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.
Highlights
Reasoning investigated the human understanding of material implications, propositional inference rules, inferences with quantifiers, and the validity of inference forms
The present paper proposes first steps toward a mental probability logic based on distributions
Four inference rules were often investigated in the psychology of human reasoning: The quartet of the MODUS PONENS, the MODUS TOLLENS and the argument forms of DENYING THE ANTECEDENT and AFFIRMING THE CONSEQUENT, here called “the MP-quartet” for short
Summary
Fifty years ago Peterson and Beach (1967) wrote a paper with the title “Man as an intuitive statistician.” In the time before the heuristics-and-biases paradigm human judgments and decisions were seen on the background of Baysian statistics. In the same time human reasoning was exclusively seen on the background of classical logic. One might have written a paper with the title “The human reasoner as an intuitive logician.”. Before that time reasoning research was exclusively done on the background of logical benchmarks, while judgment under uncertainty, was investigated on the background of probabilistic and decision theoretic benchmarks. Research on mental probability logic and the new (probabilistic) paradigm after the middle of the 1990s might have been published under the title “The human reasoner as an intuitive probabilist.”. The present paper proposes first steps toward a mental probability logic based on distributions. It employs secondorder probability distributions and some more recent concepts of modeling probabilistic dependence by copulas and vines. At the outset, give a short characterization of the beta family
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