The One-Inch Frame
Two friends argue heatedly about whether a hot dog is a sandwich. One insists it is—bread on both sides, filling in the middle. The other insists it isn’t—ask any deli. After twenty minutes, neither has moved an inch.
How to Disagree About Categories
In March 2008, Paul Graham published “How to Disagree,” introducing a hierarchy from name-calling (DH0) to refuting the central point (DH6). The essay gave people precise language to distinguish weak attacks from strong refutation.
But Graham’s hierarchy misses one common failure mode that sits between simple contradiction (DH3) and genuine refutation (DH5–6). The argument looks substantive yet produces only heat. This pattern dominates disputes over categories, definitions, and identity.
The Hot Dog Debate
Person A uses culinary taxonomy: a sandwich has bread on two opposing sides with filling between. Hot dogs fit.
Person B uses cultural practice: sandwich shops don’t sell hot dogs.
Both positions are defensible, yet the debate stalls. Why? Because “sandwich” means different things in each framework. In taxonomic terms, hot dogs qualify. In cultural practice, they don’t. Each participant treats their framework as obvious and universal while it quietly shifts beneath the conversation.
This is unmarked frame-switching—toggling between reference criteria while treating the definition as fixed. Both sides present evidence and reasoning, but progress is impossible because they’re answering different questions disguised as the same one.
The Ship of Theseus
The ancient paradox asks: if every plank of a ship is replaced over time, is the returning vessel still the same ship?
The puzzle dissolves once you fix the criterion upfront:
- If “same ship” means “same matter,” then replacement creates a different ship.
- If it means “same functional form,” then replacement preserves identity.
Both answers are correct within their frameworks. The paradox only exists when the definition slides unmarked between them. The Sorites paradox (when does removing grains turn a heap into a non-heap?) and self-referential sentences like “This sentence is false” work the same way.
Three Types of Problems
Not all contradictions arise from frame-drift. Arguments fall into three categories:
Type A (Frame-Drift): The contradiction vanishes when criteria are declared upfront. Examples: hot dogs, Ship of Theseus, Sorites paradoxes.
Type B (Broken Axioms): The contradiction persists regardless of framework because the rules themselves conflict. Examples: A company policy that says ’employees must get manager approval for all decisions’ AND ‘managers cannot make decisions without employee input.'” The rules themselves create an impossible loop.
For people immersed in math, Russell’s Paradox, certain formal treatments of the Liar sentence also qualify.
Type C (Underspecified Questions): A single grammatical question packages multiple distinct queries. “Is coffee healthy?” The answer differs depending on whether you’re asking about cardiovascular effects, sleep quality, anxiety levels, or longevity. It’s grammatically one question but actually several.
In math, The Sleeping Beauty paradox asks for “Beauty’s credence,” but the answer differs depending on whether you’re asking about objective probability (1/2) or self-locating belief (1/3).
Solutions to Type A/B/C
Different types require different fixes: explicit criteria for frame-drift, system revision for broken axioms, disambiguation for underspecified questions.
Real-World Examples
Frame-switching appears everywhere:
- Abortion debates toggle between biological markers (conception, viability, birth) and legal or moral frameworks.
- AI intelligence arguments shift between functional capability, conscious experience, and architectural similarity to human cognition.
- “Real art” disputes drift across technical skill, emotional impact, creative intent, and market value.
The frustration arises because both sides bring evidence and neither seems foolish—yet they’re not engaging the same question.
Three Diagnostic Questions
When a categorization debate stalls:
- Could both parties be correct under different criteria? → Likely frame-drift.
- Does fixing any single criterion upfront eliminate the contradiction? → Confirms frame-drift.
- Does the question embed multiple distinct queries? → Needs disambiguation.
Most heated definition disputes fail the first test. Making the framework explicit—”I’m using taxonomic structure” versus “I’m using cultural practice”—often dissolves the impasse or reveals a tractable question about which framework serves the purpose.
Why This Matters
Graham’s hierarchy flags intellectual dishonesty and sloppy reasoning. Recognizing unmarked frame-switching adds a specific diagnostic: many arguments that appear substantive actually manufacture contradiction through shifting definitions.
Declaring your framework upfront eliminates paradoxes that depend on boundary absence rather than boundary location. Many prolonged disputes persist only because participants don’t realize they’re answering different questions.
The deepest benefit isn’t just better arguments. It’s realizing when you’re not disagreeing at all.
Open Questions
When multiple frames legitimately apply simultaneously—biological, legal, and moral frameworks in abortion debates, for instance—what determines which frame should govern public policy decisions? Frame-fixing clarifies the disagreement but doesn’t settle which framework has priority.
Why do intelligent people systematically fail to notice their own frame-switches? Is this a cognitive bias, a linguistic default, or something cultural? The pattern appears too consistently to be mere carelessness.
When does frame ambiguity serve useful purposes? Are there domains where maintaining multiple frames simultaneously drives progress rather than generating confusion?
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