Formal axiomatic theories on the base of three-valued logic
Zapiski Nauchnykh Seminarov POMI, Computational complexity theory. Part VIII, Tome 304 (2003), pp. 19-74
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Formal axiomatic theories on the base of J. Lukasiewicz's three-valued logic are considered. Main notions connected with these theories are introduced, for example, the notion of a Luk-model (i.e., model of a theory in terms of J. Lukasiewicz's logic), of a Luk-consistent theory, Luk-complete theory. Logical calculi describing such theories are defined; analogues of the classical theorems on compactness and completeness are proved. Arithmetical theories based on J. Lukasewicz's logic and on its constructive (intuitionistic) variant are investigated; the theorem on effective Luk-incompleteness for a large class of arithmetical systems is proved which is a three-valued analogue of K. Goedel's famous theorem on the incompleteness of formal theories. Three-valued analogues of M. Presburger's arithmetical system are defined; it is proved that they are Luk-complete but not complete in the classical sense.
@article{ZNSL_2003_304_a3,
author = {I. D. Zaslavsky},
title = {Formal axiomatic theories on the base of three-valued logic},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {19--74},
publisher = {mathdoc},
volume = {304},
year = {2003},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/ZNSL_2003_304_a3/}
}
I. D. Zaslavsky. Formal axiomatic theories on the base of three-valued logic. Zapiski Nauchnykh Seminarov POMI, Computational complexity theory. Part VIII, Tome 304 (2003), pp. 19-74. http://geodesic.mathdoc.fr/item/ZNSL_2003_304_a3/