A Generalized Probabilistic Version of Modus Ponens

TL;DR

This paper generalizes the probabilistic Modus Ponens rule by incorporating conditional events and iterated conditionals, enabling more nuanced uncertainty management in logical inference.

Contribution

It introduces a generalized probabilistic MP using conditional events and iterated conditionals, extending the classical rule to handle nested uncertainties.

Findings

Propagation rules for bounds are consistent with non-nested probabilistic MP.

The generalized rule manages uncertainty propagation through iterated conditionals.

The approach formalizes inference with conditional random quantities.

Abstract

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.