Probabilistic argumentation systems: A new way to combine logic with probability

Probability is usually closely related to Boolean structures, i.e., Boolean algebras or propositional logic. Here we show, how probability can be combined with non-Boolean structures, and in particular non-Boolean logics. The basic idea is to describe uncertainty by (Boolean) assumptions, which may or may not be valid. The uncertain information depends then on these uncertain assumptions, scenarios or interpretations. We propose to describe information in information systems, as introduced by Scott into domain theory. This captures a wide range of systems of practical importance such as many propositional logics, first order logic, systems of linear equations, inequalities, etc. It covers thus both symbolic as well as numerical systems. Assumption-based reasoning allows then to deduce supporting arguments for hypotheses. A probability structure imposed on the assumptions permits to quantify the reliability of these supporting arguments and thus to introduce degrees of support for hypotheses. Information systems and related information algebras are formally introduced and studied in this paper as the basic structures for assumption-based reasoning. The probability structure is then formally represented by random variables with values in information algebras. Since these are in general non-Boolean structures some care must be exercised in order to introduce these random variables. It is shown that this theory leads to an extension of Dempster–Shafer theory of evidence and that information algebras provide in fact a natural frame for this theory.     

Advanced theory of mind in children with mild intellectual disability and deaf or hard of hearing children: A two-year longitudinal study in middle childhood

The present study investigates the development of advanced theory of mind (AToM) among typically developing (TD) children, children with mild intellectual disability (MID), and deaf or hard of hearing (DHH) children. The 2-year longitudinal study comprised three waves and included a large sample of children from Poland in middle childhood aged around 7.5–9.5 years (N = 779; M = 7.7, SD = 0.92 at wave 1). The analysis of children’s understanding of second-order false belief and the Faux-Pas Recognition Test showed that TD children outperformed children with MID and DHH children on both measures. At 7.5 years, almost 60% of the TD children correctly solved the second-order false belief task; correct performance at 7.5 years in children with MID and DHH children was 27 and 38% respectively. Two years later, correct performance rose to 80% (TD children), 45% (children with MID), and 63% (DHH children). Despite these differences, the speed of AToM development did not differ across the groups. The development of faux-pas recognition followed a non-linear pattern, with TD children showing no further significant development after mid-elementary school. Our findings show differences in AToM development between TD children, children with MID, and DHH children, and they suggest that children’s development of AToM may follow different developmental pathways, depending on the aspect of AToM under study.