Optimality Theory - Example

Example

As a simplified example, consider the manifestation of the English plural:

/ˈkæt/ + /z/ → (cats)(also smirks, hits, crepes)

/ˈdɒɡ/ + /z/ → (dogs)(also wugs, clubs, moms)

/ˈdɪʃ/ + /z/ → (dishes)(also classes, glasses, bushes)

Also consider the following constraint set, in descending order of domination (M: markedness, F: faithfulness):

M: *SS - Sibilant-Sibilant clusters are ungrammatical: one violation for every pair of adjacent sibilants in the output.

M: Agree(Voi) - Agree in specification of : one violation for every pair of adjacent obstruents in the output which disagree in voicing.

F: Max - Maximize all input segments in the output: one violation for each segment in the input that doesn't appear in the output (This constraint prevents deletion).

F: Dep - Output segments are dependent on having an input correspondent: one violation for each segment in the output that doesn't appear in the input (This constraint prevents insertion).

F: Ident(Voi) - Maintain the identity of the specification: one violation for each segment that differs in voicing between the input and output.

No matter how the constraints are re-ordered, the 'is' allomorph will always lose to 'iz'. This is called harmonic bounding. The violations incurred by the candidate 'dogiz' are a subset of the violations incurred by 'dogis'; specifically, if you epenthesize a vowel, changing the voicing of the morpheme is gratuitous violation of constraints. In the 'dog + z' tableau, there is a candidate 'dogz' which incurs no violations whatsoever. Within the constraint set of the problem, 'dogz' harmonically bounds all other possible candidates. This shows that a candidate does not need to be a winner in order to harmonically bound another candidate.

The tableaux from above are repeated below using the comparative tableaux format.

From the above tableau for dog + z, it can be observed that any ranking of these constraints will produce the observed output dogz. Because there are no loser-preferring comparisons, dogz wins under any ranking of these constraints; this means that no ranking can be established on the basis of this input.

The tableau for cat + z contains rows with a single W and a single L. This shows that Agree, Max, and Dep must all dominate Ident; however, no ranking can be established between those constraints on the basis of this input. Based on this tableau, the following ranking has been established:

Agree, Max, Dep >> Ident

This tableau shows that several more rankings are necessary in order to predict the desired outcome. The first row says nothing; there is no loser-preferring comparison in the first row. The second row reveals that either *SS or Agree must dominate Dep, based on the comparison between dishiz and dishz. The third row shows that Max must dominate Dep. The final row shows that either *SS or Ident must dominate Dep. From the cat + z tableau, it was established that Dep dominates Ident; this means that *SS must dominate Dep.

So far, the following rankings have been shown to be necessary:

*SS, Max >> Dep >> Ident

While it is possible that Agree can dominate Dep, it is not necessary; the ranking given above is sufficient for the observed for fishiz to emerge.

When the rankings from the tableaux are combined, the following ranking summary can be given:

*SS, Max >> Agree, Dep >> Ident

or

*SS, Max, Agree >> Dep >> Ident

There are two possible places to put Agree when writing out rankings linearly; neither is truly accurate. The first implies that *SS and Max must dominate Agree, and the second implies that Agree must dominate Dep. Neither of these are truthful, which is a failing of writing out rankings in a linear fashion like this. These sorts of problems are the reason why most linguists utilize a lattice graph to represent necessary and sufficient rankings, as shown below.

A diagram that represents necessary rankings of constraints in this style is a Hasse diagram.