The aim of the paper is to prove tha analytic completeness theorem for a logic LAs with two integral operators in the singular case. The case of absolute continuity was proved in [4]. MSC: 03B48, 03C70.

We investigate an approach for drawing logical inference from inconsistent premisses. The main idea in this approach is that the inconsistencies in the premisses should be interpreted as uncertainty of the information. We propose a mechanism, based on Kinght’s [14] study of inconsistency, for revising an inconsistent set of premisses to a minimally uncertain, probabilistically consistent one. We will then generalise the probabilistic entailment relation introduced in [15] for propositional languages to the first order case to draw logical inference from (...) a probabilistic set of premisses. We will show how this combination can allow us to limit the effect of uncertainty introduced by inconsistent premisses to only the reasoning on the part of the premise set that is relevant to the inconsistency. (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.

Evaluative studies of inductive inferences have been pursued extensively with mathematical rigor in many disciplines, such as statistics, econometrics, computer science, and formal epistemology. Attempts have been made in those disciplines to justify many different kinds of inductive inferences, to varying extents. But somehow those disciplines have said almost nothing to justify a most familiar kind of induction, an example of which is this: “We’ve seen this many ravens and they all are black, so all ravens are black.” This is (...) enumerative induction in itsfullstrength. For it does not settle with a weaker conclusion (such as “the ravens observed in the future will all be black”); nor does it proceed with any additional premise (such as the statistical IID assumption). The goal of this paper is to take some initial steps toward a justification for the full version of enumerative induction, against counterinduction, and against the skeptical policy. The idea is to explore various epistemic ideals, mathematically defined as different modes of convergence to the truth, and look for one that is weak enough to be achievable and strong enough to justify a norm that governs both the long run and the short run. So the proposal is learning-theoretic in essence, but a Bayesian version is developed as well. (shrink)

It is widely thought that chance should be understood in reductionist terms: claims about chance should be understood as claims that certain patterns of events are instantiated. There are many possible reductionist theories of chance, differing as to which possible pattern of events they take to be chance-making. It is also widely taken to be a norm of rationality that credence should defer to chance: special cases aside, rationality requires that one’s credence function, when conditionalized on the chance-making facts, should (...) coincide with the objective chance function. It is a shortcoming of a theory of chance if it implies that this norm of rationality is unsatisfiable. The primary goal of this paper is to show, on the basis of considerations concerning computability and inductive learning, that this shortcoming is more common than one would have hoped. (shrink)

We study invariant measures on compact Hausdorff spaces using finitary integral logic. For each compact Hausdorff space X and any family equation image of its continuous transformations, we find equivalent conditions for the existence of an equation image-invariant measure on X. We give two proofs of the existence of Haar measure on compact groups.

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)

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)

We study model theory of random variables using finitary integral logic. We prove definability of some probability concepts such as having F as distribution function, independence and martingale property. We then deduce Kolmogorov's existence theorem from the compactness theorem.