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Giuseppe Sanfilippo

Publications and source records attributed to Giuseppe Sanfilippo.

At least 19 recordsLinked to original sources
arXiv2023

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.

math.PR↗
arXiv2023

Probability propagation rules for Aristotelian syllogisms

We present a coherence-based probability semantics and probability propagation rules 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 bridge from our probability semantics of Aristotelian syllogisms to nonmonotonic reasoning. Then we show that reductio by conversion does not work while reductio ad impossibile can be applied in our approach. Finally, we show how the proposed probability propagation rules can be used to analyze syllogisms involving generalized quantifiers (like Most).

math.PR↗
arXiv2023

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. After that, we study some probabilistic deduction theorems, by presenting several examples. We focus on iterated conditionals and the invalidity of the Import-Export principle in the light of our Probabilistic Deduction Theorem. We use the inference from a disjunction, "$A$ or $B$", to the conditional,"if not-$A$ then $B$", as an example to show the invalidity of the Import-Export principle. We also introduce a General Import-Export principle and we illustrate it by examining some p-valid inference rules of System P. Finally, we briefly discuss some related work relevant to AI.

cs.AI↗
arXiv2023

Subjective probability, trivalent logics and compound conditionals

In this work we first illustrate the subjective theory of de Finetti. We recall the notion of coherence for both the betting scheme and the penalty criterion, by considering the unconditional and conditional cases. We show the equivalence of the two criteria by giving the geometrical interpretation of coherence. We also consider the notion of coherence based on proper scoring rules. We discuss conditional events in the trivalent logic of de Finetti and the numerical representation of truth-values. We check the validity of selected basic logical and probabilistic properties for some trivalent logics: Kleene-Lukasiewicz-Heyting-de Finetti; Lukasiewicz; Bochvar-Kleene; Sobocinski. We verify that none of these logics satisfies all the properties. Then, we consider our approach to conjunction and disjunction of conditional events in the setting of conditional random quantities. We verify that all the basic logical and probabilistic properties (included the Fréchet-Hoeffding bounds) are preserved in our approach. We also recall the characterization of p-consistency and p-entailment by our notion of conjunction.

math.PR↗
arXiv2022

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 formalized 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 fact, the set of compound conditionals can be endowed with a Boolean algebraic structure. In doing so, we pave the way to build a bridge between the long standing tradition of three-valued conditionals and a more recent proposal of looking at conditionals as elements from suitable Boolean algebras.

math.LO↗
arXiv2021

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 $\mathscr{C}(\mathcal{F})=(E_1|H_1)\wedge \cdots \wedge (E_n|H_n)$ the conjunction of the conditional events in $\mathcal{F}$. We introduce the iterated conditional $\mathscr{C}(\mathcal{F}_2)|\mathscr{C}(\mathcal{F}_1)$, where $\mathcal{F}_1$ and $\mathcal{F}_2$ are two finite families of conditional events, by showing that the prevision of $\mathscr{C}(\mathcal{F}_2)\wedge \mathscr{C}(\mathcal{F}_1)$ is the product of the prevision of $\mathscr{C}(\mathcal{F}_2)|\mathscr{C}(\mathcal{F}_1)$ and the prevision of $\mathscr{C}(\mathcal{F}_1)$. Likewise the well known equality $(A\wedge H)|H=A|H$, we show that $ (\mathscr{C}(\mathcal{F}_2)\wedge \mathscr{C}(\mathcal{F}_1))|\mathscr{C}(\mathcal{F}_1)= \mathscr{C}(\mathcal{F}_2)|\mathscr{C}(\mathcal{F}_1)$. Then, we consider the case $\mathcal{F}_1=\mathcal{F}_2=\mathcal{F}$ and we verify for the prevision $μ$ of $\mathscr{C}(\mathcal{F})|\mathscr{C}(\mathcal{F})$ that the unique coherent assessment is $μ=1$ and, as a consequence, $\mathscr{C}(\mathcal{F})|\mathscr{C}(\mathcal{F})$ coincides with the constant 1. Finally, by assuming $\mathcal{F}$ p-consistent, we deepen some previous characterizations of p-entailment by showing that $\mathcal{F}$ p-entails a conditional event $E_{n+1}|H_{n+1}$ if and only if the iterated conditional $(E_{n+1}|H_{n+1})\,|\,\mathscr{C}(\mathcal{F})$ is constant and equal to 1. We illustrate this characterization by an example related with weak transitivity.

math.PR↗
arXiv2021

Compound conditionals, Fréchet-Hoeffding bounds, and Frank t-norms

In this paper we consider compound conditionals, Fréchet-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 Fréchet-Hoeffding bounds for the prevision of conjunctions and disjunctions of $n$ 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 solutions for the linear systems; then, we analyze a selected example. We obtain a probabilistic interpretation of Frank t-norms and t-conorms as prevision of conjunctions and disjunctions of conditional events, respectively. Then, we characterize the sets of coherent prevision assessments on a family containing $n$ conditional events and their conjunction, or their disjunction, by using Frank t-norms, or Frank t-conorms. By assuming logical independence, we show that any Frank t-norm (resp., t-conorm) of two conditional events $A|H$ and $B|K$, $T_λ(A|H,B|K)$ (resp., $S_λ(A|H,B|K)$), is a conjunction $(A|H)\wedge (B|K)$ (resp., a disjunction $(A|H)\vee (B|K)$). By considering a family $\mathcal{F}$ containing three conditional events, their conjunction, and all pairwise conjunctions we give some results on Frank t-norms and coherence of the prevision assessments on $\mathcal{F}$. By assuming logical independence, we show that it is coherent to assess the previsions of all the conjunctions by means of Minimum and Product t-norms. We verify by a counterexample that, when the previsions of conjunctions are assessed by the Lukasiewicz t-norm, coherence is not assured.

math.PR↗
arXiv2021

Interpreting connexive principles in coherence-based probability logic

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 $\sim A$, then $A$", should not hold, since the conditional's antecedent $\sim A$ 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 $A|\overline{A}$ is $p(A|\overline{A})=0$. Moreover, connexive logics aim to capture the intuition that conditionals should express some "connection" between the antecedent and the consequent or, in terms of inferences, validity should require some connection between the premise set and the conclusion. This intuition is covered by a number of principles, a selection of which we analyze in our contribution. We present two approaches to connexivity within coherence-based probability logic. Specifically, we analyze connections between antecedents and consequents firstly, in terms of probabilistic constraints on conditional events (in the sense of defaults, or negated defaults) and secondly, in terms of constraints on compounds of conditionals and iterated conditionals. After developing different notions of negations and notions of validity, we analyze the following connexive principles within both approaches: Aristotle's Theses, Aristotle's Second Thesis, Abelard's First Principle and selected versions of Boethius' Theses. We conclude by remarking that coherence-based probability logic offers a rich language to investigate the validity of various connexive principles.

math.LO↗
arXiv2020

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 suitable distributivity property. Moreover, under logical independence of basic unconditional events, we give two necessary and sufficient coherence conditions. The first condition gives a geometrical characterization for the coherence of prevision assessments on a family $\mathscr{F}$ constituted by $n$ conditional events and all possible conjunctions among them. The second condition characterizes the coherence of prevision assessments defined on $\mathscr{F}\cup \mathscr{K}$, where $\mathscr{K}$ is the set of conditional constituents associated with the conditional events in $\mathscr{F}$. Then, we give a further theoretical result and we examine some examples and counterexamples. Finally, we make a comparison with other approaches and we illustrate some theoretical aspects and applications.

math.PR↗
arXiv2020

Probabilities of conditionals and previsions of iterated conditionals

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 us to formalize different kinds of latent information. We verify that for these iterated conditionals the prevision is greater than or equal to the probability of $A$. We also investigate the lower and upper bounds of the Affirmation of the Consequent inference. We conclude our study with some remarks on the supposed ''independence'' of two conditionals, and we interpret this property as uncorrelation between two random quantities.

math.PR↗
arXiv2019

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.

math.PR↗
arXiv2019

Predictive distributions that mimic frequencies over a restricted subdomain (expanded preprint version)

A predictive distribution over a sequence of $N+1$ events is said to be "frequency mimicking" whenever the probability for the final event conditioned on the outcome of the first $N$ events equals the relative frequency of successes among them. Infinitely extendible exchangeable distributions that universally inhere this property are known to have several annoying concomitant properties. We motivate frequency mimicking assertions over a limited subdomain in practical problems of finite inference, and we identify their computable coherent implications. We provide some computed examples using reference distributions, and we introduce computational software to generate any specification. The software derives from an inversion of the finite form of the exchangeability representation theorem. Three new theorems delineate the extent of the usefulness of such distributions, and we show why it may not be appropriate to extend the frequency mimicking assertions for a specified value of $N$ to any arbitrary larger size of $N$. The constructive results identify the source and structure of "adherent masses" in the limit of a sequence of finitely additive distributions. Appendices develop a novel geometrical representation of conditional probabilities which illuminate the analysis.

stat.ME↗
arXiv2018

Scoring Alternative Forecast Distributions: Completing the Kullback Distance Complex

We develop two surprising new results regarding the use of proper scoring rules for evaluating the predictive quality of two alternative sequential forecast distributions. Both of the proponents prefer to be awarded a score derived from the other's distribution rather than a score awarded on the basis of their own. A Pareto optimal exchange of their scoring outcomes provides the basis for a comparison of forecast quality that is preferred by both forecasters, and also evades a feature of arbitrariness inherent in using the forecasters' own achieved scores. The well-known Kullback divergence, used as a measure of information, is evaluated via the entropies in the two forecast distributions and the two cross-entropies between them. We show that Kullback's symmetric measure needs to be appended by three component measures if it is to characterise completely the information content of the two asserted probability forecasts. Two of these do not involve entropies at all. The resulting 'Kullback complex' supported by the 4-dimensional measure is isomorphic to an equivalent vector measure generated by the forecasters' expectations of their scores, each for one's own score and for the other's score. We foreshadow the results of a sophisticated application of the Pareto relative scoring procedure for actual sequentional observations, and we propose a standard format for evaluation.

math.PR↗
arXiv2018

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 Fréchet-Hoeffding bounds. Moreover, we study the reverse probabilistic inference from the conjunction $\mathcal{C}_{n+1}$ of $n+1$ conditional events to the family $\{\mathcal{C}_{n},E_{n+1}|H_{n+1}\}$. We consider the relation with the notion of quasi-conjunction and we examine in detail the coherence of the prevision assessments related with the conjunction of three conditional events. Based on conjunction, we also give a characterization of p-consistency and of p-entailment, with applications to several inference rules in probabilistic nonmonotonic reasoning. Finally, we examine some non p-valid inference rules; then, we illustrate by an example two methods which allow to suitably modify non p-valid inference rules in order to get inferences which are p-valid.

math.PR↗
arXiv2018

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 $(E_3|H_3)|\mathcal{C}(\mathcal{F})\neq1.$

math.PR↗
arXiv2017

Probabilistic inferences from conjoined to iterated conditionals

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 and iterations as conditional random quantities which, given some logical dependencies, may reduce to conditional events. We show how the inference to $B|A$ from $A$ and $B$ can be extended to compounds and iterations of both conditional events and biconditional events. Moreover, we determine the respective uncertainty propagation rules. Finally, we make some comments on extending our analysis to counterfactuals.

math.PR↗
arXiv2017

Reassessing Accuracy Rates of Median Decisions

We show how Bruno de Finetti's fundamental theorem of prevision has computable applications in statistical problems that involve only partial information. Specifically, we assess accuracy rates for median decision procedures used in the radiological diagnosis of asbestosis. Conditional exchangeability of individual radiologists' diagnoses is recognized as more appropriate than independence which is commonly presumed. The FTP yields coherent bounds on probabilities of interest when available information is insufficient to determine a complete distribution. Further assertions that are natural to the problem motivate a partial ordering of conditional probabilities, extending the computation from a linear to a quadratic programming problem.

stat.AP↗
arXiv2017

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 the respective bounds on the conclusion for the (non-nested) probabilistic MP.

math.PR↗