We present a propositional and a first-order logic for reasoning about higher-order upper and lower probabilities. We provide sound and complete axiomatizations for the logics and we prove decidability in the propositional case. Furthermore, we show that the introduced logics generalize some existing probability logics.

Automated reasoning about uncertain knowledge has many applications. One difficulty when developing such systems is the lack of a completely satisfactory integration of logic and probability. We address this problem directly. Expressive languages like higher-order logic are ideally suited for representing and reasoning about structured knowledge. Uncertain knowledge can be modeled by using graded probabilities rather than binary truth-values. The main technical problem studied in this paper is the following: Given a set of sentences, each having some probability of being (...) true, what probability should be ascribed to other (query) sentences? A natural wish-list, among others, is that the probability distribution (i) is consistent with the knowledge base, (ii) allows for a consistent inference procedure and in particular (iii) reduces to deductive logic in the limit of probabilities being 0 and 1, (iv) allows (Bayesian) inductive reasoning and (v) learning in the limit and in particular (vi) allows confirmation of universally quantified hypotheses/sentences. We translate this wish-list into technical requirements for a prior probability and show that probabilities satisfying all our criteria exist. We also give explicit constructions and several general characterizations of probabilities that satisfy some or all of the criteria and various (counter) examples. We also derive necessary and sufficient conditions for extending beliefs about finitely many sentences to suitable probabilities over all sentences, and in particular least dogmatic or least biased ones. We conclude with a brief outlook on how the developed theory might be used and approximated in autonomous reasoning agents. Our theory is a step towards a globally consistent and empirically satisfactory unification of probability and logic. (shrink)

We give an axiomatization of first‐order logic enriched with the almost‐everywhere quantifier over finitely additive measures. Using an adapted version of the consistency property adequate for dealing with this generalized quantifier, we show that such a logic is both strongly complete and enjoys Craig interpolation, relying on a (countable) model existence theorem. We also discuss possible extensions of these results to the almost‐everywhere quantifier over countably additive measures.

A group of philosophers led by the late John Pollock has applied a method of reasoning about probability, known as direct inference and governed by a constraint known as Reichenbach's principle, to argue in support of ‘thirdism’ concerning the Sleeping Beauty Problem. A subsequent debate has ensued about whether their argument constitutes a legitimate application of direct inference. Here I defend the argument against two extant objections charging illegitimacy. One objection can be overcome via a natural and plausible definition, given (...) here, of the binary relation ‘logically stronger than’ between two properties that can obtain even when the respective properties differ from one another in ‘arity’; given this definition, the Pollock group's argument conforms to Reichenbach's principle. Another objection prompts a certain refinement of Reichenbach's principle that is independently well-motivated. My defense of the Pollock group's argument has epistemological import beyond the Sleeping Beauty problem, because it both widens and sharpens the applicability of direct inference as a method for inferring single-case epistemic probabilities on the basis of general information of a probabilistic or statistical nature. (shrink)

This paper argues against the view that the standard explanation of phase transitions in statistical mechanics may be considered a causal explanation, a distortion that can nevertheless successfully represent causal relations.

Numerical probabilities are eliminated in favor of qualitative notions, with an eye to isolating what it is about probabilities that is essential to judgements of acceptability. A basic choice point is whether the conjunction of two propositions, each acceptable, must be deemed acceptable. Concepts of acceptability closed under conjunction are analyzed within Keisler's weak logic for generalized quantifiers — or more specifically, filter quantifiers. In a different direction, the notion of a filter is generalized so as to allow sets with (...) probability non-infinitesimally below 1 to be acceptable. (shrink)

The probabilization of a logic system consists of enriching the language (the formulas) and the semantics (the models) with probabilistic features. Such an operation is said to be exogenous if the enrichment is done on top, without internal changes to the structure, and is called endogenous otherwise. These two different enrichments can be applied simultaneously to the language and semantics of a same logic. We address the problem of studying the transference of metaproperties, such as completeness and decidability, to the (...) exogenous probabilization of an abstract logic system. First, we setup the necessary framework to handle the probabilization of a satisfaction system by proving transference results within a more general context. In this setup, we define a combination mechanism of logics through morphisms and prove sufficient condition to guarantee completeness and decidability. Then, we demonstrate that probabilization is a special case of this exogenous combination method, and that it fulfills the general conditions to obtain transference of completeness and decidability. Finally, we motivate the applicability of our technique by analyzing the probabilization of the linear temporal logic over Markov chains, which constitutes an endogenous probabilization. The results are obtained first by studying the exogenous semantics, and then by establishing an equivalence with the original probabilization given by Markov chains. (shrink)

Generics and frequency statements are puzzling phenomena: they are lawlike, yet contingent. They may be true even in the absence of any supporting instances, and extending the size of their domain does not change their truth conditions. Generics and frequency statements are parametric on time, but not on possible worlds; they cannot be applied to temporary generalizations, and yet are contingent. These constructions require a regular distribution of events along the time axis. Truth judgments of generics vary considerably across speakers, (...) whereas truth judgments of frequency statements are much more uniform. A generic may be false even if the vast majority of individuals in its domain satisfy the predicated property, whereas a frequency statement using, e.g., usually would be true. This paper argues that all these seemingly unrelated puzzles have a single underlying cause: generics and frequency statements express probability judgments, and these, in turn, are interpreted as statements of hypothetical relative frequency. (shrink)

In concrete applications of probability, statistical investigation gives us knowledge of some probabilities, but we generally want to know many others that are not directly revealed by our data. For instance, we may know prob(P/Q) (the probability of P given Q) and prob(P/R), but what we really want is prob(P/Q& R), and we may not have the data required to assess that directly. The probability calculus is of no help here. Given prob(P/Q) and prob(P/R), it is consistent with the probability (...) calculus for prob(P/Q& R) to have any value between 0 and 1. Is there any way to make a reasonable estimate of the value of prob(P/Q& R) 1 A related problem occurs when probability practitioners adopt undefended assumptions of statistical independence simply on the basis of not seeing any connection between two propositions. This is common practice, but its justification has eluded probability theorists, and researchers are typically apologetic about making such assumptions. Is there any way to defend the practice? This paper shows that on a certain conception of probability—nomic probability—there are principles of "probable probabilities" that license inferences of the above sort. These are principles telling us that although certain inferences from probabilities to probabilities are not deductively valid, nevertheless the second-order probability of their yielding correct results is 1. This makes it defeasibly reasonable to make the inferences. Thus I argue that it is defeasibly reasonable to assume statistical independence when we have no information to the contrary. And I show that there is a function Y(r, s, a) such that if prob(P/Q) = r, prob(P/R) = s, andprob(P/U) = a (where U is our background knowledge) then it is defeasibly reasonable to expect that prob(P/Q&R) = Y(r, s, a). Numerous other defeasible inferences are licensed by similar principles of probable probabilities. This has the potential to greatly enhance the usefulness of probabilities in practical application. (shrink)

A central problem in applying logical knowledge representation formalisms to traditional robotics is that the treatment of belief change is categorical in the former, while probabilistic in the latter. A typical example is the fundamental capability of localization where a robot uses its noisy sensors to situate itself in a dynamic world. Domain designers are then left with the rather unfortunate task of abstracting probabilistic sensors in terms of categorical ones, or more drastically, completely abandoning the inner workings of sensors (...) to black-box probabilistic tools and then interpreting their outputs in an abstract way. Building on a first-principles approach by Bacchus, Halpern and Levesque, and a recent continuous extension to it by Belle and Levesque, we provide an axiomatization that shows how localization can be realized wrt a basic action theory, thereby demonstrating how such capabilities can be enabled in a single logical framework. We then show how the framework can also enable localization for multiple agents, where an agent can appeal to the sensing already performed by another agent and the knowledge of their relative positions to localize itself. (shrink)

Leibniz seems to have been the first to suggest a logical interpretation of probability, but there have always seemed formidable mathematical and interpretational barriers to implementing the idea. De Finetti revived it only, it seemed, to reject it in favour of a purely decision-theoretic approach. In this paper I argue that not only is it possible to view (Bayesian) probability as a continuum-valued logic, but that it has a very close formal kinship with classical propositional logic.

The purpose of this work is to analyse the cognitive process of the domain theories in terms of the measurement theory to develop a computational machine learning approach for implementing it. As a result, the relational data mining approach, the authors proposed in the preceding books, was improved. We present the approach as an implementation of the cognitive process as the measurement theory perceived. We analyse the cognitive process in the first part of the paper and present the theory and (...) method of the logically most powerful empirical theory discovery in the second. The theory is based on the notion of law-like rules, which conform to all the properties of laws of nature, namely generality, simplicity, maximum refutability and minimum number of parameters. This notion is defined for deterministic and probabilistic cases. Based on the method, the discovery system is developed. The system was successfully applied to many practical tasks. (shrink)

A conjectural inference is proposed, aimed at producing conjectural theorems from formal conjectures assumed as axioms, as well as admitting contradictory statements as conjectural theorems. To this end, we employ Paraconsistent Informational Logic, which provides a formal setting where the notion of conjecture formulated by an epistemic agent can be defined. The paraconsistent systems on which conjectural deduction is based are sequent formulations of the C-systems presented in Carnielli-Marcos [CAR 02b]. Thus, conjectural deduction may also be considered to be a (...) tool for investigating the properties of paraconsistency in general. (shrink)

Additionally, the text shows how to develop computationally feasible methods to mesh with this framework.

In order to employ and exhibit our Simon-inspired approach to computational economics, and specifically defend our version of the view that even logically untrained humans are rational, albeit no more than “boundedly” so, we provide two models, both rooted in computational logic, of how it is that logically untrained humans perform in a seemingly irrational fashion in a particular “deviant” auction (the bi-pay auction).

Inductive probabilistic reasoning is understood as the application of inference patterns that use statistical background information to assign (subjective) probabilities to single events. The simplest such inference pattern is direct inference: from “70% of As are Bs” and “a is an A” infer that a is a B with probability 0.7. Direct inference is generalized by Jeffrey’s rule and the principle of cross-entropy minimization. To adequately formalize inductive probabilistic reasoning is an interesting topic for artificial intelligence, as an autonomous system (...) acting in a complex environment may have to base its actions on a probabilistic model of its environment, and the probabilities needed to form this model can often be obtained by combining statistical background information with particular observations made, i.e., by inductive probabilistic reasoning. In this paper a formal framework for inductive probabilistic reasoning is developed: syntactically it consists of an extension of the language of first-order predicate logic that allows to express statements about both statistical and subjective probabilities. Semantics for this representation language are developed that give rise to two distinct entailment relations: a relation ⊨ that models strict, probabilistically valid, inferences, and a relation that models inductive probabilistic inferences. The inductive entailment relation is obtained by implementing cross-entropy minimization in a preferred model semantics. A main objective of our approach is to ensure that for both entailment relations complete proof systems exist. This is achieved by allowing probability distributions in our semantic models that use non-standard probability values. A number of results are presented that show that in several important aspects the resulting logic behaves just like a logic based on real-valued probabilities alone. (shrink)

In this article, we present the probability logic LFOCP which is suitable to formalize statements about conditional probabilities of first order formulas. The logical language contains formulas such as CP≥s and CP≤s with the intended meaning ‘the conditional probability of ϕ given θ is at least s’ and ‘at most s’, respectively, where ϕ and θ are first-order formulas. We introduce a class of first order Kripke-like models that combine properties of the usual Kripke models and finitely additive probabilities. We (...) propose an infinitary axiom system and prove that it is sound and strongly complete with respect to the considered class of models. In this article, the terms finitary and infinitary concern meta language only, i.e. the logical language is countable, formulas are finite, while only proofs are allowed to be infinite. We analyse decidability of LFOCP and provide a procedure which decides satisfiability of a given conditional probability formula, in the case when the underlying first-order theory is decidable. Relationships to other systems and possible extensions of the presented approach are discussed. (shrink)