Iterated Conditionals and Characterization of P-Entailment

In this paper we deepen, in the setting of coherence, some results obtained in recent papers on the notion of p-entailment of Adams and its relationship with conjoined and iterated conditionals. We recall that conjoined and iterated conditionals are suitably defined in the framework of conditional random quantities. Given a family \(\mathcal {F}\) of n conditional events \(\{E_{1}|H_{1},\ldots , E_{n}|H_{n}\}\) we denote by \(\mathcal {C}(\mathcal {F})=(E_{1}|H_{1})\wedge \cdots \wedge (E_{n}|H_{n})\) the conjunction of the conditional events in \(\mathcal F\). We introduce the iterated conditional \(\mathcal {C}(\mathcal {F}_{2})|\mathcal {C}(\mathcal {F}_{1})\), where \(\mathcal {F}_{1}\)…

Probabilistic inferences from conjoined to iterated conditionals

Abstract There is wide support in logic, philosophy, and psychology for the hypothesis that the probability of the indicative conditional of natural language, P ( if A then B ) , is the conditional probability of B given A, P ( B | A ) . We identify a conditional which is such that P ( if A then B ) = P ( B | A ) with de Finetti's conditional event, B | A . 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 cond…

On Trivalent Logics, Compound Conditionals, and Probabilistic Deduction Theorems

In this paper we recall some results for conditional events, compound conditionals, conditional random quantities, p-consistency, and p-entailment. Then, we show the equivalence between bets on conditionals and conditional bets, by reviewing de Finetti's trivalent analysis of conditionals. But our approach goes beyond de Finetti's early trivalent logical analysis and is based on his later ideas, aiming to take his proposals to a higher level. We examine two recent articles that explore trivalent logics for conditionals and their definitions of logical validity and compare them with our approach to compound conditionals. We prove a Probabilistic Deduction Theorem for conditional events. Afte…

Generalized Logical Operations among Conditional Events

We generalize, by a progressive procedure, the notions of conjunction and disjunction of two conditional events to the case of n conditional events. In our coherence-based approach, conjunctions and disjunctions are suitable conditional random quantities. We define the notion of negation, by verifying De Morgan’s Laws. We also show that conjunction and disjunction satisfy the associative and commutative properties, and a monotonicity property. Then, we give some results on coherence of prevision assessments for some families of compounded conditionals; in particular we examine the Frechet-Hoeffding bounds. Moreover, we study the reverse probabilistic inference from the conjunction $\mathcal…

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 rece…

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 $A…

Assessment of qualitative judgements for conditional events in expert systems

Generalized probabilistic modus ponens

Modus ponens (from A and “if A then C” infer C) is one of the most basic inference rules. The probabilistic modus ponens allows for managing uncertainty by transmitting assigned uncertainties from the premises to the conclusion (i.e., from P(A) and P(C|A) infer P(C)). In this paper, we generalize the probabilistic modus ponens by replacing A by the conditional event A|H. The resulting inference rule involves iterated conditionals (formalized by conditional random quantities) and propagates previsions from the premises to the conclusion. Interestingly, the propagation rules for the lower and the upper bounds on the conclusion of the generalized probabilistic modus ponens coincide with the re…

Conjunction of Conditional Events and t-Norms

We study the relationship between a notion of conjunction among conditional events, introduced in recent papers, and the notion of Frank t-norm. By examining different cases, in the setting of coherence, we show each time that the conjunction coincides with a suitable Frank t-norm. In particular, the conjunction may coincide with the Product t-norm, the Minimum t-norm, and Lukasiewicz t-norm. We show by a counterexample, that the prevision assessments obtained by Lukasiewicz t-norm may be not coherent. Then, we give some conditions of coherence when using Lukasiewicz t-norm

Compound conditionals as random quantities and Boolean algebras

Conditionals play a key role in different areas of logic and probabilistic reasoning, and they have been studied and formalised from different angles. In this paper we focus on the de Finetti's notion of conditional as a three-valued object, with betting-based semantics, and its related approach as random quantity as mainly developed by two of the authors. Compound conditionals have been studied in the literature, but not in full generality. In this paper we provide a natural procedure to explicitly attach conditional random quantities to arbitrary compound conditionals that also allows us to compute their previsions. By studying the properties of these random quantities, we show that, in f…

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 illus…

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.…

Coherent Conditional Previsions and Proper Scoring Rules

In this paper we study the relationship between the notion of coherence for conditional prevision assessments on a family of finite conditional random quantities and the notion of admissibility with respect to bounded strictly proper scoring rules. Our work extends recent results given by the last two authors of this paper on the equivalence between coherence and admissibility for conditional probability assessments. In order to prove that admissibility implies coherence a key role is played by the notion of Bregman divergence.

Conjunction and Disjunction Among Conditional Events

We generalize, in the setting of coherence, the notions of conjunction and disjunction of two conditional events to the case of n conditional events. Given a prevision assessment on the conjunction of two conditional events, we study the set of coherent extensions for the probabilities of the two conditional events. Then, we introduce by a progressive procedure the notions of conjunction and disjunction for n conditional events. Moreover, by defining the negation of conjunction and of disjunction, we show that De Morgan’s Laws still hold. We also show that the associative and commutative properties are satisfied. Finally, we examine in detail the conjunction for a family \(\mathcal F\) of t…

Coherence Checking and Propagation of Lower Probability Bounds

In this paper we use imprecise probabilities, based on a concept of generalized coherence (g-coherence), for the management of uncertain knowledge and vague information. We face the problem of reducing the computational difficulties in g-coherence checking and propagation of lower conditional probability bounds. We examine a procedure, based on linear systems with a reduced number of unknowns, for the checking of g-coherence. We propose an iterative algorithm to determine the reduced linear systems. Based on the same ideas, we give an algorithm for the propagation of lower probability bounds. We also give some theoretical results that allow, by suitably modifying our algorithms, the g-coher…

Probabilities of conditionals and previsions of iterated conditionals

Abstract We analyze selected iterated conditionals in the framework of conditional random quantities. We point out that it is instructive to examine Lewis's triviality result, which shows the conditions a conditional must satisfy for its probability to be the conditional probability. In our approach, however, we avoid triviality because the import-export principle is invalid. We then analyze an example of reasoning under partial knowledge where, given a conditional if A then C as information, the probability of A should intuitively increase. We explain this intuition by making some implicit background information explicit. We consider several (generalized) iterated conditionals, which allow…

A Generalized Probabilistic Version of Modus Ponens

Modus ponens (\emph{from $A$ and "if $A$ then $C$" infer $C$}, short: MP) is one of the most basic inference rules. The probabilistic MP allows for managing uncertainty by transmitting assigned uncertainties from the premises to the conclusion (i.e., from $P(A)$ and $P(C|A)$ infer $P(C)$). In this paper, we generalize the probabilistic MP by replacing $A$ by the conditional event $A|H$. The resulting inference rule involves iterated conditionals (formalized by conditional random quantities) and propagates previsions from the premises to the conclusion. Interestingly, the propagation rules for the lower and the upper bounds on the conclusion of the generalized probabilistic MP coincide with …

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 $A…

Probabilistic Logic under Coherence: Complexity and Algorithms

In previous work [V. Biazzo, A. Gilio, T. Lukasiewicz and G. Sanfilippo, Probabilistic logic under coherence, model-theoretic probabilistic logic, and default reasoning in System P, Journal of Applied Non-Classical Logics 12(2) (2002) 189---213.], we have explored the relationship between probabilistic reasoning under coherence and model-theoretic probabilistic reasoning. In particular, we have shown that the notions of g-coherence and of g-coherent entailment in probabilistic reasoning under coherence can be expressed by combining notions in model-theoretic probabilistic reasoning with concepts from default reasoning. In this paper, we continue this line of research. Based on the above sem…

On the checking of g-coherence of conditional probability bounds

We illustrate an approach to uncertain knowledge based on lower conditional probability bounds. We exploit the coherence principle of de Finetti and a related notion of generalized coherence (g-coherence), which is equivalent to the "avoiding uniform loss" property introduced by Walley for lower and upper probabilities. Based on the additive structure of random gains, we define suitable notions of non relevant gains and of basic sets of variables. Exploiting them, the linear systems in our algorithms can work with reduced sets of variables and/or constraints. In this paper, we illustrate the notions of non relevant gain and of basic set by examining several cases of imprecise assessments d…

Quasi conjunction and p-entailment in nonmonotonic reasoning

We study, in the setting of coherence, the extension of a probability assessment defined on n conditional events to their quasi conjunction. We consider, in particular, two special cases of logical dependencies; moreover, we examine the relationship between the notion of p-entailment of Adams and the inclusion relation of Goodman and Nguyen. We also study the probabilistic semantics of the QAND rule of Dubois and Prade; then, we give a theoretical result on p-entailment.

Canonical Extensions of Conditional Probabilities and Compound Conditionals

In this paper we show 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. We do this by first proving that the canonical extension of a full conditional probability on a finite algebra of events to the corresponding algebra of conditionals is compatible with taking subalgebras of events.

Compound conditionals, Fr\'echet-Hoeffding bounds, and Frank t-norms

Abstract In this paper we consider compound conditionals, Frechet-Hoeffding bounds and the probabilistic interpretation of Frank t-norms. By studying the solvability of suitable linear systems, we show under logical independence the sharpness of the Frechet-Hoeffding bounds for the prevision of conjunctions and disjunctions of n conditional events. In addition, we illustrate some details in the case of three conditional events. We study the set of all coherent prevision assessments on a family containing n conditional events and their conjunction, by verifying that it is convex. We discuss the case where the prevision of conjunctions is assessed by Lukasiewicz t-norms and we give explicit s…

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 $n$ 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 de…

Probabilistic Logic under Coherence‚ Model−Theoretic Probabilistic Logic‚ and Default Reasoning in System P

We study probabilistic logic under the viewpoint of the coherence principle of de Finetti. In detail, we explore how probabilistic reasoning under coherence is related to model-theoretic probabilistic reasoning and to default reasoning in System P. In particular, we show that the notions of g-coherence and of g-coherent entailment can be expressed by combining notions in model-theoretic probabilistic logic with concepts from default reasoning. Moreover, we show that probabilistic reasoning under coherence is a generalization of default reasoning in System P. That is, we provide a new probabilistic semantics for System P, which neither uses infinitesimal probabilities nor atomic bound (or bi…

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 $n$ 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 and we prove a …

Transitive reasoning with imprecise probabilities

We study probabilistically informative (weak) versions of transitivity, by using suitable definitions of defaults and negated defaults, in the setting of coherence and imprecise probabilities. We represent p-consistent sequences of defaults and/or negated defaults by g-coherent imprecise probability assessments on the respective sequences of conditional events. Finally, we prove the coherent probability propagation rules for Weak Transitivity and the validity of selected inference patterns by proving the p-entailment for the associated knowledge bases.

Conjunction of Conditional Events and T-norms

We study the relationship between a notion of conjunction among conditional events, introduced in recent papers, and the notion of Frank t-norm. By examining different cases, in the setting of coherence, we show each time that the conjunction coincides with a suitable Frank t-norm. In particular, the conjunction may coincide with the Product t-norm, the Minimum t-norm, and Lukasiewicz t-norm. We show by a counterexample, that the prevision assessments obtained by Lukasiewicz t-norm may be not coherent. Then, we give some conditions of coherence when using Lukasiewicz t-norm.

Computational aspects in checking of coherence and propagation of conditional probability bounds

In this paper we consider the problem of reducing the computational difficulties in g-coherence checking and propagation of imprecise conditional probability assessments. We review some theoretical results related with the linear structure of the random gain in the betting criterion. Then, we propose a modi ed version of two existing algorithms, used for g-coherence checking and propagation, which are based on linear systems with a reduced number of unknowns. The reduction in the number of unknowns is obtained by an iterative algorithm. Finally, to illustrate our procedure we give some applications.

Transitive Reasoning with Imprecise Probabilities

We study probabilistically informative (weak) versions of transitivity by using suitable definitions of defaults and negated defaults in the setting of coherence and imprecise probabilities. We represent \(\text{ p-consistent }\) sequences of defaults and/or negated defaults by g-coherent imprecise probability assessments on the respective sequences of conditional events. Finally, we present the coherent probability propagation rules for Weak Transitivity and the validity of selected inference patterns by proving p-entailment of the associated knowledge bases.

Efficient checking of coherence and propagation of imprecise probability assessments

We consider the computational difficulties in the checking of coherence and propagation of imprecise probability assessments. We examine the linear structure of the random gain in betting criterion and we propose a general methodology which exploits suitable subsets of the set of values of the random gain. In this way the checking of coherence and propagation amount to examining linear systems with a reduced number of unknowns. We also illustrate an example.

Conditional Random Quantities and Iterated Conditioning in the Setting of Coherence

We consider conditional random quantities (c.r.q.’s) in the setting of coherence. Given a numerical r.q. X and a non impossible event H, based on betting scheme we represent the c.r.q. X|H as the unconditional r.q. XH + μH c , where μ is the prevision assessed for X|H. We develop some elements for an algebra of c.r.q.’s, by giving a condition under which two c.r.q.’s X|H and Y|K coincide. We show that X|HK coincides with a suitable c.r.q. Y|K and we apply this representation to Bayesian updating of probabilities, by also deepening some aspects of Bayes’ formula. Then, we introduce a notion of iterated c.r.q. (X|H)|K, by analyzing its relationship with X|HK. Our notion of iterated conditiona…

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…

Transitivity in coherence-based probability logic

We study probabilistically informative (weak) versions of transitivity by using suitable definitions of defaults and negated defaults in the setting of coherence and imprecise probabilities. We represent p-consistent sequences of defaults and/or negated defaults by g-coherent imprecise probability assessments on the respective sequences of conditional events. Moreover, we prove the coherent probability propagation rules for Weak Transitivity and the validity of selected inference patterns by proving p-entailment of the associated knowledge bases. Finally, we apply our results to study selected probabilistic versions of classical categorical syllogisms and construct a new version of the squa…

Quasi Conjunction and Inclusion Relation in Probabilistic Default Reasoning

We study the quasi conjunction and the Goodman & Nguyen inclusion relation for conditional events, in the setting of probabilistic default reasoning under coherence. We deepen two recent results given in (Gilio and Sanfilippo, 2010): the first result concerns p-entailment from a family F of conditional events to the quasi conjunction C(S) associated with each nonempty subset S of F; the second result, among other aspects, analyzes the equivalence between p-entailment from F and p-entailment from C(S), where S is some nonempty subset of F. We also characterize p-entailment by some alternative theorems. Finally, we deepen the connections between p-entailment and the Goodman & Nguyen inclusion…

Centering and Compound Conditionals under Coherence

There is wide support in logic , philosophy , and psychology for the hypothesis that the probability of the indicative conditional of natural language, \(P(\textit{if } A \textit{ then } B)\), is the conditional probability of B given A, P(B|A). We identify a conditional which is such that \(P(\textit{if } A \textit{ then } B)= P(B|A)\) with de Finetti’s conditional event, B|A. 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 a…

On general conditional prevision assessments

In this paper we consider general conditional random quantities of the kind $X|Y$, where $X$ and $Y$ are finite discrete random quantities. Then, we introduce the notion of coherence for conditional prevision assessments on finite families of general conditional random quantities. Moreover, we give a compound prevision theorem and we examine the relation between the previsions of $X|Y$ and $Y|X$. Then, we give some results on random gains and, by a suitable alternative theorem, we obtain a characterization of coherence. We also propose an algorithm for the checking of coherence. Finally, we briefly examine the case of imprecise conditional prevision assessments by introducing the notions of…

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 cohere…

Probabilistic Logic under Coherence, Model-Theoretic Probabilistic Logic, and Default Reasoning

We study probabilistic logic under the viewpoint of the coherence principle of de Finetti. In detail, we explore the relationship between coherence-based and model-theoretic probabilistic logic. Interestingly, we show that the notions of g-coherence and of g-coherent entailment can be expressed by combining notions in model-theoretic probabilistic logic with concepts from default reasoning. Crucially, we even show that probabilistic reasoning under coherence is a probabilistic generalization of default reasoning in system P. That is, we provide a new probabilistic semantics for system P, which is neither based on infinitesimal probabilities nor on atomic-bound (or also big-stepped) probabil…