Verovatnosni računi sekvenata i klasifikacija neklasičnih logika zasnovana na entropiji

Abstract

Posle kratkog uvodnog pregleda, rad je podeljen na dva dela. Prvi deo se bavi prisustvom verovatnoće u logici (v. [16], [17], [18], [19], [22], [23] i [24]), a drugi je posvećen primeni entropije u klasifikaciji polivalentnih logika (v. [14], [15], [20], [21] i [25]). Osnovna ideja koja dominira prvim delom rada jeste obogaćivanje Gentzen- ovog računa sekvenata klasične logike iskaza jednim verovatnosnim operatorom definisanim nad sekventima Γ ⊢ Δ kako bi se izrazila činjenica da "verovatnoća istinitosti sekventa Γ ⊢ Δ pripada intervalu [a, b] ⊂ [0, 1]". Uvodimo sledeće sisteme: LKprob, LKprob(ε), NKprob i LKfuzz. Osnovna forma sekevnata u sistemu LKprob je Γ ⊢ b, a Δ sa gore datim značenjem. Sistem LKprob(ε) se fokusira na Suppes-ove forme Γ ⊢ n Δ koje omogućavaju formalizaciju rečenice "verovatnoća istinitosti sekventa Γ ⊢ Δ pripada intervalu [1 - nε,1] ⊆ [0, 1]", za neki n ∈ N. Sistem NKprob predstavlja prirodno-dedukcijski analogon računu sekvenata LKprob. Modeli zasnovani na Carnap–Popper–Leblance-ovoj semantici definisani su za svaki od ovih računa uz odgovarajuće rezultate saglasnosti i potpunosti. Konačno, račun LKfuzz je uveden sa opdžtijom formom sekvenata Γ ⊢ h Δ, gde je h element konačne mreže, sa ciljem da se opiše jedno rasplinuće računa LK. Značenje sekventa Γ ⊢ h Δ je "da je h mera rasplinuća sekventa Γ ⊢ Δ". Modeli za LKfuzz su dati sa rezultatima saglasnosti i potpunosti, a dokaz-teoretski tretman računa LKfuzz uključuje i teoremu eliminacije sečenja. Drugi deo rada istražuje činjenicu da svaki logički sistem povezan sa particijom indukovanom odgovarajućom Lindenbaum–Tarski–jevom algebrom omogućava definisanje njegove entropije. Definišemo entropiju logičkog sistema baziranoj na geometrijskoj raspodeli mera nad odgovarajućom particijom skupa formula. Ova definicija omogućava klasifikaciju polivalentnih iskaznih logika u odnosu na njihovu entropiju. Asimptotske aproksimacije entropije nekih beskonačnovalentnih logika su takođe date. Razmotreni primeri uključuju Lukasiewicz-evu, Kleene-jevu i Priest-ovu trovalentnu logiku, Belnap-ovu četvorovalentnu logiku, Gödel-ove i McKay-eve m-valentne logike, i Heyting-ovu i Dummett-ovu beskonačnovalentnu logiku.

After a brief introductory survey, this work is divided into two parts. The first part deals with presence of probability in logic (v. [16], [17], [18], [19], [22], [23] and [24]), and the second one is devoted to the application of entropy in classification of many–valued logics (v. [14], [15], [20], [21] and [25]). The basic idea, dominant in the first part of the work, is to enrich the Gentzen’s sequent calculus LK for propositional classical logic by a kind of probability operator defined over the sequents Γ ⊢ Δ in order to express the fact that ”the truthfulness probability of Γ ⊢ Δ belongs to the interval [a, b] ⊂ [0, 1]”. We introduce the following four systems: LKprob, LKprob(ε), NKprob and LKfuzz. The basic form of sequents in LKprob is Γ ⊢ b, a Δ with the above given intended meaning. The system LKprob(ε) is focused on the Suppes-forms Γ ⊢ n Δ enabling to formalize the sentence ”the truthfulness probability of Γ ⊢ Δ belongs to the interval [1 - nε,1] ⊆ [0, 1]”, for some n ∈ N. The system NKprob presents a natural deduction counterpart of the sequent calculus LKprob. The models founded on Carnap– Popper–Leblance probability semantics are defined for each of these calculi and accompanied by the corresponding soundness and completeness results. Finally, the calculus LKfuzz is introduced with a more general form of the sequents Γ ⊢ х Δ, where x is an element of a finite lattice, with the aim to describe a fuzzification of LK. The meaning of Γ ⊢ х Δ is that ”x is the fuzziness measure of Γ ⊢ Δ". Models for LKfuzz are given with soundness and completeness results, and a proof– theoretical treatment of LKfuzz includes the cut–elimination theorem. The second part of the work explores the fact that each logical system associated with the partition induced by the corresponding Lindenbaum–Tarski algebra makes it possible to define its entropy. We define the entropy of a logical system based on geometric distribution of measures over matching partition of set of formulae. This definition enables the classification of many–valued propositional logics according to their entropies. Asymptotic entropy approximations for some infinite–valued logics are proposed as well. The considered examples include Lukasiewicz’s, Kleene’s and Priest’s three–valued logics, Belnap’s four–valued logic, Gödel’s and McKay’s m– valued logics, and Heyting’s and Dummett’s infinite–valued logics.