What are the nine possible proposition-type combinations in arguments?
This explores where the 'nine combinations' come from in formal argument classification — specifically the proposition-type pairings inside Wagemans's Periodic Table of Arguments, where every argument links two claims and each claim can be one of a few basic kinds.
This explores where the 'nine combinations' come from in formal argument classification. The short version: arguments don't just have a premise and a conclusion floating in the abstract — each of those is a *kind* of proposition, and when you pair the kinds, the math gives you a fixed grid of possibilities. That grid is one of the three axes in Wagemans's Periodic Table of Arguments Can argument schemes be organized by formal principles instead of lists?.
Here's the move that makes it click. Wagemans treats every argument as a link between two propositions — the thing you're arguing *from* and the thing you're arguing *to*. Each proposition is one of three types depending on what it asserts: a *fact* (is something the case), a *value* (is something good or bad), or a *policy* (should something be done). Three types on the premise side times three types on the conclusion side gives 3 × 3 = nine proposition-type combinations. That pairing is exactly the third of the three orthogonal axes that organize the whole scheme space Can three axes organize all possible argument schemes?, sitting alongside the subject-vs-predicate axis and the first-order-vs-second-order axis.
What's genuinely surprising is *why* this matters. The old way of cataloguing arguments — Walton's list of 60-plus schemes — grew by family resemblance, one scheme at a time, with no principle saying when the list was complete. The combinatorial approach flips that: because the axes are finite and orthogonal, the space of possible arguments is *closed*. You can locate every argument by its coordinates, and — like gaps in the chemical periodic table — you can spot combinations nobody has studied yet because the cell exists even if no one has filled it Can argument schemes be organized by formal principles instead of lists?. The nine proposition pairings aren't a tidy summary of what people happen to argue; they're a prediction of what's *possible* to argue.
The thing the reader might not expect: this elegant closed structure runs straight into messy reality the moment you try to apply it. The same passage of text can be reconstructed as different arguments by different readers, with no ground truth to settle it — the formalization is underdetermined by the words themselves Why do different people reconstruct the same argument differently?. So even with a perfect nine-cell grid, *which* cell a given real-world argument lands in is partly a judgment call. And when machines try to do the sorting, they stumble: classifying schemes demands integrating inferential patterns across scattered parts of a text, which carries far higher cognitive load than tagging surface features Why does argument scheme classification stumble where other NLP tasks succeed?.
If you want to go deeper, the periodic-table notes Can argument schemes be organized by formal principles instead of lists? and Can three axes organize all possible argument schemes? are the core; the reconstruction note Why do different people reconstruct the same argument differently? is the necessary counterweight on why a clean theory of nine combinations doesn't make argument analysis a solved problem.
Sources 4 notes
Wagemans shows that three orthogonal axes generate a closed, finite classification space for all argument types, replacing the family-resemblance logic behind Walton's 60+ schemes. This mirrors the chemical periodic table's shift from contingent lists to predictive structure.
Wagemans's Periodic Table maps all argument schemes onto coordinates across three axes: subject-predicate structure, first-order versus second-order reasoning, and proposition-type pairings. This combinatorial approach replaces Walton's open-ended list with a closed, systematic space enabling computational analysis and discovery of unstudied scheme types.
Multiple valid argument reconstructions exist for the same text with no ground truth. This is not annotation error but an inherent feature of the task—different formalization schemas are each internally valid.
Scheme classification requires recognizing inferential patterns across distributed text spans, not local surface features. Models plateau at F1 0.55–0.65 while the same systems exceed 0.80 on component tagging and stance, suggesting the integrative reasoning demand is fundamentally different.