Uncertain Linear Logic via Fibring of Probabilistic and Fuzzy Logic

TL;DR

This paper introduces an uncertain linear logic framework derived from probabilistic and fuzzy logic, grounded in a simple semantics based on counting observations, and interprets linear logic as a logic of resource conservation.

Contribution

It unifies probabilistic and fuzzy logic within a linear logic framework using a novel semantics based on evidence counting, revealing linear logic as a resource-sensitive reasoning system.

Findings

Probabilistic and fuzzy logic correspond to different heuristic assumptions.

Two sets of formulas for propagating truth values are derived.

Linear logic rules emerge naturally from the semantics.

Abstract

Beginning with a simple semantics for propositions, based on counting observations, it is shown that probabilistic and fuzzy logic correspond to two different heuristic assumptions regarding the combination of propositions whose evidence bases are not currently available. These two different heuristic assumptions lead to two different sets of formulas for propagating quantitative truth values through lattice operations. It is shown that these two sets of formulas provide a natural grounding for the multiplicative and additive operator-sets in linear logic. The standard rules of linear logic then emerge as consequences of the underlying semantics. The concept of linear logic as a ``logic of resources" is manifested here via the principle of ``conservation of evidence" -- the restrictions to weakening and contraction in linear logic serve to avoid double-counting of evidence (beyond any double-counting incurred via use of heuristic truth value functions).