Probabilistic stit logic and its decomposition

Decisions on which climate risks to accept and on how to use energy in the future affect every citizen's life. Public participation on these issues is necessary to find policies supported by the public. It is especially important because uncertainties about climate issues allow only descriptions in terms of subjective probabilities, invariably linked to subjective preferences. Subjective probabilities and preferences of experts can be balanced by those of ordinary citizens. IA-focus groups are a promising research method to explore citizens' views on climate change. We discuss this method and its application involving approx. 100 citizens in Switzerland. Three findings are presented. First, the discussion groups emphasize the moral dimension of man-made climate change. Second, they tend to see a future characterized by high levels of energy use as nightmarish, while they tend to see a future characterized by low levels of energy use as an attractive option. Third, while they see energy reduction as desirable, the willingness to engage in high-cost options like a hard-hitting carbon tax is practically nil. This apparent contradiction can be interpreted as follows: the possibility of major climate impacts ignites the public imagination, but subjective probabilities associated with it are rather low. Under these conditions, support for a low-energy society indicates that citizens expect such an option to be achievable at low cost. Rather than dismissing this expectation offhand as unrealistic, we argue that research on how low-cost options could lead to significant reduction in energy use would meet an important social need.

Reports an error in "Bridging the gap between subjective probability and probability judgments: The quantum sequential sampler" by Jiaqi Huang, Jerome R. Busemeyer, Zo Ebelt and Emmanuel M. Pothos (Psychological Review, 2025[Jul], Vol 132[4], 916-955; see record 2025-25219-001). The authors noticed a typo in Equation 17. In the matrix shown, the entry in the fourth row should be "(0 0 β+ … . .)" instead of "(0 β+ … . .)". A zero is missing from the fourth row. The online version of this article has been corrected. (The following abstract of the original article appeared in record 2025-25219-001.) One of the most important challenges in decision theory has been how to reconcile the normative expectations from Bayesian theory with the apparent fallacies that are common in probabilistic reasoning. Recently, Bayesian models have been driven by the insight that apparent fallacies are due to sampling errors or biases in estimating (Bayesian) probabilities. An alternative way to explain apparent fallacies is by invoking different probability rules, specifically the probability rules from quantum theory. Arguably, quantum cognitive models offer a more unified explanation for a large body of findings, problematic from a baseline classical perspective. This work addresses two major corresponding theoretical challenges: first, a framework is needed which incorporates both Bayesian and quantum influences, recognizing the fact that there is evidence for both in human behavior. Second, there is empirical evidence which goes beyond any current Bayesian and quantum model. We develop a model for probabilistic reasoning, seamlessly integrating both Bayesian and quantum models of reasoning and augmented by a sequential sampling process, which maps subjective probabilistic estimates to observable responses. Our model, called the Quantum Sequential Sampler, is compared to the currently leading Bayesian model, the Bayesian Sampler (J. Zhu et al., 2020) using a new experiment, producing one of the largest data sets in probabilistic reasoning to this day. The Quantum Sequential Sampler embodies several new components, which we argue offer a more theoretically accurate approach to probabilistic reasoning. Moreover, our empirical tests revealed a new, surprising systematic overestimation of probabilities. (PsycInfo Database Record (c) 2026 APA, all rights reserved).

Authors like Keynes, H. Jeffreys and Carnap advocated using a concept of "logical probability". Logical probability had the following properties: (a) it was representable as a function from potential states of full belief (or "evidence") to states of subjective or credal probability judgment. (b) Such functions were alleged to be constrained by principles of probability logic. (c) All rational agents were supposed to be obliged to adopt the standard function that probability logic prescribed. In this essay, it is argued that these three requirements could be satisfied only if probability logic prescribed that credal probability should be numerically determinate. Keynes denied that it should numerically determinate and Carnap abandoned the idea that probability logic could supply a determinate function from states of full belief to numerically determinate credal states that all rational agents ought to adopt. The paper explains that once this is conceded, logical probability ought to be interpreted rather differently than it is customarily is.

Muth’s Rational Expectations Hypothesis, That “…The Subjective Probability Distributions Are Distributed Around an Objective Probability Distribution, for a given Information Set…” is an Oxymoron. It Can Be Corrected by Amending It to Read “…The Subjective Probability Distributions Are Distributed Around a Subjective Consensus Probability Distribution, for a given Information Set…”

This handbook provides a systematic survey of the central areas of philosophical logic. It is divided into four volumes, each devoted to a major sub-field within the disciplines. The articles should be of value to general philosophers, linguists, logicians, mathematicians and computer scientists. Volume 1, Elements of Classical Logic, deals with the background to what has come to be considered the standard formulation of predicate logic - both as far as its semantics and proof theory are concerned. The central chapter on predicate logic is followed by chapters outlining various alternative, but essentially equivalent ways of constructing the semantics for first-order logic as well as its proof theory. In addition, this volume contains a discussion of higher-order extensions of first-order logic and a compendium of the algorithmic and decision-theoretic prerequisites in the study of logical systems. Volume 2, Extensions of Classical Logic, surveys the most significant extensions of predicate logic and their applications to various philosophical fields of inquiry. The 12 chapters in this volume together provide a succinct introduction to a variety of intensional frameworks, a discussion of the most well-known logical systems, as well as an overview of major applications and of the open problems in the respective fields. Volume 3, Alternatives to Classical Logic, consists of a series of surveys of some of the alternatives to the basic assumptions of classical logic. These include many-valued logic, partial logic, free logic, relevance and entailment logics, dialogue logic, quantum logic and intuitionism. Volume 4, Topics in the Philosophy of Language, presents a panorama of the applications of logical tools and methods in the formal analysis of natural language. Since a number of developments in philosophical logic were originally stimulated by concern arising in the semantic analysis of natural language discourse, the chapters in this volume provide some criteria of evaluation of the applications of work in philosophical logic. In revealing both the adequacies and inadequacies of logical investigations in the semantic structures of natural discourse, these chapters also point the way to future developments in philosophical logic in general and thus close again the circle of inquiry relating logic and language.

1 Modal logic: a semantic perspective

Probabilistic Logic over Paths

The handling of uncertain information is of crucial importance for the success of expert systems. This paper gives an overview on logic-based approaches to probabilistic reasoning and goes into more details about recent developments for relational, respectively first-order, probabilistic methods like Markov logic networks, and Bayesian logic programs. In particular, we feature the maximum entropy approach as a powerful and elegant method that combines convenience with respect to knowledge representation with excellent inference properties. While comparing the different approaches is a difficult task due to the variety of the available concepts and to the absence of a common interface, we address this problem from both a conceptual and practical point of view. On a conceptual layer we propose and discuss several criteria by which first-order probabilistic methods can be distinguished, and apply these criteria to a series of approaches. On the practical layer, we briefly describe some systems for probabilistic reasoning, and go into more details on the KReator system as a versatile toolbox for various approaches to first-order probabilistic relational learning, modelling, and reasoning. Moreover, we illustrate applications of probabilistic logics in various scenarios.

One-counter processes are pushdown processes over a singleton stack alphabet (plus a stack-bottom symbol). We study the problems of model checking asynchronous products of one-counter processes against 1) first-order logic FO(R) with reachability predicate, 2) the finite variable fragments FO k(R) (k ≥ 2) of FO(R), 3) EF-logic which is a fragment of FO 2(R), and 4) all these logics extended with simple component-wise synchronizing predicates. We give a rather complete picture of their combined, expression, and data complexity. To this end, we show that these problems are poly-time reducible to two syntactic restrictions of Presburger Arithmetic, which are equi-expressive with first-order modulo counting theory of (ℕ, <), for which we give optimal quantifier elimination procedures. In particular, these problems are all shown to be in PSPACE, which is in sharp contrast to the closely related problem of model checking FO(R) over pushdown processes (with one stack) which has nonelementary complexity. Finally, we apply our proof method to give a fixed automatic (and so rational) graph whose modal logic theory has nonelementary complexity, solving a recently posed open question.KeywordsModel CheckLinear OrderAtomic FormulaMembership ProblemModel Check ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

As a survey of many technical results in probability theory and probability logic, this monograph by two widely respected scholars offers a valuable compendium of the principal aspects of the formal study of probability. Hugues Leblanc and Peter Roeper explore probability functions appropriate for propositional, quantificational, intuitionistic, and infinitary logic and investigate the connections among probability functions, semantics, and logical consequence. They offer a systematic justification of constraints for various types of probability functions, in particular, an exhaustive account of probability functions adequate for first-order quantificational logic. The relationship between absolute and relative probability functions is fully explored and the book offers a complete account of the representation of relative functions by absolute ones. The volume is designed to review familiar results, to place these results within a broad context, and to extend the discussions in new and interesting ways. Authoritative, articulate, and accessible, it will interest mathematicians and philosophers at both professional and post-graduate levels.

Approaches for extending logic to deal with uncertainty immanent to many real-world problems are often on the one side purely qualitative, such as modal logics, or on the other side quantitative, such as probabilistic logics. Research on combinations of qualitative and quantitative extensions to logic which put qualitative constraints on probability distributions, has mainly remained theoretical until now. In this paper, we propose a practically useful logic, which supports qualitative as well as quantitative uncertainty and can be extended with modalities with varying level of quantitative precision. This language has a solid semantic foundation based on imprecise probability theory. While in general imprecise probabilistic inference is much harder than the precise case, this is the first expressive imprecise probabilistic formalism for which probabilistic inference is shown to be as hard as corresponding precise probabilistic problems. A second contribution of this paper is an inference algorithm for this language based on the translation to a weighted model counting (WMC) problem, an approach also taken by state-of-the-art probabilistic inference methods for precise problems.

Bayesian belief networks (also known asbelief networks,causal probabilistic networks,causal nets,graphical probability networks,probabilistic cause–effect models, andprobabilistic influence diagrams) provide decision‐support for a wide range of problems involving uncertainty and probabilistic reasoning. The underlying theory of BBNs is Bayesian probability theory and the notion of propagation. Although this has been around for a long time it is only since the late 1980s that efficient algorithms and tools to implement them have been developed to enable propagation in networks with a reasonable number of variables. The recent explosion of interest in BBNs is due to these developments, which mean that for the first time realistic size problems can be solved. These developments make BBNs an excellent method for reasoning about uncertainty. BBNs have proved useful in applications such as medical diagnosis and diagnosis of mechanical failures. Within the software engineering community the interest in BBNs is motivated by the need to integrate diverse types of evidence to provide rigorous quantitative dependability/safety arguments. BBNs enable us to incorporate not just the results from testing but also other evidence about the process and product; this evidence can be both objective and subjective. The BBN forces the assessor to expose all assumptions about the impact of different forms of evidence and hence provides a visible and auditable dependability or safety argument. This article provides a very brief introduction to BBNs and explains how they can be used for assessing critical systems.

The chapter is an attempt to systematize and to provide consistency and inconsistency results for axiomatic modal theories that arise in an expressively rich context. The study is carried out in diagonal modal logic, that is, modal operator logic with fixed points for boxed formulas, as studied by Smorynski. In modal operator logic the interdependencies of the various modal logics can be exploited, which allows for a systematization of the consistencies and inconsistencies of the modal systems depending on the modal properties they assume the modal notion to have. Toward the end of the chapter the focus is on the consistency and inconsistency of multimodal theories. In particular, it is investigated whether multimodal paradoxes are genuinely new paradoxes or whether they are reducible in some way to the unimodal paradoxes. To this end a notion of reducibility is developed and it is shown that there are both reducible and irreducible paradoxes.

We study the expressivity and the complexity of various logics in probabilistic team semantics with the Boolean negation. In particular, we study the extension of probabilistic independence logic with the Boolean negation, and a recently introduced logic first-order theory of random variables with probabilistic independence. We give several results that compare the expressivity of these logics with the most studied logics in probabilistic team semantics setting, as well as relating their expressivity to a numerical variant of second-order logic. In addition, we introduce novel entropy atoms and show that the extension of first-order logic by entropy atoms subsumes probabilistic independence logic. Finally, we obtain some results on the complexity of model checking, validity and satisfiability of our logics.

21 Modal logic and philosophy