📈 economics

Measurable Majorities Are Not Finitely Axiomatizable

This paper proves that strict majority reasoning in finite social decision frames is not finitely axiomatizable by demonstrating that no bounded finite fragment can replace the Moss-Pedersen coherence criterion, as the shortest coherence violation can be arbitrarily long.

Original authors: Lawrence S. Moss, Arthur Paul Pedersen

Published 2026-06-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Lawrence S. Moss, Arthur Paul Pedersen

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: The "Rulebook" Problem

Imagine you are trying to write a rulebook for a voting system. Your goal is to create a set of simple, finite rules (axioms) that can perfectly describe every possible situation where a "majority" makes sense.

If a voting situation follows these rules, we call it "measurable" (meaning it can be represented by a fair probability number, like saying "there is a 60% chance this group wins"). If it breaks the rules, it is "incoherent" (a structural contradiction where the majority logic falls apart).

The Paper's Main Discovery:
The authors prove that you cannot write a finite rulebook for this. No matter how many rules you write down, there will always be a tricky, complex voting scenario that follows all your rules but is still logically broken. To catch every possible broken scenario, you would need an infinite list of rules.

The Core Concept: The "Incoherence Index"

To understand why, the authors introduce a concept called the Incoherence Index. Think of this as the "length of the shortest trap."

The Trap: A voting trap is a specific sequence of groups (blocs) that look like they should form a majority, but when you add them all up, they cancel each other out perfectly, leaving no one with a clear winner.
The Index: This is the number of groups needed to build that trap.
- A short trap (Index 2) is easy to spot.
- A long trap (Index 100) is very hard to spot.

The paper asks: Is there a maximum length for these traps?

Hypothesis: Maybe if you check all traps up to length 10, you've caught them all.
Reality: The authors prove no. For any number you pick (say, 100), they can construct a voting system where the shortest trap is actually length 102.

The Analogy: The "Perfectly Balanced See-Saw"

Imagine a giant see-saw with many seats.

The Goal: You want to place groups of people on the see-saw so that it stays perfectly balanced (neither side goes down).
The Rule: In a "measurable" world, you shouldn't be able to balance the see-saw unless you are using groups that are exactly tied (50/50 splits).
The Trap: The authors found a way to arrange groups of people (voting blocs) that are not tied, yet when you put them all on the see-saw, it balances perfectly. This is a logical contradiction.

The paper shows that you can build these "impossible balancing acts" using longer and longer chains of groups.

You can make a chain of 4 groups that balances.
You can make a chain of 6 groups that balances.
You can make a chain of 1,000 groups that balances.

The longer the chain, the harder it is to detect the error. The authors prove that there is no limit to how long these chains can get.

How They Proved It: The "Geometric Construction"

Instead of using complex combinatorics (counting every possible combination), the authors used geometry.

The Map: They turned every voting group into a point in a multi-dimensional space (like a map with thousands of directions).
The Core: They built a special, highly symmetrical "core" of voting groups. Imagine a perfect star shape where every point is connected to every other point in a specific way.
The Magic Vector: They found a special "laser beam" (a vector) that passes through the center of this star.
- This laser hits the "core" groups exactly at a 90-degree angle (it ignores them).
- However, it hits every other possible group at a slant (it sees them as positive or negative).
The Result: Because of this laser, they could prove that the only way to balance the see-saw (create a zero-sum sequence) is to use the specific groups in their "core." And the shortest way to do that requires exactly $2k + 2$ groups.

By making the "universe" of voters bigger and bigger, they forced the shortest possible trap to get longer and longer.

Why This Matters for Logic

The paper concludes with a punchline about logic and language:

In the logic system the authors study (Moss-Pedersen logic), there is a rule called the "Coherence Scheme." This scheme says, "If you have a sequence of groups of length $N$ that balances, then..."
Because the authors proved that traps can be arbitrarily long, this rule must be an infinite list.
You cannot replace this infinite list with a finite set of sentences. No matter how many sentences you write, there will always be a "long trap" that slips through the cracks.

Summary in One Sentence

The authors proved that the logical complexity of "strict majority" voting is infinite; you can always construct a voting scenario that is so complex it requires a longer and longer list of rules to detect its internal contradictions, meaning no finite rulebook can ever perfectly describe all valid majority systems.

Technical Summary: Measurable Majorities Are Not Finitely Axiomatizable

Problem Statement
The paper investigates the finite axiomatizability of strict majority reasoning within finite social decision frames. Specifically, it addresses whether the coherence criterion characterizing when qualitative majority judgments are representable by a finitely additive measure can be replaced, in the finite setting, by any bounded finite fragment. The central question is whether the "incoherence index"—the minimal length of a sequence of voting blocs required to expose a structural contradiction (incoherence)—is uniformly bounded across all finite frames. This problem resolves Conjecture 5.7 from Moss and Pedersen (2026), which posited that no such uniform finite bound exists.

Methodology
The authors provide a purely geometric proof of the unboundedness of the incoherence index, offering an alternative to the recent combinatorial approach by Blanco (2026). The methodology proceeds as follows:

Geometric Mapping: The study maps the properties of subset selection directly into the geometry of rational vector spaces ( $\mathbb{Q}^{W_k}$ ). Subsets are represented by bipolar indicator vectors ( $x_A \in \{-1, 1\}^{W_k}$ ), where the condition of a "perfectly balanced" sequence (covering the universe exactly $m/2$ times) translates to a zero-sum condition ( $\sum x_{A_i} = 0$ ).
Core Construction: For a fixed integer $k \geq 1$ , the authors construct a specific universe of voters $W_k$ consisting of all $(k+1)$ -element subsets of a base set $V_k = \{1, \dots, 2k+2\}$ . The size of this universe is the central binomial coefficient $\binom{2k+2}{k+1}$ .
Dictator Blocs and Linear Independence: A "core" set of $2k+2$ "dictator blocs" is defined, where each bloc contains all voters possessing a specific base element. The authors prove that any non-empty perfectly balanced sequence formed exclusively from these blocs must have a length that is a multiple of $2k+2$ . This establishes a lower bound on the length of any potential coherence violation derived from this core.
Generic Vector Separation: Using a "Finite-Union Lemma" (Bia lynicki-Birula et al., 1959), the authors construct a generic rational vector $u^*_k$ that is orthogonal to the linear span of the core vectors but non-orthogonal to all other balanced vectors in the middle layer of the power set. This vector is used to define a maximal standard frame that includes the core vectors but excludes any shorter balanced obstructions.

Key Contributions and Results

Unbounded Incoherence Index: The paper proves that for every $k \geq 1$ , there exists a maximal standard social decision frame $M_k$ whose incoherence index is exactly $2k + 2$ . Consequently, there is no uniform finite bound on the incoherence index of social decision frames.
Resolution of Conjecture 5.7: This result confirms Conjecture 5.7 from Moss and Pedersen (2026), demonstrating that the coherence scheme required to characterize measurable majorities cannot be truncated to a finite set of instances.
Explicit Middle-Layer Construction: The construction provides an explicit infinite sequence of universe sizes ( $2n_k = \binom{2k+2}{k+1}$ ) satisfying the properties predicted by Conjecture B.25. Specifically, it constructs a complement-free family in the middle layer (subsets of size $n_k$ ) that contains a balanced subfamily of size $2k+2$ but no balanced subfamilies of sizes $2, 4, \dots, 2k$ .
Non-Finite Axiomatizability: By combining the unboundedness result with the soundness and completeness theorem for the Moss-Pedersen minimal logic, the authors establish that the class of measurable social decision frames is not finitely axiomatizable in the Moss-Pedersen language. Any finite set of sentences would only capture coherence instances up to some finite length $M$ , leaving the constructed frame $M_k$ (with index $2k+2 > M$ ) as a counterexample that satisfies the finite axioms but fails representability.

Significance
The paper asserts that the inherent complexity of strict majority reasoning is a deep combinatorial reality embedded in the geometry of finite vector spaces, rather than merely an artifact of measure theory. While Moss and Pedersen (2026) established that representability is characterized by coherence, this work demonstrates that the boundary between representable and non-representable frames has unbounded finite complexity. Every obstruction to representability is finite, but there is no finite ceiling on the length of the obstruction required.

The authors note that their geometric construction yields an electorate size that scales exponentially with the index ( $O(4^k/\sqrt{k})$ ), whereas the combinatorial proof by Blanco (2026) achieves quadratic scaling. However, the geometric approach is presented as a self-contained mechanism derived from first principles, isolating a symmetric family of half-sized voting blocs and extending it to a maximal frame where every shorter balanced obstruction is excluded. This confirms that the infinitude of the coherence scheme in the Moss-Pedersen logic is inescapable.