Probabilistic Logic: Many-valuedness and Intensionality

TL;DR

This paper revises existing probabilistic logics, embedding them into a many-valued logic and a 2-valued intensional first-order logic, enhancing formal reasoning about probabilities and applying these results to Probabilistic Logic Programming.

Contribution

It provides a complete revision of Nilsson's probabilistic structures and introduces a unified logical framework for reasoning about probabilities using many-valued and intensional logics.

Findings

Embedding Nilsson's structure yields a many-valued logic.

Probabilistic reasoning can be embedded into a 2-valued intensional first-order logic.

Theoretical results are applied to Probabilistic Logic Programming.

Abstract

The probability theory is a well-studied branch of mathematics, in order to carry out formal reasoning about probability. Thus, it is important to have a logic, both for computation of probabilities and for reasoning about probabilities, with a well-defined syntax and semantics. Both current approaches, based on Nilsson's probability structures/logics, and on linear inequalities in order to reason about probabilities, have some weak points. In this paper we have presented the complete revision of both approaches. We have shown that the full embedding of Nilsson'probabilistic structure into propositional logic results in a truth-functional many-valued logic, differently from Nilsson's intuition and current considerations about propositional probabilistic logic. Than we have shown that the logic for reasoning about probabilities can be naturally embedded into a 2-valued intensional FOL with intensional abstraction, by avoiding current ad-hoc system composed of two different 2-valued logics: one for the classical propositional logic at lower-level, and a new one at higher-level for probabilistic constraints with probabilistic variables. The obtained theoretical results are applied to Probabilistic Logic Programming.