Self-reference and the theory of concept

Ova doktorska disertacija bavi se samoreferencijom kao formalnim svojstvom intenzionalnogznacenja jezickih izraza, pre svega predikata. Formalna svojstva intenzionalnog znacenja predikatatrebalo bi da budu predmet nove logicke teorije koju je zamislio i u kratkim crtama opisaojedan od najznacajnijih logicara i matematicara prošlog veka, ali i uopšte - Kurt Gedel (KurtGödel). Pošto je koristio termin pojam za svojstva ili relacije koje cine intenzionalno znacenjepredikata, Gedel je tu teoriju nazvao teorija pojmova. Brojne, ali kratke i cesto zagonetne,Gedelove napomene svedoce o njegovom ocekivanju da ta teorija postane za logiku centralna.Osobina pojmova koja dobija poseban znacaj u kontekstu zasnivanja formalne teorije koja senjima bavi jeste upravo njihova samoreferencija, tj. mogucnost primene pojma na samog sebe,kao i ucestvovanja pojma u gra enju sopstvenog znacenja. Glavni zadatak ovog rada je boljerazumevanje uloge ta dva oblika samoreferencije u zasnivanju i razvoju teorije pojmova. Videcemoda ona može biti trostruka. Pre svega, samoreferencija u odnosu na odre eni objekat mogucaje upravo zahvaljujuci intenzionalnom znacenju tog objekta. Ona tako ukazuje na formalnekarakteristike intenzionalnog znacenja na kojima se zasniva njena mogucnost i koje ga suštinskirazlikuju od ekstenzionalnog znacenja. Osim toga, samoreferencija može imati i znacajnu uloguu definisanju pojmova i dokazivanju njihovih svojstava. Konacno, odre ene instance pojmovnesamoreferencije vode paradoksima koji mogu ciniti teoriju u kojoj su formulisani protivrecnom.Rešavanje tih paradoksa bi trebalo da omoguci postavljanje osnova teorije pojmova.Ova disertacija podeljena je u cetiri dela. U uvodnom delu razjasnicemo osnovne termine kojecemo koristiti u disertaciji. Tako e cemo u kratkim crtama opisati istoriju samoreferencije iuticaj koji je imala na razvoj matematike. Taj uticaj je u najvecoj meri ostvaren preko njeneuloge u formulisanju paradoksa unutar matematickih teorija. Potreba da se ti paradoksi izbegnudovela je do promene u razumevanju osnovnih matematickih pojmova, kojom je samoreferencijastavljena izvan polja matematickih istraživanja.U drugom delu rada srešcemo se sa drugacijim gledanjem na samoreferenciju i njene potencijalnoparadoksalne posledice. U tom delu bavimo se teorijama u kojima je samoreferencijapostala standardno metodološko sredstvo, a njene paradoksalne posledice osnova za dolaženjedo važnih, tipicno negativnih, rezultata. Videcemo da razliciti oblici samoreferencije koji se utim oblastima javljaju imaju zajednicko jezgro koje objašnjava njihovu ulogu u dolaženju do tihrezultata ali i u formulisanju paradoksa.Treci deo rada predstavlja pokušaj da se približimo razumevanju odnosa samoreferencije iintenzionalnog znacenja objekta u odnosu na koji se ona javlja. Time bi trebalo da se približimoi razumevanju formalnih osobina pojmova. U ovom delu rada tako e cemo detaljnije opisatiparadokse kojima pojmovna samoreferencija vodi. Branicemo rešenje tih paradoksa koje sezasniva na ogranicavanju smislene primenljivosti pojma i redefinisanju komplementa pojma.Pokušacemo da pokažemo da to rešenje, ako je formulisano u okviru intuicionisticke logike,predstavlja solidnu osnovu teorije pojmova. Zakljucni deo rada sumira rezultate istraživanja i objašnjava Gedelovu filozofsku poziciju injenu vezu sa teorijom pojmovaThis doctoral dissertation is a study of self-reference as a formal property that characterizesthe intensional meaning of some linguistic expressions, such as predicates. Formal propertiesof intensional meanings of predicates are supposed to be the subject matter of a new logicaltheory that is envisioned and briefly described by one of the most important logicians andmathematicians of the last century, and in general - Kurt Gödel. Since he used the term conceptto denote the properties or relations that form the intensional meaning of predicates, Gödeltermed such a theory the theory of concepts. His numerous, but short and often enigmatic,remarks show that he expected this theory to become central to logic.A property of concepts that assumes a special importance in the context of establishing aformal theory that studies them, is just their self-reference, i.e. the possibility of an applicationof a concept to itself, or of its containment in its own meaning. Better understanding of therole that both kinds of self-reference have in establishing and developing the theory of conceptswill be the main goal of this work. We will see that this role can be threefold. First of all, thepossibility of self-reference with respect to some object depends on its intensional meaning. Itcan thus expose some formal properties of intensional meaning on which it rests and whichcrucially distinguish it from the extensional meaning. Besides that, the self-reference can have asignificant methodological use in the future theory, where it can participate in the definitions ofconcepts and in the proofs of their properties. Finally, some instances of self-reference lead toparadoxes, that can make the theory in which they are formulated inconsistent. An appropriatesolution to these paradoxes should set the ground for the theory of concepts.This dissertation is divided into four parts. In the first part we explain the terms we will beusing throughout the work. We also briefly describe the history of self-reference and the influenceit had on the development of mathematics. This influence is accomplished mainly through therole that self-reference had in the formulation of paradoxes inside mathematical theories. Theneed to avoid these paradoxes has led to a change in the way some basic mathematical notionsare understood, which put the considerations of self-reference outside the field of mathematicalinvestigation.In the second part we see quite a different attitude towards the self-reference and its consequences,that became a standardly used tool in particular areas of logic. The apparentlyparadoxical consequences of self-reference are inside these theories transformed into some important,mostly negative, results. We will see that different forms of self-reference that appear inthese areas share a common core which explains their role in achieving these results, but also inderiving different paradoxes.The third part is an attempt at approaching the question of the connection between theself-reference with respect to some entity and its intensional understanding. This should bring uscloser to understanding the formal properties of concepts that embody this intensional meaning.This part also tries to trace an adequate solution to the intensional paradoxes, guided by theremarks Gödel made on the subject. We argue in favor of one particular solution accomplished byrestricting the meaningful applicability of concepts, and redefining the notion of a complementof some concept. We will try to show that, if developed inside intuitionistic logic, this solutionmakes for a solid basis of the future theory.vIn the final part we review the results of the investigation and describe Gödel’s philosophicalposition which explains his interest in the theory of concepts