English auxiliaries

In analysing the English auxiliary system, we have noticed a number of coöccurrence restrictions, summarized by the flowchart below.

Reading this flowchart from left to right, we can see that it expresses the generalizations that:

1. (present) tense and modals (will) are first, and in complementary distribution
2. Next, perfective have may appear, if it does, then whatever is next will be in the perfective form (en)
3. Next, progressive be may appear, if it does then whatever is next will be in the progressive form (ing)

The flowchart notation will be used frequently in this post, and so I will explain how to interpret it in some detail. The flowchart is a graph, where the nodes (shown as boxes) represent grammatical categories, and the edges (directed arrows) represent lexical items. Formally, an edge \(w\) exists from node \(u\) to node \(v\) just in case the first feature of the lexical item represented by \(w\) is \(\bullet u\), and the first negative feature of this lexical item is \(v\). There is an edge will from node v to node s in the above graph because the lexical item \(\textsf{will}\mathrel{::}\bullet v.k\bullet.s\) has as its first feature \(\bullet v\) and its first negative feature is \(s\). It would make the most sense to label edges with lexical items, instead of just their morpho-phonological form. It is for example not explicitly indicated, which of the silent lexical items in our lexicon is connecting the node y to the node v (the e is my current best approximation to an ε). However, looking at the final lexicon from the last post, we see that there are exactly three silent lexical items, and only one begins with feature \(\bullet y\) and has \(v\) as its first negative feature.

The analysis represented in figure 1 was arrived at in a mechanical way, by decomposing lexical items in order to try to eliminate redundancies in the lexicon. However, the analysis is only for a fragment of English (even of the English auxiliary system), and we will try to extend it in a similarly mechanical way.

Before we continue, I will modify our lexicon so as to collapse the distinction between the categories v and y by renaming them both to y. This will make the lexical item represented as an arrow from y to v a silent loop, which we will want to eliminate. This lexical item was introduced when decomposing the forms of the word eat, but has outlived its usefulness. I will then, now that v is unused, change the category named V to v.

Tense and modals

We have in our fragment only a single tense, the present, and a single modal, will. We would like to extend our fragment so as to deal with more sentences, such as the below.

1. John ate
2. John would eat
3. John can have eaten
4. John must have been eating
5. John could be eating
6. John should eat

As before, we can analyze these sentences as being composed out of whole words (given dependency structures), and decompose them to identify regularites. To save time, I will simply report the results of this process in figure 1.

\(\textsf{d}\mathrel{::}\underline{\bullet y}.k\bullet.s\)	\(\textsf{would}\mathrel{::}\bullet y. k\bullet.s\)	\(\textsf{can}\mathrel{::}\bullet y. k\bullet.s\)
\(\textsf{must}\mathrel{::}\bullet y. k\bullet.s\)	\(\textsf{could}\mathrel{::}\bullet y. k\bullet.s\)	\(\textsf{should}\mathrel{::}\bullet y. k\bullet.s\)
\(ate = eat \oplus d\)

These lexical items all fall into familiar position classes; namely d is parallel to s, and would, could, can, must and should are parallel to will in the sense that they have the same feature bundles (and thus the same distribution). We can see this by looking at the flowchart for our lexicon.

There is one major loci of redundancy in the grammar: all of these tense and modal elements share a feature sequence: \(k\bullet.s\). Accordingly, we extract a shared functional head \(\epsilon\mathrel{::}\underline{\bullet t}.k\bullet.s\) from our tense/modal elements, as shown excerpted below in figure 2.

\(\epsilon\mathrel{::}\underline{\bullet t}.k\bullet.s\)	\(\textsf{will}\mathrel{::}\bullet y.t\)
	\(\textsf{s}\mathrel{::}\underline{\bullet y}.t\)
\(\textsf{will} = \textsf{will} \oplus \epsilon\)	\(\textsf{s} \mapsto \textsf{s}\oplus\epsilon\)

In this table is an unfamiliar notation: \(\textsf{s}\mapsto\textsf{s}\oplus\epsilon\). This is to be understood as saying that in all of the morphological equations in our grammar, we replace the present tense s (the left hand side) with the present tense s plus the head hosting the subject position. Of course, we need other equations for the other modals, and another substitution for the past tense d, which are not shown here.

This gives rise to a lexicon displayed in the graph below in figure 4. In this lexicon, each one of the eight tense and modal elements have one fewer features, and in exchange we have added a new lexical item with three features, making for a net decrease of five features.

There are no longer any featural redundancies in our lexicon, but we can reduce the number of lexical entries if we view pairs like will and would (and can and could) as being inflected forms of a single lexeme.¹ We can decompose the tenses out from those lexical items, as well as from our current tense lexical items, and obtain the lexical items below.

\(\textsf{d}\mathrel{::}\underline{\bullet m}.t\)	\(\epsilon\mathrel{::}\underline{\bullet y}.m\)
	\(\textsf{will}\mathrel{::}\bullet y.m\)
\(\textsf{d} \mapsto \epsilon\oplus\textsf{d}\)	\(\textsf{would} = \textsf{will}\oplus \textsf{d}\)

The lexicon we arrive at is easier to make sense of as a graph, as shown below.

Negation and do

The English affixal negation n’t has been argued rather convincingly (by Zwicky and Pullum) to be an inflectional affix rather than a cliticized form of the negative word not. We begin again with a set of sentences we would like to extend our grammar developed thus far to account for.

1. John won’t have been eating
2. John hasn’t eaten
3. John isn’t eating
4. John mustn’t eat

Beginning once again with a whole word analysis, we can successively decompose the novel forms (won’t, hasn’t, isn’t, and mustn’t) until we obtain the negative head nt, and the new category \(pol\).

\(\textsf{nt}\mathrel{::}\underline{\bullet t}.pol\)	\(\epsilon\mathrel{::}\underline{\bullet t}.pol\)
\(\textsf{won’t} = \textsf{will}\oplus\textsf{s}\oplus\textsf{nt}\)	\(\textsf{hasn’t} = \textsf{have}\oplus\epsilon\oplus\textsf{s}\oplus\textsf{nt}\)

The resulting lexicon is perhaps better viewed as a flowchart.

Sentence 7 shows us that n’t must be introduced above have and be. Sentence 10 shows us that n’t must not interact with the category \(m\). The options then for the placement of n’t are restricted to

1. \(y\)
2. \(t\)
3. \(s\)

We have chosen the second option, that of splitting \(t\) into two categories, \(t\) and \(pol\), with n’t connecting the two. Choosing the first option, of splitting what is currently \(y\) in to \(y\) and \(pol\) would be formally possible, but as a consequence we would need to abandon the generalization that the ‘head’ of the morphological word is introduced syntactically low; then n’t would be the first head making up the word won’t, and thus will would need to be reanalyzed as an affix. This is well within the realm of analytical possibilities, but departs wildly from linguistic dogma. The third option, of splitting the category \(s\) into \(s\) and \(pol\) would require us to make negation weak - it could not support spellout of the morphological word it is part of. This is because the subject position is introduced by the empty head leading in to \(s\), and subjects are pronounced to the left of negative word forms in our data. Thus far, we have not had to worry about the strength of our lexical items; we have simply assumed them all to be strong.² Our choice of the second option (above tense, but below the subject position) is in line with much syntactic work, which places negation between the subject position (in AgrSP) and above tense (see e.g. Haegeman’s book (pdf)). Many of these works seem to me not to be making a distinction between affixal and whole-word negation however, and so it is not clear to me to what extent this convergence is of interest.

Our analysis predicts the syntactic existence of forms like eatsn’t and aten’t, which of course do not exist. These unattested forms are predicted by our analysis because negation combines with tense, irrespective of what may have come before. This is a standard problem, and a standard move in transformational grammar is to impose a filter, that blocks those structures in which the verb has combined with affixal negation. One way to do this is to prohibit the verb from raising through negation, i.e. by banning formation of a complex head containing both negation and a main verb. Auxiliaries and modals are exempt from such a filter. I have nothing insightful to say about this. As I see it, there are two formal options:

1. Ban derivations which contain a sequence of heads n’t-T-e-e-e-V.
2. Do not have a morphological realization rule for any sequence of heads n’t-T-e-e-e-V.

These options look to me functionally identical, and I don’t see any principled way of choosing between them. I will adopt the second approach here, because we have been making reference to morphology for a while, and thus do not need to introduce a new kind of mechanism (a syntactic filter).

Instead of using affixal negation on main verbs, English makes use of the periphrastic alternatives below.

1. John doesn’t eat
2. John didn’t eat

Again we begin with a whole-word analysis (the details of which are again suppressed), and end up with the additional lexical item below.

\(\textsf{do}\mathrel{::}\bullet v.m\)

Lexical do has been assigned the feature bundle it has because it

1. combines only with an uninflected lexical verb (\(\bullet v\))
2. is compatible with tense \(m\) but not with modals, or auxiliaries

It is as usual more enlightening to view the flowchart of our lexicon.

This analysis of do does not yet account for the intuition behind do-support, which is that do is in complmentary distribution with inflected main verbs. There are five constructions in which this obtains:

VP ellipsis

Mary ate, and John did too.

Negative imperatives

Don’t eat!

Negative sentences without auxiliaries or modals

John didn’t eat.

Subject auxiliary inversion without auxiliaries or modals

Did John eat?

Verum focus sentences without auxiliaries or modals

John DID eat.

We have not discussed (nor will we in this course) ellipsis (though see this paper for what I take to be a promising perspective), and so this will be left unexplored. I view the negative imperatives as covered from whatever account suffices for negative sentences. Negative sentences have already been discussed, and we accounted for the impossibility of negated main verbs by refusing to realize them morphologically. This will be our strategy for all of the further distributional limitations of these expressions. I will simply give suggestions about how to implement these.

Verum focus

We can add a silent lexical item (which we can nonetheless interpret as phonological focus marking) around the polarity head, along with its neutral counterpart:

\(\textsf{FOC}\mathrel{::}\underline{\bullet pol}.foc\)	\(\epsilon\mathrel{::}\underline{\bullet pol}.foc\)

We also require that main verbs cannot be interpreted morphologically with FOC.

SAI

We add a silent lexical item above the subject position head, which is necessarily strong.

\(\epsilon\mathrel{::}\underline{\bullet s}.q\)

We require that main verbs cannot be interpreted morphologically with this head.

These suggestions restrict the distribution of main verbs, but the other half of the do-support phenomenon is that do is restricted in a complementary way. Slightly more formally the generalization is:

do appears iff we have at least one of the following heads

• \(\epsilon\mathrel{::}\underline{\bullet s}.q\)
• \(\textsf{FOC}\mathrel{::}\underline{\bullet pol}.foc\)
• \(\textsf{nt}\mathrel{::}\underline{\bullet t}.pol\)

Therefore, do should be blocked if none of those heads appear. This can be done by the morphology refusing to interpret the following sequences of heads: \[\textsf{do}\oplus\textbf{T}\oplus\epsilon_{\neg NEG}\oplus\epsilon_{\neg FOC}\]

The lexicon we have arrived at can be viewed as the following graph.

This graph corresponds to the lexicon below, which assigns strength to lexical items in such a manner as to realize verbs low in the structure, or to the variant of this lexicon where all heads are uniformly strong.³

\(\epsilon\mathrel{::}\underline{\bullet s}.q\)	\(\epsilon\mathrel{:}\underline{\bullet foc}.k\bullet.s\)	\(\textsf{FOC}\mathrel{:}\underline{\bullet pol}.foc\)	\(\epsilon\mathrel{:}\underline{\bullet pol}.foc\)
\(\textsf{nt}\mathrel{:}\underline{\bullet t}.pol\)	\(\epsilon\mathrel{:}\underline{\bullet t}.pol\)	\(\textsf{must}\mathrel{:}\bullet y.t\)	\(\textsf{should}\mathrel{:}\bullet y.t\)
\(\textsf{s}\mathrel{:}\underline{\bullet m}.t\)	\(\textsf{d}\mathrel{:}\underline{\bullet m}.t\)	\(\epsilon\mathrel{:}\underline{\bullet y}.m\)
\(\textsf{will}\mathrel{:}\bullet y.m\)	\(\textsf{can}\mathrel{:}\bullet y.m\)	\(\textsf{do}\mathrel{::}\bullet v.m\)
\(\textsf{have}\mathrel{:}\bullet perf. y\)	\(\textsf{en}\mathrel{:}\underline{\bullet x}.perf\)	\(\epsilon\mathrel{:}\underline{\bullet x}.y\)
\(\textsf{be}\mathrel{:}\bullet prog. x\)	\(\textsf{ing}\mathrel{:}\underline{\bullet V}.prog\)	\(\epsilon\mathrel{:}\underline{\bullet v}.x\)

---

1. Some modals, like must, doesn’t appear to have a ‘tense’ distinction. Instead of listing this fact in the morphology (\(\textsf{must} = \textsf{must}\oplus\textsf{s}\) and \(\textsf{must} = \textsf{must}\oplus\textsf{d}\)) I have opted for making must ‘skip over’ the tense head. The modal should could be thought to be the past form of shall, but because we do not have shall in our lexicon at the moment I have decided to treat it like must. ↩︎

2. This is far from a knock down argument. In fact, most people seem to want lexical verbs to be pronounced rather low in the structure, which would require the following strength settings on our lexical items (speaking in the language of figure /posts/englishauxiliaries/): the empty lexical items between \(v\) and \(x\), between \(x\) and \(y\), and between \(y\) and \(m\) must be weak, as too must the tense morphemes. Then the polarity morphemes that we have introduced should also be weak (whether they are placed at \(t\) or at \(s\)). ↩︎

3. The lexical items have been made weak by default. This makes every lexeme be realized in its base position, unless of course the subject-auxiliary inversion head is present. There is no strength setting on heads in this analysis that will make auxiliaries be pronounced uniformly in T, and lexical verbs in V. ↩︎