Conditional Probability Logic over Conditional Probability Spaces

2015 ◽  
pp. 49-70
Author(s):  
Mauricio S. C. Hernandes
Keyword(s):  
Conditional Probability ◽  
Probability Logic ◽  
Probability Spaces

A first-order conditional probability logic

2011 ◽  
Vol 20 (1) ◽  
pp. 235-253 ◽  
Author(s):  
M. Milosevic ◽  
Z. Ognjanovic
Keyword(s):  
Conditional Probability ◽  
Probability Logic ◽  
First Order

Quantifying over events in probability logic: an introduction

2016 ◽  
Vol 27 (8) ◽  
pp. 1581-1600
Author(s):  
STANISLAV O. SPERANSKI
Keyword(s):  
Computational Complexity ◽  
Boolean Algebras ◽  
Probability Logic ◽  
Discrete Probability ◽  
Complexity Results ◽  
Probability Spaces

In this article we describe a bunch of probability logics with quantifiers over events, and develop primary techniques for proving computational complexity results (in terms of m-degrees) about these logics, mainly over discrete probability spaces. Also the article contains a comparison with some other probability logics and a discussion of interesting analogies with research in the metamathematics of Boolean algebras, demonstrating a number of attractive features and intuitive advantages of the present proposal.

A probabilistic interpolation theorem

1985 ◽  
Vol 50 (3) ◽  
pp. 708-713 ◽  
Author(s):  
Douglas N. Hoover
Keyword(s):  
Interpolation Theorem ◽  
Order Logic ◽  
First Order Logic ◽  
Probability Logic ◽  
Admissible Set ◽  
First Order ◽  
Probability Spaces ◽  
Image Position ◽  
Dedekind Cuts ◽  
Exact Definition

The probability logic is a logic with a natural interpretation on probability spaces (thus, a logic whose model theory is part of probability theory rather than a system for putting probabilities on formulas of first order logic). Its exact definition and basic development are contained in the paper [3] of H. J. Keisler and the papers [1] and [2] of the author. Building on work in [2], we prove in this paper the following probabilistic interpolation theorem for .Let L be a countable relational language, and let A be a countable admissible set with ω ∈ A (in this paper some probabilistic notation will be used, but ω will always mean the least infinite ordinal). is the admissible fragment of corresponding to A. We will assume that L is a countable set in A, as is usual in practice, though all that is in fact needed for our proof is that L be a set in A which is wellordered in A.Theorem. Let ϕ(x) and ψ(x) be formulas of LAP such thatwhere ε ∈ [0, 1) is a real in A (reals may be defined in the usual way as Dedekind cuts in the rationals). Then for any real d > ε¼, there is a formula θ(x) of (L(ϕ) ∩ L(ψ))AP such thatand

scholarly journals A logic with higher order conditional probabilities

2007 ◽  
pp. 141-154 ◽  
Author(s):  
Zoran Ognjanovic ◽  
Nebojsa Ikodinovic
Keyword(s):  
Conditional Probability ◽  
Probability Space ◽  
Possible Worlds ◽  
Axiomatic System ◽  
Probability Logic ◽  
Conditional Probabilities ◽  
Unconditional Probability ◽  
Possible World ◽  
Intended Meaning ◽  
Probability Operators

We investigate probability logic with the conditional probability operators This logic, denoted LCP, allows making statements such as: P?s?, CP?s(? | ?) CP?0(? | ?) with the intended meaning "the probability of ? is at least s" "the conditional probability of ? given ? is at least s", "the conditional probability of ? given ? at most 0". A possible-world approach is proposed to give semantics to such formulas. Every world of a given set of worlds is equipped with a probability space and conditional probability is derived in the usual way: P(? | ?) = P(?^?)/P(?), P(?) > 0, by the (unconditional) probability measure that is defined on an algebra of subsets of possible worlds. Infinitary axiomatic system for our logic which is sound and complete with respect to the mentioned class of models is given. Decidability of the presented logic is proved.

scholarly journals A first-order conditional probability logic with iterations

2013 ◽  
Vol 93 (107) ◽  
pp. 19-27 ◽  
Author(s):  
Milos Milosevic ◽  
Zoran Ognjanovic
Keyword(s):  
Conditional Probability ◽  
Completeness Theorem ◽  
Axiomatic System ◽  
Probability Logic ◽  
Intended Meaning ◽  
First Order

We investigate a first-order conditional probability logic with equality, which is, up to our knowledge, the first treatise of such logic. The logic, denoted LFPOIC=, allows making statements such as: CP?s(?, ?), and CP?s(?, ?), with the intended meaning that the conditional probability of ? given ? is at least (at most) s. The corresponding syntax, semantic, and axiomatic system are introduced, and Extended completeness theorem is proven.

Probability Spaces

2021 ◽  
pp. 147-153
Author(s):  
James Davidson
Keyword(s):  
Conditional Probability ◽  
General Context ◽  
Probability Measures ◽  
Product Spaces ◽  
Probability Spaces ◽  
Product Measures

This chapter defines probability measures and probability spaces in a general context, as a case of the concepts introduced in Chapter 3. The axioms of probability are explained, and the important concepts of conditional probability and independence are introduced and linked to the role of product spaces and product measures.

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