136 research outputs found
Precise Propagation of Upper and Lower Probability Bounds in System P
In this paper we consider the inference rules of System P in the framework of
coherent imprecise probabilistic assessments. Exploiting our algorithms, we
propagate the lower and upper probability bounds associated with the
conditional assertions of a given knowledge base, automatically obtaining the
precise probability bounds for the derived conclusions of the inference rules.
This allows a more flexible and realistic use of System P in default reasoning
and provides an exact illustration of the degradation of the inference rules
when interpreted in probabilistic terms. We also examine the disjunctive Weak
Rational Monotony of System P+ proposed by Adams in his extended probability
logic.Comment: 8 pages -8th Intl. Workshop on Non-Monotonic Reasoning NMR'2000,
April 9-11, Breckenridge, Colorad
Probabilistic entailment in the setting of coherence: The role of quasi conjunction and inclusion relation
In this paper, by adopting a coherence-based probabilistic approach to
default reasoning, we focus the study on the logical operation of quasi
conjunction and the Goodman-Nguyen inclusion relation for conditional events.
We recall that quasi conjunction is a basic notion for defining consistency of
conditional knowledge bases. By deepening some results given in a previous
paper we show that, given any finite family of conditional events F and any
nonempty subset S of F, the family F p-entails the quasi conjunction C(S);
then, given any conditional event E|H, we analyze the equivalence between
p-entailment of E|H from F and p-entailment of E|H from C(S), where S is some
nonempty subset of F. We also illustrate some alternative theorems related with
p-consistency and p-entailment. Finally, we deepen the study of the connections
between the notions of p-entailment and inclusion relation by introducing for a
pair (F,E|H) the (possibly empty) class K of the subsets S of F such that C(S)
implies E|H. We show that the class K satisfies many properties; in particular
K is additive and has a greatest element which can be determined by applying a
suitable algorithm
Subjective probability, trivalent logics and compound conditionals
In this work we first illustrate the subjective theory of de Finetti. We
recall the notion of coherence for both the betting scheme and the penalty
criterion, by considering the unconditional and conditional cases. We show the
equivalence of the two criteria by giving the geometrical interpretation of
coherence. We also consider the notion of coherence based on proper scoring
rules. We discuss conditional events in the trivalent logic of de Finetti and
the numerical representation of truth-values. We check the validity of selected
basic logical and probabilistic properties for some trivalent logics:
Kleene-Lukasiewicz-Heyting-de Finetti; Lukasiewicz; Bochvar-Kleene; Sobocinski.
We verify that none of these logics satisfies all the properties. Then, we
consider our approach to conjunction and disjunction of conditional events in
the setting of conditional random quantities. We verify that all the basic
logical and probabilistic properties (included the Fr\'{e}chet-Hoeffding
bounds) are preserved in our approach. We also recall the characterization of
p-consistency and p-entailment by our notion of conjunction
On compound and iterated conditionals
We illustrate the notions of compound and iterated conditionals introduced, in recent papers, as suitable conditional random quantities, in the framework of coherence. We motivate our definitions by examining some concrete examples. Our logical operations among conditional events satisfy the basic probabilistic properties valid for unconditional events. We show that some, intuitively acceptable, compound sentences on conditionals can be analyzed in a rigorous way in terms of suitable iterated conditionals. We discuss the Import-Export principle, which is not valid in our approach, by also examining the inference from a material conditional to the associated conditional event. Then, we illustrate the characterization, in terms of iterated conditionals, of some well known p-valid and non p-valid inference rules
Quasi Conjunction, Quasi Disjunction, T-norms and T-conorms: Probabilistic Aspects
We make a probabilistic analysis related to some inference rules which play
an important role in nonmonotonic reasoning. In a coherence-based setting, we
study the extensions of a probability assessment defined on conditional
events to their quasi conjunction, and by exploiting duality, to their quasi
disjunction. The lower and upper bounds coincide with some well known t-norms
and t-conorms: minimum, product, Lukasiewicz, and Hamacher t-norms and their
dual t-conorms. On this basis we obtain Quasi And and Quasi Or rules. These are
rules for which any finite family of conditional events p-entails the
associated quasi conjunction and quasi disjunction. We examine some cases of
logical dependencies, and we study the relations among coherence, inclusion for
conditional events, and p-entailment. We also consider the Or rule, where quasi
conjunction and quasi disjunction of premises coincide with the conclusion. We
analyze further aspects of quasi conjunction and quasi disjunction, by
computing probabilistic bounds on premises from bounds on conclusions. Finally,
we consider biconditional events, and we introduce the notion of an
-conditional event. Then we give a probabilistic interpretation for a
generalized Loop rule. In an appendix we provide explicit expressions for the
Hamacher t-norm and t-conorm in the unitary hypercube
Probabilistic entailment and iterated conditionals
In this paper we exploit the notions of conjoined and iterated conditionals,
which are defined in the setting of coherence by means of suitable conditional
random quantities with values in the interval . We examine the iterated
conditional , by showing that p-entails if and only if
. Then, we show that a p-consistent family
p-entails a conditional event if
and only if , or for some nonempty
subset of , where is the quasi
conjunction of the conditional events in . Then, we examine the
inference rules , , , and of System~P
and other well known inference rules ( , ,
). We also show that , where
is the conjunction of the conditional events in
. We characterize p-entailment by showing that
p-entails if and only if .
Finally, we examine \emph{Denial of the antecedent} and \emph{Affirmation of
the consequent}, where the p-entailment of from does
not hold, by showing that $(E_3|H_3)|\mathcal{C}(\mathcal{F})\neq1.
Conditional Random Quantities and Compounds of Conditionals
In this paper we consider finite conditional random quantities and
conditional previsions assessments in the setting of coherence. We use a
suitable representation for conditional random quantities; in particular the
indicator of a conditional event is looked at as a three-valued quantity
with values 1, or 0, or , where is the probability of . We
introduce a notion of iterated conditional random quantity of the form
defined as a suitable conditional random quantity, which coincides
with when . Based on a recent paper by S. Kaufmann, we
introduce a notion of conjunction of two conditional events and then we analyze
it in the setting of coherence. We give a representation of the conjoined
conditional and we show that this new object is a conditional random quantity.
We examine some cases of logical dependencies, by also showing that the
conjunction may be a conditional event; moreover, we introduce the negation of
the conjunction and by De Morgan's Law the operation of disjunction. Finally,
we give the lower and upper bounds for the conjunction and the disjunction of
two conditional events, by showing that the usual probabilistic properties
continue to hold
On conditional probabilities and their canonical extensions to Boolean algebras of compound conditionals
In this paper we investigate canonical extensions of conditional probabilities to Boolean algebras of conditionals. Before entering into the probabilistic setting, we first prove that the lattice order relation of every Boolean algebra of conditionals can be characterized in terms of the well-known order relation given by Goodman and Nguyen. Then, as an interesting methodological tool, we show that canonical extensions behave well with respect to conditional subalgebras. As a consequence, we prove that a canonical extension and its original conditional probability agree on basic conditionals. Moreover, we verify that the probability of conjunctions and disjunctions of conditionals in a recently introduced framework of Boolean algebras of conditionals are in full agreement with the corresponding operations of conditionals as defined in the approach developed by two of the authors to conditionals as three-valued objects, with betting-based semantics, and specified as suitable random quantities. Finally we discuss relations of our approach with nonmonotonic reasoning based on an entailment relation among conditionals
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