# Nonmonotonic Logics and Semantics

### Daniel Lehmann

Hebrew University, Jerusalem, Israel

### Abstract

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 all the elements
of A hold.
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, where the notion of
"preferred" satisfies a set of natural properties.
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.

Eyal Amir
Last modified: Wed Sep 9 11:18:55 PDT 1998