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.

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

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, which can be interpreted by introducing a suitable distributive 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 F constituted by n conditional events and all possible conjunctions among them. The second condition characterizes the coherence of prevision assessments defined on $F\cup K$, where $K$ is the set of conditional constituents associated with the conditional events in $F$. Then, we give some further theoretical results and we examine some examples and counterexamples. Finally, we make a comparison with other approaches and we illustrate some theoretical aspects and applications

On the role of deduction in reasoning from uncertain premises

The probabilistic approach to reasoning hypothesizes that most reasoning, both in everyday life and in science, takes place in contexts of uncertainty. The central deductive concepts of classical logic, consistency and validity, can be generalised to cover uncertain degrees of belief. Binary consistency can be generalised to coherence, where the probability judgments for two statements are coherent if and only if they respect the axioms of probability theory. Binary validity can be generalised to probabilistic validity (p-validity), where an inference is p-valid if and only if the uncertainty of its conclusion cannot be coherently greater than the sum of the uncertainties of its premises. But the fact that this generalisation is possible in formal logic does not imply that people will use deduction in a probabilistic way. The role of deduction in reasoning from uncertain premises was investigated across ten experiments and 23 inferences of differing complexity. The results provide evidence that coherence and p-validity are not just abstract formalisms, but that people follow the normative constraints set by them in their reasoning. It made no qualitative difference whether the premises were certain or uncertain, but certainty could be interpreted as the endpoint of a common scale for degrees of belief. The findings are evidence for the descriptive adequacy of coherence and p-validity as computational level principles for reasoning. They have implications for the interpretation of past findings on the roles of deduction and degrees of belief. And they offer a perspective for generating new research hypotheses in the interface between deductive and inductive reasoning. Keywords: Reasoning; deduction; probabilistic approach; coherence; p-validit

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(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 conditional random quantities, which sometimes reduce to conditional events, given logical dependencies. We also show, for the first time, how to extend the inference of centering for conditional events, inferring B|A from the conjunction A and B, to compounds and iterations of both conditional events and biconditional events, B||A, and generalize it to n-conditional event

On the role of deduction in reasoning from uncertain premises

The probabilistic approach to reasoning hypothesizes that most reasoning, both in everyday life and in science, takes place in contexts of uncertainty. The central deductive concepts of classical logic, consistency and validity, can be generalised to cover uncertain degrees of belief. Binary consistency can be generalised to coherence, where the probability judgments for two statements are coherent if and only if they respect the axioms of probability theory. Binary validity can be generalised to probabilistic validity (p-validity), where an inference is p-valid if and only if the uncertainty of its conclusion cannot be coherently greater than the sum of the uncertainties of its premises. But the fact that this generalisation is possible in formal logic does not imply that people will use deduction in a probabilistic way. The role of deduction in reasoning from uncertain premises was investigated across ten experiments and 23 inferences of differing complexity. The results provide evidence that coherence and p-validity are not just abstract formalisms, but that people follow the normative constraints set by them in their reasoning. It made no qualitative difference whether the premises were certain or uncertain, but certainty could be interpreted as the endpoint of a common scale for degrees of belief. The findings are evidence for the descriptive adequacy of coherence and p-validity as computational level principles for reasoning. They have implications for the interpretation of past findings on the roles of deduction and degrees of belief. And they offer a perspective for generating new research hypotheses in the interface between deductive and inductive reasoning. Keywords: Reasoning; deduction; probabilistic approach; coherence; p-validit