Probability Semantics for Aristotelian Syllogisms

We present a coherence-based probability semantics for (categorical) Aristotelian syllogisms. For framing the Aristotelian syllogisms as probabilistic inferences, we interpret basic syllogistic sentence types A, E, I, O by suitable precise and imprecise conditional probability assessments. Then, we define validity of probabilistic inferences and probabilistic notions of the existential import which is required, for the validity of the syllogisms. Based on a generalization of de Finetti's fundamental theorem to conditional probability, we investigate the coherent probability propagation rules of argument forms of the syllogistic Figures I, II, and III, respectively. These results allow to show, for all three Figures, that each traditionally valid syllogism is also valid in our coherence-based probability semantics. Moreover, we interpret the basic syllogistic sentence types by suitable defaults and negated defaults. Thereby, we build a knowledge bridge from our probability semantics of Aristotelian syllogisms to nonmonotonic reasoning. Finally, we show how the proposed semantics can be used to analyze syllogisms involving generalized quantifiers

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

Algorithms for the extension of precise and imprecise conditional probability assessments: an implementation with maple V

In this paper, we illustrate an implementation with Maple V of some procedures which allow to exactly propagate precise and imprecise probability assessments. The extension of imprecise assessments is based on a suitable generalization of the concept of coherence of de Finetti. The procedures described are supported by some examples and relevant cases

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

B

A process model of the understanding of uncertain conditionals

To build a process model of the understanding of conditionals we extract a common core of three semantics of if-then sentences: (a) the conditional event interpretation in the coherencebased probability logic, (b) the discourse processingtheory of Hans Kamp, and (c) the game-theoretical approach of Jaakko Hintikka. The empirical part reports three experiments in which each participant assessed the probability of 52 if-then sentencesin a truth table task. Each experiment included a second task: An n-back task relating the interpretation of conditionals to working memory, a Bayesian bookbag and poker chip task relating the interpretation of conditionals to probability updating, and a probabilistic modus ponens task relating the interpretation of conditionals to a classical inference task. Data analysis shows that the way in which the conditionals are interpreted correlates with each of the supplementary tasks. The results are discussed within the process model proposed in the introduction

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.

Decision-Making in the Context of Imprecise Probabilistic Beliefs

Coherent imprecise probabilistic beliefs are modelled as incomplete comparative likelihood relations admitting a multiple-prior representation. Under a structural assumption of Equidivisibility, we provide an axiomatization of such relations and show uniqueness of the representation. In the second part of the paper, we formulate a behaviorally general axiom relating preferences and probabilistic beliefs which implies that preferences over unambiguous acts are probabilistically sophisticated and which entails representability of preferences over Savage acts in an Anscombe-Aumann-style framework. The motivation for an explicit and separate axiomatization of beliefs for the study of decision-making under ambiguity is discussed in some detail.