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.     

Subgames within Large Games and the Heuristic of Imitation

We study repeated normal form games where the number of players is large. We argue that it is interesting to look at such games as being divided into subgames, each of which we call a neighbourhood. The structure of such a game is given by a graph G whose nodes are players and edges denote visibility. The neighbourhoods are maximal cliques in G. The game proceeds in rounds where in each round the players of every clique X of G play a strategic form game among each other. A player at a node v strategises based on what she can observe, i.e., the strategies and the outcomes in the previous round of the players at vertices adjacent to v. Based on this, the player may switch strategies in the same neighbourhood, or migrate to another neighbourhood. We are interested in addressing questions regarding the eventual stability of such games. We incrementally impose constraints on the ‘types’ of the players. First, we look at players who are unconstrained in their strategising abilities, in that, players who may use unbounded memory. We then consider the case of memoryless players. We show that in both these cases the eventual stability of the game can be characterised in terms of potentials. We then introduce a simple modal logic in which the types of the players can be specified. We show that when the players play according to these specified types, it can be effectively decided whether the game stabilises. Finally, we look at the important heuristic of imitation. Simple imitative strategies can be specified in the logic introduced by us. We show that in a population of optimisers and imitators, we can decide how ‘worse-off’ the imitators are by playing imitative strategies rather than optimal ones.     

Superordinate shape classification using natural shape statistics

This paper investigates the classification of shapes into broad natural categories such as animal or leaf. We asked whether such coarse classifications can be achieved by a simple statistical classification of the shape skeleton. We surveyed databases of natural shapes, extracting shape skeletons and tabulating their parameters within each class, seeking shape statistics that effectively discriminated the classes. We conducted two experiments in which human subjects were asked to classify novel shapes into the same natural classes. We compared subjects’ classifications to those of a naive Bayesian classifier based on the natural shape statistics, and found good agreement. We conclude that human superordinate shape classifications can be well understood as involving a simple statistical classification of the shape skeleton that has been “tuned” to the natural statistics of shape.     

A rule-based presentation order facilitates category learning

We investigated the mechanisms by which concepts are learned from examples by manipulating the presentation order in which the examples were presented to subjects. We introduce the idea of a rule-based presentation order, which is a sequence that respects the internal organization of the examples within a category. We find that such an order substantially facilitates learning, as compared with previously known beneficial orders, such as a similarity-based order. We discuss this result in light of the central distinction between rule-based and similaritybased learning models.     

Adaptation of CBT for Traumatized Egyptians: Examples from Culturally Adapted CBT (CA-CBT)

In this article we illustrate how CBT can be adapted to a traumatized Egyptian population with Islamic beliefs, giving examples from our adaptation of Culturally Adapted–CBT (CA-CBT) for this cultural group. We discuss a culturally sensitive assessment measure of local somatic complaints and cultural syndromes that was devised based on clinical experience with traumatized Egyptians. We also demonstrate how to normalize symptoms, create positive expectancy about the treatment, and educate about trauma. We give examples of how mindfulness can be adapted for an Egyptian Islamic population, and we describe local religious strategies, such as dhikr (religious chanting), salah (ritualistic prayer), and dua (supplication), that may be used to promote attentional shift from rumination topics and to teach attentional control. We describe how “loving kindness” can be adapted for this group. We outline how to modify culturally generated catastrophic cognitions and how to conduct interoceptive exposure and to create positive re-associations in a culturally sensitive manner. We describe how worry themes are explored and addressed based on a heuristic panic attack–PTSD model; how to teach anger management in a culturally sensitive way; and how to address sleep-related problems in this population. We suggest using cultural transitional “rituals” at the end of the treatment to give patients a sense of closure and a positive feeling of transformation. A case example is presented to illustrate cultural challenges associated with delivering CA-CBT to an Egyptian population. We introduce certain concepts such as cultural grounding and explanatory model bridging, both therapeutic techniques that increase adherence, positive expectancy, and cultural consonance.     

Agreement Theorems in Dynamic-Epistemic Logic

This paper introduces Agreement Theorems to dynamic-epistemic logic. We show first that common belief of posteriors is sufficient for agreement in epistemic-plausibility models, under common and well-founded priors. We do not restrict ourselves to the finite case, showing that in countable structures the results hold if and only if the underlying plausibility ordering is well-founded. We then show that neither well-foundedness nor common priors are expressible in the language commonly used to describe and reason about epistemic-plausibility models. The static agreement result is, however, finitely derivable in an extended modal logic. We provide the full derivation. We finally consider dynamic agreement results. We show they have a counterpart in epistemic-plausibility models, and provide a new form of agreements via public announcements.     

A Calculus of Substitutions for DPL

We consider substitutions in order sensitive situations, having in the back of our minds the case of dynamic predicate logic (DPL) with a stack semantics. We start from the semantic intuition that substitutions are move instructions on stacks: the syntactic operation [y/x] is matched by the instruction to move the value of the y-stack to the x-stack. We can describe these actions in the positive fragment of DPLE. Hence this fragment counts as a logic for DPL-substitutions. We give a calculus for the fragment and prove soundness and completeness.     

Mathematical concepts for analyzing ambiguous situations

Abstract

We examine some mathematical tools for dealing with ambiguous situations. The main tool is the use of non-standard logic with truth-values in what is called a locale. This approach is related to fuzzy set theory, which we briefly discuss. We also consider probabilistic concepts. We include specific examples and describe the way a researcher can set up a suitable locale to analyse a concrete situation.     

Forgetting of Foreign‐Language Skills: A Corpus‐Based Analysis of Online Tutoring Software

We explore the nature of forgetting in a corpus of 125,000 students learning Spanish using the Rosetta Stone^® foreign‐language instruction software across 48 lessons. Students are tested on a lesson after its initial study and are then retested after a variable time lag. We observe forgetting consistent with power function decay at a rate that varies across lessons but not across students. We find that lessons which are better learned initially are forgotten more slowly, a correlation which likely reflects a latent cause such as the quality or difficulty of the lesson. We obtain improved predictive accuracy of the forgetting model by augmenting it with features that encode characteristics of a student's initial study of the lesson and the activities the student engaged in between the initial and delayed tests. The augmented model can predict 23.9% of the variance in an individual's score on the delayed test. We analyze which features best explain individual performance.     

A New Perspective Concerning Experiments on Semantic Intuitions

Machery, Mallon, Nichols, and Stich [2004; forthcoming] use experimental methods to raise a spectre of doubt about reliance on intuitions in developing theories of reference which are then deployed in philosophical arguments outside the philosophy of language. Machery et al. ran a cross-cultural survey asking Western and East Asian participants about a famous case from the philosophical literature on reference (Kripke's Gödel example). They interpret their results as indicating that there is significant variation in participants' intuitions about semantic reference for that case. We argue that this interpretation is mistaken. We detail a type of ambiguity found in Machery et al.'s probe but not yet noted in the response literature. We argue that this epistemic ambiguity could have affected their results. We do not stop there, however: Rather than rest content with a possibility claim, we ran four studies to test the impact of this ambiguity on participants' responses. We found that this accounts for much of the variation in Machery et al.'s original experiment. We conclude that in the light of our new data, their argument is no longer convincing.     

Modelling simultaneous games in dynamic logic

We make a proposal for formalizing simultaneous games at the abstraction level of player’s powers, combining ideas from dynamic logic of sequential games and concurrent dynamic logic. We prove completeness for a new system of ‘concurrent game logic’ CDGL with respect to finite non-determined games. We also show how this system raises new mathematical issues, and throws light on branching quantifiers and independence-friendly evaluation games for first-order logic.     

A New Game Equivalence,its Logic and Algebra

We present a new notion of game equivalence that captures basic powers of interacting players. We provide a representation theorem, a complete logic, and a new game algebra for basic powers. In doing so, we establish connections with imperfect information games and epistemic logic. We also identify some new open problems concerning logic and games.

     

Reasoning Continuously: A Formal Construction of Continuous Proofs

We begin with the idea that lines of reasoning are continuous mental processes and develop a notion of continuity in proof. This requires abstracting the notion of a proof as a set of sentences ordered by provability. We can then distinguish between discrete steps of a proof and possibly continuous stages, defining indexing functions to pick these out. Proof stages can be associated with the application of continuously variable rules, connecting continuity in lines of reasoning with continuously variable reasons. Some examples of continuous proofs are provided. We conclude by presenting some fundamental facts about continuous proofs, analogous to continuous structural rules and composition. We take this to be a development on its own, as well as lending support to non-finitistic constructionism.

     

Embedding Logics into Product Logic

We construct a faithful interpretation of ukasiewicz's logic in product logic (both propositional and predicate). Using known facts it follows that the product predicate logic is not recursively axiomatizable.We prove a completeness theorem for product logic extended by a unary connective of Baaz [1]. We show that Gödel's logic is a sublogic of this extended product logic.We also prove NP-completeness of the set of propositional formulas satisfiable in product logic (resp. in Gödel's logic).     

The “Sprinter effect”: When self-control and involvement stand in the way of sequential performance

This research examines the joint effect of dispositional self-control and situational involvement on performance in two successive resource-demanding tasks. We demonstrate that being highly involved and having high self-control facilitates high performance in the first task but, contrary to intuition, may jeopardize performance in a second, unexpected task. We term this the “sprinter effect” and demonstrate it in both lab and field settings. We further explore how a “marathon” mindset can debias this effect.     

Bayesian inference for psychology,part III: Parameter estimation in nonstandard models

We demonstrate the use of three popular Bayesian software packages that enable researchers to estimate parameters in a broad class of models that are commonly used in psychological research. We focus on WinBUGS, JAGS, and Stan, and show how they can be interfaced from R and MATLAB. We illustrate the use of the packages through two fully worked examples; the examples involve a simple univariate linear regression and fitting a multinomial processing tree model to data from a classic false-memory experiment. We conclude with a comparison of the strengths and weaknesses of the packages. Our example code, data, and this text are available via https://osf.io/ucmaz/.