The Impotence of Binary Systems

Or Sometimes I See Repeated Patterns and Can’t Stop Thinking About Them

According to a November 2025 POLITICO poll, 59 percent of Americans say political polarization is worse than it was five years ago. Nearly half — 49 percent — say the country’s best times are behind us. Fifty-two percent say “radical change” is necessary to make life better in America.

When the government shuts down over budget impasses we blame polarization. We blame tribalism.

We’re looking in the wrong place. The problem isn’t who’s in charge (even when it’s That Fucking Guy) — it’s the math. Binary systems don’t fail because people are unreasonable. They fail because two is structurally insufficient for productive governance.

Photo by Jon Tyson on Unsplash

Consider how many fundamental frameworks of conscience require three elements, not two. Sigmund Freud’s structural model of the psyche, detailed in “The Ego and the Id”, divides the mind into Id, Ego, and Superego — impulsive desire, moral conscience, and the mediating reality principle that navigates between them. Aristotle’s “Rhetoric,” written in the 4th century B.C., identifies three modes of persuasion (which is where this article started!): Pathos, Logos, and Ethos — emotional appeal, logical argument, and ethical credibility working in concert. The Christian Trinity solved the theological problem of unity-in-multiplicity with Father, Son, and Holy Spirit. The American founders designed government with Legislative, Executive, and Judicial branches (Three, count ‘em three!) to check each other’s power.

And I’d be remiss as a sci-fi guy if I didn’t bring up Fritz Lang’s 1926 film Metropolis, in which it is shown that “The mediator between the head and the hands must be the heart” - arguing that capital and labor need a third force to broker between them.

This reflects a structural principle: two creates opposition, three creates negotiation. Binary systems offer only conflict or stalemate. Add a third element and suddenly coalition-building becomes possible. Any two elements can outvote the third, but no single element can dominate alone. Three is polarization proof!

The mathematics is unforgiving. In any decision-making system with three participants, you need two for a majority. This incentivizes alliance-building and compromise. The Ego must sometimes ally with the Id against the Superego’s harsh judgment, sometimes with the Superego against the Id’s destructive impulses. Effective rhetoric combines emotional resonance with logical structure, or ethical authority with passionate appeal. Each element holds potential swing-vote power. Five or seven work mathematically but sacrifice the cognitive elegance that makes three so powerful. Three hits the sweet spot: complex enough to enable coalition dynamics, simple enough to see at a glance.

In spite of what you learned from Schoolhouse Rock, there’s nothing magic about three. It’s simply the minimum non-binary system. What matters isn’t achieving exactly three forces, but escaping the paralysis of two. Parliamentary democracies with four or five viable parties function because coalition requirements force negotiation. According to political scientists G. Bingham Powell Jr. and Arend Lijphart, multiparty proportional representation systems typically produce more stable policy outcomes and higher citizen satisfaction than two-party majoritarian systems. No single party commands a majority; governing requires assembling support across factional lines. This prevents both permanent deadlock and unchecked dominance.

Deadlock

American politics has exactly two stable positions today. According to Pew Research Center analysis from 2022, there are zero Republicans in Congress more liberal than the most conservative Democrat, and zero Democrats more conservative than the most liberal Republican — complete partisan sorting unprecedented in modern American history. Every issue becomes binary: our side versus their side, win or lose, zero-sum conflict.

This wasn’t always true. When President Ronald Reagan and House Speaker Tip O’Neill negotiated the Social Security reforms of 1983, they were able to build coalitions because their parties weren’t ideologically pure. The 1983 amendments passed with significant bipartisan support: 243-102 in the House and 58-14 in the Senate. Conservative Southern Democrats could defect to support Republican initiatives; moderate Northern Republicans could vote with Democrats. Neither party leader commanded a uniform bloc. Political scientists call this era’s Congress “cross-pressured,” with four, five, six meaningful factions that could form different coalitions on different issues, as documented by Keith Poole and Howard Rosenthal’s DW-NOMINATE scoring system. That wasn’t a trinity, but it was non-binary, which is what mattered.

The ideological sorting of parties eliminated this flexibility. As political scientists Nolan McCarty, Keith Poole and Howard Rosenthal document in “Polarized America” (2006), the Civil Rights Act realignment, geographic sorting, nationalization of politics, and primary systems rewarding ideological purity have created two internally homogeneous parties. The structure that once enabled cross-party coalition-building has collapsed into pure binary opposition.

And binary systems, when governing high-stakes questions, produce predictable pathology. There’s no swing vote, no coalition incentive, no escape valve. When power splits between parties, nothing moves. When one party controls all branches, they maximize extraction before inevitable loss—dominance. Both states are dysfunctional. A non-binary system would naturally prevent both through coalition requirements.

Winner-take-all electoral rules, combined with what political scientist Maurice Duverger called “Duverger’s Law”—the principle that plurality-rule elections tend to favor two-party systems — create structural barriers to third parties. Ballot access restrictions, debate participation thresholds requiring 15% polling support (set by the Commission on Presidential Debates), and campaign finance structures all prevent third parties from emerging. The two parties have every incentive to maintain their duopoly and sufficient power to enforce it. Fixing the problem requires the cooperation of those who benefit from it. So, tough luck.

The Founders designed a government of threes—three branches checking each other through institutional loyalty. What they got was a government of twos — two parties capturing all three branches, collapsing institutional checks into team loyalty. They assumed diverse regional and economic interests would create cross-cutting allegiances, as Madison argued in Federalist No. 10. They never imagined such complete binary sorting.

We keep constructing binary systems and wondering why they produce paralysis. The mathematics is clear: two creates deadlock, more than two creates motion. American politics doesn’t need three parties; it just needs release from exactly two.

Rob Greene was a poli-sci major in under-grad, where he wrote such brilliant papers as, “What Would Spider-Man Do?” concerning the Supreme Court, and “Leadership According to Rob and Rost.”