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.
This paper is a sequel to the joint publication of Scott and Krauss [8] in which the first aspects of a mathematical theory are developed which might be called “First Order Probability Logic”. No attempt will be made to present this additional material in a self-contained form. We will use the same notation and terminology as introduced and explained in Scott and Krauss [8], and we will frequently refer to the theorems stated and proved in the preceding paper.
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
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.
This paper derives the theoretical underpinnings behind the following observed empirical facts in credit risk modeling: The probability of default, the seniority, the thickness of the tranche, the debt cushion, and macroeconomic factors are the important determinants of the conditional probability density function of the recovery rate given default (RGD) of a firm’s debt and its tranches. In a portfolio of debt securities, the conditional probability density functions of the recovery rate given default of tranches have point probability masses near zero and one, and the expected value of the recovery rate given default increases as the seniority or debt cushion increases. The paper derives other results as well, such as the fact that the conditional probability distribution function associated with any senior tranche dominates that of any junior tranche by first-order. The standard deviation of the recovery rate given default of a senior security need not be greater than that of a junior security. It is proved that the expected value of the recovery rate given default need not increase as the proportional thickness of the tranche increases.
A first-order conditional probability logic with iterations
