1,721,052 research outputs found
Triangular norms and conjunction of conditional events
No full textWe 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
Algebraic aspects and coherence conditions for conjunctions among conditional events
Full text linkWe deepen the study of a notion of conjunction among conditional events, introduced in previous papers in theframework of coherence. This notion of conjunction, differently from other approaches, is given in the setting ofconditional random quantities. We show that some well known properties which are satisfied by conjunctionsof unconditional events are also satisfied by conjunctions of conditional events. In particular we examine anadditive property and a decomposition formula, by also obtaining a generalized inclusion-exclusion formula. Then,by exploiting the notion of conjunction, we introduce the set of constituents generated bynconditional events.Moreover, under logical independence, we give a necessary and sufficient condition of coherence for the previsionassessments on a familyFconstituted bynconditional events and all possible conjunctions among some of them.This condition of coherence has a simple geometrical characterization in terms of a suitable convex hull. Such acharacterization amounts to the solvability of a linear system as in the case of unconditional events. Then, weillustrate the set of all coherent assessments on the familyFby a list of linear inequalities on the componentsof the prevision assessment. Finally, given a coherent assessmentMonF, we show that every possible value ofthe random vector associated withFis itself a particular coherent assessment onF
Probabilistic interpretations of the square of opposition
No full textWe investigate the square of opposition from a probabilistic point of view. Probability allows for dealing with exceptions and uncertainty. We will interpret the corners of the square by means of (precise or imprecise) conditional probability assessments. They will be defined within the framework of coherence, which originally goes back to de Finetti. In this framework probabilities are conceived as degrees of belief, where conditional probability is defined as a primitive concept. Coherence allows for dealing with partial and imprecise assessments. Moreover, the coherence approach is especially suitable for dealing with zero antecedent probabilities (i.e., here conditioning events may have probability zero): This is relevant for studying different probabilistic interpretations of the existential import.
In this talk, we will discuss probabilistic notions of the existential import and present probabilistic interpretations of universally affirmative and negative as well as particular affirmative and negative propositions. After choosing appropriate probabilistic constraints for defining the four basic types of propositions and the existential import, we will present a probabilistic version of the traditional square of opposition. We will discuss in what sense the traditional relations—contradictories, contraries, sub-contraries, and sub-alternations— are also contained in the probabilistic square of opposition. Moreover, we will generalize our probabilistic interpretation of the basic syllogistic concepts to construct probabilistic versions of selected syllogisms. We will also relate them to inference rules in nonmonotonic reasoning. Finally, we will discuss how probabilistic syllogisms could serve as a rationality framework for human reasoning about quantifiers within the so-called “new psychology of reasoning”
Probabilistic inference and syllogisms
No full textTraditionally, syllogisms are arguments with two premises and one conclusion which are constructed by propositions of the form “All S are P ” and “At least one S is P ” and their respective negated versions. We will discuss probabilistic notions of the existential import and the basic sentences type. We will develop an intuitively plausible version of the syllogisms that is able to deal with uncertainty, exceptions and nonmonotonicity. We will develop a new semantics for categorical syllogisms that is based on subjective probability. Specifically, we propose de Finetti’s principle of coherence and its generalization to lower and upper conditional probabilities as the fundamental corner stones for the new semantics. Coherence allows for dealing with partial and imprecise assessments. Moreover, it is especially suitable for handling zero antecedent probabilities (i.e., here conditioning events may have probability zero): This is relevant for studying the probabilistic interpretation of the existential import. Then, we will generalize our probabilistic interpretation of the basic syllogistic concepts to construct probabilistic versions of selected syllogisms. Finally, we will relate them to inference rules in nonmonotonic reasoning
Transitivity in coherence-based probability logic
Full text linkWe 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 square of opposition in terms of defaults and negated defaults
On general conditional random quantities
No full textIn the first part of this paper, recalling a general discussion on iterated conditioning given by de Finetti in the appendix of his book, vol. 2, we give a representation of a conditional random quantity as . In this way, we obtain the classical formula \pr{(XH|K)} =\pr{(X|HK)P(H|K)}, by simply using linearity of prevision. Then, we consider the notion of general conditional prevision \pr(X|Y), where and are two random quantities, introduced in 1990 in a paper by Lad and Dickey. After recalling the case where is an event, we consider the case of discrete finite random quantities and we make some critical comments and examples. We give a notion of coherence for such more general conditional prevision assessments; then, we obtain a strong generalized compound prevision theorem. We study the coherence of a general conditional prevision assessment \pr(X|Y) when has no negative values and when has no positive values. Finally, we give some results on coherence of \pr(X|Y) when assumes both positive and negative values. In order to illustrate critical aspects and remarks we examine several examples
Efficient checking of coherence and propagation of imprecise probability assessments
No full textWe 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
A General Probabilistic Database Model
No full textA new model for probabilistic databases, using interval-valued conditional probability assessments, is proposed. The concept of coherence adopted in our approach is based on a suitable generalization of the coherence principle of de Finetti and can be related to similar definitions given for lower and upper probabilities by other authors. A corresponding probabilistic relational algebra is introduced and some new oper- ators are defined. Finally, some simple examples are given
Generalized coherence and connection property of imprecise conditional previsions.
No full textIn this paper we consider imprecise conditional prevision assessments on random quantities with finite set of possible values. We use a notion of generalized coherence which is based on the coherence principle of de Finetti. We consider the checking of g-coherence, by extending some previous results obtained for imprecise conditional probability assessments. Then, we study a connection property of interval-valued gcoherent prevision assessments, by extending a result given in a previous paper for precise assessments
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