## Contents

Probability logic & non-monotonic reasoning from an evolutionary perspective

Modeling human inference within the framework of probability logic

Thinking about counterfactual possibilities in middle-childhood

## Overall description

In the last decades many new approaches were developed in logic, artificial intelligence, probability theory and statistics, and in cognitive science. They were motivated by the fact that classical logic and traditional probabilistic methods were insufficient to describe knowledge about complex and uncertain environments. Most of these new approaches are “hybrid”, combining different disciplines. Moreover, many of them—like nonmonotonic logics—were “inspired” by the speculation that artificial systems should mimic human intelligence.

Psychology started to incorporate parts of these systems into theories on human reasoning. Typical examples are (i) the use of Bayesian networks in combination with structural equations to model human causal reasoning, the use of Bayesian probability to model human understanding of conditionals and inference, or Bayesian inference in perceptual object recognition. Only recently Bayesian models of cognition have become an important new approach in psychology. For those who want to mimic human intelligence by computers it is relevant to understand how humans are really making inferences and which principles govern the underlying processes.

Hybrid approaches make it difficult to investigate problems without expertise from different fields. The project therefore connects expertise from logic, mathematics, and cognitive psychology to investigate the following topics.

- Probability logic and nonmonotonic reasoning and its application to human inference.
- Game theoretic evaluation of nonmonotonic reasoning.
- Foundations of causality and causal reasoning.
- The representation of complex knowledge by (probabilistic) conditional independence structures (graphical versus compositional non-graphical models), local inference in complex systems, and noisy connectives.
- Counterfactual reasoning, its relevance in the context of causal reasoning, its treatment in probabilistic structures.
- The development of reasoning in children, especially the understanding of counterfactuals and its relationships to the development of a theory of mind.

The CRP brings together two logically (Schurz, Kistler), two mathematically (Jirousek, Gilio), and three psychologically (Kleiter, Perner, Beck) oriented individual projects. The applicants are from six countries, Germany, France, Czech Republic, Austria, Great Britain (Beck), and Italy (Gilio). What is common to them is the striving for a deeper understanding of general principles of reasoning. Central are theoretical and empirical questions of

- Rationality: coherence in probability (Gilio, Schurz, Kleiter) and in causal reasoning (Kistler); game-theoretic simulation of evolutionary efficiency (Schurz, Kleiter).
- Complexity: global versus local knowledge representation (Jirousek); causal relations in complex multi-level systems (Kistler); enumeration of essential graphs and structuring the according model space (Kleiter).
- Weak logical systems: probability logic, nonmonotonic reasoning (Schurz, Kleiter, Gilio), noisy connectives (Jirousek), possibilistic and belief function models (Jirousek), relationships to fuzzy approaches and t-norms (Gilio)
- Empirical status: Experimental studies of counterfactual reasoning in children (Perner, Beck), inference with argument forms central to probability logic and system P in adults (Schurz, Kleiter).

The individual projects are concerned with reasoning, inference, and knowledge representation. The “dimensions” along which the various parts investigate reasoning are

The general structure of knowledge and belief and the foundational principles of reasoning are investigated in the philosophical parts. Problems of implementation and algorithms are treated in the mathematical and computer science parts. Finally, human reasoning is experimentally investigated in children and adults in the psychological parts. Thus, there are three levels at which the “dimensions” are investigated,

The project has two kinds of aims and objectives:

- To deepen and to elaborate the status of research in several of these domains. The progress will be decisively facilitated by learning from neighboring domains. That is, collaboration will improve the deepening of the individual parts of the project.
- To broaden the research fields by elaborating the “higher level” topics that connect several dimensions and domains in the scheme given above. Such higher level topics are

## Probability logic and non-monotonic reasoning from an evolutionary perspective (Schurz)

Prof. Dr. Gerhard Schurz, Department of Philosophy, University of Düsseldorf, Germany, gerhard.schurz[AT]phil-fak.uni-duesseldorf.de

Based on our earlier work [23, 24, 25, 26, 27, 28, 29] the following three hypotheses shall be investigated:

- (1) The NMR-hypothesis
- Human reasoning with conditionals is in good accordance with the core rules of non-monotonic reasoning (NMR for short), especially with certain (fundamental or derived) rules of the system P. These rules have an underlying probabilistic semantics and, thus, are probabilistically reliable [1, 23, 24, 20]. Also other NMR-systems, weaker or stronger than P, shall be evaluated according to their fit with human conditional reasoning; especially stronger systems which generate default assumptions of relevance and irrelevance (e.g. [10, 23]).
- (2) Evolutionary Explanation of the NMR-Hypothesis
- The natural and social environment of human beings consists of self-regulatory systems which have been gradually selected by evolution, and whose behavior is not described by strict conditionals, but by normic conditionals of the form “if A, then normally B”, which entail high conditional probability assertions of the form P(B∣A) = high [25, 26, 27]. On this reason, humans’ conditional reasoning is well adapted to reasoning with uncertain and implicitly probabilistic conditionals in general. These reasoning mechanisms are not domain-specific but quite content-independent.
- (3) Evolutionary explanation of exceptions to theses (1+2)
- In the domain of human actions, when the goal is the detection of rule-breakings, humans reasoning tends to follow the monotonic rules of strict conditionals (e.g. Modus Tollens) to a high degree [8]. Similar findings have been made in other domains of deontic reasoning [21]. This fact has an evolutionary explanation in terms of the selection of humans’ ability to control the satisfaction of normative social requirements.

Theses (1+2) together with (3) are in support of the dual process theory of reasoning defended by Over [21, 122], which explains human reasoning by the double effect of content-and purpose-specific modules and higher cognitive abilities which are content- and purpose-independent. The three hypotheses shall be investigated both theoretically and experimentally.

## Causation, causal judgments, and causal reasoning (Kistler)

Prof. Dr. Max Kistler, Department of Philosophy, University of Grenoble, France, kistler[AT]free.fr

The project will bring together recent contributions on causation from philosophy, cognitive science, statistics, econometrics, computer science, and the social sciences. Its specific aims are:

- To inquire whether the graph-theoretic model of causal reasoning in terms of Bayesian networks can help clarify the philosophical debate on causation. We start from the recent synthesis of that work provided by Spirtes, Glymour, Scheines [30], Pearl [22], and Halpern [13].
- To investigate whether recent philosophical accounts of causation inspired by the graph-theoretic model (in particular, Woodward’s [32]) version of the manipulationist account, and Hitchcock’s [14], Yablo’s [33, 34], and Menzies’ [16] versions of a contrastive account of causation) can yield plausible results.
- To check whether the concept of causation can be applied to processes crossing levels of organization in complex systems.

## Conditional independence models (Jiroušek)

Prof. Dr. Radim Jiroušek, Czech Academy of Sciences, Czech Republic,
radim[AT]utia.cas.cz

Dr. Jiři Vomlel (collaborator), same department vomlel[AT]utia.cas.cz

We have shown that with a compositional operator graphical Markov models (Bayesian networks, for example) can be represented by sequences of probability distributions. We will extend our approach to achieve the following goals:

- We will further generalize our theoretical approach to wider classes of uncertainty representation. This will be done with respect to probabilistic models, to belief networks, and to possibilistic approaches.
- We will develop computational tools for the application compositional models. The representation of conditional independence models by sequences offers advantages for efficient algorithms to infer marginal distributions. For model construction, we will design heuristic data-based algorithms to construct compositional models from large data-bases.
- For conditional probability distributions with a local structure corresponding to noisy logical connectives we develop compact representations that allow efficient inference.
- We will develop tools that describe local knowledge in user friendly and simple ways for human processing.

## Modeling human inference within the framework of probability logic (Kleiter)

Prof. Dr. Gernot Kleiter, Department of Psychology, University of Salzburg, Austria,
gernot.kleiter[AT]sbg.ac.at, 0043-(0)676-4241570

Dr. Andy Fugard (postdoc), same department andy.fugard[AT]sbg.ac.at

Dr. Niki Pfeifer (collaborator), same department niki.pfeifer[AT]sbg.ac.at

See also the related project Mental Probability Logic (Austrian Research Fund, FWF).

We strive for a theory of human reasoning. The theory should provide both computational competence and descriptive validity. For computational competence we rely on probability logic. For the description of cognitive representations and processes we use principles of semantic networks and graphical models .

We will

- elaborate the theoretical foundations of the normative and descriptive parts of the approach,
- run experiments on human reasoning within the framework of coherence based mental probability logic,
- investigate probabilistic and logical dependencies/independencies in conditional independence models, and
- compare different models of reasoning in game-theoretic simulations.

The probabilistic approach to deductive reasoning claims that even “purely logical” tasks are solved as if they were tasks belonging to probability theory. The well-known Wason Selection Task is solved as if the human subjects would maximize information gain [18]. Syllogisms are solved as if the subjects would process Bayesian probabilities [19]. And argument forms like the modus ponens or the modus tollens are solved as if the subjects were experts in probability logic.

- The proposal takes probability logic [2, 12] based on the coherence approach of subjective probability as the basic reference theory. There are, however, different kinds of coherence, (“simple”) coherence [7], g-coherence [5] (weak coherence) and total coherence (strong coherence) [9, 17, 31]. It must be clarified which kind of coherence is relevant for which kind of psychological model building.
- Our main goals is the development of a psychological theory of reasoning.
The theory should explain the competence of human inference, avoiding ad
hoc principles or studying errors and biases only. Inferential competence is
systematically documented in logical systems. Therefor our motivation to
have a logical system in the background. Such a theory, however, should
also specify details about the cognitive representation and processing of
reasoning tasks.
Many properties of the propositional structure of logical inference can be represented by graphical networks. We assume that conditionals are a central building block in such networks (so that we have directed graphs), that the direction of inferences is important, and that affirmation of propositions facilitates and negation complicates [15] processing.

## Counterfactual reasoning in children (Perner)

Prof. Dr. Josef Perner, Department of Psychology, University of Salzburg, Austria,
josef.perner[AT]sbg.ac.at

Mag. Eva Rafetseder, same department eva.rafetseder[AT]sbg.ac.at

- Theoretical analysis of the difference between hypothetical reasoning (assuming something is the case, what else will be the case) and counterfactual reasoning (assuming that something else had been the case than was the case, then what else would be the case).
- Application of theoretical analysis to explain developmental differences in when children become able to reason counterfactually: They master different aspects of it.
- Linking the development of counterfactual reasoning to developments in causal reasoning and theory of mind (mental causation).

## Thinking about counterfactual possibilities in middle childhood (Beck)

Dr. Sarah Beck, Department of Psychology, University of Birmingham, UK,
s.r.beck[AT]bham.ac.uk

Dr. Kevin Riggs (collaborator), London Metropolitan University, London, UK, k.riggs[AT]londonmet.ac.uk

Dr. Patrick Burns (post-doc), Department of Psychology, University of Birmingham, UK, P.Burns.2[AT]bham.ac.uk

Our aim is to test a developmental account of counterfactual thinking. Broadly our thinking is that 3-4 year olds can think about alternative worlds, at 5 years children think of counterfactuals as possibilities, and still later that children can make comparisons between counterfactual and actual worlds (which give rise to counterfactual emotions).

When we miss trains or have to work late to meet a deadline, thoughts of how the world might have been otherwise often come to us. In the last decade developmental psychologists have explored when children under 4 years old imagine alternatives to reality. More recently, research has addressed developments in middle-childhood (between 5 and 10 years). We focus on how children come to understand counterfactuals as possible events that could have replaced the actual event. This relationship between the counterfactual and actual world is a central feature of adult counterfactual thinking [6]. In order to relate representations of two possible worlds, first, one needs to hold in mind two different world models (actual and counterfactual) and second, one needs to make comparisons between them, i.e. see where the actual and counterfactual worlds diverge. There are developments in children’s counterfactual thinking which may map onto these abilities:

- Holding in mind multiple possibilities
- Beck et al. [4] used a task in which children had to hold in mind what actually happened and what could have happened. The standard counterfactual requires only that children think about a single alternative outcome, not that they recognize that at one point in the past both were possible. This “late” development in counterfactual thinking, thinking of counterfactuals as possibilities, involves holding in mind multiple possible worlds.
- Comparisons between possibilities
- A second possibly later development in counterfactual thinking is understanding counterfactual emotions, such as regret and relief. These experiences rely on a comparison between reality and what could have happened. Children start to show an understanding of these counterfactual emotions between the ages of 6 and 9 [3]. Guttentag and Ferrell [11] read children short stories in which two characters experienced the same outcome (e.g. falling off a bike). One could have easily avoided the outcome (by taking a different route) whereas the other could not (they always took that route). Children under 7 did not recognize that the one for whom the counterfactual world contained a better outcome would feel worse.

## Probabilistic reasoning under coherence (Gilio)

Prof. Dr. Angelo Gilio, Dipartimento di Metodi e Modelli, Facolta’ di Ingegeneria, University di Roma “La Sapienza”, gilio[AT]dmmm.uniroma1.it

The contribution will deepen important aspects, related to probability logic and to human reasoning under uncertainty.

- It will investigate the probabilistic interpretation of inference rules in nonmonotonic reasoning.
- It will generalize some well known inference rules to the case of conditional random quantities.
- It will study the connection property of the sets of, respectively, coherent/g-coherent precise/imprecise probability and prevision assessments on arbitrary families of conditional events and conditional random quantities.
- Interpretation of the lower and upper bounds in the propagation of probability assessments in terms of t-norms and t-conorms.

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