Ranking constraints

I suppose it is tempting, if the only tool you have is a hammer, to treat everything as if it were a nail."

Abraham Maslow

The selection problem and the ranking problem

There are two common tasks for the OT analyst:
- Selection problem: Select the winner. Given a ranking, find the mappings that are generated by it.
- Ranking problem: Discover a ranking. Given a mapping or mappings, find which constraint rankings are necessary to generate it (or them).
The tableaux we’ve seen – Violation Tableaux (VTs), with asterisks, exclamation marks and shading – are very useful for the selection problem, but not ideal for the ranking problem.

Crucial ranking and non-crucial rankings

There will always be constraints that we can’t rank with the data we have.
A crucial ranking is one must hold for the optimal candidate to win. If the constraints were flipped, the winner would change.
If two contraints are not crucially ranked, the winner remains the same reagardless of their order.
Sometimes, this difference is represented by different vertical lines between the constraint columns of a tableau.
- A solid line between constraints represents a crucial ranking, while a dashed line represents
- A dashed line represents a non-crucial ranking.
- Dashed and solid lines cannot represent the full space of crucial rankings, as we’ll discover below.¹
A ranking argument demonstrates a crucial ranking, usually in the form of a tableau.
- The smallest ranking argument possible has two candidates and two constraints.
- The two constraints must conflict.
- The two candidates must be from the same candidate set.
- One of the candidates must be a winner.
- For the ranking argument to be sound, there can’t be a third constraint that favors the winner over the loser.

Let’s make tableaux showing ranking arguments for the Dutch problem from last time.
You should be able to copy-paste the tableau below into your favorite spreadsheet program.

You can use a Hasse diagram to show the crucial constraint rankings in a language:
- A vertical line means the constraint at the top of the line is crucially ranked above constraint at the bottom of the line.

Let’s list the crucial rankings shown in the diagram.
- Let’s enumerate some rankings that are compatible with the ranking requirements
- There are ranking requirements that can be expressed with a Hasse diagram but not dashed and solid lines in a violation tableau.
Hasse diagrams cannot easily represent the full space of crucial rankings, as we’ll discover below, but there are some sets of crucial rankings that can accurately represented with Hasse diagrams.

The violation tableau (with violation marks, exclamation marks, and shading) is a good way to illustrate the selection problem, but is not a good way to address the ranking problem.
The comparative tableau compares the winning candidate to different losing candidates, and is a good way to address the ranking problem.

Example of a comparative tableau
	Input	Winner ~ Loser	*CC	Dep–V
a.	/at-ka/	[atəka] ~ [atka]	W	L
b.	/at-a/	[ata] ~ [atəa]	e	W

Anatomy of a comparative tableau:
- Each row compares the winning candidate to one losing candidate.
- Each cell indicates whether a constraint prefers the Winner or Loser
- Indifferent constraints can be indicated with (e).
- Let’s make a comparative tableau for Dutch.
Each row of Ws and Ls encodes certain ranking requirements, sometimes called Elementary Ranking Condition (ERC [ɚk]: Prince 2002)
- For the desired winner to be optimal, at least one constraint assessing W (preferring the targeted optimum) must dominate all constraints assessing L (those preferring the targeted suboptimum).
  - Every L in a row must have a W to (somewhere) to its left!
  - Memorize this! Chant this! Write a song for this!
- For example <W, L, L, e> entails that the first constraint dominates the second, and the first constraint also dominates the third.
- What ranking requirements (if any) does <W, W, L, e> entail?

What are the crucial rankings for each the tableaux below?
Harder question: what ranking information that can be represented with ERCs that can’t be represented with dashed and solid lines?
A similar question: what ranking information that can be represented with ERCs that can’t be represented in a Hasse diagram?

Tableau 1
		Winner ~ Loser	Con–X	Con–Y	Con–Z
	a.	cand-1 ~ cand-2	W	L
	b.	cand-1 ~ cand-3		W	L

Tableau 2
		Winner ~ Loser	Con–X	Con–Y	Con–Z
	a.	cand-1 ~ cand-2	W	W	L
	b.	cand-1 ~ cand-3		L	W

Tableau 3
		Winner ~ Loser	Con–X	Con–Y	Con–Z
	a.	cand-1 ~ cand-2	L	W	L
	b.	cand-1 ~ cand-3			W

Tableau 4
		Winner ~ Loser	Con–X	Con–Y	Con–Z
	a.	cand-1 ~ cand-2	W	L	L
	b.	cand-1 ~ cand-3	L	W

Tableau 5
		Winner ~ Loser	Con–X	Con–Y	Con–Z
	a.	cand-1 ~ cand-2	W	L
	b.	cand-1 ~ cand-3			W

The best or worst of both worlds, depending on who you ask.
- The good: indicates both violations and constraint preferences
- The bad: not as clear as either VTs or CTs (collectively harmonically bounded), columns and rows don’t mean the same things in CTs and VTs (but in a HT they’re treated as equivalent)

Things to note:
- Subscripts make it easier to tell apart the VT part and the CT part.
- There are never Ws or Ls in the row containing the winner.
Going forward:
- Use CTs when you’re trying to show crucial rankings, discover the ranking, or verify that a ranking is possible.
- Use VTs when we’re trying to show how the constraint ranking selects the winner.
- Being able to read and write hybrid tableaux can be useful, but they’re never the best tool for the job.
  - Whenever I see a VT in a paper or on a conference handout to argue for a particular ranking, I immediately annotate it with Ws and Ls to verify the author’s claims.

ERC fusion is a way to combine two different ERCs, giving ranking information that entailed by the ERCs together.
- <W, L, W> ○ <e, W, L> = <W, L ,L>
- <W, L, e> ○ <W, W, W> = <W, L, W>
- <W, L, L> ○ <L, W, W> = <L, L, L>
ERCs can be combined using the following three rules:
- L dominance: W ○ L = L
- e identity: X ○ e = X
- Self identity: X ○ X = X
This is useful for:
- 1. Inconsistency detection (if fusion results in a W-less ERC)
- 2. Reasoning through disjunctive arguments

If you want to read more about this beyond what we discuss, a great place is Prince (2017) “Representing Grammars”, which discusses the merits of different representational devices at length. In the “Don’t Go There” category are violation tableaux with dashed lines and hybrid tableaux. ↩