How to combine diagrammatic logics

This paper is a submission to the contest: How to combine logics? at the World Congress and School on Universal Logic III, 2010. We claim that combining “things”, whatever these things are, is made easier if these things can be seen as the objects of a category. We define the category of diagrammatic logics, so that categorical constructions can be used for combining diagrammatic logics. As an example, a combination of logics using an opfibration is presented, in order to study computational side-effects due to the evolution of the state during the execution of an imperative program.

💡 Research Summary

The paper proposes a categorical framework for combining logics by treating each logic as an object in a specially constructed category called the “category of diagrammatic logics” (DiagLog). A diagrammatic logic consists of a signature, a set of inference rules, and a model functor, all packaged as a diagram. Morphisms between such logics are structure‑preserving translations that respect signatures, rules, and models simultaneously. By organizing logics and their translations into DiagLog, the authors can import the full machinery of category theory to perform systematic combinations.

Key categorical constructions are examined. Pushouts in DiagLog provide a principled way to merge two logics that share a common sub‑logic: given logics L₁ and L₂ with a shared fragment L₀, the pushout L₁ ⊔_{L₀} L₂ yields a new logic that contains both original rule sets while preserving the common part. Dually, pullbacks extract the intersection of two logics, giving a minimal common sub‑logic. These constructions subsume traditional combination techniques such as fusion, extension, and restriction, but do so in a uniform, mathematically rigorous manner.

Beyond static combination, the paper introduces an opfibration‑based method to handle dynamic aspects, specifically computational side‑effects arising from mutable state in imperative programs. The base category S models the state space, while the total category L models program commands as diagrams. An opfibration p : L → S assigns to each command its state transition. The opfibration’s lifting property guarantees that composing commands respects state evolution, thereby integrating side‑effect semantics with the underlying logical reasoning. As a concrete illustration, the authors encode a small IMP‑style language, represent its syntax, operational semantics, and state updates as a diagrammatic logic, and then use the opfibration to combine this with a pure logical framework. This yields a unified semantics where both logical inference and state mutation coexist coherently.

The authors discuss advantages: (1) a single categorical setting accommodates a wide variety of logics, enhancing reuse; (2) rich categorical operations allow simultaneous treatment of static and dynamic features; (3) proofs of combination properties become abstract categorical arguments rather than ad‑hoc syntactic manipulations. They also acknowledge limitations, such as the prerequisite categorical expertise and the need to verify the existence of pushouts for particular logic pairs.

In conclusion, the paper positions the category of diagrammatic logics as a universal platform for logic combination, and outlines future work including automated pushout computation, exploration of other fibred structures, and applications to security, privacy, and quantum logics.