Weakening cartography: On the formal foundation of functional hierarchies

SICOGG 2022 e-poster (August 12–14, 2022)

Chenchen (Julio) Song, cjs021 AT zju.edu.cn
School of International Studies, Zhejiang University

➊ Classical cartography, formally

Hierarchies of syntactic categories, each given rise to by a binary relation \(\mathbf{R}_\mathcal{A}\) on categories of a major part of speech \(\mathcal{A}\), such that \(\forall X, Y, Z \in \mathcal{A}\)

Thus, a classical cartographic hierarchy is a strict total order. The binary relation in question is usually deemed one of selection:

The classical view is multiply problematic.

➋ Problems of classical cartography

1. Transitivity failure

One could argue away some or even all of the above cases by derivational means (e.g., movement, Zwart 2009), but the problem of transitivity is more than counterexamples:

Either transitivity or the selection-based definition of \(\mathbf{R}\) is wrong.

2. Totality failure

Previous concerns about the foundation of cartography mostly target transitivity, but Song (2019) further notices that the totality condition on \(\mathbf{R}\) is also problematic.

Some categories belong to the same hierarchy but do not co-occur by design.

However \(\mathbf{R}\) is defined, it should have room for incomparable elements:

➌ Saving by weakening

Two previous attempts to “save” cartography by weakening its order relation:

1. Song (2019): partial order
reflexive, transitive, antisymmetric

2. Larson (2021): total preorder
reflexive, transitive, total

The different ways of weakening are due to the authors' different empirical foci (totality vs. transitivity failure). Larson reshapes the classical view more substantially, while Song merely adjusts its individual components.

Shared merit: freed from “selection pitfall”
transitivity   reflexivity

➍ A middle-way proposal

Definition 1 Weak Cartographic Hypothesis
All functional hierarchies are preorders. Some of them are furthermore total preorders, partial orders, or linear orders.
Remark 1 The above definition utilizes the “strength” relation between order relations: relation between order relations

R = reflexive, Tr = transitive, To = total, Ant = antisymmetric

Definition 2 Cartographic Relation (\(\sqsubseteq\))
\(\forall X, Y \in \mathcal{A},\) if \(Y\) functionally selects \(X\) in derivation, then \(X\) can fall in the scope of \(Y\) in the background ontology, written \(X \sqsubseteq Y\). The latter criterion defines functional hierarchies.
Remark 2 This ordering criterion is inherited from Song (2019). The binary relation \(\sqsubseteq\) is read “has a scope smaller than or equal to.” It is clearly reflexive/transtive and also evades the “problem of plenitude.”

➎ Possible functional hierarchies

Notation: \(X \rightarrow Y \equiv X \sqsubseteq Y,\) \(\{X, Y\} \equiv \) \(X\) and \(Y\) are incomparable.

The chain (linear order):

\[\dots X \rightarrow Y \rightarrow Z \rightarrow W \rightarrow V\dots\]

The connected digraph, with incomparable elements (preorder):

\[\dots X \rightarrow Y \leftrightarrows Z \rightarrow \{W_1, W_2\} \rightarrow V\dots\]

The connected digraph, w/o incomparable elements (total preorder):

\[\dots X \rightarrow Y \leftrightarrows Z \rightarrow W \leftrightarrows V\dots\]

The DAG (partial order):

\[\dots X \rightarrow \{Y_1, Y_2, Y_3\} \rightarrow Z \rightarrow \{W_1, W_2\} \rightarrow V\dots\]
Remark 3 Functional (sub)hierarchies are typically chains, especially if we take into account the subtle differences between iterable categories (e.g., the multiple Topic heads). Hence, the classical view is fine in many or even most cases, and linguists whose immediate concerns are order-theoretically nonexceptional (i.e., no incomparable categories or cycles) may conveniently stick to classical cartography.

➏ Bigger picture

The middle-way proposal can be extended from individual hierarchies to the entire functional category inventory. Consider two hierarchies defined by \(\mathbf{R}_\mathcal{A}\) and \(\mathbf{R}_\mathcal{B}\):

\[\mathcal{A}: \dots X \rightarrow \{Y_1, Y_2\} \rightarrow Z \rightarrow W\dots\] \[\mathcal{B}: \dots X \leftrightarrows Y \rightarrow Z \rightarrow W\dots \]

The two combined is still a preorder and may be viewed as a “superhierarchy.” One may also study the formal relations (e.g., monotone functions) across such hierarchies. Song (2019) explores such metatheoretical issues with the aid of mathematical category theory.

➐ Remaining issues

For future research: