Self-reference and the theory of concept

[J. B. Kostić ]

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Filozofija - Filozofija logike / Philosophy - Philosophy of logic

Ova doktorska disertacija bavi se samoreferencijom kao formalnim svojstvom intenzionalnog znacenja jezickih izraza, pre svega predikata. Formalna svojstva intenzionalnog znacenja predikata trebalo bi da budu predmet nove logicke teorije koju je zamislio i u kratkim crtama opisao jedan od najznacajnijih logicara i matematicara prošlog veka, ali i uopšte - Kurt Gedel (Kurt Gödel). Pošto je koristio termin pojam za svojstva ili relacije koje cine intenzionalno znacenje predikata, 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 se njima bavi jeste upravo njihova samoreferencija, tj. mogucnost primene pojma na samog sebe, kao i ucestvovanja pojma u graženju sopstvenog znacenja. Glavni zadatak ovog rada je bolje razumevanje uloge ta dva oblika samoreferencije u zasnivanju i razvoju teorije pojmova. Videcemo da ona može biti trostruka. Pre svega, samoreferencija u odnosu na odreženi objekat moguca je upravo zahvaljujuci intenzionalnom znacenju tog objekta. Ona tako ukazuje na formalne karakteristike intenzionalnog znacenja na kojima se zasniva njena mogucnost i koje ga suštinski razlikuju od ekstenzionalnog znacenja. Osim toga, samoreferencija može imati i znacajnu ulogu u definisanju pojmova i dokazivanju njihovih svojstava. Konacno, odrežene instance pojmovne samoreferencije 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 koje cemo koristiti u disertaciji. Takože cemo u kratkim crtama opisati istoriju samoreferencije i uticaj koji je imala na razvoj matematike. Taj uticaj je u najvecoj meri ostvaren preko njene uloge u formulisanju paradoksa unutar matematickih teorija. Potreba da se ti paradoksi izbegnu dovela je do promene u razumevanju osnovnih matematickih pojmova, kojom je samoreferencija stavljena izvan polja matematickih istraživanja. U drugom delu rada srešcemo se sa drugacijim gledanjem na samoreferenciju i njene potencijalno paradoksalne posledice. U tom delu bavimo se teorijama u kojima je samoreferencija postala standardno metodološko sredstvo, a njene paradoksalne posledice osnova za dolaženje do važnih, tipicno negativnih, rezultata. Videcemo da razliciti oblici samoreferencije koji se u tim oblastima javljaju imaju zajednicko jezgro koje objašnjava njihovu ulogu u dolaženju do tih rezultata ali i u formulisanju paradoksa. Treci deo rada predstavlja pokušaj da se približimo razumevanju odnosa samoreferencije i intenzionalnog znacenja objekta u odnosu na koji se ona javlja. Time bi trebalo da se približimo i razumevanju formalnih osobina pojmova. U ovom delu rada takože cemo detaljnije opisati paradokse kojima pojmovna samoreferencija vodi. Branicemo rešenje tih paradoksa koje se zasniva 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 i njenu vezu sa teorijom pojmova

This doctoral dissertation is a study of self-reference as a formal property that characterizes the intensional meaning of some linguistic expressions, such as predicates. Formal properties of intensional meanings of predicates are supposed to be the subject matter of a new logical theory that is envisioned and briefly described by one of the most important logicians and mathematicians of the last century, and in general - Kurt Gödel. Since he used the term concept to denote the properties or relations that form the intensional meaning of predicates, Gödel termed 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 a formal theory that studies them, is just their self-reference, i.e. the possibility of an application of a concept to itself, or of its containment in its own meaning. Better understanding of the role that both kinds of self-reference have in establishing and developing the theory of concepts will be the main goal of this work. We will see that this role can be threefold. First of all, the possibility of self-reference with respect to some object depends on its intensional meaning. It can thus expose some formal properties of intensional meaning on which it rests and which crucially distinguish it from the extensional meaning. Besides that, the self-reference can have a significant methodological use in the future theory, where it can participate in the definitions of concepts and in the proofs of their properties. Finally, some instances of self-reference lead to paradoxes, that can make the theory in which they are formulated inconsistent. An appropriate solution 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 be using throughout the work. We also briefly describe the history of self-reference and the influence it had on the development of mathematics. This influence is accomplished mainly through the role that self-reference had in the formulation of paradoxes inside mathematical theories. The need to avoid these paradoxes has led to a change in the way some basic mathematical notions are understood, which put the considerations of self-reference outside the field of mathematical investigation. 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 apparently paradoxical 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 in these areas share a common core which explains their role in achieving these results, but also in deriving different paradoxes. The third part is an attempt at approaching the question of the connection between the self-reference with respect to some entity and its intensional understanding. This should bring us closer 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 the remarks Gödel made on the subject. We argue in favor of one particular solution accomplished by restricting the meaningful applicability of concepts, and redefining the notion of a complement of some concept. We will try to show that, if developed inside intuitionistic logic, this solution makes for a solid basis of the future theory. v In the final part we review the results of the investigation and describe Gödel’s philosophical position which explains his interest in the theory of concepts.

Kurt Gedel, samoreferencija, intenzionalno znacenje, pojam, teorija pojmova

Kurt Gödel, self-reference, intensional meaning, concept, theory of concepts

161.1(043.3) 81:1(043.3)

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