Uncertain Linear Logic via Fibring of Probabilistic and Fuzzy Logic
This work provides a theoretical foundation for linear logic as a logic of resources, addressing researchers in logic and AI, but it is incremental as it builds on existing probabilistic and fuzzy frameworks.
The paper tackled the problem of grounding linear logic in probabilistic and fuzzy semantics by showing that their heuristic assumptions for combining propositions lead to formulas that naturally correspond to linear logic's multiplicative and additive operators, with the rules emerging from a semantics based on counting observations and conservation of evidence.
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).