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11Probability propagation rules for Aristotelian syllogismsAnnals of Pure and Applied Logic. forthcoming.
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22Connexive Logic, Probabilistic Default Reasoning, and Compound ConditionalsStudia Logica 112 (1): 167-206. 2023.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, thenA, 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 $${A| \overline{A}}$$ A | A ¯ is $${p(A| \over…Read more
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21Square of opposition under coherenceIn M. B. Ferraro, P. Giordani, B. Vantaggi, M. Gagolewski, P. Grzegorzewski, O. Hryniewicz & María Ángeles Gil (eds.), Soft Methods for Data Science, . pp. 407-414. 2017.Various semantics for studying the square of opposition have been proposed recently. So far, only [14] studied a probabilistic version of the square where the sentences were interpreted by (negated) defaults. We extend this work by interpreting sentences by imprecise (set-valued) probability assessments on a sequence of conditional events. We introduce the acceptability of a sentence within coherence-based probability theory. We analyze the relations of the square in terms of acceptability and s…Read more
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25Centering and compound conditionals under coherenceIn M. B. Ferraro, P. Giordani, B. Vantaggi, M. Gagolewski, P. Grzegorzewski, O. Hryniewicz & María Ángeles Gil (eds.), Soft Methods for Data Science, . pp. 253-260. 2017.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 illustra…Read more
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26Generalized probabilistic modus ponensIn A. Antonucci, L. Cholvy & O. Papini (eds.), Symbolic and Quantitative Approaches to Reasoning with Uncertainty (Lecture Notes in Artificial Intelligence, vol. 10369). pp. 480-490. 2017.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 previ…Read more
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24Probabilistic semantics for categorical syllogisms of Figure IIIn D. Ciucci, G. Pasi & B. Vantaggi (eds.), Scalable Uncertainty Management. pp. 196-211. 2018.A coherence-based probability semantics for categorical syllogisms of Figure I, which have transitive structures, has been proposed recently (Gilio, Pfeifer, & Sanfilippo [15]). We extend this work by studying Figure II under coherence. Camestres is an example of a Figure II syllogism: from Every P is M and No S is M infer No S is P. We interpret these sentences by suitable conditional probability assessments. Since the probabilistic inference of ~????|???? from the premise set {????|????, …Read more
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16Probabilistic entailment and iterated conditionalsIn S. Elqayam, Igor Douven, J. St B. T. Evans & N. Cruz (eds.), Logic and uncertainty in the human mind: a tribute to David E. Over, Routledge. pp. 71-101. 2020.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 F={E1|H1, E2|H2} p-entails a conditional event E3|H3 if and only if E3|H3= 1, or (E3|H3)|QC(S) = 1 for some nonempty subset S of F, whe…Read more
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15Probabilistic squares and hexagons of opposition under coherenceInternational Journal of Approximate Reasoning 88 282-294. 2017.Various semantics for studying the square of opposition and the hexagon of opposition have been proposed recently. We interpret sentences by imprecise (set-valued) probability assessments on a finite sequence of conditional events. We introduce the acceptability of a sentence within coherence-based probability theory. We analyze the relations of the square and of the hexagon in terms of acceptability. Then, we show how to construct probabilistic versions of the square and of the hexagon of oppos…Read more
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27Probabilistic inferences from conjoined to iterated conditionalsInternational Journal of Approximate Reasoning 93 103-118. 2018.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…Read more
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74We 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 fundame…Read more
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21Interpreting connexive principles in coherence-based probability logic.In J. Vejnarová & J. Wilson (eds.), Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU 2021, LNAI 12897), . pp. 672-687. 2021.We present probabilistic approaches to check the validity of selected connexive principles within the setting of coherence. Connexive logics emerged from the intuition that conditionals of the form If ∼A, then A, should not hold, since the conditional’s antecedent ∼A contradicts its consequent A. Our approach covers this intuition by observing that for an event A the only coherent probability assessment on the conditional event A|~A is p(A|~A)=0 . Moreover, connexive logics aim to capt…Read more
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29Probabilities of conditionals and previsions of iterated conditionalsInternational Journal of Approximate Reasoning 121. 2020.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 Cas information, the probabil…Read more
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77Transitivity in coherence-based probability logicJournal of Applied Logic 14 46-64. 2016.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 pro…Read more
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307Algebraic aspects and coherence conditions for conjoined and disjoined conditionalsInternational Journal of Approximate Reasoning 126 98-123. 2020.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 const…Read more
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240Probabilistic inferences from conjoined to iterated conditionalsInternational Journal of Approximate Reasoning 93 103-118. 2018.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 iterat…Read more
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348Generalized logical operations among conditional eventsApplied Intelligence 49 79-102. 2019.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…Read more
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76Probabilistic logic under coherence, model-theoretic probabilistic logic, and default reasoning in System PJournal of Applied Non-Classical Logics 12 (2): 189-213. 2002.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 . 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 und…Read more
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1236Conjunction, disjunction and iterated conditioning of conditional eventsIn R. Kruse (ed.), Advances in Intelligent Systems and Computing, Springer. 2013.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 th…Read more
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325Conditional Random Quantities and Compounds of ConditionalsStudia Logica 102 (4): 709-729. 2014.In this paper we consider conditional random quantities (c.r.q.’s) in the setting of coherence. Based on betting scheme, a c.r.q. X|H is not looked at as a restriction but, in a more extended way, as \({XH + \mathbb{P}(X|H)H^c}\) ; in particular (the indicator of) a conditional event E|H is looked at as EH + P(E|H)H c . This extended notion of c.r.q. allows algebraic developments among c.r.q.’s even if the conditioning events are different; then, for instance, we can give a meaning to the sum X|…Read more