Imprecise Uncertain Reasoning: A Distributional Approach

Abstract

The contribution proposes to model imprecise and uncertain reasoning by a mental probability logic that is based on probability distributions. It shows how distributions are combined with logical operators and how distributions propagate in inference rules. It discusses a series of examples like the Linda task, the suppression task, Doherty's pseudodiagnosticity task, and some of the deductive reasoning tasks of Rips. It demonstrates how to update distributions by soft evidence and how to represent correlated risks. The probabilities inferred from different logical inference forms may be so similar that it will be impossible to distinguish them empirically in a psychological study. Second-order distributions allow to obtain the probability distribution of being coherent. The maximum probability of being coherent is a second-order criterion of rationality. Technically the contribution relies on beta distributions, copulas, vines, and stochastic simulation.

A Quantitative Symbolic Approach to Individual Human Reasoning

“…(Chater & Oaksford, 1999) or attach the probabilities to different types of inferences, e.g. (Kleiter, 2018). We also use a slightly different approach than Johnson-Laird (1999) by considering the relationship on counting the values in the truth table that evaluate to true, but according to the answer set semantics.…”

Propensities and Second Order Uncertainty: A Modified Taxi Cab Problem

“…How would this affect our decision over which bet to take? This uncertainty about our first order uncertainty is known as second order uncertainty (e.g., Kleiter, 2018), and we currently know little about how classic findings in the judgment and decision-making literature apply under such conditions. Kahneman and Varey (1990) divided uncertainty along another dimension: internal uncertainty and external uncertainty (see also Juanchich et al, 2017).…”

“…Similarly, while second order uncertainty has been written about in the context of causal Bayesian reasoning within the judgment and decision-making literature (e.g., Gigerenzer and Hoffrage, 1995;Welsh and Navarro, 2012;Kleiter, 2018), reasoning under these conditions has rarely been studied, and experiments aiming to study real world reasoning have also typically done this using problems with only first order uncertainty. For example, in the classic taxi cab problem (Tversky and Kahneman, 1974;Bar-Hillel, 1980), solvers are asked to reason about whether a cab involved in a hit and run accident was from the 'blue' company (as opposed to the 'green') in light of a population base rate (which suggests green cabs are more common) and an eye witness report (which claims a blue cab was involved).…”

“…We created such a distribution in the ' AgenaRisk' Bayesian network program, which can be seen in Figure 1. We use a beta distribution (Kleiter, 2018) based upon the two nodes above it: FIGURE 1 | A beta distribution with mean 80.0% and standard deviation 16.3% created using the Agenarisk software.…”