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 $E|H$ is looked at as a three-valued quantity with values 1, or 0, or $p$, where $p$ is the probability of $E|H$. We introduce a notion of iterated conditional random quantity of the form $(X|H)|K$ defined as a suitable conditional random quantity, which coincides with $X|HK$ when $H \subseteq K$. 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

Connexive Logic, Probabilistic Default Reasoning, and Compound Conditionals

We present two approaches to investigate the validity of connexive principles and related formulas and properties within coherence-based probability logic. Connexive logic emerged from the intuition that conditionals of the form if not-A, then A, should not hold, since the conditional’s antecedent not-A contradicts its consequent A. Our approaches cover this intuition by observing that the only coherent probability assessment on the conditional event A | not-A is p(A | not-A) = 0. In the first approach we investigate connexive principles within coherence-based probabilistic default reasoning, by interpreting defaults and negated defaults in terms of suitable probabilistic constraints on conditional events. In the second approach we study connexivity within the coherence framework of compound conditionals, by interpreting connexive principles in terms of suitable conditional random quantities. After developing notions of validity in each approach, we analyze the following connexive principles: Aristotle’s theses, Aristotle’s Second Thesis, Abelard’s First Principle, and Boethius’ theses. We also deepen and generalize some principles and investigate further properties related to connexive logic (like non-symmetry). Both approaches satisfy minimal requirements for a connexive logic. Finally, we compare both approaches conceptually

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

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

A probabilistic analysis of selected notions of iterated conditioning under coherence

It is well know that basic conditionals satisfy some desirable basic logical and probabilistic properties, such as the compound probability theorem, but checking the validity of these becomes trickier when we switch to compound and iterated conditionals. We consider de Finetti's notion of conditional as a three-valued object and as a conditional random quantity in the betting framework. We recall the notions of conjunction and disjunction among conditionals in selected trivalent logics. First, in the framework of specific three-valued logics we analyze the notions of iterated conditioning introduced by Cooper-Calabrese, de Finetti and Farrell, respectively. We show that the compound probability theorem and other basic properties are not preserved by these objects, by also computing some probability propagation rules. Then, for each trivalent logic we introduce an iterated conditional as a suitable random quantity which satisfies the compound prevision theorem and some of the desirable properties. We also check the validity of two generalized versions of Bayes' Rule for iterated conditionals. We study the p-validity of generalized versions of Modus Ponens and two-premise centering for iterated conditionals. Finally, we observe that all the basic properties are satisfied only by the iterated conditional mainly developed in recent papers by Gilio and Sanfilippo in the setting of conditional random quantities

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 $[0,1]$. We examine the iterated conditional $(B|K)|(A|H)$, by showing that $A|H$ p-entails $B|K$ if and only if $(B|K)|(A|H) = 1$. Then, we show that a p-consistent family $\mathcal{F}=\{E_1|H_1,E_2|H_2\}$ p-entails a conditional event $E_3|H_3$ if and only if $E_3|H_3=1$, or $(E_3|H_3)|QC(\mathcal{S})=1$ for some nonempty subset $\mathcal{S}$ of $\mathcal{F}$, where $QC(\mathcal{S})$ is the quasi conjunction of the conditional events in $\mathcal{S}$. Then, we examine the inference rules $And$, $Cut$, $Cautious$ $Monotonicity$, and $Or$ of System~P and other well known inference rules ($Modus$ $Ponens$, $Modus$ $Tollens$, $Bayes$). We also show that $QC(\mathcal{F})|\mathcal{C}(\mathcal{F})=1$, where $\mathcal{C}(\mathcal{F})$ is the conjunction of the conditional events in $\mathcal{F}$. We characterize p-entailment by showing that $\mathcal{F}$ p-entails $E_3|H_3$ if and only if $(E_3|H_3)|\mathcal{C}(\mathcal{F})=1$. Finally, we examine \emph{Denial of the antecedent} and \emph{Affirmation of the consequent}, where the p-entailment of $(E_3|H_3)$ from $\mathcal{F}$ does not hold, by showing that $(E_3|H_3)|\mathcal{C}(\mathcal{F})\neq1.

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

A probabilistic analysis of selected notions of iterated conditioning under coherence

It is well known that basic conditionals satisfy some desirable basic logical and probabilistic properties, such as the compound probability theorem. However checking the validity of these becomes trickier when we switch to compound and iterated conditionals. Herein we consider de Finetti's notion of conditional both in terms of a three-valued object and as a conditional random quantity in the betting framework. We begin by recalling the notions of conjunction and disjunction among conditionals in selected trivalent logics. Then we analyze the notions of iterated conditioning in the frameworks of the specific three-valued logics introduced by Cooper-Calabrese, by de Finetti, and by Farrel. By computing some probability propagation rules we show that the compound probability theorem and other important properties are not always preserved by these formulations. Then, for each trivalent logic we introduce an iterated conditional as a suitable random quantity which satisfies the compound prevision theorem as well as some other desirable properties. We also check the validity of two generalized versions of Bayes' Rule for iterated conditionals. We study the p-validity of generalized versions of Modus Ponens and two-premise centering for iterated conditionals. Finally, we observe that all the basic properties are satisfied within the framework of iterated conditioning followed in recent papers by Gilio and Sanfilippo in the setting of conditional random quantities