Practical Grammar 4: Difference between revisions

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== Exercise 4 ==
<font face="Arial, Helvetica, sans-serif">
<font size="3">


♣ Add the following test sentences to your grammar and tell the program that they are ungrammatical:
== Verbs and VPs ==


(1) *John [disappeared the hospital].<br>
Next, we want to associate verbs and VPs with reasonable f-structures. As with NPs, we do this in two steps:
(2) *Martha [stayed the hospital].<br>
(3) *Fred [resides].<br>
(4) *Joe [saw Fred John].<br>
(5) *John [sent Martha to a check].<br>
(6) *We [gave Fred].


♣ Parse each sentence.<br>
a. We add appropriate features to the lexical entries of verbs, and<br>
♣ Does the grammar make the right prediction?<br>
b. we add the correct annotation to the VP-rules, so that each node in the VP tree gets the desired f-structure.
♣ If not, formulate in grammatical terms what the problem seems to be.


<font face="Arial, Helvetica, sans-serif">
'''Features of verbs'''
<font size="3">
 
Let us look at the following sentences:
 
(1) She ''is'' happy.<br>
(2) They ''are'' happy.<br>
(3) He ''was'' happy.
 
<span style="color: blue>Exercise 4.1</span>
 
<span style="color: blue>
State which verbal feature the constrast between (1) and (3) motivates. What are the feature's possible values?
</span>
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your answer
<div class="mw-collapsible-content">
The feature is '''TENSE''' and its possible values are pres, past, future.
</div>
</div><br>
 
<span style="color: blue>Exercise 4.2</span>
 
<span style="color: blue>
State which verbal feature the constrast between (1) and (2) motivates. What are the feature's possible values?
</span>
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your answer
<div class="mw-collapsible-content">
The feature is '''NUMBER''' (NUM), whose possible values are sg and pl.
</div>
</div><br>
 
<span style="color: blue>Exercise 4.3</span>
# Go to <span class="newwin">https://xlfg.labri.fr/</span> and log in.<br>
 
Then you locate our workshop:
 
# Choose Workshops|All Workshops.
# Look for the workshop "Syntax 1" for the current semester.
# In the column "Action", click the button "Consult the workshop".
 
Now you find a project of the workshop:
 
# On the right side of the screen is the list of projects of our workshop.
# Look for "Grammar 3" and click on "Create your own version".
# Now click on "Edit your version".
 
* Add appropriate NUM features to all the nouns in the lexicon.
* Add appropriate TENSE and NUM features to all the verbs in the lexicon.
 
== Annotating the rule VP -> V NP ==
 
Let us now turn to the rule that combines a V and an NP into a VP:
 
VP -> V NP;
 
In the previous unit, we formulated the following annotated c-structure rule for combining a D and an N into an NP:
 
1. NP -> D N
2. {
3.  ↑=↓1;
4.  ↑=↓2;
5. }
 
'''Reminder:'''
 
# ↑=↓1; in the rule above means that the NP and its first daugher share an f-structure.
# ↑=↓2; in the rule above means that the NP and its second daugher share an f-structure.
# ↑=↓1; and ↑=↓2; means that there is a single f-structure that the NP and its two daughers share.
# As a result if the feature values of D and N are compatible, then the f-structure of the NP will be the unification of the f-structures of D and N.<br>
 
<span style="color: blue>Exercise 4.4</span>
 
* In Grammar 3, give the VP rule an annotation that requires the f-structure of the V to be the same as the f-structure of the VP.
* Annotate the two NP rules as well.
* Add
 
(1) broke the bottle <br>
(2) broke the bottles
 
to the test sentence list and mark them as grammatical.
* Parse.
* Inspect the f-structures of the two VPs.
 
Next, we add subjects before the two VPs to turn them into sentences:
 
* Change the test sentences to
 
(3) Lilly broke the bottle <br>
(4) Lilly broke the bottles
<br>
 
* Parse and note what happens. Why?
 
* Click on '''Declarations''' to see the problem.
* Change the start symbol of the grammar from VP to S. Save the grammar and parse again.


== Valence ==
== Valence ==


'''Valence''' is the representation of the knowledge speakers have about what other kinds of constituents a word needs to combine with. You will remember from traditional grammar the distinction between '''intransitive''' and '''transitive''' verbs. These are just names for those verbs, respectively, which do not need a direct object (i.e. the verbs ''appear'' and ''cough'') and those which do (like ''have'' and ''trust''). Objects are called '''complements''' in our theory; so, to express that the verb ''have'' needs a direct object, its representation would contain the line
<span style="color: blue>Exercise 4.6</span>
 
* Add the following test sentences to your grammar, telling the program that they are ungrammatical:
 
(5) *John disappeared Fred.<br>
(6) *Lilly saw.<br>
 
* Add the words in (5)-(6) to the lexicon, if necessary.
* Parse each sentence.<br>
* Does the grammar make the right prediction?<br>
* If not, formulate in grammatical terms what the problem seems to be.


[PRED 'HAVE<SUBJ,OBJ>'],
To solve this problem, we need to add 2 important concepts to our grammatical theory: '''grammatical functions''' and the '''PRED feature'''.


which translated into people speech means that the expression needs one and only one complement and that the part of speech of this complement needs to be NP.
== Grammatical functions ==


Correspondingly,
Currently, our VP rule takes the following form:


COMPS <>
1. VP -> V NP
2. {
3.  ↑=↓1;
4. }


means that the expression does not need and, in fact, is not allowed to combine with any complement. That is correct for intransitive verbs like ''appear'', since you cannot say such things as <span style="color: blue>*Lilly appears the cake.</span>.
In words, the rule says 2 things:


As we will see, words cannot only select complements, but also subjects and determiners. This is what the two attribute SPR is for. Verbs are marked as
(a) In c-structure, a VP consists of a V daughter followed by an NP daughter.<br>
(b) In f-structure, the V and the VP share one and the same f-structure. Another way of saying this is that the V is '''the head''' of the VP.


SPR <NP>,  
'''Generalization'''
'''If A is the head of B, then the f-structure of A is also the f-structure of B.'''


i.e. they must have a subject NP and so-called common nouns (i.e. ''cat'' or ''student'') are listed as
The '''head-of''' relationship (marked by the annotation '''↑=↓1;''') is a '''grammatical function''' or a '''grammatical relation''' between 2 elements in a sentence.


SPR <D>,
There are other grammatical functions that you already know informally: '''subject-of''' and '''object-of'''. These are grammatical functions of NPs to the '''predicate''' of a sentence.


which means that they must combine with a determiner like ''a'' or ''the'' in order to function as a subject or an object.
There are 2 NPs and a verb in the sentence ''Lilly broke the bottle.''  
* State which grammatical function these 3 elements bear to each other and to the sentence!


With this background about the 2 valence attributes, look at the lexical entries for ''Lilly'' and ''Fido'' again. You see that both expressions are marked as
The functions '''subject''' and '''object''' are coded by word order in English.  


SPR <><br>
* Fill in the blank in the following 2 sentences:
COMPS <>.


From this we conclude that both words do not need to be combined with a specifier or a complement in order to function as the subject or object of a sentence. And this is correct, as the two sentences below illustrate:
(a) In English, the subject of a verb __________ the verb in linear order.
(b) In English, the object of a verb __________ the verb in linear order.


(1) <span style="color: blue>Lilly snores.</span><br>
We will now formally introduce grammatical functions into our grammar beginning with the object function.
(2) <span style="color: blue>I like Fido.</span>


In (1), the word ''Lilly'' serves as the specifier (= subject) of the verb ''snores'' and can do so all by itself. In (2), the word ''Fido'' is the direct object complement of the transitive verb ''likes'' and again it can do so all by itself. Compare this with what happens, when we substitute a common noun for the names ''Lilly'' and ''Fido'' in the sentences above:
To identify the postverbal NP as an '''object''' in the following c-structure rule, we add line 4 to the annotation of the rule:


(3) <span style="color: blue>*Student snores.</span><br>
1. VP -> V NP
(4) <span style="color: blue>*I like cat.</span>
2. {
3.  ↑=↓1;
4. '''(↑ OBJ) =↓2;'''
5. }


Both sentences become ungrammatical! The reason is simple: as already mentioned above, common nouns like ''student'' or ''cat'' are [SPR <D>], which means that they first need to combine with a determiner in order to serve as the subject or object of a sentence. Thus, (3) and (4) can be made grammatical by putting determiners in front of the two common nouns:
In words, the amended rule now says 3 things:


(5) <span style="color: blue>'''The''' student snores.</span><br>
(a) In c-structure, a VP consists of a V daughter followed by an NP daughter.<br>
(6) <span style="color: blue>I like '''any''' cat.</span>
(b) In f-structure, the V is '''the head''' of the VP.<br>
(c) In f-structure, the NP is the object of the VP (and hence also of the V, since the V and the VP are the same in f-structure).  


== Self Test Exercises on Verb Valence ==
* Add line 4 to your current version of Grammar 3.
<br>
* Parse and inspect the f-structures of


(1) <span style="color: blue>Kim <u>snores</u>.</span>
(7) Lilly broke the bottle<br>
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
(8) Lilly broke the bottles
Check your answer
<div class="mw-collapsible-content">
{|
|-
| SPR || <NP[''nom'']>
|-
| COMPS || < >
|}
</div>
</div><br>


2. <span style="color: blue>Kim <u>likes</u> Robin.</span>
<span style="color: blue>Exercise 4.5</span>
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your answer
<div class="mw-collapsible-content">
{|
|-
| SPR  || <NP[''nom'']>
|-
| COMPS || <NP[''acc'']>
|}
</div>
</div><br>


3. <span style="color: blue>Kim <u>talked</u> to the student.</span>
* Now, it is easy to identify the preverbal NP as the '''subject''' of the sentence. Add the required line to the c-structure rule that introduces the subject-NP into the sentence.
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
* Parse.
Check your answer
* Inspect the result and interpret it.
<div class="mw-collapsible-content">
{|
|-
| SPR    || <NP[''nom'']>
|-
| COMPS || < PP >
|}
</div>
</div><br>


4. <span style="color: blue>Kim <u>showed</u> Hessen to Robin.</span>
Look at the f-structures of the 2 ungrammatical sentences
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your answer
<div class="mw-collapsible-content">
{|
|-
| SPR    || <NP[''nom'']>
|-
| COMPS || <NP[''acc''], PP>
|}
</div>
</div><br>


5. <span style="color: blue>Kim <u>handed</u> Robin the apple.</span>
(5) *John disappeared Fred.<br>
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
(6) *Lilly saw.
Check your answer
<div class="mw-collapsible-content">
{|
|-
| SPR    || <NP[''nom'']>
|-
| COMPS || <NP[''acc''], NP[''acc'']>
|}
</div>
</div><br>


6. <span style="color: blue>Kim <u>told</u> Robin that the student likes her.</span>
* Inspect the result and interpret it.
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your answer
<div class="mw-collapsible-content">
{|
|-
| SPR    || <NP[''nom'']>
|-
| COMPS || <NP[''acc''], S>
|}
</div>
</div><br>


7. <span style="color: blue>Kim <u>mentioned</u> to Robin that the student likes her.</span>
== The PRED feature ==
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your answer
<div class="mw-collapsible-content">
{|
|-
| SPR    || <NP[''nom'']>
|-
| COMPS || <PP, S>
|}
</div>
</div><br>


8. <span style="color: blue>Kim <u>wants</u> [to eat an apple.]</span>
'''Valence''' is the representation of the knowledge speakers have about what other kinds of constituents a word needs to combine with. You will remember from traditional grammar the distinction between '''intransitive''' and '''transitive''' verbs. These are just names for those verbs, respectively, which do not need a direct object (i.e. the verb ''disappear'') and those which do (like ''see'').
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your answer
<div class="mw-collapsible-content">
{|
|-
| SPR    || <NP[''nom'']>
|-
| COMPS || <VP[''inf'']>
|}
</div>
</div><br>


9. <span style="color: blue>Kim <u>believes</u> Robin to like the student.</span>
In Lexical-Functional Grammar, the valence of a word is expressed by the value of its PRED(icate) feature. The PRED features of the 2 verbs ''disappeared'' and ''saw'' are written under the next 2 sentences:
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your answer
<div class="mw-collapsible-content">
{|
|-
| SPR    || <NP[''nom'']>
|-
| COMPS || <NP[''acc''], VP[''inf'']>
|}
</div>
</div><br>


10. <span style="color: blue>Kim <u>seems</u> to Robin to like the student.</span>
(9) <br>
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
<span style="color: blue>John</span> disappeared.<br>
Check your answer
[PRED &nbsp; 'DISAPPEAR<<span style="color: blue>SUBJ</span>>']<br>
<div class="mw-collapsible-content">
{|
|-
| SPR    || <NP[''nom'']>
|-
| COMPS || <NP[''acc''], VP[''inf'']>
|}
</div>
</div><br>


11. <span style="color: blue>Kim <u>is</u> intelligent.</span>
(10) <br>
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
<span style="color: blue>Joe</span> saw <span style="color: red>Fred</span>.<br>
Check your answer
[PRED &nbsp; 'SEE<<span style="color: blue>SUBJ</span>, <span style="color: red>OBJ</span>>']<br>
<div class="mw-collapsible-content">
{|
|-
| SPR    || <NP[''nom'']>
|-
| COMPS || <AP>
|}
</div>
</div><br>


12. <span style="color: blue>Kim <u>considers</u> the student intelligent.</span>
<span style="color: blue>Exercise 4.7</span>
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your answer
<div class="mw-collapsible-content">
{|
|-
| SPR    || <NP[''nom'']>
|-
| COMPS || <NP[''acc''], AP>
|}
</div>
</div><br>


13. <span style="color: blue>Kim <u>talked</u> to Robin about the student.</span>
* Open Grammer 4
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
* Add annotated grammar rules for S, VP, and NP to parse sentences (3)-(6). Do not change the lexicon yet!
Check your answer
* Add (9)-(10) as test sentences with their grammaticality status.
<div class="mw-collapsible-content">
* Parse.
{|
* Add the correct PRED feature to the lexical entries of the verbs.
|-
* Parse.
| SPR    || <NP[''nom'']>
* Inspect the f-structures of all 4 sentences carefully.
|-
| COMPS || <PP, PP>
|}
</div>
</div><br>


14. <span style="color: blue>Kim <u>saw</u> Robin talk to the student.</span>
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
Check your answer
<div class="mw-collapsible-content">
{|
|-
| SPR    || <NP[''nom'']>
|-
| COMPS || <NP[''acc''], VP[''base'']>
|}
</div>
</div><br>


== Homework for next week ==
<span style="color: blue>Homework</span>


Go to the [http://141.2.159.95:7029/wt/ Online Grammar] and do the following:
Read the following pages in the textbook:
section 2.2: p. 13-17
section 2.3: p. 22-25
section 2.3.2: everything


* Click on the words in the lexicon one by one.
Note: You will probably not understand everything in these pages! Do your best on the basis of what we've done in class so far.
* Before you click on a word, predict what label the grammar will give to that word.
* Predict for which words the grammar will give more than one label and what the labels are going to be. You can check this by clicking on the numbers in the upper righthand corner.  


<br>
<br>


<div align="center">
  [[Syntax_1_Wiki |'''Main page''']]|[[Syntax_1_Wiki:_Week_1| '''Week 1''']] |[[Syntax_1_Wiki:_Week_2| '''Week 2''']]| '''Week 3'''
</div>


<!--
<!--
| '''Week 3''']]|[[Syntax_1_Wiki:_Week_4| '''Week 4''']]| [[Syntax_1_Wiki:_Week_5| '''Week 5''']]|[[Syntax_1_Wiki:_Week_6| '''Week 6''']]|[[Syntax_1_Wiki:_Week_7| '''Week 7''']]
(2) <br>
-->
<span style="color: blue>The bottle</span> broke.</span><br>
[PRED &nbsp; 'BREAK<<span style="color: blue>SUBJ</span>>']<br>
 
(4) <br>
<span style="color: blue>Alice</span> broke <span style="color: red>the bottle</span>.<br>
[PRED &nbsp; 'SEE<<span style="color: blue>SUBJ</span>,<span style="color: red>OBJ</span>>']<br>


(5) <br>
<span style="color: blue>John</span> sent <span style="color: red>Martha</span> <span style="color: green>a check</span>.<br>
[PRED &nbsp; 'SEE<<span style="color: blue>SUBJ</span>,<span style="color: red>OBJ</span>,<span style="color: green>OBJ-TH</span>>']<br>


(6) <br>
<span style="color: blue>We</span> gave <span style="color: red>Fred</span> <span style="color: green>a wastebasket</span>.<br>
[PRED &nbsp; 'SEE<<span style="color: blue>SUBJ</span>,<span style="color: red>OBJ</span>,<span style="color: green>OBJ-TH</span>>']
-->


[[Practical_Grammar_2 | Page 2]]   [[Practical_Grammar_3 | Page 3]]
<br>
<br>
<font face="Arial, Helvetica, sans-serif">
<font size="2">
<div align="center">
    [[Practical_Grammar|'''Main page''']]  [[Practical_Grammar_2|'''Week 2''']] [[Practical_Grammar_3|'''Week 3''']]  '''Week 4'''  [[Practical_Grammar_5|'''Week 5''']]  [[Practical_Grammar_6|'''Week 6''']]  [[Practical_Grammar_7_new|'''Week 7''']]  [[Practical_Grammar_8|'''Week 8''']]  [[Practical_Grammar_9|'''Week 9''']]
</div>

Latest revision as of 11:50, 7 April 2025

Verbs and VPs

Next, we want to associate verbs and VPs with reasonable f-structures. As with NPs, we do this in two steps:

a. We add appropriate features to the lexical entries of verbs, and
b. we add the correct annotation to the VP-rules, so that each node in the VP tree gets the desired f-structure.

Features of verbs

Let us look at the following sentences:

(1) She is happy.
(2) They are happy.
(3) He was happy.

Exercise 4.1

State which verbal feature the constrast between (1) and (3) motivates. What are the feature's possible values?

Check your answer

The feature is TENSE and its possible values are pres, past, future.


Exercise 4.2

State which verbal feature the constrast between (1) and (2) motivates. What are the feature's possible values?

Check your answer

The feature is NUMBER (NUM), whose possible values are sg and pl.


Exercise 4.3

  1. Go to https://xlfg.labri.fr/ and log in.

Then you locate our workshop:

  1. Choose Workshops|All Workshops.
  2. Look for the workshop "Syntax 1" for the current semester.
  3. In the column "Action", click the button "Consult the workshop".

Now you find a project of the workshop:

  1. On the right side of the screen is the list of projects of our workshop.
  2. Look for "Grammar 3" and click on "Create your own version".
  3. Now click on "Edit your version".
  • Add appropriate NUM features to all the nouns in the lexicon.
  • Add appropriate TENSE and NUM features to all the verbs in the lexicon.

Annotating the rule VP -> V NP

Let us now turn to the rule that combines a V and an NP into a VP:

VP -> V NP;

In the previous unit, we formulated the following annotated c-structure rule for combining a D and an N into an NP:

1. NP -> D N
2. {
3.  ↑=↓1;
4.  ↑=↓2;
5. }

Reminder:

  1. ↑=↓1; in the rule above means that the NP and its first daugher share an f-structure.
  2. ↑=↓2; in the rule above means that the NP and its second daugher share an f-structure.
  3. ↑=↓1; and ↑=↓2; means that there is a single f-structure that the NP and its two daughers share.
  4. As a result if the feature values of D and N are compatible, then the f-structure of the NP will be the unification of the f-structures of D and N.

Exercise 4.4

  • In Grammar 3, give the VP rule an annotation that requires the f-structure of the V to be the same as the f-structure of the VP.
  • Annotate the two NP rules as well.
  • Add

(1) broke the bottle
(2) broke the bottles

to the test sentence list and mark them as grammatical.

  • Parse.
  • Inspect the f-structures of the two VPs.

Next, we add subjects before the two VPs to turn them into sentences:

  • Change the test sentences to

(3) Lilly broke the bottle
(4) Lilly broke the bottles

  • Parse and note what happens. Why?
  • Click on Declarations to see the problem.
  • Change the start symbol of the grammar from VP to S. Save the grammar and parse again.

Valence

Exercise 4.6

  • Add the following test sentences to your grammar, telling the program that they are ungrammatical:

(5) *John disappeared Fred.
(6) *Lilly saw.

  • Add the words in (5)-(6) to the lexicon, if necessary.
  • Parse each sentence.
  • Does the grammar make the right prediction?
  • If not, formulate in grammatical terms what the problem seems to be.

To solve this problem, we need to add 2 important concepts to our grammatical theory: grammatical functions and the PRED feature.

Grammatical functions

Currently, our VP rule takes the following form:

1. VP -> V NP
2. {
3.  ↑=↓1;
4. }

In words, the rule says 2 things:

(a) In c-structure, a VP consists of a V daughter followed by an NP daughter.
(b) In f-structure, the V and the VP share one and the same f-structure. Another way of saying this is that the V is the head of the VP.

Generalization
If A is the head of B, then the f-structure of A is also the f-structure of B.

The head-of relationship (marked by the annotation ↑=↓1;) is a grammatical function or a grammatical relation between 2 elements in a sentence.

There are other grammatical functions that you already know informally: subject-of and object-of. These are grammatical functions of NPs to the predicate of a sentence.

There are 2 NPs and a verb in the sentence Lilly broke the bottle.

  • State which grammatical function these 3 elements bear to each other and to the sentence!

The functions subject and object are coded by word order in English.

  • Fill in the blank in the following 2 sentences:
(a) In English, the subject of a verb __________ the verb in linear order.
(b) In English, the object of a verb __________ the verb in linear order.

We will now formally introduce grammatical functions into our grammar beginning with the object function.

To identify the postverbal NP as an object in the following c-structure rule, we add line 4 to the annotation of the rule:

1. VP -> V NP
2. {
3.  ↑=↓1;
4.  (↑ OBJ) =↓2;
5. }

In words, the amended rule now says 3 things:

(a) In c-structure, a VP consists of a V daughter followed by an NP daughter.
(b) In f-structure, the V is the head of the VP.
(c) In f-structure, the NP is the object of the VP (and hence also of the V, since the V and the VP are the same in f-structure).

  • Add line 4 to your current version of Grammar 3.
  • Parse and inspect the f-structures of

(7) Lilly broke the bottle
(8) Lilly broke the bottles

Exercise 4.5

  • Now, it is easy to identify the preverbal NP as the subject of the sentence. Add the required line to the c-structure rule that introduces the subject-NP into the sentence.
  • Parse.
  • Inspect the result and interpret it.

Look at the f-structures of the 2 ungrammatical sentences

(5) *John disappeared Fred.
(6) *Lilly saw.

  • Inspect the result and interpret it.

The PRED feature

Valence is the representation of the knowledge speakers have about what other kinds of constituents a word needs to combine with. You will remember from traditional grammar the distinction between intransitive and transitive verbs. These are just names for those verbs, respectively, which do not need a direct object (i.e. the verb disappear) and those which do (like see).

In Lexical-Functional Grammar, the valence of a word is expressed by the value of its PRED(icate) feature. The PRED features of the 2 verbs disappeared and saw are written under the next 2 sentences:

(9)
John disappeared.
[PRED   'DISAPPEAR<SUBJ>']

(10)
Joe saw Fred.
[PRED   'SEE<SUBJ, OBJ>']

Exercise 4.7

  • Open Grammer 4
  • Add annotated grammar rules for S, VP, and NP to parse sentences (3)-(6). Do not change the lexicon yet!
  • Add (9)-(10) as test sentences with their grammaticality status.
  • Parse.
  • Add the correct PRED feature to the lexical entries of the verbs.
  • Parse.
  • Inspect the f-structures of all 4 sentences carefully.


Homework

Read the following pages in the textbook:
section 2.2: p. 13-17 
section 2.3: p. 22-25
section 2.3.2: everything

Note: You will probably not understand everything in these pages! Do your best on the basis of what we've done in class so far.