6 research outputs found
Conjunction, disjunction and iterated conditioning of conditional events
Starting from 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, whose set of possible values normally contains the probabilities assessed for the two conditional events. We examine some cases of logical dependencies, where the conjunction is a conditional event; moreover, we give the lower and upper bounds on the conjunction. We also examine an apparent paradox concerning stochastic independence which can actually be explained in terms of uncorrelation. We briefly introduce the notions of disjunction and iterated conditioning and we show that the usual probabilistic properties still hold
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
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
Probabilistic inferences from conjoined to iterated conditionals
There is wide support in logic, philosophy, and psychology for the hypothesis that the probability of the indicative conditional of natural language, , is the conditional probability of given , . We identify a conditional which is such that with de Finetti's conditional event, . An objection to making this identification in the past was that it appeared unclear how to form compounds and iterations of conditional events. In this paper, we illustrate how to overcome this objection with a probabilistic analysis, based on coherence, of these compounds and iterations. We interpret the compounds and iterations as conditional random quantities which, given some logical dependencies, may reduce to conditional events.
We show how the inference to from and can be extended to compounds and iterations of both conditional events and biconditional events.
Moreover, we determine the respective uncertainty propagation rules.
Finally, we make some comments on extending our analysis to counterfactuals
Algebraic aspects and coherence conditions for conjoined and disjoined conditionals
We deepen the study of conjoined and disjoined conditional events in the setting of coherence. These objects, differently from other approaches, are defined in the framework of conditional random quantities. We show that some well known properties, valid in the case of unconditional events, still hold in our approach to logical operations among conditional events. In particular we prove a decomposition formula and a related additive property. Then, we introduce the set of conditional constituents generated by conditional events and we show that they satisfy the basic properties valid in the case of unconditional events. We obtain a generalized inclusion-exclusion formula, which can be interpreted by introducing a suitable distributive property. Moreover, under logical independence of basic unconditional events, we give two necessary and sufficient coherence conditions.
The first condition gives a geometrical characterization for
the coherence of prevision assessments on a family F constituted by n conditional events and all possible conjunctions among them.
The second condition characterizes the coherence of prevision assessments defined on , where is the set of conditional constituents associated with the conditional events in .
Then, we give some further theoretical results and we examine some examples and counterexamples.
Finally, we make a comparison with other approaches and we illustrate some theoretical aspects and applications