Sonja Smets

A qualitative theory of dynamic interactive belief revision

We present a logical setting that incorporates a belief-revision mechanism within Dynamic-Epistemic logic. As the “static” basis for belief revision, we use epistemic plausibility models, together with a modal language based on two epistemic operators: a “knowledge” modality K (the standard S5, fully introspective, notion), and a “safe belief” modality □ (“weak”, non-negatively-introspective, notion, capturing a version of Lehrer’s “indefeasible knowledge”). To deal with “dynamic” belief revision, we introduce action plausibility models, representing various types of “doxastic events”. Action models “act” on state models via a modified update product operation: the “Action-Priority” Update. This is the natural dynamic generalization of AGM revision, giving priority to the incoming information (i.e., to “actions”) over prior beliefs. We completely axiomatize this logic, and show how our update mechanism can “simulate”, in a uniform manner, many different belief-revision policies.

Keep ‘hoping’ for rationality: a solution to the backward induction paradox

We formalise a notion of dynamic rationality in terms of a logic of conditional beliefs on (doxastic) plausibility models. Similarly to other epistemic statements (e.g. negations of Moore sentences and of Muddy Children announcements), dynamic rationality changes its meaning after every act of learning, and it may become true after players learn it is false. Applying this to extensive games, we “simulate” the play of a game as a succession of dynamic updates of the original plausibility model: the epistemic situation when a given node is reached can be thought of as the result of a joint act of learning (via public announcements) that the node is reached. We then use the notion of “stable belief”, i.e. belief that is preserved during the play of the game, in order to give an epistemic condition for backward induction: rationality and common knowledge of stable belief in rationality. This condition is weaker than Aumann’s and compatible with the implicit assumptions (the “epistemic openness of the future”) underlying Stalnaker’s criticism of Aumann’s proof. The “dynamic” nature of our concept of rationality explains why our condition avoids the apparent circularity of the “backward induction paradox”: it is consistent to (continue to) believe in a player’s rationality after updating with his irrationality.

Probabilistic dynamic belief revision

We investigate the discrete (finite) case of the Popper–Renyi theory of conditional probability, introducing discrete conditional probabilistic models for knowledge and conditional belief, and comparing them with the more standard plausibility models. We also consider a related notion, that of safe belief, which is a weak (non-negatively introspective) type of “knowledge”. We develop a probabilistic version of this concept (“degree of safety”) and we analyze its role in games. We completely axiomatize the logic of conditional belief, knowledge and safe belief over conditional probabilistic models. We develop a theory of probabilistic dynamic belief revision, introducing probabilistic “action models” and proposing a notion of probabilistic update product, that comes together with appropriate reduction laws.

A Quantum Structure Description of the Liar Paradox

In this article we propose an approach thatmodels the truth behavior of cognitive entities (i.e.,sets of connected propositions) by taking into accountin a very explicit way the possible influence of the cognitive person (the one that interactswith the considered cognitive entity). Hereby wespecifically apply the mathematical formalism of quantummechanics because of the fact that this formalism allows the description of real contextual influences,i.e., the influence of the measuring apparatus on thephysical entity. We concentrated on the typicalsituation of the liar paradox and have shown that (1) the true-false state of this liar paradox canbe represented by a quantum vector of the nonproducttype in a finite-dimensional complex Hilbert space andthe different cognitive interactions by the actions of the corresponding quantum projections, (2)the typical oscillations between false and true —the paradox — is now quantum dynamically describedby a Schrodinger equation. We analyze possiblephilosophical implications of this result.

Quantum logic as a dynamic logic

We address the old question whether a logical understanding of Quantum Mechanics requires abandoning some of the principles of classical logic. Against Putnam and others (Among whom we may count or not E. W. Beth, depending on how we interpret some of his statements), our answer is a clear “no”. Philosophically, our argument is based on combining a formal semantic approach, in the spirit of E. W. Beth’s proposal of applying Tarski’s semantical methods to the analysis of physical theories, with an empirical–experimental approach to Logic, as advocated by both Beth and Putnam, but understood by us in the view of the operational- realistic tradition of Jauch and Piron, i.e. as an investigation of “the logic of yes–no experiments” (or “questions”). Technically, we use the recently-developed setting of Quantum Dynamic Logic (Baltag and Smets 2005, 2008) to make explicit the operational meaning of quantum-mechanical concepts in our formal semantics. Based on our recent results (Baltag and Smets 2005), we show that the correct interpretation of quantum-logical connectives is dynamical, rather than purely propositional. We conclude that there is no contradiction between classical logic and (our dynamic reinterpretation of) quantum logic. Moreover, we argue that the Dynamic-Logical perspective leads to a better and deeper understanding of the “non-classicality” of quantum behavior than any perspective based on static Propositional Logic.

A Dynamic-Logical Perspective on Quantum Behavior

In this paper we show how recent concepts from Dynamic Logic, and in particular from Dynamic Epistemic logic, can be used to model and interpret quantum behavior. Our main thesis is that all the non-classical properties of quantum systems are explainable in terms of the non-classical flow of quantum information. We give a logical analysis of quantum measurements (formalized using modal operators) as triggers for quantum information flow, and we compare them with other logical operators previously used to model various forms of classical information flow: the “test” operator from Dynamic Logic, the “announcement” operator from Dynamic Epistemic Logic and the “revision” operator from Belief Revision theory. The main points stressed in our investigation are the following: (1) The perspective and the techniques of “logical dynamics” are useful for understanding quantum information flow. (2) Quantum mechanics does not require any modification of the classical laws of “static” propositional logic, but only a non-classical dynamics of information. (3) The main such non-classical feature is that, in a quantum world, all information-gathering actions have some ontic side-effects. (4) This ontic impact can affect in its turn the flow of information, leading to non-classical epistemic side-effects (e.g. a type of non-monotonicity) and to states of “objectively imperfect information”. (5) Moreover, the ontic impact is non-local: an information-gathering action on one part of a quantum system can have ontic side-effects on other, far-away parts of the system.

The Liar-paradox in a Quantum Mechanical Perspective

In this paper we concentrate on the nature of the liar paradox asa cognitive entity; a consistently testable configuration of properties. We elaborate further on a quantum mechanical model (Aerts, Broekaert and Smets, 1999) that has been proposed to analyze the dynamics involved, and we focus on the interpretation and concomitant philosophical picture. Some conclusions we draw from our model favor an effective realistic interpretation of cognitive reality.

Belief revision as a truth-tracking process

We analyze the learning power of iterated belief revision methods, and in particular their universality: whether or not they can learn everything that can be learnt. We look in particular at three popular methods: conditioning, lexicographic revision and minimal revision. Our main result is that conditioning and lexicographic revision are universal on arbitrary epistemic states, provided that the observational setting is sound and complete (only true data are observed, and all true data are eventually observed) and provided that a non-standard (non-well-founded) prior plausibility relation is allowed. We show that a standard (well-founded) belief-revision setting is in general too narrow for this. We also show that minimal revision is not universal. Finally, we consider situations in which observational errors (false observations) may occur. Given a fairness condition (saying that only finitely many errors occur, and that every error is eventually corrected), we show that lexicographic revision is still universal in this setting, while the other two methods are not.

On the Origin of Probabilities in Quantum Mechanics: Creative and Contextual Aspects

At the beginning of this century, a lot of changes as well in the field of politics, art and sciences have led to a change of paradigms and ways in which people think and interact ‘(the/their) world’. That climate of change lasted the whole century, due to the inertia of old ideas and the required time people needed to build ‘new images’ (or even, new world views) that incorporate the new findings. Nonetheless, as soon as old paradigms get overruled, new ones appear. This is something which also happened after the ‘invention’ of quantum physics. The modernist deterministic world guided by Laplace’s prime intelligence had to make place for one in which appear probabilities that have a mysterious status, as we will explain now. The probabilities of classical statistical theories [1, 3], e.g., statistical, mechanics, thermodynamics, classical probability calculus, have never been considered to be an obscure subject, because they can be explained as being due to a lack of knowledge about an eventual deterministic underlying reality. So, these classical probabilities are only a mathematical formalization of the lack of knowledge about the system under study. When quantum mechanics was born as an intrinsic probabilistic theory, the question was raised rapidly of whether these quantum probabilities [5, 10, 13, 17] can also be explained as due to a lack of knowledge. The field of research investigating this problem was referred to as the search for hidden variable theories, the hidden variables describing this so called deterministic underlying reality. During the years many theorems (e.g., the famous no-go theorem of J. von Neumann [4], or its elaborations [6, 7, 8].) have shown that hidden variable theories for quantum mechanics are impossible, indicating that quantum probabilities are of a fundamentally different nature than classical probabilities and seemingly not due to a lack of knowledge. Some physicists formulated very clearly their opinion: quantum mechanical probabilities are ontologically present in reality itself. These ontological (or objective) probabilities destroyed the classical picture of the world in such a way that the search for an image of what really happens in the ‘physical world’ had been abandoned, and still is so in many fields of micro-physics. As such, a large quantity of the contemporary community of physicists consider ‘real’ physics as something definitely complementary to anything to be understood as possibly eligible by ‘realism’. Without going into any debate on this, we will show that it is indeed possible to find a picture of quantum entities where these ‘strange’ probabilities are explained. In fact, even formally, we only have to introduce a specific new concept within the theory of physics, that was not explicitly present before, namely a model of aspects of creation.

The Dynamic Turn in Quantum Logic

In this paper we show how ideas coming from two areas of research in logic can reinforce each other. The first such line of inquiry concerns the “dynamic turn” in logic and especially the formalisms inspired by Propositional Dynamic Logic (PDL); while the second line concerns research into the logical foundations of Quantum Physics, and in particular the area known as Operational Quantum Logic, as developed by Jauch and Piron (Helve Phys Acta 42:842–848, 1969), Piron (Foundations of Quantum Physics, 1976). By bringing these areas together we explain the basic ingredients of Dynamic Quantum Logic, a new direction of research in the logical foundations of physics.