3 research outputs found
Nonmonotonic Logics and Semantics
Tarski gave a general semantics for deductive reasoning: a formula a may be
deduced from a set A of formulas iff a holds in all models in which each of the
elements of A holds. A more liberal semantics has been considered: a formula a
may be deduced from a set A of formulas iff a holds in all of the "preferred"
models in which all the elements of A hold. Shoham proposed that the notion of
"preferred" models be defined by a partial ordering on the models of the
underlying language. A more general semantics is described in this paper, based
on a set of natural properties of choice functions. This semantics is here
shown to be equivalent to a semantics based on comparing the relative
"importance" of sets of models, by what amounts to a qualitative probability
measure. The consequence operations defined by the equivalent semantics are
then characterized by a weakening of Tarski's properties in which the
monotonicity requirement is replaced by three weaker conditions. Classical
propositional connectives are characterized by natural introduction-elimination
rules in a nonmonotonic setting. Even in the nonmonotonic setting, one obtains
classical propositional logic, thus showing that monotonicity is not required
to justify classical propositional connectives.Comment: 28 pages. Misprint corrected 15/04/0
Nonmonotonic inference operations
A. Tarski proposed the study of infinitary consequence operations as the
central topic of mathematical logic. He considered monotonicity to be a
property of all such operations. In this paper, we weaken the monotonicity
requirement and consider more general operations, inference operations. These
operations describe the nonmonotonic logics both humans and machines seem to be
using when infering defeasible information from incomplete knowledge. We single
out a number of interesting families of inference operations. This study of
infinitary inference operations is inspired by the results of Kraus, Lehmann
and Magidor on finitary nonmonotonic operations, but this paper is
self-contained.Comment: 54 pages. A short version appeared in Studia Logica, Vol. 53 no. 2
(1994) pp. 161-20
Deductive inference operations
A. Tarski [21] proposed the study of infinitary consequence operations as the central topic of mathematical logic. He considered monotonicity to be a property of all such operations. In this paper, we weaken the monotonicity requirement and consider more general operations, inference operations. These operations describe the nonmonotonic logics both humans and machines seem to be using when infering defeasible information from incomplete knowledge. We single out a number of interesting families of inference operations. This study of infinitary inference operations is inspired by the results of [11] on finitary nonmonotonic operations, but this paper is self-contained.