A First-order Conditional Probability Logic with Iterations

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(φ, θ), and CP6s(φ, θ), with the intended meaning that the conditional probability of φ given θ is at least (at most) s. The corresponding syntax, semantic, and axiomatic system are introduced, and Extended completeness theorem is proven. 1. Syntax and semantics The recent papers [1, 3, 6], discuss conditional probability extensions of classic propositional logic, while [2] introduces a first-order conditional probability logic in which iterations of conditional probability operators are not allowed. In this paper, we abandon that restriction and also extend logical language by adding equality, which causes changes in the corresponding syntax and semantics. Solving those issues is the main novelty presented in this paper. Let [0, 1]Q denote the set of all rational numbers from the interval [0, 1]. The language L of the LFOICP-logic consists of countable sets of variables V ar = {x1, x2, . . .}, relation symbols R m i , the relation symbol = which is, of course, interpreted rigidly as equality, and function symbols F j , where m and n are arities of these symbols, logical connectives ∧ and ¬, the quantifier ∀, and binary conditional probability operators CP>s and CP6t for all s ∈ [0, 1]Q, t ∈ [0, 1)Q. Constants are function symbols whose arity is 0. Terms and atomic formulas are defined as in the first-order classical logic with equality. The set of formulas ForFOICP= is the smallest set containing atomic formulas and closed under the following formation rules: if φ and θ are formulas, then ¬φ, CP>s(φ, θ), CP6t(φ, θ), φ∧θ and (∀x)φ are formulas. We use the standard abbreviations for other connectives, while P>s(φ) denotes CP>s(φ,⊤). A formula 2010 Mathematics Subject Classification: 03B48, 03B042, 03B45. Partially supported by Ministarstvo prosvete i nauke Republike Srbije under grants III44006 and ON174026.