alhjotimèc,

Orismìc 8.2 ('Oroc)

DojeÐshc mÐac gl¸ssac L:

1. K�je stajer� thc L eÐnai ènac L-ìroc.

2. K�je metablht eÐnai ènac L-ìroc.

3. TÐpota �llo den eÐnai L-ìroc.

'Enac basikìc ìroc (ground term) eÐnai ènac ìroc o opoÐoc denperilamb�nei metablhtèc.ParadeÐgmata ìrwn:

I Φωτης, Ερμης (basikoÐ ìroi).

I x , y , x56 (mh basikoÐ ìroi).

Argìtera ja exet�soume kai ta sÔmbola sunart sewn.

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8.3 TÔpoi Kathgorhmatik c Logik cOrismìc 8.3 ( TÔpoc (formula) )

'Estw mÐa gl¸ssa L.

1. An R eÐnai èna sÔmbolo kathgor matoc bajmoÔ n thc L,kai t1, . . . , tn eÐnai L-ìroi, tìte to R(t1, . . . , tn) eÐnai ènac

atomikìc L-tÔpoc.

2. An t, t ′ eÐnai L-ìroi tìte t=t ′ eÐnai ènac atomikìc L-tÔpoc.

3. Oi ⊥,> eÐnai atomikoÐ L-tÔpoi.

4. An A,B eÐnai L-tÔpoi, tìte kai oi (¬A), (A ∧ B), (A ∨ B),(A→ B), (A↔ B) eÐnai L-tÔpoi.

5. An h A eÐnai ènac L-tÔpoc kai x mÐa metablht , tìte oi

(∀x A) kai (∃x A) eÐnai L-tÔpoi.

6. TÐpota �llo den eÐnai L-tÔpoc.

H proteraiìthta twn sundèsmwn eÐnai ìpwc sthn Protasiak Logik . Oi posodeÐktec ∀x kai ∃x eÐnai tìso isquroÐ ìso kai h¬.

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ParadeÐgmata tÔpwn

1. αγορασε(Φωτης, x)“O F¸thc agìrase to x”.

2. ∃x αγορασε(Φωτης, x)“O F¸thc agìrase k�ti”.

3. ∀x (καθηγητης(x)→ ανθρωπος(x))“K�je kajhght c eÐnai �njrwpoc”.

4. ∀x (αγορασε(Τασος, x)→ Sun(x))“'O,ti agìrase o T�soc eÐnai Sun”.

AnaptÔssoume to dèndro (formation tree) enìc tÔpou kaigr�foume touc upo-tÔpouc tou (subformulas) ìpwc sthnProtasiak Logik .

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ParadeÐgmata tÔpwn

5. ∀x (αγορασε(Τασος, x)→ αγορασε(Σοφια, x))“H SofÐa agìrase ìti agìrase kai o T�soc”.

6. ∀x αγορασε(Τασος, x)→ ∀x αγορασε(Σοφια, x)“An o T�soc agìrase ta p�nta, tìte kai h SofÐaagìrase ta p�nta”.

7. ∀x∃y αγορασε(x , y)“Ta p�nta agìrasan k�ti”.

8. ∃y∀x αγορασε(x , y)“K�ti agor�sthke apì ta p�nta”.

9. ∃x∀y αγορασε(x , y)“K�ti agìrase ta p�nta”.

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9. ShmasiologÐa Kathgorhmatik c Logik c'Opwc sthn Protasiak Logik , ètsi kai t¸ra prèpei nadiatup¸soume

I ti eÐnai mÐa kat�stash sthn Kathgorhmatik Logik

I p¸c na ermhneÔoume touc tÔpouc thc Kathgorhmatik cLogik c se mÐa dedomènh kat�stash

9.1 Domèc (katast�seic sthn Kathgorhmatik Logik )

Orismìc 9.1

'Estw mÐa gl¸ssa L. MÐa L-dom (h opoÐa merikèc forèc

onom�zetai montèlo) M:

I èqei èna mh-kenì sÔnolo apì antikeÐmena ta opoÐa h M“gnwrÐzei”. To sÔnolo autì onom�zetai to pedÐo tim¸n(domain) sÔmpan (universe) thc M, kai gr�fetai wc

dom(M).

I diatup¸nei ti shmaÐnoun ta sÔmbola thc L se sqèsh me ta

parap�nw antikeÐmena.

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Domèc

H apìdosh (interpretation) mÐac stajer�c sth M eÐnai ènaantikeÐmeno sto dom(M).

H apìdosh (interpretation) enìc sumbìlou kathgor matoc sthM eÐnai mÐa sqèsh sto dom(M).

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Par�deigma dom cGia thn gl¸ssa L pou qrhsimopoioÔme wc par�deigma, mÐaL-dom prèpei na diatup¸nei:

I poia antikeÐmena eÐnai sto sÔmpan thcI poia apì ta antikeÐmena thc eÐnai o T�soc, h SofÐa, klpI poia antikeÐmena eÐnai Sun, kajhghtèc, �njrwpoiI poia antikeÐmena agìrasan �lla antikeÐmena

Parak�tw up�rqei èna di�gramma miac sugkekrimènhc L-dom c,thn opoÐa onom�zoume M.Up�rqoun 12 antikeÐmena sto sÔmpan thc M.'Eqoume b�lei etikètec (pq “F¸thc”) se merik� antikeÐmena giana deÐxoume to nìhma twn stajer¸n thc L (pq Φωτης).Oi apodìseic (ènnoiec) twn kathgorhm�twn Sun, ανθρωποςanaparÐstantai wc perioqèc sto di�gramma.H apìdosh (ènnoia) tou kathgor matoc καθηγητηςanaparÐstatai me maÔrec koukkÐdec.H apìdosh (ènnoia) tou kathgor matoc αγορασε anaparÐstataime kateujunìmenec grammèc metaxÔ antikeimènwn.

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H dom M

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T�soc Tasoc

Den prèpei na up�rqei sÔgqush metaxÔ tou antikeimènou • methn etikèta “T�soc”, tou sÔmpantoc thc M, me to sÔmbolo(stajer�) Τασος thc gl¸ssac L.

(Qrhsimopoi¸ diaforetikèc grammatoseirèc gia na diakrÐnwmetaxÔ twn sumbìlwn mÐac gl¸ssac kai twn antikeimènwn tousÔmpantoc mÐac dom c thc gl¸ssac.)

Aut� eÐnai diaforetik� pr�gmata. To sÔmbolo Τασος eÐnaisuntaktikì, en¸ to • eÐnai shmasiologikì. Gia th dom M, ΤασοςeÐnai èna ìnoma gia to antikeÐmeno • me thn etikèta “T�soc”.

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T�soc Tasoc

Gia na apofÔgoume thn sÔgqush ja uiojet soume thnparak�tw sÔmbash:

SÔmbash 9.2

'Estw M mÐa L-dom kai c mÐa stajer� thc L. Gr�foume cM

gia thn apìdosh thc c sth M. To antikeÐmeno tou sÔmpantoc

thc M eÐnai autì pou onom�zei h c sth M.

'Ara ΤασοςM=to antikeÐmeno • me etikèta “T�soc”.

Se mÐa �llh dom , to sÔmbolo Τασος mporeÐ na onom�zei (nashmaÐnei) k�ti �llo.

H ènnoia mÐac stajer�c c eÐnai to antikeÐmeno cM to opoÐo èqeiapodojeÐ sto c apì th M. MÐa stajer� (kai k�je sÔmbolo thcL) èqei tìsec ènnoiec ìsec kai o arijmìc twn L-dom¸n.

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Ta upìloipa sÔmbola

H gl¸ssa L pou qrhsimopoioÔme wc par�deigma èqei stajerèckai kathgor mata pr¸tou kai deutèrou bajmoÔ.Gia na anaparast soume grafik� mÐa dom M aut c thcgl¸ssac:

I zwgrafÐsame èna sÔnolo apì antikeÐmena (to sÔmpan thcM)

I dhl¸same poia antikeÐmena thc M onom�zontai apì poiecstajerèc thc L

I dhl¸same poia antikeÐmena ikanopoioÔn ta kathgor matapr¸tou bajmoÔ (Sun, ανθρωπος, καθηγητης) sÔmfwna me thM

I zwgrafÐsame kateujunìmenec grammèc metaxÔ twnantikeimènwn pou ikanopoioÔn to kathgìrhma deutèroubajmoÔ (αγορασε) sÔmfwna me th M. H kateÔjunsh thcgramm c èqei shmasÐa.

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Ta upìloipa sÔmbola

An up rqan perissìtera apì èna kathgor mata deutèroubajmoÔ sthn L tìte ja tan aparaÐthto na b�loume etikètecstic kateujunìmenec grammèc.

Genik�, den up�rqei eÔkoloc trìpoc na anaparast soumegrafik� apodìseic kathgorhm�twn trÐtou megalÔteroubajmoÔ.

Kathgor mata bajmoÔ 0 eÐnai ta Ðdia me ta �toma thcProtasiak c Logik c.

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9.2 Alhjotimèc se mÐa dom (mh tupik perigraf )Pìte eÐnai ènac tÔpoc qwrÐc posodeÐktec alhj c se mÐa dom ;

I O tÔpoc Sun(Ερμης) eÐnai alhj c sth M, giatÐ ΕρμηςM eÐnaièna antikeÐmeno ◦ to opoÐo h M lèei ìti eÐnai Sun.Autì to gr�foume M � Sun(Ερμης).To diab�zoume wc “h M lèei ìti Sun(Ερμης)”.Prosoq : Aut eÐnai mÐa diaforetik qr sh tou � apìaut tou OrismoÔ 3.1.To sÔmbolo � qrhsimopoieÐtai gia dÔo diaforetikoÔcskopoÔc.

I OmoÐwc, o tÔpoc αγορασε(Σοφια, Αρης) eÐnai alhj c sth M.Sumbolik� M � αγορασε(Σοφια, Αρης).

I O tÔpoc αγορασε(Σοφια, Σοφια) eÐnai yeud c sth M, giatÐ hM den lèei ìti h stajer� Σοφια onom�zei èna antikeÐmeno •to opoÐo αγορασε ton eautì tou.Sumbolik� M 2 αγορασε(Σοφια, Σοφια).

I OmoÐwc, M 2 Sun(Τασος) ∨ αγορασε(Φωτης, Αρης).156 / 232

MÐa �llh dom Parak�tw eÐnai mÐa �llh L-dom , h M ′.

Up�rqoun mìno 10 antikeÐmena sto sÔmpan thc M ′.157 / 232

Sqetik� me th M ′

I M ′ 2 αγορασε(Σοφια, Αρης).I M ′ � Σοφια=Τασος.

I M ′ � ανθρωπος(Τριτων) ∧ Sun(Τριτων).

I M ′ � αγορασε(Τασος, Ερμης) ∧ αγορασε(Ερμης, c).

Poia h alhjotim twn parak�tw tÔpwn;

I αγορασε(Σοφια, Αρης)→ ανθρωπος(Αρης).I αγορασε(c, Ερμης)→ Sun(Αρης) ∨ ¬ανθρωπος(Τριτων).

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ErmhneÔontac tÔpouc me posodeÐktec (mh tupik perigraf )

Pìte eÐnai ènac tÔpoc me posodeÐktec alhj c se mÐa dom ;

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ErmhneÔontac posodeÐktecP¸c exet�zoume an o tÔpoc ∃x αγορασε(x , Ερμης) eÐnai alhj csth M;Sumbolik�, M � ∃x αγορασε(x , Ερμης);Sth fusik gl¸ssa, “uposthrÐzei h M ìti k�ti agìrase tonErm ;”

Gia na eÐnai alhj c o parap�nw tÔpoc, prèpei na up�rqei ènaantikeÐmeno x sto sÔmpan thc M ètsi ¸ste

M � αγορασε(x , Ερμης).

Me �lla lìgia, h M prèpei na uposthrÐzei ìti

αγορασε(x , ◦), ìpou ◦=ΕρμηςM .

Up�rqei tètoio antikeÐmeno: blèpoume sto di�gramma ìti to xmporeÐ na eÐnai, pq, ΤασοςM .

Sunep¸c, M � ∃x αγορασε(x , Ερμης).160 / 232

Par�deigma:M � ∀x(agorase(Tasoc, x)→ agorase(Sofia, x));

“Gia k�je antikeÐmeno x sto sÔmpan thc M, eÐnai o tÔpocαγορασε(Τασος, x)→ αγορασε(Σοφια, x) alhj c sth M”;

Sth M up�rqoun 12 pijan� x . Prèpei na elègxoume an o tÔpocαγορασε(Τασος, x)→ αγορασε(Σοφια, x) eÐnai alhj c sth M giak�je èna apì aut�.

O tÔpoc αγορασε(Τασος, x)→ αγορασε(Σοφια, x) ja eÐnaialhj c sth M gia k�je èna antikeÐmeno x tètoio ¸ste o tÔpocαγορασε(Τασος, x) eÐnai yeud c sth M. (GiatÐ;) 'Ara qrei�zetaina elègxoume ta x gia ta opoÐa o tÔpoc αγορασε(Τασος, x)eÐnai alhj c.

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Par�deigma:M � ∀x(agorase(Tasoc, x)→ agorase(Sofia, x));

O tÔpoc αγορασε(Τασος, x) eÐnai alhj c mìno gia èna x , gia toΕρμης

M .

Gia to antikeÐmeno ◦=ΕρμηςM , o tÔpoc αγορασε(Σοφια, ◦) eÐnaiepÐshc alhj c sthn M. 'Ara o tÔpocαγορασε(Τασος, ◦)→ αγορασε(Σοφια, ◦) eÐnai alhj c sth M.

'Ara o tÔpoc αγορασε(Τασος, x)→ αγορασε(Σοφια, x) eÐnaialhj c sth M gia k�je antikeÐmeno x thc M. Sunep¸c,M � ∀x(αγορασε(Τασος, x)→ αγορασε(Σοφια, x)).

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'Askhsh: Poia eÐnai alhj sth M ;

∃x(Sun(x) ∧ αγορασε(Φωτης, x))∀x(καθηγητης(x)→ ανθρωπος(x))∃x(Sun(x) ∧ ∃y αγορασε(y , x))

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ErmhneÔontac tÔpouc (mh tupik perigraf )Gia èna sÔnjeto tÔpo ìpwc ∃x(Sun(x) ∧ ∃y αγορασε(y , x)):BreÐte p¸c metafr�zetai k�je upo-tÔpoc sth fusik gl¸ssa,xekin¸ntac apì touc atomikoÔc upo-tÔpouc (ta fÔlla toudèndrou (formation tree) ) ft�nontac mèqri thn arqik prìtash(rÐza tou dèndrou).

Autìc eÐnai suqn� ènac kalìc trìpoc na ermhneÔoume tÔpouc.Pq, o parap�nw tÔpoc anafèrei ìti up�rqei èna x to opoÐoeÐnai Sun kai k�ti to agìrase (eÐnai dhlad h kat�lhxh mÐackateujunìmenhc gramm c). 'Ara elègqoume tic grammèc.

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9.3 Alhjotimèc se mÐa dom (tupik perigraf )

EÐdame p¸c na ermhneÔoume orismènouc tÔpouc se mÐa dom koit¸ntac to di�gramma thc dom c.

Qreiazìmaste ìmwc èna pio tupikì/majhmatikì trìpo gia naermhneÔoume ìlouc touc tÔpouc thc Kathgorhmatik c Logik cse domèc.

Sth Protasiak Logik , upologÐzame thn alhjotim mÐacprìtashc se mÐa kat�stash upologÐzontac tic alhjotimèc twnupo-prot�sewn thc, xekin¸ntac apì tic atomikèc upo-prot�seic(ta fÔlla tou dèndrou thc (formation tree) ) ft�nontac mèqri thrÐza tou dèndrou.

Sth Kathgorhmatik Logik ta pr�gmata den eÐnai tìsoapl�...

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'Ena prìblhmaO tÔpoc ∀x(αγορασε(Τασος, x)→ Sun(x)) eÐnai alhj c sth dom thc selÐdac 163. To dèndro tou eÐnai:

MporoÔme na ermhneÔsoume autìn ton tÔpo xekin¸ntac apì tafÔlla, katal gontac sth rÐza;

EÐnai o tÔpoc αγορασε(Τασος, x) alhj c sth M;EÐnai o tÔpoc Sun(x) alhj c sth M;

Den eÐnai ìloi oi tÔpoi thc Kathgorhmatik c Logik c alhjeÐc

yeudeÐc se mÐa dom !166 / 232

EleÔjerec kai desmeumènec metablhtèc

Prèpei na elègxoume p¸c emfanÐzontai oi metablhtèc stouctÔpouc.

Orismìc 9.3

'Estw A ènac tÔpoc.

1. H emf�nish mÐac metablht c x se èna atomikì upo-tÔpo

tou A lègetai desmeumènh (bound) an brÐsketai k�tw apì

èna posodeÐkth ∀x ∃x sto dèndro (formation tree) tou A.

2. Alli¸c h emf�nish thc metablht c lègetai eleÔjerh (free).

3. Oi eleÔjerec metablhtèc enìc tÔpou A eÐnai oi metablhtèc

oi opoÐec èqoun eleÔjerec emfanÐseic ston A.

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Par�deigma∀x(R(x , y) ∧ R(y , z)→ ∃z(S(x , z) ∧ R(z , y)))

Oi eleÔjerec metablhtèc tou tÔpou eÐnai oi y , z .ShmeÐwsh: h z èqei eleÔjerec kai desmeumènec emfanÐseic.

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Prot�seic

Orismìc 9.4

MÐa prìtash (sentence) eÐnai ènac tÔpoc qwrÐc eleÔjerecmetablhtèc.

ParadeÐgmata:

I O tÔpoc ∀x(αγορασε(Τασος, x)→ Sun(x)) eÐnai prìtash.I Oi upo-tÔpoi tou:

I αγορασε(Τασος, x)→ Sun(x)I αγορασε(Τασος, x)I Sun(x)

den eÐnai prot�seic.

Poiec eÐnai prot�seic;

I αγορασε(Φωτης, Τριτων)I αγορασε(Σοφια, x)I x=xI ∀x(∃y(y=x)→ x=y)I ∀x∀y(x=y → ∀z(R(x , z)→ R(y , z)))

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Pr¸to prìblhma: eleÔjerec metablhtècOi prot�seic eÐnai alhjeÐc yeudeÐc se mÐa dom .All� oi tÔpoi pou den eÐnai prot�seic den eÐnai alhjeÐc yeudeÐc se mÐa dom !'Enac tÔpoc me eleÔjerec metablhtèc den eÐnai alhj c yeud cse mÐa dom M, giatÐ oi eleÔjerec metablhtèc tou den èqounnìhma sth M. EÐnai san na rwt�me “eÐnai to x=7 alhjèc;”Den mporoÔme na ermhneÔsoume èna tÔpo A thcKathgorhmatik c Logik c se mÐa dom ìpwc ermhneÔameprot�seic thc Protasiak c Logik c se katast�seic, giatÐ hdom den eÐnai mÐa “oloklhrwmènh” kat�stash: den dÐnei ènnoiecstic eleÔjerec metablhtèc tou A.

'Ara prèpei na d¸soume timèc stic eleÔjerec metablhtèc prinermhneÔsoume èna tÔpo.Autì prèpei na to k�noume akìma kai ìtan oi timèc autèc denephre�zoun thn ermhneÐa tou tÔpou (pq, ìtan o tÔpoc eÐnaix=x).

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Ekq¸rhsh tim¸n stic metablhtèc (assignment)MÐa ermhneutik sun�rthsh metablht¸n (ESM) dÐnei timèc sticeleÔjerec metablhtèc.

MÐa ESM eÐnai gia tic metablhtèc ì,ti mÐa dom gia ticstajerèc.

Orismìc 9.5

'Estw M mÐa dom . MÐa ermhneutik sun�rthsh metablht¸n(ESM) h sto plaÐsio thc M antistoiqeÐ se k�je metablht èna

antikeÐmeno apì to sÔmpan thc M:

h : V → dom(M)

(V eÐnai to sÔnolo twn metablht¸n kai dom(M) eÐnai to sÔnolo

twn antikeimènwn (sÔmpan) thc M.)

Gia mÐa ESM h kai mÐa metablht x , gr�foume h(x) gia to

antikeÐmeno to opoÐo èqei antistoiqhjeÐ sth x apì thn h.

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Ekq¸rhsh tim¸n stic metablhtèc (assignment)

DojeÐshc mÐac L-dom c M kai mÐac ESM h sto plaÐsio thc M,èqoume mÐa “oloklhrwmènh” kat�stash. Sunep¸c, mporoÔme naermhneÔsoume:

I opoiod pote L-ìro. H tim tou ja eÐnai èna antikeÐmeno sto

sÔmpan thc M.

I opoiod pote atomikì ( mh) tÔpo mÐac gl¸ssac L. H tim tou ja eÐnai alhj c yeud c.

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ErmhneÔontac ìrouc

MÐa dom kai mÐa ESM dÐnoun mÐa “oloklhrwmènh” kat�stash.Mac dÐnoun tic alhjotimèc ìlwn twn atomik¸n tÔpwn.Ja deÐxoume pr¸ta p¸c ermhneÔoume ìrouc kai met� p¸cermhneÔoume tÔpouc.

Orismìc 9.6

'Estw mÐa gl¸ssa L, M mÐa L-dom , kai h mÐa ESM sto

plaÐsio thc M.

Tìte gia k�je L-ìro t, h tim tou t sth M sÔmfwna me thn heÐnai to antikeÐmeno thc M to opoÐo èqei antistoiqhjeÐ ston tapì:

I Th M, an to t eÐnai mÐa stajer�. Se aut thn perÐptwsh h

tim tou t eÐnai tM .

I Thn h, an to t eÐnai mÐa metablht . Se aut thn perÐptwsh

h tim tou t eÐnai h(t).

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ErmhneÔontac ìrouc: par�deigma(1) H tim tou ìrou Τασος sth M sÔmfwna me thn h eÐnai toantikeÐmeno • me etikèta “T�soc”. (Apì ed¸ kai sto ex c ja toanaparist¸ wc “T�soc” ΤασοςM , all� ìqi wc Τασος.)(2) H tim thc x sth M sÔmfwna me thn h eÐnai Erm c.

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ShmasiologÐa tÔpwn qwrÐc posodeÐktecMporoÔme t¸ra na ermhneÔsoume opoiod pote tÔpo qwrÐcposodeÐktec.'Estw mÐa L-dom M kai mÐa ESM h. Gr�foume M, h � A an otÔpoc A eÐnai alhj c sth M sÔmfwna me thn h. Alli¸cgr�foume M, h 2 A.

Orismìc 9.7

1. 'Estw ìti R eÐnai èna sÔmbolo kathgor matoc bajmoÔ nmÐac gl¸ssac L, kai t1, . . . , tn eÐnai L-ìroi. Ac upojèsoumeìti h tim tou ti sth M sÔmfwna me thn h eÐnai ai , gia

k�je i=1, . . . , n (deÐte ton Orismì 9.6).

Tìte M, h � R(t1, . . . , tn) an h M uposthrÐzei ìti h di�taxh

(a1, . . . , an) èqei th sqèsh R . Alli¸c, M, h 2 R(t1, . . . , tn).

2. An t, t ′ eÐnai ìroi, tìte M, h � t=t ′ an ta t kai t ′ èqoun thn

Ðdia tim sth M sÔmfwna me thn h. An den èqoun thn Ðdia

tim tìte M, h 2 t=t ′.

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ShmasiologÐa tÔpwn qwrÐc posodeÐktec

Orismìc 9.7 (sunèqeia)

3. M, h � > kai M, h 2 ⊥.4. M, h � A ∧ B an M, h � A kai M, h � B . Alli¸c,

M, h 2 A ∧ B .

5. OmoÐwc orÐzoume thn ermhneÐa twn tÔpwn ¬A, A ∨ B ,

A→ B kai A↔ B .

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ErmhneÔontac tÔpouc qwrÐc posodeÐktec: par�deigma

I M, h � ανθρωπος(z)I M, h � x=ΕρμηςI M, h 2 αγορασε(Σοφια, v) ∨ z=Φωτης

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DeÔtero prìblhma: desmeumènec metablhtècGnwrÐzoume p¸c na dÐnoume timèc se eleÔjerec metablhtèc: memÐa ESM. Me autì ton trìpo mporèsame na ermhneÔsoumeìlouc touc tÔpouc qwrÐc posodeÐktec.All� up�rqoun tÔpoi pou èqoun desmeumènec metablhtèc. Oitimèc aut¸n twn metablht¸n den dÐnontai, kai den prèpei nadÐnontai, apì thn kat�stash, kaj¸c oi metablhtèc autècelègqontai apì posodeÐktec.P¸c upologÐzoume tic timèc desmeumènwn metablht¸n;DÔo trìpoi:

1. Elègqoume tic pijanèc ESM:

I Για τον υπαρξιακό ποσοδείκτη ∃ χρειαζόμαστε κάποιαΕΣΜ για να είναι αληθής ο τύπος.

I Για τον καθολικό ποσοδείκτη ∀ χρειαζόμαστε όλες τις ΕΣΜγια να είναι αληθής ο τύπος.

2. QrhsimopoioÔme paiqnÐdia (Hintikka games), ta opoÐa macbohjoÔn na katal�boume tic epidr�seic twn posodeikt¸n,kai eidikìtera twn emfwleumènwn (nested) posodeikt¸n.

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ShmasiologÐa tÔpwn me posodeÐktec

Orismìc 9.7 (sunèqeia)Ac upojèsoume ìti gnwrÐzoume p¸c na ermhneÔsoume èna tÔpo Asth M sÔmfwna me opoiad pote ESM. 'Estw x mÐa

opoiad pote metablht , kai h mÐa opoiad pote ESM. Tìte:

6. M, h � ∃xA an up�rqei k�poia ESM g h opoÐa sumfwneÐ

me thn h gia ìlec tic metablhtèc ektìc, pijan¸c, apì th x ,kai isqÔei M, g � A. An den up�rqei tètoia ESM tìte

M, h 2 ∃xA.

7. M, h � ∀xA an M, g � A gia k�je ESM g h opoÐa sumfwneÐ

me thn h gia ìlec tic metablhtèc ektìc, pijan¸c, apì th x .Alli¸c, M, h 2 ∀xA.

“H g sumfwneÐ me thn h gia ìlec tic metablhtèc ektìc, pijan¸c,apì th x” shmaÐnei ìti g(y)=h(y) gia ìlec tic metablhtèc yektìc thc x . (Autì den apokleÐei thn perÐptwsh g(x)=h(x).)

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ErmhneÔontac tÔpouc me posodeÐktec: par�deigma

I M, h � ∃x ανθρωπος(x)giatÐ, pq M, g � ανθρωπος(x) ìpou g(x)=F¸thc.

I M, h 2 ∀x ανθρωπος(x)giatÐ, pq M, g 2 ανθρωπος(x) ìpou g(x)='Arhc.

I M, h � ∀x(αγορασε(Σοφια, x)→ ¬ανθρωπος(x))(Elègxte sugkekrimèna th g me g(x)=Erm c kai th g meg(x)='Arhc.)

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SÔmbash gia eleÔjerec metablhtècH parak�tw sÔmbash eÐnai qr simh gia na gr�foume kaiermhneÔoume tÔpouc.Sta biblÐa suqn� blèpoume ekfr�seic thc morf c:

“'Estw A(x1, . . . , xn) ènac tÔpoc. ”

H parap�nw èkfrash dhl¸nei ìti oi eleÔjerec metablhtèc touA eÐnai metaxÔ twn x1, . . . , xn.ShmeÐwsh: Oi x1, . . . , xn prèpei na eÐnai ìlec diaforetikèc.Epiplèon, den eÐnai aparaÐthto na èqoun ìlec oi x1, . . . , xn

eleÔjerec emfanÐseic ston A.Par�deigma: 'Estw o tÔpoc C :

∀x(R(x , y)→ ∃yS(y , z))

Ja mporoÔsame na gr�youme ton tÔpo wc:I C (y , z)I C (x , z , v , y)I C (an den qrhsimopoi soume aut th sÔmbash)

all� ìqi wc C (x).181 / 232

SÔmbash gia Ermhneutikèc Sunart seic Metablht¸nGia k�je tÔpo A, an isqÔei M, h � A ìqi, exart�tai apì thn

h(x) mìno gia tic metablhtèc x oi opoÐec èqoun eleÔjerec

emfanÐseic ston A.'Ara gia èna tÔpo A(x1, . . . , xn), an h(x1)=a1, . . . , h(xn)=an, tìteeÐnai OK na gr�youme M � A(a1, . . . , an) antÐ gia M, h � A.

I Ac upojèsoume ìti mac èqoun d¸sei èna tÔpo C (y , z) tètoio¸ste

∀x(R(x , y)→ ∃yS(y , z))

An pq h(y)=a, h(z)=b, tìte mporoÔme na gr�youme

M � C (a, b), M � ∀x(R(x , a)→ ∃yS(y , b))

antÐ gia M, h � C . ShmeÐwsh: mìno oi eleÔjerec emfanÐseictou y ston C èqoun antikatastajeÐ apì to a. Oidesmeumènec emfanÐseic tou y paramènoun wc èqoun.

I Gia mÐa prìtash S , an isqÔei M, h � S ìqi, den exart�taikajìlou apì thn h. 'Ara mporoÔme na gr�youme M � S .

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P¸c na ermhneÔoume tÔpouc;

EÐdame ton orismì thc al jeiac se mÐa dom (Alfred Tarski,dekaetÐa 1950).Poioc ìmwc eÐnai o kalÔteroc trìpoc na elègqoume an isqÔeiM � A;

I Se merikèc peript¸seic mporoÔme na metafr�zoume ton Asth fusik gl¸ssa kai na elègqoume an eÐnai alhj c sthM (deÐte selÐda 164).

I QrhsimopoioÔme ton Orismì 9.7 kai elègqoume ìlec ticpijanèc ESM. Pq, gia ton tÔpo ∀x(καθηγητης(x)→ Sun(x))elègqoume, gia ìla ta x , an to x eÐnai kajhght c tìteeÐnai kai Sun.

I QrhsimopoioÔme ta paiqnÐdia Hintikka, ta opoÐa macbohjoÔn na katal�boume tic epidr�seic twn posodeikt¸n.Me ta paiqnÐdia Hintikka den ja asqolhjoÔme se autì tom�jhma.

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10. Sqèsh tou KathgorhmatikoÔ LogismoÔ me th fusik gl¸ssa

10.1 Apì thn tupik gl¸ssa sth fusik H met�frash twn prot�sewn thc Kathgorhmatik c Logik c sthfusik gl¸ssa den eÐnai polÔ duskolìterh apì th met�frashtwn prot�sewn thc Protasiak c Logik c sth fusik gl¸ssa.Oi metablhtèc prèpei apaloifjoÔn: sth fusik gl¸ssa den ticqrhsimopoioÔme.

∀x(καθηγητης(x) ∧ ¬(x=Φωτης)→ αγορασε(x , Τριτων))“Gia k�je x , an to x eÐnai ènac kajhght c kai to x den eÐnai oF¸thc tìte to x agìrase ton TrÐtwn.”“K�je kajhght c ektìc apì ton F¸th agìrase ton TrÐtwn.”(MporeÐ kai o F¸thc na ton agìrase.)

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Apì thn tupik gl¸ssa sth fusik

∃x∃y∃z(αγορασε(x , y) ∧ αγορασε(x , z) ∧ ¬(y=z))“Up�rqoun x , y , z tètoia ¸ste to x agìrase to y , to xagìrase to z , kai to y den eÐnai to z .”“K�ti agìrase toul�qiston dÔo diaforetik� pr�gmata.”

∀x(∃y∃z(αγορασε(x , y) ∧ αγορασε(x , z) ∧ ¬(y=z))→ x=Τασος)“Gia k�je x , an to x agìrase dÔo diaforetik� pr�gmata, tìteto x eÐnai o T�soc.”“Otid pote agìrase dÔo diaforetik� pr�gmata eÐnai o T�soc.”Prosoq : h parap�nw prìtash den lèei ìti o T�soc agìrasedÔo diaforetik� pr�gmata, all� ìti kanènac �lloc denagìrase dÔo diaforetik� pr�gmata.

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10.2 TupopoÐhsh thc fusik c gl¸ssacEkfr�ste tic upo-ènnoiec sth logik . 'Epeita sunjèste ticupo-ènnoiec se mÐa logik prìtash.

I Upo-ènnoia: “to x agor�sthke”/ “to x èqeiagorast ”:∃y αγορασε(y , x).

I Otid pote agor�sthke den eÐnai �njrwpoc:∀x(∃y αγορασε(y , x)→ ¬ανθρωπος(x)).Prosoq : h met�frash se∀x∃y(αγορασε(y , x)→ ¬ανθρωπος(x)) den eÐnai swst .

I K�je Sun agor�sthke: ∀x(Sun(x)→ ∃y αγορασε(y , x)).I K�poio Sun èqei èna agorast :∃x(Sun(x) ∧ ∃y αγορασε(y , x)).

I 'Oloi oi agorastèc eÐnai �njrwpoi kai kajhghtèc:∀x(∃y αγορασε(x , y)︸︷︷︸

o x eÐnai agorast c

→ ανθρωπος(x) ∧ καθηγητης(x)).

I Kanènac kajhght c den agìrase èna Sun:¬∃x(καθηγητης(x) ∧ ∃y(αγορασε(x , y) ∧ (Sun(y))︸︷︷︸

o x agìrase èna Sun

).

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TupopoÐhsh thc fusik c gl¸ssacSuqn� qrei�zetai na poÔme:

I “'Oloi oi kajhghtèc eÐnai �njrwpoi”:∀x(καθηγητης(x)→ ανθρωπος(x)).OQI ∀x(καθηγητης(x) ∧ ανθρωπος(x)).OQI ∀x καθηγητης(x)→ ∀x ανθρωπος(x).

I “'Oloi oi kajhghtèc eÐnai �njrwpoi Sun”:∀x(καθηγητης(x)→ ανθρωπος(x) ∨ Sun(x)).

I “TÐpota den eÐnai Sun kai kajhght c”:¬∃x(Sun(x) ∧ καθηγητης(x)), ∀x(Sun(x)→ ¬καθηγητης(x)).

Sunep¸c o tÔpoc ∀x(A→ B) eÐnai polÔ suqnìc.EpÐshc oi ∀x(A ∧ B), ∀x(A ∨ B), ∃x(A ∧ B), ∃x(A ∨ B)qrei�zontai suqn�: lène ìti ta p�nta/k�ti eÐnai A kai/ B .

O tÔpoc ∃x(A→ B), eidik� ìtan h x èqei eleÔjerh emf�nishston A, eÐnai sp�nioc.An gr�yete autìn ton tÔpo elègxte an èqete k�nei l�joc.

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I Up�rqei toul�qiston èna Sun: ∃x Sun(x).

I Up�rqoun toul�qiston dÔo Sun:∃x∃y(Sun(x) ∧ Sun(y) ∧ x 6=y), ∀x∃y(Sun(y) ∧ y 6=x).

I Up�rqoun toul�qiston trÐa Sun:∃x∃y∃z(Sun(x) ∧ Sun(y) ∧ Sun(z) ∧ x 6=y ∧ y 6=z ∧ x 6=z), ∀x∀y∃z(Sun(z) ∧ z 6=x ∧ z 6=y).

I Den up�rqei Sun: ¬∃x Sun(x).I Up�rqei to polÔ èna Sun:

1. ¬∃x∃y(Sun(x) ∧ Sun(y) ∧ x 6=y)δηλαδή “¬(υπάρχουν τουλάχιστον δύο Sun)”

2. ∀x∀y(Sun(x) ∧ Sun(y)→ x=y).3. ∃x∀y(Sun(y)→ x=y).

I Up�rqei akrib¸c èna Sun:1. “Υπάρχει τουλάχιστον ένα Sun”∧“Υπάρχει το πολύ ένα

Sun”.2. ∃x∀y(Sun(y)↔ x=y).

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11. SÔmbola sunart sewn

Sthn arijmhtik qrhsimopoioÔme sunart seic ìpwc pq +, −, ×,√x .

Sthn Kathgorhmatik Logik mporoÔme na ekfr�soumesunart seic.'Ena sÔmbolo sun�rthshc eÐnai san èna sÔmbolo kathgor matoc mÐa stajer�, all� apodÐdetai se mÐa dom wc mÐa sun�rthsh.K�je sÔmbolo sun�rthshc èqei èna sugkekrimèno bajmì (arity).

SumbolÐzoume sun jwc ta sÔmbola sunart sewn me f , g .Apì ed¸ kai sto ex c uiojetoÔme thn parak�tw epèktash touOrismoÔ 8.1:

Orismìc 11.1

MÐa gl¸ssa eÐnai èna sÔnolo apì stajerèc, kai sÔmbola

kathgorhm�twn kai sunart sewn sugkekrimènou bajmoÔ.

Prosoq : den prèpei na up�rqei sÔgqush metaxÔ mÐac ESM hsto plaÐsio mÐac dom c M, me mÐa sun�rthsh f mÐac gl¸ssac L.

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'Oroi me sÔmbola sunart sewnT¸ra mporoÔme epekteÐnoume ton Orismì 8.2:

Orismìc 11.2 ('Oroc)

DojeÐshc mÐac gl¸ssac L:

1. K�je stajer� thc L eÐnai ènac L-ìroc.

2. K�je metablht eÐnai ènac L-ìroc.

3. An f eÐnai èna sÔmbolo sun�rthshc bajmoÔ n thc L, kait1, . . . , tn eÐnai L-ìroi, tìte o f (t1, . . . , tn) eÐnai ènac L-ìroc.

4. TÐpota �llo den eÐnai L-ìroc.

Par�deigma:

'Estw ìti h L èqei mÐa stajer� c, èna sÔmbolo sun�rthshc fpr¸tou bajmoÔ, kai èna sÔmbolo sun�rthshc g deutèroubajmoÔ. Tìte ta parak�tw eÐnai L-ìroi:

I cI f (c)I g(x , x) (wc sun jwc h x eÐnai metablht )I g(f (c), g(x , x)) 190 / 232

ShmasiologÐa sumbìlwn sunart sewnPrèpei epiplèon na epekteÐnoume ton Orismì 9.1: an h L èqeisÔmbola sunart sewn, mÐa L-dom prèpei na diatup¸nei tonìhma touc.Gia k�je sÔmbolo sun�rthshc f bajmoÔ n thc L, mÐa L-dom Mprèpei na lèei poio antikeÐmeno (apì to sÔmpan thc)antistoiqeÐ h f se k�je di�taxh (a1, . . . , an), ìpou a1, . . . , an

eÐnai antikeÐmena tou sÔmpantoc thc M.SumbolÐzoume wc f M(a1, . . . , an) to antikeÐmeno tou sÔmpantocthc M pou h f antistoiqeÐ sth di�taxh (a1, . . . , an).'Ena sÔmbolo sun�rthshc bajmoÔ 0 jewreÐtai mÐa stajer�.

Par�deigma

Sthn arijmhtik , h M mporeÐ na uposthrÐzei ìti oi sunart seic+ kai × ekfr�zoun thn prìsjesh kai ton pollaplasiasmìarijm¸n: to 5 èqei antistoiqhjeÐ me to 2+3, to 8 me to 2×4, klp.Den eÐnai aparaÐthto h M na to uposthrÐzei autì. MporeÐ nauposthrÐzei enallaktik� ìti h sun�rthsh × ekfr�zei thnprìsjesh. 191 / 232

ErmhneÔontac ìrouc me sÔmbola sunart sewnT¸ra mporoÔme na epekteÐnoume ton Orismì 9.6:

Orismìc 11.3

H tim enìc L-ìrou t se mÐa L-dom M sÔmfwna me mÐa ESM hsto plaÐsio thc M orÐzetai wc ex c:

I An o t eÐnai stajer� tìte h tim tou eÐnai to antikeÐmeno

tM to opoÐo èqei antistoiqhjeÐ ston t apì th M.

I An o t eÐnai metablht tìte h tim tou eÐnai to antikeÐmeno

h(t) to opoÐo èqei antistoiqhjeÐ ston t apì thn h.

I An o t eÐnai sun�rthsh thc morf c f (t1, . . . , tn), kai oitimèc twn ìrwn t1, . . . , tn sth M sÔmfwna me thn h eÐnai

antÐstoiqa a1, . . . , an, tìte h tim tou t eÐnai to antikeÐmeno

f M(a1, . . . , an).

Sunep¸c h tim enìc ìrou sth M sÔmfwna me thn h eÐnai p�ntaèna antikeÐmeno tou sÔmpantoc thc M. Me �lla lìgia ènac

ìroc den èqei alhjotim .

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ShmasiologÐa tÔpwn me sÔmbola sunart sewn

O orismìc 9.7 den qrei�zetai allag , pèra apì to ìtiqrhsimopoieÐtai plèon me ton Orismì 11.3 (kai ìqi me tonOrismì 9.6).

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ArijmhtikoÐ ìroi'Estw mÐa gl¸ssa L gia arijmhtik kai gia progr�mmatalogismikoÔ pou qrhsimopoioÔn arijmoÔc, h opoÐa perilamb�nei:

I tic stajerèc 0, 1, 2, . . . (upogrammÐzw aut� ta sÔmbola giana ta xeqwrÐsw apì touc fusikoÔc arijmoÔc 0, 1, 2, . . . ),

I ta sÔmbola sunart sewn deutèrou bajmoÔ +,−,×, kaiI ta sÔmbola kathgorhm�twn deutèrou bajmoÔ <,≤, >,≥.

ApodÐdoume ta parap�nw sÔmbola se mÐa dom me sÔmpan tosÔnolo {0, 1, . . . }, ìpou to 0 apodÐdetai sto 0, to 1 sto 1, klp,to +(x , y) ekfr�zei x+y , to <(x , y) ekfr�zei x<y , klp.ShmeÐwsh: h tim tou −(34, 55) den mporeÐ na problefjeÐ,mporeÐ na eÐnai opoiosd pote arijmìc.

ParadeÐgmata ìrwn:+(x , 1), +(2, +(x , 5)), ×(+(3, 7), −(5, 2)). 'Oqi to +(x , y , z)ParadeÐgmata tÔpwn:>(×(3, x), 0), ∀y(>(y , 0)→ >(×(y , y), y)).

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SunoyÐzontac

MÐa L-dom M perilamb�nei:

I èna mh-kenì sÔnolo, to dom(M)

I gia k�je stajer� c ∈ L, èna stoiqeÐo cM ∈ dom(M)

I gia k�je sÔmbolo sun�rthshc f ∈ L bajmoÔ n, mÐasun�rthsh f M : dom(M)n → dom(M)

I gia k�je sÔmbolo kathgor matoc R ∈ L bajmoÔ n, mÐasqèsh RM sto dom(M), dhlad RM ⊆ dom(M)n.

(Gia èna sÔnolo S , to Sn eÐnai to

n forèc︷︸︸︷S × S × · · · × S .)

ShmeÐwsh: den prèpei na up�rqei sÔgqush metaxÔ mÐac ESM hsto plaÐsio mÐac dom c M, ìpou h : V → dom(M), me mÐasun�rthsh f mÐac gl¸ssac L, ìpou f M : dom(M)n → dom(M).

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12. Epiqeir mata, egkurìthtaH Kathgorhmatik Logik eÐnai pio ekfrastik apì thnProtasiak Logik . All� h empeirÐa mac apì thn Protasiak Logik mac bohj�ei na diatup¸soume ton orismì thcegkurìthtac sthn Kathgorhmatik Logik .

Orismìc 12.1 ('Egkuro epiqeÐrhma)

'Estw L mÐa gl¸ssa kai A1, . . . ,An,B tÔpoi thc L.'Ena epiqeÐrhma “A1, . . . ,An, �ra B” eÐnai ègkuro an se k�je

L-dom M kai ESM h sto plaÐsio thc M ìpou

M, h � A1,M, h � A2, . . . kai M, h � An,, tìte M, h � B .

Se aut thn perÐptwsh gr�foume A1, . . . ,An � B .

Me �lla lìgia, se k�je kat�stash (dom + ESM) sthn opoÐaoi tÔpoi A1, . . . ,An eÐnai ìloi alhjeÐc, o B prèpei epÐshc naeÐnai alhj c.Eidik perÐptwsh: n=0. Se aut thn perÐptwsh gr�foume � B .ShmaÐnei ìti o B eÐnai alhj c se k�je L-dom sÔmfwna me k�jeESM.

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Egkurìthta, Ikanopoihsimìthta, IsodunamÐa

'Estw mÐa gl¸ssa L.

Orismìc 12.2 ('Egkuroc TÔpoc)

'Enac L-tÔpoc eÐnai (logik�) ègkuroc an gia k�je L-dom M kai

ESM h sto plaÐsio thc M, èqoume M, h � A.Gr�foume “ � A”an o A eÐnai ègkuroc.

Orismìc 12.3 (Ikanopoi simoc TÔpoc)

'Enac L-tÔpoc eÐnai ikanopoi simoc an gia k�poia L-dom M kai

ESM h sto plaÐsio thc M, èqoume M, h � A.

Orismìc 12.4 (IsodÔnamoi TÔpoi)

Oi L-tÔpoi A,B eÐnai logik� isodÔnamoi an gia k�je L-dom Mkai ESM h sto plaÐsio thc M, èqoume M, h � A an kai mìno an

M, h � B .

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Egkurìthta, Ikanopoihsimìthta, IsodunamÐa

Oi sqèseic metaxÔ aut¸n twn ennoi¸n (selÐda 43) isqÔoun kaigia thn Kathgorhmatik Logik .

'Ara, pq oi ènnoiec tou ègkurou, ikanopoi simou tÔpou, kai thcisodunamÐac, mporoÔn na ekfrastoÔn me b�sh thn ènnoia touègkurou epiqeir matoc.

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Poia epiqeir mata eÐnai ègkura;

Merik� paradeÐgmata apì ègkura epiqeir mata:

I ègkura epiqeir mata Protasiak c Logik c: pq A ∧ B � A.

I poll� kainoÔrgia: pq∀x(καθηγητης(x)→ ανθρωπος(x)),∃x(καθηγητης(x) ∧ αγορασε(x , Τριτων)),� ∃x(ανθρωπος(x) ∧ αγορασε(x , Τριτων))

Elègqontac an èna epiqeÐrhma A1, . . . ,An � B eÐnai ègkuro eÐnaigenik� p�ra polÔ dÔskolo.Den mporoÔme na elègxoume ìti o B eÐnai alhj c se ìlec ticL-domèc + ESM stic opoÐec oi A1, . . . ,An eÐnai ìloi alhjeÐc(ìpwc k�name me touc pÐnakec al jeiac).Autì sumbaÐnei giatÐ up�rqoun �peirec L-domèc.

Je¸rhma 12.5 (Church, 1935)

Den up�rqei algìrijmoc o opoÐoc mporeÐ na entopÐsei akrib¸c

ta ègkura epiqeir mata thc Kathgorhmatik c Logik c.

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Trìpoi gia na elègqoume thn egkurìthta twnepiqeirhm�twn

Par� to Je¸rhma 12.5, sthn pr�xh mporoÔme suqn� naelègqoume an èna epiqeÐrhma thc Kathgorhmatik c Logik ceÐnai ègkuro. Autì mporoÔme na to petÔqoume me toucparak�tw trìpouc:

I �mesh epiqeirhmatologÐa

I isodunamÐec

I sust mata apodeÐxewn ìpwc to sÔsthma kanìnwn fusik csumperasmatologÐac

QrhsimopoioÔme touc parap�nw trìpouc kai gia na deÐxoumeìti ènac tÔpoc eÐnai ègkuroc, dhlad � A.

Oi pÐnakec al jeiac den eÐnai qr simoi giatÐ up�rqoun �peirecdomèc.

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12.1 'Amesh epiqeirhmatologÐaAc deÐxoume ìti∀x(καθηγητης(x)→ ανθρωπος(x)),∃x(καθηγητης(x) ∧ αγορασε(x , Τριτων))� ∃x(ανθρωπος(x) ∧ αγορασε(x , Τριτων)).

Ac p�roume mÐa opoiad pote L-dom M (ìpou L eÐnai h gl¸ssapou qrhsimopoioÔme wc par�deigma). Ac upojèsoume ìti(1) M � ∀x(καθηγητης(x)→ ανθρωπος(x)).(2) M � ∃x(καθηγητης(x) ∧ αγορασε(x , Τριτων)).'Ara prèpei na broÔme èna antikeÐmeno α sth M tètoio ¸steM � ανθρωπος(α) ∧ αγορασε(α, Τριτων).SÔmfwna me thn upìjesh (2) up�rqei èna α sth M tètoio ¸steM � καθηγητης(α) ∧ αγορασε(α, Τριτων).'Ara M � καθηγητης(α).SÔmfwna me thn upìjesh (1),M � καθηγητης(α)→ ανθρωπος(α).'Ara M � ανθρωπος(α).'Ara M � ανθρωπος(α) ∧ αγορασε(α, Τριτων). 201 / 232

Par�deigmaAc deÐxoume ìti∀x(ανθρωπος(x)→ καθηγητης(x)),∀x(Sun(x)→ καθηγητης(x)),∀x(Sun(x) ∨ ανθρωπος(x))� ∀x καθηγητης(x).

Ac p�roume opoiad pote dom M tètoia ¸ste:(1) M � ∀x(ανθρωπος(x)→ καθηγητης(x)),(2) M � ∀x(Sun(x)→ καθηγητης(x)),(3) M � ∀x(Sun(x) ∨ ανθρωπος(x))Prèpei na deÐxoume ìti M � ∀x καθηγητης(x).Ac p�roume èna opoiod pote antikeÐmeno α thc M. Jèloume naisqÔei M � καθηγητης(α).SÔmfwna me thn (3) èqoume M � Sun(α) ∨ ανθρωπος(α).An M � ανθρωπος(α), tìte apì thn (1) isqÔei M � καθηγητης(α).Alli¸c M � Sun(α). Tìte apì th (2) isqÔei M � καθηγητης(α).'Ara kai apì tic 2 peript¸seic èqoume M � καθηγητης(α).

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'Amesh epiqeirhmatologÐa me isìthtaAc deÐxoume ìti o tÔpoc ∀x∀y(x=y ∧ ∃zR(x , z)→ ∃uR(y , u))eÐnai ègkuroc.Ac p�roume mÐa opoiad pote dom M, kai antikeÐmena α, β stosÔmpan thc M. Prèpei na deÐxoume ìti

M � α=β ∧ ∃zR(α, z)→ ∃uR(β, u)

'Ara prèpei na deÐxoume ìtiAN M � α=β ∧ ∃zR(α, z) TOTE M � ∃uR(β, u).

All� AN M � α=β ∧ ∃zR(α, z), tìte ta α, β eÐnai to ÐdioantikeÐmeno.Sunep¸c M � ∃zR(β, z).

'Ara up�rqei èna antikeÐmeno γ sto sÔmpan thc M tètoio ¸steM � R(β, γ).

Sunep¸c M � ∃uR(β, u). Telei¸same.203 / 232

12.2 IsodunamÐec

H Kathgorhmatik Logik perilamb�nei tic isodunamÐec pou jadoÔme parak�tw kaj¸c KAI tic isodunamÐec thc Protasiak cLogik c.'Estw A,B tuqaÐoi tÔpoi.

28. O tÔpoc ∀x∀yA eÐnai logik� isodÔnamoc me ton ∀y∀xA.

29. O tÔpoc ∃x∃yA eÐnai (logik�) isodÔnamoc me ton ∃y∃xA.

30. O tÔpoc ¬∀xA eÐnai isodÔnamoc me ton ∃x¬A.

31. O tÔpoc ¬∃xA eÐnai isodÔnamoc me ton ∀x¬A.

32. O tÔpoc ∀x(A ∧ B) eÐnai isodÔnamoc me ton ∀xA ∧ ∀xB .

33. O tÔpoc ∃x(A ∨ B) eÐnai isodÔnamoc me ton ∃xA ∨ ∃xB .

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IsodunamÐec me desmeumènec metablhtèc

34. An h x den èqei eleÔjerh emf�nish ston A, tìte oi tÔpoi∀xA kai ∃xA eÐnai logik� isodÔnamoi me ton A.Pq oi ∀x ∃xP(x)︸︷︷︸

A

kai ∃x ∃xP(x)︸︷︷︸A

eÐnai isodÔnamoi me ∃xP(x)︸︷︷︸A

.

35. An h x den èqei eleÔjerh emf�nish ston A, tìteo tÔpoc ∃x(A ∧ B) eÐnai isodÔnamoc me ton A ∧ ∃xB , kai otÔpoc ∀x(A ∨ B) eÐnai isodÔnamoc me ton A ∨ ∀xB .

36. An h x den èqei eleÔjerh emf�nish ston A, tìteo tÔpoc ∀x(A→ B) eÐnai isodÔnamoc me ton A→ ∀xB , kaio tÔpoc ∃x(A→ B) eÐnai isodÔnamoc me ton A→ ∃xB .

37. ShmeÐwsh: An h x den èqei eleÔjerh emf�nish ston B , tìteo tÔpoc ∀x(A→ B) eÐnai isodÔnamoc me ton ∃xA→ B , kaio tÔpoc ∃x(A→ B) eÐnai isodÔnamoc me ton ∀xA→ B .Prosoq : O posodeÐkthc all�zei!

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Metonom�zontac tic desmeumènec metablhtèc

38. Ac upojèsoume ìti:

I το Q είναι ∀ ή ∃,I η y είναι μία μεταβλητή η οποία δεν έχει εμφάνιση στον A,I ο B προέρχεται από τον A αντικαθιστώντας όλες τιςελεύθερες εμφανίσεις της x στον A με την y .

Tìte o tÔpoc QxA eÐnai isodÔnamoc me ton QyB .

Pq o tÔpoc ∀x∃y αγορασε(x , y) eÐnai isodÔnamoc me ton∀z∃u αγορασε(z , u).

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IsodunamÐec/egkurìthta me isìthta

39. O tÔpoc t=t eÐnai ègkuroc gia k�je ìro t.

40. Gia opoiod pote ìrouc t, uo tÔpoc t=u eÐnai isodÔnamoc me ton u=t.

41. (Arq tou Leibniz) An A eÐnai ènac tÔpoc ston opoÐo h xèqei eleÔjerh emf�nish, h y den emfanÐzetai kajìlou stonA, kai o B proèrqetai apì ton A antikajist¸ntac mÐa perissìterec eleÔjerec emfanÐseic thc x apì thn y , tìte otÔpoc

x=y → (A↔ B)

eÐnai ègkuroc.Par�deigma: o tÔpoc x=y → (∀zR(x , z)↔ ∀zR(y , z)) eÐnaiègkuroc.

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Par�deigma

Ac deÐxoume ìti an h x den eÐnai eleÔjerh metablht tou A tìteo tÔpoc ∀x(∃x¬B → ¬A) eÐnai isodÔnamoc me ton ∀x(A→ B).Oi parak�tw tÔpoi eÐnai isodÔnamoi:

I ∀x(∃x¬B → ¬A)

I ∃x¬B → ¬A (isodunamÐa 34, kaj¸c h x den eÐnai eleÔjerhmetablht tou ∃x¬B → ¬A)

I ¬∀xB → ¬A (isodunamÐa 30)

I A→ ∀xB (par�deigma sel. 61)

I ∀x(A→ B) (isodunamÐa 36, kaj¸c h x den eÐnai eleÔjerhmetablht tou A)

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TÔpoi pou DEN eÐnai logik� isodÔnamoi

Gia orismènouc tÔpouc A,B , oi parak�tw tÔpoi DEN eÐnailogik� isodÔnamoi. (P�nta ìmwc isqÔeipr¸toc tÔpoc � deÔteroc tÔpoc.)

I ∀x(A→ B) kai ∀xA→ ∀xB .

I ∃x(A ∧ B) kai ∃xA ∧ ∃xB .

I ∀xA ∨ ∀xB kai ∀x(A ∨ B).

MporeÐte na breÐte mÐa dom M kai tÔpouc A,B ìpouM � deÔteroc tÔpoc, all� M 2 pr¸toc tÔpoc;

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13. Fusik sumperasmatologÐa gia thn Kathgorhmatik Logik

To sÔsthma kanìnwn fusik c sumperasmatologÐac (naturaldeduction (ND)) gia thn Kathgorhmatik Logik perilamb�neitouc kanìnec gia thn Protasiak Logik kai nèouc kanìnec giatouc posodeÐktec ∀,∃ kai thn isìthta =.

Prin kataskeÔasoume mÐa apìdeixh me to sÔsthma ND eÐnaiqr simo na thn anaptÔxoume pr¸ta me �meshepiqeirhmatologÐa kai èpeita na thn metatrèyoume se mÐaapìdeixh ND.

Oi apodeÐxeic prot�sewn thc Kathgorhmatik c Logik c me tosÔsthma ND mporeÐ na eÐnai pio dÔskolec apì tic apodeÐxeicprot�sewn thc Protasiak c Logik c, giatÐ t¸ra èqoume nèouckanìnec oi opoÐoi mac dÐnoun perissìterec epilogèc, kai mporeÐarqik� na k�noume l�joc epilogèc.

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Eisagwg ∃, ∃-introduction, ∃IGia na apodeÐxoume mÐa prìtash thc morf c ∃xA, prèpei naapodeÐxoume thn A(t) gia k�poio basikì ìro t thc epilog c mac.

...1 A(t) to apodeÐxame autì . . .2 ∃xA ∃I (1)

SÔmbash 13.1

Apì ed¸ kai sto ex c A(t) eÐnai h prìtash h opoÐa proèrqetai

apì ton tÔpo A(x) antikajist¸ntac ìlec tic eleÔjerec

emfanÐseic thc x me ton ìro t.

JumhjeÐte ìti ènac basikìc ìroc eÐnai ènac ìroc o opoÐoc denperilamb�nei metablhtèc. Sunep¸c perilamb�nei stajerèc kaisÔmbola sun�rthshc.An se k�poia dom M h prìtash A(t) eÐnai alhj c, tìte kai h∃xA eÐnai alhj c giatÐ up�rqei èna antikeÐmeno sth M (h tim tou t sth M) to opoÐo k�nei thn A alhj .

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Eisagwg ∃, ∃-introduction, ∃I

Epilègontac ton kat�llhlo ìro t mporeÐ na eÐnai dÔskolo. Gi'autì eÐnai qr simo na anaptÔssoume mÐa apìdeixh pr¸ta me�mesh epiqeirhmatologÐa kai èpeita na thn metatrèpoume semÐa apìdeixh ND.

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Apaloif ∃, ∃-elimination, ∃E'Estw A(x) ènac tÔpoc. An èqoume apodeÐxei thn prìtash ∃xA,tìte mporoÔme na apodeÐxoume mÐa prìtash B :

I upojètontac thn A(c), ìpou h c eÐnai mÐa nèa stajer� hopoÐa den qrhsimopoieÐtai sthn B sthn apìdeixh wct¸ra, kai

I apodeiknÔontac thn B apì thn parap�nw upìjesh.

Kat� thn di�rkeia thc apìdeixhc thc B mporoÔme naqrhsimopoi soume otid pote èqoume dh apodeÐxei.All� ìtan apodeÐxoume thn B den mporoÔme naqrhsimopoi soume argìtera kanèna mèroc thc apìdeixhc thc B ,sumperilambanomènhc kai thc c. Sunep¸c kleÐnoume thnapìdeixh thc B apì thn A(c) se èna koutÐ:

1 ∃xA to èqoume apodeÐxei autì

2 A(c) upìjeshh apìdeixh

3 B to apodeÐxame

4 B ∃E (1, 2, 3)213 / 232

Apaloif ∃, ∃-elimination, ∃E

H stajer� c lègetai stajer� Skolem.

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Aitiolìghsh kanìna ∃E

An h prìtash ∃xA eÐnai alhj c se mÐa dom M, tìte up�rqeièna antikeÐmeno α sto sÔmpan thc M tètoio ¸ste M � A(α).

To α mporeÐ na mhn onom�zetai apì k�poia stajer� sth M.MporoÔme ìmwc na prosjèsoume mÐa nèa stajer�, pq thnstajer� c, gia na to onom�sei. Epiplèon, prosjètoume thnplhroforÐa sth M ìti h c onom�zei to α.

H stajer� c prèpei na eÐnai kainoÔrgia giatÐ oi upìloipecstajerèc mporeÐ na mhn onom�zoun to α sth M.

'Ara h prìtash A(c) gia k�poia nèa stajer� c eÐnai san thnprìtash ∃xA. An mporoÔme na apodeÐxoume thn prìtash B apìthn upìjesh A(c), tìte eÐnai san na apodeiknÔoume thn B apìthn ∃xA.

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Par�deigma qr shc kanìnwn ∃I kai ∃EAc deÐxoume ìti ∃x(P(x) ∧ Q(x)) ` ∃xP(x) ∧ ∃xQ(x).

1 ∃x(P(x) ∧ Q(x)) dedomèno

2 P(c) ∧ Q(c) upìjesh3 P(c) ∧E (2)4 ∃xP(x) ∃I (3)5 Q(c) ∧E (2)6 ∃xQ(x) ∃I (5)7 ∃xP(x) ∧ ∃xQ(x) ∧I (4, 6)

8 ∃xP(x) ∧ ∃xQ(x) ∃E (1, 2, 7)

Sth fusik gl¸ssa: JewroÔme ìti ∃x(P(x) ∧ Q(x)). Tìteup�rqei èna α tètoio ¸ste P(α) ∧ Q(α). 'Ara P(α) kai Q(α).'Ara ∃xP(x) kai ∃xQ(x). Sunep¸c ∃xP(x) ∧ ∃xQ(x).

ShmeÐwsh: Mìno prot�seic up�rqoun se mÐa apìdeixh ND. Seautèc tic apodeÐxeic den prèpei na up�rqoun tÔpoi me eleÔjerecmetablhtèc.

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Eisagwg ∀, ∀-introduction, ∀IGia na eis�goume mÐa prìtash thc morf c ∀xA, gia k�poiotÔpo A(x), prèpei na eis�goume mÐa nèa stajer�, pq c, h opoÐaden èqei qrhsimopoihjeÐ wc t¸ra sthn apìdeixh, kai naapodeÐxoume thn prìtash A(c).Kat� thn di�rkeia thc apìdeixhc thc A(c) mporoÔme naqrhsimopoi soume otid pote èqoume dh apodeÐxei.All� ìtan apodeÐxoume thn A(c) den mporoÔme naqrhsimopoi soume argìtera thn c. Sunep¸c kleÐnoume thnapìdeixh thc A(c) se èna koutÐ:

1 c ∀I stajer�h apìdeixh

2 A(c) to apodeÐxame

3 ∀xA ∀I (1, 2)

Aut eÐnai h monadik for� sto sÔsthma ND ìpou gr�foumemÐa gramm (gramm 1) h opoÐa perilamb�nei èna ìro kai ìqimÐa prìtash. Epiplèon, eÐnai h monadik for� ìpou èna koutÐden arqÐzei me mÐa gramm h opoÐa onom�zetai “upìjesh”. 217 / 232

Aitiolìghsh Kanìna ∀I

Gia na deÐxoume ìti M � ∀xA, prèpei na deÐxoume ìti M � A(α)gia k�je antikeÐmeno α sto sÔmpan thc M.

Sunep¸c dialègoume èna tuqaÐo α, prosjètoume mÐa nèastajer� c h opoÐa onom�zei to α, kai apodeiknÔoume thnprìtash A(c). Epeid to α eÐnai tuqaÐo, èqoume deÐxei ìti ∀xA.

H stajer� c prèpei na eÐnai kainoÔrgia giatÐ oi stajerèc pouèqoun dh qrhsimopoihjeÐ mporeÐ na mhn onom�zoun to α.

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Apaloif ∀, ∀-elimination, ∀E'Estw ènac tÔpoc A(x). An èqoume katafèrei na gr�youme thnprìtash ∀xA, tìte mporoÔme na gr�youme kai thn A(t) giaopoiod pote basikì ìro t thc epilog c mac (poion ìro t jaepilèxoume;).

...1 ∀xA to apodeÐxame autì . . .2 A(t) ∀E (1)

Aitiolìghsh: an h ∀xA eÐnai alhj c se mÐa dom , tìte kai h A(t)ja eÐnai opwsd pote alhj c, gia opoiod pote basikì ìro t.

H epilog tou kat�llhlou t mporeÐ na eÐnai dÔskolh. Gi' autìeÐnai qr simo na anaptÔssoume mÐa apìdeixh pr¸ta me �meshepiqeirhmatologÐa kai èpeita na thn metatrèpoume se mÐaapìdeixh ND.

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Par�deigma qr shc kanìnwn ∀I kai ∀EAc deÐxoume ìti P → ∀xQ(x) ` ∀x(P → Q(x)).Se aut thn perÐptwsh to P eÐnai sÔmbolo kathgor matocbajmoÔ 0, dhlad eÐnai èna �tomo.

1 P → ∀xQ(x) dedomèno

2 c ∀I stajer�3 P upìjesh4 ∀xQ(x) → E (3, 1)5 Q(c) ∀E (4)6 P → Q(c) → I (3, 5)

7 ∀x(P → Q(x)) ∀I (2, 6)

Sth fusik gl¸ssa: JewroÔme ìti P → ∀xQ(x). Tìte gia k�jeantikeÐmeno α, an P tìte ∀xQ(x), �ra Q(α).'Ara gia k�je antikeÐmeno α, an P , tìte Q(α).Me �lla lìgia, gia k�je antikeÐmeno α, èqoume P → Q(α).'Ara ∀x(P → Q(x)).

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Par�deigma qr shc ìlwn twn kanìnwn twn posodeikt¸n

Ac deÐxoume ìti ∃x∀yG (x , y) ` ∀y∃xG (x , y).

1 ∃x∀yG (x , y) dedomèno

2 d ∀I stajer�3 ∀yG (c, y) upìjesh4 G (c, d) ∀E (3)5 ∃xG (x , d) ∃I (4)6 ∃xG (x , d) ∃E (1, 3, 5)

7 ∀y∃xG (x , y) ∀I (2, 6)

Sth fusik gl¸ssa: JewroÔme ìti ∃x∀yG (x , y). Tìte up�rqeik�poio antikeÐmeno α tètoio ¸ste ∀yG (α, y).'Ara gia opoiod pote antikeÐmeno β, èqoume ìti G (α, β), �rakai ∃xG (x , β).Epeid to β tan èna tuqaÐo antikeÐmeno, èqoume ∀y∃xG (x , y).

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L�joc qr sh kanìnwnPrèpei na gnwrÐzoume wc t¸ra ìti ∀x∃y <(x , y) 2 ∃y∀x <(x , y).Pq stouc fusikoÔc arijmoÔc h prìtash ∀x∃y <(x , y) eÐnaialhj c, en¸ h ∃y∀x <(x , y) eÐnai yeud c.'Ara h parak�tw apìdeixh prèpei na eÐnai LAJOS:

1 ∀x∃y <(x , y) dedomèno

2 c ∀I stajer�3 ∃y <(c, y) ∀E (1)4 <(c, d) upìjesh5 <(c, d)

√(4)

6 <(c, d) ∃E (3, 4, 5) ← LAJOS

7 ∀x <(x , d) ∀I (2, 6)8 ∃y∀x <(x , y) ∃I (7)

H stajer� Skolem d, thn opoÐa eis�game sth gramm 4, denprèpei na emfanÐzetai sto sumpèrasma (grammèc 5, 6). DeÐtethn selÐda 213. 'Ara h qr sh tou kanìna ∃E sth gramm 6 eÐnail�joc.

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AnakÔptwn Kanìnac: ∀→EAutìc o kanìnac eÐnai san ton kanìna PC: mei¸nei thn apìdeixhkat� mÐa gramm . EÐnai qr simoc kanìnac all� den eÐnaiaparaÐthtoc.'Estw ìti èqoume apodeÐxei tic prot�seic ∀x(A(x)→ B(x)) kaiA(t), gia k�poiouc tÔpouc A(x), B(x), kai k�poio basikì ìro t.GnwrÐzoume ìti apì aut� sumperaÐnoume ìti B(t):

1 ∀x(A(x)→ B(x)) to apodeÐxame autì. . .2 A(t) . . . kai autì3 A(t)→ B(t) ∀E (1)4 B(t) → E (2, 3)

O kanìnac ∀→E :

1 ∀x(A(x)→ B(x)) to apodeÐxame autì. . .2 A(t) . . . kai autì3 B(t) ∀→E (2, 1)

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Par�deigma qr shc kanìna ∀→E

Ac deÐxoume ìti∀x∀y(P(x , y)→ Q(x , y)), ∃xP(x , a) ` ∃yQ(y , a).

1 ∀x∀y(P(x , y)→ Q(x , y)) dedomèno2 ∃xP(x , a) dedomèno

3 P(c, a) upìjesh4 Q(c, a) ∀→E (3, 1)5 ∃yQ(y , a) ∃I (4)

6 ∃yQ(y , a) ∃E (2, 3, 5)

Qrhsimopoi same apeujeÐac ton kanìna ∀→E se dÔoposodeÐktec ∀. Autì eÐnai akìma pio qr simo.

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Kanìnec gia isìthta

I Anaklastik idiìthta isìthtac (reflexivity refl).Se opoiad pote gramm mÐac apìdeixhc mporeÐte naeis�gete mÐa prìtash thc morf c t=t, gia opoiod potebasikì L-ìro t kai gia opoiad pote gl¸ssa L.

...1 t=t refl

Aitiolìghsh: opoiad pote L-dom k�nei thn prìtash t=talhj (o t eÐnai basikìc ìroc).

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Kanìnec gia isìthta

I Antikat�stash Ðswn ìrwn (=substitution =sub).An A(x) eÐnai ènac tÔpoc, oi t, u eÐnai basikoÐ ìroi, èqeteapodeÐxei A(t) kai t=u u=t, tìte mporeÐte na gr�yeteA(u).

1 A(t) to apodeÐxame autì. . .

2...

3 t=u . . . kai autì4 A(u) =sub(1, 3)

Aitiolìghsh: an oi t, u eÐnai Ðsoi tìte mporoÔme naantikatast soume ton t me ton u wc proc th metablht xston tÔpo A.

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Par�deigma me isìthta

Ac deÐxoume ìti c=d ` d=c. Oi c, d eÐnai stajerèc.

1 c=d dedomèno2 d=d refl3 d=c =sub(2, 1)

Autì eÐnai suqn� qr simo, gi' autì mporoÔme na toqrhsimopoioÔme wc anakÔptonta kanìna:

1 c=d dedomèno2 d=c =sym(1)

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Ac deÐxoume ìti ` ∀x∃y(y=f (x)).

1 c ∀I stajer�2 f (c)=f (c) refl3 ∃y(y=f (c)) ∃I (2)

4 ∀x∃y(y=f (x)) ∀I (1, 3)

Sth fusik gl¸ssa: Gia opoiod pote antikeÐmeno α, èqoumef (α)=f (α).'Ara gia opoiod pote α, up�rqei k�ti Ðso me to f (α) (to Ðdio tof (α)).'Ara gia opoiod pote α, èqoume ∃y(y=f (α)).Epeid to α tan èna tuqaÐo antikeÐmeno, èqoume ∀x∃y(y=f (x)).

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Ac deÐxoume ìti ∃x∀y(P(y)→ y=x), ∀xP(f (x)) ` ∃x(x=f (x)).

1 ∃x∀y(P(y)→ y=x) dedomèno2 ∀xP(f (x)) dedomèno

3 ∀y(P(y)→ y=c) upìjesh4 P(f (c)) ∀E (2)5 f (c)=c ∀→E (4, 3)6 c=f (c) =sym(5)7 ∃x(x=f (x)) ∃I (6)

8 ∃x(x=f (x)) ∃E (1, 3, 7)

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DidaqjeÐsa Ôlh

Protasiak Logik

I Suntaktikì

I ShmasiologÐa

I Met�frash Logik c se fusik gl¸ssa kai antÐstrofa

I Epiqeir mata, egkurìthta

I †πίνακες αλήθειαςI άμεση επιχειρηματολογία

I ισοδυναμίες, †κανονικές μορφέςI σύστημα κανόνων φυσικής συμπερασματολογίας

I †σύστημα Beth αποδείξεων

Kathgorhmatik Logik

Ta Ðdia me thn Protasiak Logik , ektìc apì †.230 / 232

Merik� apì ta jèmata pou den kalÔyame

I apodeÐxeic orjìthtac kai plhrìthtac twn susthm�twnkanìnwn fusik c sumperasmatologÐac kai Beth apodeÐxewn

I je¸rhma sump�geiac

I je¸rhma tou Godel

I mh klassik� eÐdh Logik c, pq Tropik Logik (ModalLogic), Qronik Logik (Temporal Logic), Grammik Logik (Linear Logic)

I peperasmènec domèc kai upologistik poluplokìthta

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Logik se ereunhtikì epÐpedo

Sthn Plhroforik gÐnetai qr sh diafìrwn eid¸n Logik c, ìpwcProtasiak c, Kathgorhmatik c, Tropik c, Qronik c kaiDunamik c Logik c. Up�rqoun efarmogèc sta pedÐa teqnht cnohmosÔnhc, b�sewn dedomènwn, par�llhlwn kaikatanemhmènwn susthm�twn, polu-praktorik¸n susthm�twn,anapar�stashc gn¸shc, diatÔpwshc kai epal jeushcdiergasi¸n, klp. H Jewrhtik Plhroforik qrei�zetai thLogik .

H Logik apoteleÐ antikeÐmeno melèthc kai gia �llouc tomeÐc,ìpwc ta Majhmatik�, thn FilosofÐa, kai th GlwssologÐa.

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dom tou maj matoc · dom tou maj matoc i protasiak logik ( propositional logic). i kathgorhmatik...

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