Complementarity in Social Measurement: A Partition-Logic Approach

Abstract

Partition logics -- non-Boolean event structures obtained by pasting Boolean algebras -- provide a natural language for situations in which a system has a definite latent state but can be accessed and resolved only through mutually incompatible coarse-grained modes of observation. We show that this structure arises in a range of social-science settings by constructing six explicit examples from personnel assessment, survey framing, clinical diagnosis, espionage coordination, legal pluralism, and organizational auditing. For each case we identify the latent state space, the observational contexts as partitions, and the shared atoms that intertwine contexts, yielding instances of the $L_{12}$ bowtie, triangle, pentagon, and automaton partition logics. These examples make precise a notion of social complementarity: different modes of inquiry can be incompatible even though the underlying system remains fully value-definite. Complementarity in this sense does not entail contextuality or ontic indeterminacy. We further compare the classical probabilities generated by convex mixtures of dispersion-free states with the quantum-like Born probabilities available when the same exclusivity graph admits a faithful orthogonal representation. The framework thus separates logical structure from probabilistic realization and suggests empirically testable benchmarks for quantum-cognition models.

Figures (5)

• Figure 1: Greechie orthogonality diagram (hypergraph) of the $L_{12}$ personnel-assessment logic. Each line represents one context---one available mode of observation. The written-test context$\mathcal{C}_W$ (left) carries three observed categories: the singleton $\{2\}$ (Analyst, recognizable by adequate test performance), the pair $\{3,4\}$ (Communicator and Developing, conflated because both score poorly on the analytical task), and the singleton $\{1\}$ (Star, top scorer). The interview context$\mathcal{C}_I$ (right) carries a different triple of observed categories: $\{3\}$ is now resolved (Communicator, identifiable by social poise) but $\{2,4\}$ is conflated (Analyst and Developing, both weak in interpersonal tasks). The observed category $\{1\}$ is the intertwining atom: it appears in both contexts because the Star's latent social profile---excellence on both dimensions---makes this profile identifiable regardless of which instrument is chosen. The five observed categories, together with their lattice-theoretic joins and meets, generate a $12$-element non-Boolean logic, hence the name $L_{12}$.
• Figure 2: Pentagon survey logic. Each color corresponds to a context (policy issue) corresponding to stances on immigration, healthcare, taxation, education, and defense spending (clockwise from top right). The cyclic structure reflects the pattern of framing overlaps among the five issues.
• Figure 3: Greechie diagram of the triangle logic applied to clinical diagnosis. Each edge represents one context (diagnostic instrument). The three yellow vertices are the intertwining atoms---the singleton observed categories, each identifiable under two adjacent instruments. The three white vertices are the conflated observed categories, each merging $4$ (Subclinical) with a different partner. $4$ is the "stealth" subtype: it never appears as a singleton under any instrument, yet it has a fully determinate latent social profile. The six observed categories, together with their lattice-theoretic operations, form a non-Boolean logic that cannot be embedded in any single Boolean algebra.
• Figure 4: Hypergraph of the legal-pluralism logic. Three legal codes (Statutory, Religious, Customary) form three contexts. Yellow vertices are intertwining atoms: $\{a_1,a_2\}$ (forest clearing---prohibited under both statutory and religious law) and $\{a_4,a_6\}$ (commercial use---set apart by both customary and religious law). White vertices represent groupings specific to one code. The non-Boolean pasting reflects the impossibility of a single "master code" that respects all three systems' distinctions simultaneously.
• Figure 5: The four organizational states arranged in a $2\times 2$ grid. The financial audit (directive $a$, red dashed) partitions the grid horizontally: it reveals the row (financial health) but conflates columns (compliance status). The operational inspection (directive $b$, blue dotted) partitions vertically: it reveals the column but conflates rows. The two partitions share no atoms---this is maximal complementarity.