Many tasks in statistical and causal inference can be construed as problems of entailment in a suitable formal language. We ask whether those problems are more difficult, from a computational perspective, for causal probabilistic languages than for pure probabilistic (or “associational”) languages. Despite several senses in which causal reasoning is indeed more complex—both expressively and inferentially—we show that causal entailment (or satisfiability) problems can be systematically and robustly reduced to purely probabilistic problems. Thus there is no jump in computational complexity. (...) Along the way we answer several open problems concerning the complexity of well-known probability logics, in particular demonstrating the ${\exists \mathbb {R}}$ -completeness of a polynomial probability calculus, as well as a seemingly much simpler system, the logic of comparative conditional probability. (shrink)

In this paper I argue that de Finetti provided compelling reasons for rejecting countable additivity. It is ironical therefore that the main argument advanced by Bayesians against following his recommendation is based on the consistency criterion, coherence, he himself developed. I will show that this argument is mistaken. Nevertheless, there remain some counter-intuitive consequences of rejecting countable additivity, and one in particular has all the appearances of a full-blown paradox. I will end by arguing that in fact it is no (...) paradox, and that what it shows is that conditionalisation, often claimed to be integral to the Bayesian canon, has to be rejected as a general rule in a finitely additive environment. (shrink)

We describe a dual-process theory of how individuals estimate the probabilities of unique events, such as Hillary Clinton becoming U.S. President. It postulates that uncertainty is a guide to improbability. In its computer implementation, an intuitive system 1 simulates evidence in mental models and forms analog non-numerical representations of the magnitude of degrees of belief. This system has minimal computational power and combines evidence using a small repertoire of primitive operations. It resolves the uncertainty of divergent evidence for single events, (...) for conjunctions of events, and for inclusive disjunctions of events, by taking a primitive average of non-numerical probabilities. It computes conditional probabilities in a tractable way, treating the given event as evidence that may be relevant to the probability of the dependent event. A deliberative system 2 maps the resulting representations into numerical probabilities. With access to working memory, it carries out arithmetical operations in combining numerical estimates. Experiments corroborated the theory's predictions. Participants concurred in estimates of real possibilities. They violated the complete joint probability distribution in the predicted ways, when they made estimates about conjunctions: P, P, P, disjunctions: P, P, P, and conditional probabilities P, P, P. They were faster to estimate the probabilities of compound propositions when they had already estimated the probabilities of each of their components. We discuss the implications of these results for theories of probabilistic reasoning. (shrink)

We consider the problem of induction over languages containing binary relations and outline a way of interpreting and constructing a class of probability functions on the sentences of such a language. Some principles of inductive reasoning satisfied by these probability functions are discussed, leading in turn to a representation theorem for a more general class of probability functions satisfying these principles.

In a well-known paper, Timothy Williamson claimed to prove with a coin-flipping example that infinitesimal-valued probabilities cannot save the principle of Regularity, because on pain of inconsistency the event ‘all tosses land heads’ must be assigned probability 0, whether the probability function is hyperreal-valued or not. A premise of Williamson’s argument is that two infinitary events in that example must be assigned the same probability because they are isomorphic. It was argued by Howson that the claim of isomorphism fails, but (...) a more radical objection to Williamson’s argument is that it had been, in effect, refuted long before it was published. (shrink)

The paper focuses on extending to the first order case the semantical program for modalities first introduced by Dana Scott and Richard Montague. We focus on the study of neighborhood frames with constant domains and we offer in the first part of the paper a series of new completeness results for salient classical systems of first order modal logic. Among other results we show that it is possible to prove strong completeness results for normal systems without the Barcan Formula (like (...) FOL + K)in terms of neighborhood frames with constant domains. The first order models we present permit the study of many epistemic modalities recently proposed in computer science as well as the development of adequate models for monadic operators of high probability. Models of this type are either difficult of impossible to build in terms of relational Kripkean semantics [40].We conclude by introducing general first order neighborhood frames with constant domains and we offer a general completeness result for the entire family of classical first order modal systems in terms of them, circumventing some well-known problems of propositional and first order neighborhood semantics (mainly the fact that many classical modal logics are incomplete with respect to an unmodified version of either neighborhood or relational frames). We argue that the semantical program that thus arises offers the first complete semantic unification of the family of classical first order modal logics. (shrink)

My dissertation explores the ways in which Rudolf Carnap sought to make philosophy scientific by further developing recent interpretive efforts to explain Carnap’s mature philosophical work as a form of engineering. It does this by looking in detail at his philosophical practice in his most sustained mature project, his work on pure and applied inductive logic. I, first, specify the sort of engineering Carnap is engaged in as involving an engineering design problem and then draw out the complications of design (...) problems from current work in history of engineering and technology studies. I then model Carnap’s practice based on those lessons and uncover ways in which Carnap’s technical work in inductive logic takes some of these lessons on board. This shows ways in which Carnap’s philosophical project subtly changes right through his late work on induction, providing an important corrective to interpretations that ignore the work on inductive logic. Specifically, I show that paying attention to the historical details of Carnap’s attempt to apply his work in inductive logic to decision theory and theoretical statistics in the 1950s and 1960s helps us understand how Carnap develops and rearticulates the philosophical point of the practical/theoretical distinction in his late work, offering thus a new interpretation of Carnap’s technical work within the broader context of philosophy of science and analytical philosophy in general. (shrink)

In this article we present a p-adic valued probabilistic logic equation image which is a complete and decidable extension of classical propositional logic. The key feature of equation image lies in ability to formally express boundaries of probability values of classical formulas in the field equation image of p-adic numbers via classical connectives and modal-like operators of the form Kr, ρ. Namely, equation image is designed in such a way that the elementary probability sentences Kr, ρα actually do have their (...) intended meaning—the probability of propositional formula α is in the equation image-ball with the center r and the radius ρ. Due to modal nature of the operators Kr, ρ, it was natural to use the probability Kripke like models as equation image-structures, provided that probability functions range over equation image instead of equation image or equation image. (shrink)

Pruss uses an example of Lester Dubins to argue against the claim that appealing to hyperreal-valued probabilities saves probabilistic regularity from the objection that in continuum outcome-spaces and with standard probability functions all save countably many possibilities must be assigned probability 0. Dubins’s example seems to show that merely finitely additive standard probability functions allow reasoning to a foregone conclusion, and Pruss argues that hyperreal-valued probability functions are vulnerable to the same charge. However, Pruss’s argument relies on the rule of (...) conditionalisation, but I show that in examples like Dubins’s involving nonconglomerable probabilities, conditionalisation is self-defeating. (shrink)

‘Probability logic’ might seem like an oxymoron. Logic traditionally concerns matters immutable, necessary and certain, while probability concerns the uncertain, the random, the capricious. Yet our subject has a distinguished pedigree. Ramsey begins his classic “Truth and Probability” with the words: “In this essay the Theory of Probability is taken as a branch of logic. … “speaks of “the logic of the probable.” And more recently, regards probabilities as estimates of truth values, and thus probability theory as a natural outgrowth (...) of two‐valued logic—what he calls “probability logic.” However the point is put, probability theory and logic are clearly intimately related. This chapter explores some of the multifarious connections between probability and logic, and focuses on various philosophical issues in the foundations of probability theory. (shrink)

The working assumption of this paper is that noncommuting variables are irreducibly interdependent. The logic of such dependence relations is the author's independence-friendly (IF) logic, extended by adding to it sentence-initial contradictory negation ¬ over and above the dual (strong) negation ∼. Then in a Hilbert space ∼ turns out to express orthocomplementation. This can be extended to any logical space, which makes it possible to define the dimension of a logical space. The received Birkhoff and von Neumann "quantum logic" (...) can be interpreted by taking their "disjunction" to be ¬(∼A & ∼ B). Their logic can thus be mapped into a Boolean structure to which an additional operator ∼ has been added. (shrink)

Kyburg’s opposition to the subjective Bayesian theory, and in particular to its advocates’ indiscriminate and often questionable use of Dutch Book arguments, is documented and much of it strongly endorsed. However, it is argued that an alternative version, proposed by both de Finetti at various times during his long career, and by Ramsey, is less vulnerable to Kyburg’s misgivings. This is a logical interpretation of the formalism, one which, it is argued, is both more natural and also avoids other, widely-made (...) objections to Bayesianism. (shrink)

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)

Exhumation and study of the 1945 paradox of confirmation brings out the defect of its formulation. In the context of quantifier conditional-probability logic it is shown that a repair can be accomplished if the truth-functional conditional used in the statement of the paradox is replaced with a connective that is appropriate to the probabilistic context. Description of the quantifier probability logic involved in the resolution of the paradox is presented in stages. Careful distinction is maintained between a formal logic language (...) and its semantics, as the same language may be outfitted with different semantics. An acquaintance with sections 1? 5 of Hailperin (2006) covering the sentential aspects of probability logic is assumed as background information for quantifier probability logic. (shrink)

Cylindric set algebras are algebraizations of certain logical semantics. The topic surveyed here, i.e. probabilities defined on cylindric set algebras, is closely related, on the one hand, to probability logic (to probabilities defined on logical formulas), on the other hand, to measure theory. The set algebras occuring here are associated, in particular, with the semantics of first order logic and with non-standard analysis. The probabilities introduced are partially continous, they are continous with respect to so-called cylindric sums.

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)

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)

This paper is a contribution to the algebraic logic of probabilistic models of Łukasiewicz predicate logic. We study the MV-states defined on polyadic MV-algebras and prove an algebraic many-valued version of Gaifman’s completeness theorem.

Borel's concept of denumerable probability is described by means of three of his illustrative problems and their solution. These problems are then reformulated in contemporary terms and solved from the viewpoint of probability logic. A section compares Kolmogorov set-theoretic probability with probability logic. The concluding section describes a highly adverse criticism of Borel's conception for its not using something like Kolmogorov theory and, in support of Borel, this criticism is countered from the standpoint of quantifier probability logic.

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.

This essay describes a variety of contributions which relate to the connection of probability with logic. Some are grand attempts at providing a logical foundation for probability and inductive inference. Others are concerned with probabilistic inference or, more generally, with the transmittance of probability through the structure (logical syntax) of language. In this latter context probability is considered as a semantic notion playing the same role as does truth value in conventional logic. At the conclusion of the essay two fully (...) elaborated semantically based constructions of probability logic are presented. (shrink)

This volume concerns Rational Agents - humans, players in a game, software or institutions - which must decide the proper next action in an atmosphere of partial information and uncertainty. The book collects formal accounts of Uncertainty, Rationality and Agency, and also of their interaction. It will benefit researchers in artificial systems which must gather information, reason about it and then make a rational decision on which action to take.