A physical system is synergistic if it cannot be reduced to its constituents. Intuitively this is paraphrased into the common statement that 'the whole is greater than the sum of its parts'. In this manner, many basic elements in combination may give rise to some unexpected collective behavior. A paradigmatic example of such phenomenon is information. Several sources, which are already known individually, may provide some new knowledge when joined together. Here we take the trivial case of discrete random variables and explore whether and how it is possible get more information out of lesser parts. Our approach is inspired by set theory as the fundamental description of part-whole relations. If taken unaltered, synergistic behavior is forbidden by the set theoretical axioms. Indeed, the union of sets cannot contain extra elements not found in any particular one of them. However, random variables are not a perfect analogy of sets. We formalise the distinction, finding a single broken axiom - union/intersection distributivity. Nevertheless, it remains possible to describe information using Venn-type diagrams. We directly connect the existence of synergy to the failure of distributivity for random variables. When compared to the partial information decomposition framework (PID), our technique fully reproduces previous results while resolving the self-contradictions that plagued the field and providing additional constraints on the solutions. This opens the way towards quantifying emergence in large systems.

翻译：如果一个物理系统无法被简化为其组成部分，则称其具有协同性。直观上，这一概念常被表述为“整体大于部分之和”。通过这种方式，许多基本元素组合时可能产生某些意想不到的集体行为。信息便是此类现象的典型示例：若干已知的独立信源在结合时可能提供新的知识。本文以离散随机变量这一简单情形为对象，探究是否以及如何能够从较少部分中获取更多信息。我们的研究方法受集合论启发，因其是对部分-整体关系的基本描述。若直接套用集合论，协同行为会被其公理所禁止——确实，集合的并集不可能包含任何单个集合中不存在的额外元素。然而，随机变量并非集合的完美类比。我们通过形式化区分两者，发现唯一被违背的公理是并/交运算的分配律。尽管如此，我们仍能使用维恩图式图表描述信息。我们直接将协同性的存在与随机变量分配律的失效联系起来。与部分信息分解框架相比，我们的方法在完全复现已有结果的同时，解决了长期困扰该领域的自相矛盾问题，并为解集提供了额外约束。这为量化大型系统中的涌现现象开辟了新途径。