Following some ideas of Roberto Magari, we propose trial and error probabilistic functions, i.e. probability measures on the sentences of arithmetic that evolve in time by trial and error. The set ℐ of the sentences that get limit probability 1 is a Π3—theory, in fact ℐ can be a Π3—complete set. We prove incompleteness results for this setting, by showing for instance that for every k > 0 there are true Π3—sentences that get limit probability less than 1/2k. No (...) set ℐ as above can contain the set of all true Π3—sentences, although we exhibit some ℐ containing all the true Σ2—sentences. We also consider an approach based on the notions of inner probability and outer probability, and we compare this approach with the one based on trial and error probabilistic functions. Although the two approaches are shown to be different, we single out an important case in which they are equivalent. (shrink)

Systems of logico-probabilistic (LP) reasoning characterize inference from conditional assertions interpreted as expressing high conditional probabilities. In the present article, we investigate four prominent LP systems (namely, systems O, P, Z, and QC) by means of computer simulations. The results reported here extend our previous work in this area, and evaluate the four systems in terms of the expected utility of the dispositions to act that derive from the conclusions that the systems license. In addition to conforming to the (...) dominant paradigm for assessing the rationality of actions and decisions, our present evaluation complements our previous work, since our previous evaluation may have been too severe in its assessment of inferences to false and uninformative conclusions. In the end, our new results provide additional support for the conclusion that (of the four systems considered) inference by system Z offers the best balance of error avoidance and inferential power. Our new results also suggest that improved performance could be achieved by a modest strengthening of system Z. (shrink)

The book can also be used as an excellent text for graduate-level courses in AI, operations research, or applied probability.

We study probabilistic logic under the viewpoint of the coherence principle of de Finetti. In detail, we explore how probabilistic reasoning under coherence is related to model- theoretic probabilistic reasoning and to default reasoning in System . In particular, we show that the notions of g-coherence and of g-coherent entailment can be expressed by combining notions in model-theoretic probabilistic logic with concepts from default reasoning. Moreover, we show that probabilistic reasoning under coherence is a (...) generalization of default reasoning in System . That is, we provide a new probabilistic semantics for System , which neither uses infinitesimal probabilities nor atomic bound (or big-stepped) probabilities. These results also provide new algorithms for probabilistic reasoning under coherence and for default reasoning in System , and they give new insight into default reasoning with conditional objects. (shrink)

A new probabilistic-informational concept, earlier constructed by Mugur-Schächter, is further developed. Associated with Jaynes's principle, this concept permits one to define a measure for the distance between the state of a system evolving under stable constraints and the equilibrium with these constraints. An illustration is given for a gas evolving in a thermostated box. It appears that the free energy of the gas estimates the distance to equilibrium, the estimation being defined in abstract informational-probabilistic terms.

Los's probability semantics are used to identify the appropriate probability conditional for use in probabilistic explanations. This conditional is shown to have applications to probabilistic reasoning in expert systems. The reasoning scheme of the system MYCIN is shown to be probabilistically invalid; however, it is shown to be "close" to a probabilistically valid inference scheme.

Probabilistic reasoning is traditionally represented by inferences of the following form (also called probabilistic explanations):where A and B are one-place predicates in a first order language, P(A | B) is the conditional probability of observing A among individuals having property B, and q is close to one.This argument is not logically valid, as the premises may be true while the conclusion is false. Moreover, as it stands, the premises do not even make the conclusion plausible. It may be (...) the case that 90% of the population dies before the age of 85, and that individual j is a member of the population, but this in itself does not make it plausible that individual j will die before the age of 85. (shrink)

The paper describes and proves completeness theorems for a series of proof systems formalizing common sense reasoning about uncertain knowledge in the case where this consists of sets of linear constraints on a probability function.

In this paper, we present a probabilistic-based method to predict malaria disease at an early stage. Malaria is a very dangerous disease that creates a lot of health problems. Therefore, there is a need for a system that helps us to recognize this disease at early stages through the visual symptoms and from the environmental data. In this paper, we proposed a Bayesian network model to predict the occurrences of malaria disease. The proposed BN model is built on (...) different attributes of the patient’s symptoms and environmental data which are divided into training and testing parts. Our proposed BN model when evaluated on the collected dataset found promising results with an accuracy of 81%. One the other hand, F1 score is also a good evaluation of these probabilistic models because there is a huge variation in class data. The complexity of these models is very high due to the increase of parent nodes in the given influence diagram, and the conditional probability table also becomes more complex. (shrink)

The paper describes and proves completeness theorems for a series of proof systems formalizing common sense reasoning about uncertain knowledge in the case where this consists of sets of linear constraints on a probability function.

Volume 18, Issue 5, October 2019, Page 475-486.

Remarkable progress in the mathematics and computer science of probability has led to a revolution in the scope of probabilistic models. In particular, ‘sophisticated’ probabilistic methods apply to structured relational systems such as graphs and grammars, of immediate relevance to the cognitive sciences. This Special Issue outlines progress in this rapidly developing field, which provides a potentially unifying perspective across a wide range of domains and levels of explanation. Here, we introduce the historical and conceptual foundations of the (...) approach, explore how the approach relates to studies of explicit probabilistic reasoning, and give a brief overview of the field as it stands today. (shrink)

Conceptual combination performs a fundamental role in creating the broad range of compound phrases utilised in everyday language. This article provides a novel probabilistic framework for assessing whether the semantics of conceptual combinations are compositional, and so can be considered as a function of the semantics of the constituent concepts, or not. While the systematicity and productivity of language provide a strong argument in favor of assuming compositionality, this very assumption is still regularly questioned in both cognitive science and (...) philosophy. Additionally, the principle of semantic compositionality is underspecifi ed, which means that notions of both "strong" and "weak" compositionality appear in the literature. Rather than adjudicating between different grades of compositionality, the framework presented here contributes formal methods for determining a clear dividing line between compositional and non-compositional semantics. In addition, we suggest that the distinction between these is contextually sensitive. Compositionality is equated with a joint probability distribution modeling how the constituent concepts in the combination are interpreted. Marginal selectivity is introduced as a pivotal probabilistic constraint for the application of the Bell/CH and CHSH systems of inequalities. Non-compositionality is equated with a failure of marginal selectivity, or violation of either system of inequalities in the presence of marginal selectivity. This means that the conceptual combination cannot be modeled in a joint probability distribution, the variables of which correspond to how the constituent concepts are being interpreted. The formal analysis methods are demonstrated by applying them to an empirical illustration of twenty-four non-lexicalised conceptual combinations. (shrink)

We introduce a probabilistic extension of propositional intuitionistic logic. The logic allows making statements such as P≥sα, with the intended meaning “the probability of truthfulness of α is at least s”. We describe the corresponding class of models, which are Kripke models with a naturally arising notion of probability, and give a sound and complete infinitary axiomatic system. We prove that the logic is decidable.

This is part A of a paper in which we defend a semantics for counterfactuals which is probabilistic in the sense that the truth condition for counterfactuals refers to a probability measure. Because of its probabilistic nature, it allows a counterfactual ‘ifAthenB’ to be true even in the presence of relevant ‘Aand notB’-worlds, as long such exceptions are not too widely spread. The semantics is made precise and studied in different versions which are related to each other by (...) representation theorems. Despite its probabilistic nature, we show that the semantics and the resulting system of logic may be regarded as a naturalistically vindicated variant of David Lewis’ truth-conditional semantics and logic of counterfactuals. At the same time, the semantics overlaps in various ways with the non-truth-conditional suppositional theory for conditionals that derives from Ernest Adams’ work. We argue that counterfactuals have two kinds of pragmatic meanings and come attached with two types of degrees of acceptability or belief, one being suppositional, the other one being truth based as determined by our probabilistic semantics; these degrees could not always coincide due to a new triviality result for counterfactuals, and they should not be identified in the light of their different interpretation and pragmatic purpose. However, for plain assertability the difference between them does not matter. Hence, if the suppositional theory of counterfactuals is formulated with sufficient care, our truth-conditional theory of counterfactuals is consistent with it. The results of our investigation are used to assess a claim considered by Hawthorne and Hájek, that is, the thesis that most ordinary counterfactuals are false. (shrink)

According to Bayesian epistemology, rational learning from experience is consistent learning, that is learning should incorporate new information consistently into one's old system of beliefs. Simon M. Huttegger argues that this core idea can be transferred to situations where the learner's informational inputs are much more limited than Bayesianism assumes, thereby significantly expanding the reach of a Bayesian type of epistemology. What results from this is a unified account of probabilistic learning in the tradition of Richard Jeffrey's 'radical (...) probabilism'. Along the way, Huttegger addresses a number of debates in epistemology and the philosophy of science, including the status of prior probabilities, whether Bayes' rule is the only legitimate form of learning from experience, and whether rational agents can have sustained disagreements. His book will be of interest to students and scholars of epistemology, of game and decision theory, and of cognitive, economic, and computer sciences. (shrink)

This paper proposes an approach to investigate norm-governed learning agents which combines a logic-based formalism with an equation-based counterpart. This dual formalism enables us to describe the reasoning of such agents and their interactions using argumentation, and, at the same time, to capture systemic features using equations. The approach is applied to norm emergence and internalisation in systems of learning agents. The logical formalism is rooted into a probabilistic defeasible logic instantiating Dung’s argumentation framework. Rules of this logic are (...) attached with probabilities to describe the agents’ minds and behaviours as well as uncertain environments. Then, the equation-based model for reinforcement learning, defined over this probability distribution, allows agents to adapt to their environment and self-organise. (shrink)

The paper offers a formalization of reasoning about distributed multi-agent systems. The presented propositional probabilistic temporal epistemic logic |$\textbf {PTEL}$| is developed in full detail: syntax, semantics, soundness and strong completeness theorems. As an example, we prove consistency of the blockchain protocol with respect to the given set of axioms expressed in the formal language of the logic. We explain how to extend |$\textbf {PTEL}$| to axiomatize the corresponding first-order logic.

In previous work, we studied four well known systems of qualitative probabilistic inference, and presented data from computer simulations in an attempt to illustrate the performance of the systems. These simulations evaluated the four systems in terms of their tendency to license inference to accurate and informative conclusions, given incomplete information about a randomly selected probability distribution. In our earlier work, the procedure used in generating the unknown probability distribution (representing the true stochastic state of the world) tended to (...) yield probability distributions with moderately high entropy levels. In the present article, we present data charting the performance of the four systems when reasoning in environments of various entropy levels. The results illustrate variations in the performance of the respective reasoning systems that derive from the entropy of the environment, and allow for a more inclusive assessment of the reliability and robustness of the four systems. (shrink)

Probabilistic argumentation and neuro-argumentative systems offer new computational perspectives for the theory and applications of argumentation, but their principled construction involves two entangled problems. On the one hand, probabilistic argumentation aims at combining the quantitative uncertainty addressed by probability theory with the qualitative uncertainty of argumentation, but probabilistic dependences amongst arguments as well as learning are usually neglected. On the other hand, neuro-argumentative systems offer the opportunity to couple the computational advantages of learning and massive parallel computation (...) from neural networks with argumentative reasoning and explanatory abilities, but the relation of probabilistic argumentation frameworks with these systems has been ignored so far. Towards the construction of neuro-argumentative systems based on probabilistic argumentation, we associate a model of abstract argumentation and the graphical model of Boltzmann machines in order to (i... (shrink)

We study the existence of efficient approximation methods to verify quantitative specifications of probabilistic systems. Models of such systems are labelled discrete time Markov chains and checking specifications consists of computing satisfaction probabilities of linear temporal logic formulas. We prove that, in general, there is no polynomial time randomized approximation scheme with relative error for probabilistic verification. However, in many applications, specifications can be expressed by monotone formulas or negation of monotone formulas and randomized approximation schemes with absolute (...) error are sufficient. We show how some simple randomized approximation algorithms can improve the efficiency of verification of such probabilistic specifications and can be implemented in a probabilistic model checker. (shrink)

The pragmatic character of the Dutch book argument makes it unsuitable as an "epistemic" justification for the fundamental probabilist dogma that rational partial beliefs must conform to the axioms of probability. To secure an appropriately epistemic justification for this conclusion, one must explain what it means for a system of partial beliefs to accurately represent the state of the world, and then show that partial beliefs that violate the laws of probability are invariably less accurate than they could be (...) otherwise. The first task can be accomplished once we realize that the accuracy of systems of partial beliefs can be measured on a gradational scale that satisfies a small set of formal constraints, each of which has a sound epistemic motivation. When accuracy is measured in this way it can be shown that any system of degrees of belief that violates the axioms of probability can be replaced by an alternative system that obeys the axioms and yet is more accurate in every possible world. Since epistemically rational agents must strive to hold accurate beliefs, this establishes conformity with the axioms of probability as a norm of epistemic rationality whatever its prudential merits or defects might be. (shrink)

The aim of the paper is to develop the notion of partial probability distributions as being more realistic models of belief systems than the standard accounts. We formulate the theory of partial probability functions independently of any classical semantic notions. We use the partial probability distributions to develop a formal semantics for partial propositional calculi, with extensions to predicate logic and higher order languages. We give a proof theory for the partial logics and obtain soundness and completeness results.

We present new probabilistic generalizations of Pearl’s entailment in System Z and Lehmann’s lexicographic entailment, called Zλ- and lexλ-entailment, which are parameterized through a value λ ∈ [0,1] that describes the strength of the inheritance of purely probabilistic knowledge. In the special cases of λ = 0 and λ = 1, the notions of Zλ- and lexλ-entailment coincide with probabilistic generalizations of Pearl’s entailment in System Z and Lehmann’s lexicographic entailment that have been recently introduced (...) by the author. We show that the notions of Zλ- and lexλ-entailment have similar properties as their classical counterparts. In particular, they both satisfy the rationality postulates of System P and the property of Rational Monotonicity. Moreover, Zλ-entailment is weaker than lexλ-entailment, and both Zλ- and lexλ-entailment are proper generalizations of their classical counterparts. (shrink)

There is a generic way to add any new feature to a system. It involves identifying the basic units which build up the system and introducing the new feature to each of these basic units. In the case where the system is argumentation and the feature is probabilistic we have the following. The basic units are: the nature of the arguments involved; the membership relation in the set S of arguments; the attack relation; and the choice (...) of extensions. Generically to add a new aspect to an argumentation network \ can be done by adding this feature to each component. This is a brute-force method and may yield a non-intuitive or meaningful result. A better way is to meaningfully translate the object system into another target system which does have the aspect required and then let the target system endow the aspect on the initial system. In our case we translate argumentation into classical propositional logic and get probabilistic argumentation from the translation. Of course what we get depends on how we translate. In fact, in this paper we introduce probabilistic semantics to abstract argumentation theory based on the equational approach to argumentation networks. We then compare our semantics with existing proposals in the literature including the approaches by M. Thimm and by A. Hunter. Our methodology in general is discussed in the conclusion. (shrink)

One of the many intriguing features of the axiomatic systems of probability investigated in Popper (1959), appendices _iv, _v, is the different status of the two arguments of the probability functor with regard to the laws of replacement and commutation. The laws for the first argument, (rep1) and (comm1), follow from much simpler axioms, whilst (rep2) and (comm2) are independent of them, and have to be incorporated only when most of the important deductions have been accomplished. It is plain that, (...) in the presence of (comm1), the principle (sub), which says that terms that are intersubstitutable in the first argument are intersubstitutable also in the second argument, implies (comm2), and in Popper’s systems the converse implication obtains. It is naturally asked what is needed in an axiomatic theory of probability in order to enforce this equivalence. Leblanc (1981) offered a rather weak set of axioms, containing (comm1) and (comm2), that suffice for the derivation of (sub). In this paper Leblanc’s result is improved in a number of different ways. Three weaker systems, one of which is incomparable with the other two, are shown to admit the same implication. DOI:10.5007/1808-1711.2011v15n2p271. (shrink)

The long history of ergodic and quasi-ergodic hypotheses provides the best example of the attempt to supply non-probabilistic justifications for the use of statistical mechanics in describing mechanical systems. In this paper we reverse the terms of the problem. We aim to show that accepting a probabilistic foundation of elementary particle statistics dispenses with the need to resort to ambiguous non-probabilistic notions like that of (in)distinguishability. In the quantum case, starting from suitable probability conditions, it is possible (...) to deduce elementary particle statistics in a unified way. Following our approach Maxwell-Boltzmann statistics can also be deduced, and this deduction clarifies its status.Thus our primary aim in this paper is to give a mathematically rigorous deduction of the probability of a state with given energy for a perfect gas in statistical equilibrium; that is, a deduction of the equilibrium distribution for a perfect gas. A crucial step in this deduction is the statement of a unified statistical theory based on clearly formulated probability conditions from which the particle statistics follows. We believe that such a deduction represents an important improvement in elementary particle statistics, and a step towards a probabilistic foundation of statistical mechanics.In this Part I we first present some history: we recall some results of Boltzmann and Brillouin that go in the direction we will follow. Then we present a number of probability results we shall use in Part II. Finally, we state a notion of entropy referring to probability distributions, and give a natural solution to Gibbs' paradox. (shrink)

A formulation of probabilistic causality is given in terms of the theory of abstract dynamical systems. Causal factors are identified as invariants of motion of a system. Repetition of an experiment leads to the notion of stationarity, and causal factors yield a decomposition of the stationary probability law of the experiment into ergodic components. In these, statistical behaviour is uniform. Control of identified causal factors leads to a corresponding statistical law for the events, which is offered as a (...) notion of probabilistic causality. After a suggestion by Feller, randomization is identified as mixing, formulated in above terms. (shrink)

Several theoretical publications on the Dirac equation published during the last decades have shown that, an interpretation is possible, which ascribes the origin of electron spin and magnetic moment to an autonomous circular motion of the point-like charged particle around a fixed centre. In more recent publications an extension of the original so called “Zitterbewegung Interpretation” of quantum mechanics was suggested, in which the spin results from an average of instantaneous spin vectors over a Zitterbewegung period. We argue that, the (...) corresponding autonomous motion of the electron should, if it is real, determine non-relativistic spin measurements. Such a direct connection with the established formal quantum mechanical description of spin measurements, into which spin is introduced as a “non-classical” quantity has, to our knowledge, not been reported. In the present work we show that, under certain “model assumptions” concerning the proposed autonomous motion, results of spin measurements, including measurements of angular correlations in singlet systems, can indeed be correctly described using classical probabilities. The success of the model is evidence for the “reality” of the assumed autonomous motion. The resulting model violates the Bell—inequalities to the same extent as quantum mechanics. (shrink)

In this paper we exploit the notions of conjoined and iterated conditionals, which are defined in the setting of coherence by means of suitable conditional random quantities with values in the interval [0,1]. We examine the iterated conditional (B|K)|(A|H), by showing that A|H p-entails B|K if and only if (B|K)|(A|H) = 1. Then, we show that a p-consistent family F={E1|H1, E2|H2} p-entails a conditional event E3|H3 if and only if E3|H3= 1, or (E3|H3)|QC(S) = 1 for some nonempty subset S (...) of F, where QC(S) is the quasi conjunction of the conditional events in S. Then, we examine the inference rules And, Cut, Cautious Monotonicity, and Or of System P and other well known inference rules (Modus Ponens, Modus Tollens, Bayes). We also show that QC(F)|C(F) = 1, where C(F) is the conjunction of the conditional events in F. We characterize p-entailment by showing that F p-entails E3|H3 if and only if (E3|H3)|C(F) = 1. Finally, we examine Denial of the antecedent and Affirmation of the consequent, where the p-entailment of (E3|H3) from F does not hold, by showing that (E3|H3)|C(F) is not equal to 1. (shrink)

We propose a combination of probabilistic reasoning from conditional constraints with approaches to default reasoning from conditional knowledge bases. In detail, we generalize the notions of Pearl's entailment in system Z, Lehmann's lexicographic entailment, and Geffner's conditional entailment to conditional constraints. We give some examples that show that the new notions of z-, lexicographic, and conditional entailment have similar properties like their classical counterparts. Moreover, we show that the new notions of z-, lexicographic, and conditional entailment are proper (...) generalizations of both their classical counterparts and the classical notion of logical entailment for conditional constraints. (shrink)