Semantics: Notes 4

Emmon Bach, SOAS, UMass(Amherst)
Oxford: 5 February 2008
contact: ebach@linguist.umass.edu
Copyright Emmon Bach 2008. All rights reserved.
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(8 February) Fragment II; Relations between syntax and semantics

Preclass: Recaps

Exercises on Functions and Lambdas

L2: Extensions of L1

We will look at the extensions first by way of examples. Then we will give our new grammar and its interpretation in a more formal way. Additions: common nouns (CN), determiners (DET), DP (term phrases), IV adverbs.

Common Nouns: CN

We want to be able to represent the meanings of words and phrases like these:

apple
horse
unicorn
person who lives in London

And we want to be able to incorporate them in full DP's (term-phrases) like these:

the unicorn
every person who lives in London
an apple
some horse

So in the first place we add a category CN to accomodate common nouns. And we need a category of items which will combine with them to form DP's.

As usual, the general rule going from BA to PA for all A is in effect.

We first look at the treatment in PTQ. Then we will look at some alternatives. For simplicity, we will still stick with extensional interpretations.

Reminder: we skated past this very quickly. Philosophers distinguish between two kinds of meaning: intensions and extensions (cf. Frege's Sinn `sense' and Bedeutung `reference.' For now: sticking to extensions means we just think of the meanings as the things, truth conditions, etc. as they apply in this world. Intensions bring other ways things could be, in short, various other possible worlds.
To build in full DP's (term-phrases) that contain CN's we want to have determiners (Det) as well.

  • CN: The category of common nouns is interpreted as containing expressions that denote sets, just like intransitive verb phrases.

    • They are syntactically distinct: so the reason you can't say:

         *Pat unicorn.
           or
         *The sings is here.

    is a matter of grammar, or syntax.

    This point needs to be stressed as it is sometime said that the relation between syntax and interpretation in Montague Grammar is a one-one function. It is not. It is rather a many-one relation, or function if we take a disambiguated language for a base.

    So the type of CN's is <e,t>. This is a nice kind of meaning to serve as a basis for quantification, as we will now see.

  • DP: The category of term-phrases is interpreted as containing expressions that denote generalized quantifiers: families of sets (extensional interpretation) or families of properties (intensional interpretation).

    • We can accomodate the expressions we called N (names) in our first grammar, by invoking a type-lifting (TL) operation:

    N => DP : e => <<e,t>,t>

    (In a bit, we will consider another way of thinking about this treatment of DP's, N's amd CN's.)

  • Endocentric modifiers

    • Categorial grammars project an infinite number of categories, and carry with them a natural classification of expressions.

    1. arguments: A
    2. endocentric modifers: A/A, A\A
    3. exocentric constructions: A/B, A\B (A ≠ B)
    (It is interesting to compare this classification to other popular classifications, notably those that invoke the notion head. We will maybe do that later.)

    PTQ contained several kinds of endocentric modifiers: using the common abbreviations IV for instransitive, one is the category of intransitive verb phrase modifiers:

    IV\IV

    This includes the basic lexical items: rapidly, slowly, voluntarily, allegedly... it also includes some prepositional phrase built up from prepositions plus DP's.

    Exercise: what type would these prepositions have?

    Excursus: Two views of adverbs. The traditional category of adverb is a sort of wastebasket or "etc" class. It unites quite disparate expressions as seen in these examples with "adverbs" underlined:

    1. He answered politely.
    2. She is very angry.
    3. Undoubtedly, that is correct.
    4. He politely answered.
    5. He hadn't politely answered.
    6. He politely hadn't answered.
    7. He hadn't answered politely.
    8. He politely hadn't answered rudely
    9. You have behaved badly.
    Montague's PTQ deals with only one of these kinds, the one in (i), considered as an endocentric modifier, using the abbreviation IV for the category of intransitive verb (IVP for phrases) it would be (in our categorial notation) IV\IV, as mentioned above. This is semantically but not syntactically equivalent to another endocentric category: IV/IV for control verbs like try to.

    This view of (one kind of) adverb as of higher functional type is to be contrasted with analyses that follow the ideas of Donald Davidson (1967, 1980), which treat (some kinds) of adverbs as encoding attributes of events, with events taken as a central and necessary part of the models used for natural language. We will return to this topic in a later session in some detail.

    One more feature of the framework is worth commenting on. The last example (ix) illustrates a frequent pattern of categories in natural language. The verb behave is usually taken to require a manner adverb as a complement. If we accept this idea then it shows that functor categories can themselves act as arguments. So in general functors need to be satisfied -- "Frege's projection principle" -- or licensed by functors that take them as their argument category. There has been quite a lot of discussion in the literature about the problem of determining when a certain construction involves (endocentric) modification or argument satisfaction -- that is, when the item in question is acting as a complement. (McConnel-Ginet and Dowty have both discussed this question at length. More discussion and references later in our course.

    Exercise: Give an informal explanation of what kind of modifiers might be involved in the other examples. If any of the examples seem to be impossible, comment on why they seem bad.

    Excursus: Montague as a linguist.

    Excursus: Montague as a linguist. Montague's general heuristic seemed to be something like "take your language seriously," a sort of WYSIWYG strategy. In other words, unlike other generative grammarians of his time -- as well as later! , he tended to try to give direct interpretations for surface-true English expressions. This strategy was elevated to a limitation on analyses called the Wellformedness Constraint by Partee among others (Partee 1979, 1984). The strategy has also become a part of many non-transformational theories. As a constraint, it then has to evaluated as an empirical hypothesis. [Discussion on this.]

    Montague actually used an order-free notation for his functors, so you have to go to the rules to see what is being claimed, in this case PTQ's rule S10. The operation involved is F7: F7(δ,β) = βδ

    Montague's analysis of the internal structure of Term Phrases (DP's) in PTQ differs from what we have followed so far. Note these points:

    1. What we have called BN here includes proper names: John, Mary, Bill, ninety, he0, he1, he2,...
      2. That is, the category includes a countably infinite set of abstract "pronouns" (which will be treated as individual variables). They will be crucial for the treatment of quantification.

    2. They follow the general "basic to phrasal" analysis. (Montague uses the labels BT and PT: think Terms and Termphrases.)

    3. There is no category of determiners. The determiners are introduced directly in rules (a £100 word for this is syncategorematically). Compare the introduction of and in our first fragment. More satisfying for most linguists is the addition of a category for determiners.

      4. Exercise: What would that category be? It has the job of taking common noun phrases and turning them into generalized quantifier expressions, so (staying extensional) its result type is to be <<e,t>,t> and its argument type will be the type of CN's <e,t> (we have to postpone continuation until we get more information about the extensions of the CATegories of our syntax).
      Historical excursus on nominals.

      History: early phrase structure grammars as bases for transformational grammar followed a quite traditional analysis. Words like Sally, Chicago, horse, horses, ... were all considered subspecies of Nouns. Noun Phrases had (following English) the canonical shape Det + Noun, and nouns of the sorts just listed were subcategorized according to whether they had to have Determiners as sisters to form NP's. Between NP's and N was an uneasy hierarchy of possibilities, named N-Bar (N̅), Nom or the like. The status of pronouns was (to my recollection) not very clear. Postal (1969) argued for considering "socalled pronouns" a subcategory of Determiners. All along in this time, there was lurking the notion of a head. The NP picture was changed around the end of the 1980's with Abney's (Abney 1987) arguments for considering Determiners to be the heads of the categories in question and his labeling of the old NP's as DP's has been widely adopted.

      The idea that terms and term phrases are of the same species as common nouns is quite foreign to the logical tradition, which is rooted in the logic of first-order languages like the predicate calculus. Montague follows this tradition but expands the possibilities with the introduction of generalized quantifier expressions for natural language (English) consatituents of the sort under consideration here. The DP hypothesis of Abney offers a nice way of bringing the two traditions together. We are following that way here.

      Matthews (2007) includes a critical review of the whole question and history of the whole N - NP - DP matter.

      So more formally, lets extend our grammar explicitly by adding these categories. I'll list the semantic types immediately after the category here.

      G2 for L2: The new categories are these (the lists are in general not those of PTQ):

      1. CN : <e,t>
        2. BCN = {horse, person, koala, unicorn, hurricane, temperature,...}
      2. DP/CN : <<e,t>,<<e,t>,t>>
        3. BDP/CN = {the, every, some,...}

        We redefine several categories to bring them into line with the exposition of something like Montague's PTQ system, but departing in certain significant ways.

      3. S/DP : <dp,t> (abbreviation: IV)
        4. (using a simplified notation: a stands for the semantic type assigned to the category A (NB: significant use of lower vs higher case). Exercise: write out the unabbreviated syntactic categories and semantic types here.

        BS/DP = {run, jump, laugh, rise,...}

        This analysis departs from PTQ in taking subject DP's to be arguments of intransitive verb phrases, rather than the other way around. Discussion about this departure below under the Status of Subjects.

        I follow PTQ here in taking the basic lexical items verb forms to be uninflected bare verb stems, and the same for transitive verbs below, assuming some treatment of inflectional morphology more in line with linguistic realities.

      4. IV\IV : <iv,iv>
        5. BIV\IV = {slowly, rapidly, allegedly,...}
        Exercise: write out the unabbreviated category and type.

      5. (S\DP)/DP : <dp,<dp,t>> (abbreviation: TV
        6. B(S\DP)/DP = {love, detest, admire, see,...}

        Exercise: write out the unabbreviated category and type.

        Again, we need to provide for inflection or case-marking on the objects of the transitive verbs.

    The Status of Subjects

    The grammar of PTQ assumes that Subject DP's act as functors taking IVP's as arguments: so their category is just that of a generalized quantifier expression: for extensional verbs that gets us the equivalence that we noted at when we were first looking at the uniform treatment of DP's in English:

        : John'(run') is true iff run'(j) is true, where John' stands for λM[M(j)], translated into English: the set of sets that John belongs to contains the set of runners just in case John is a runner.

    In Montague's Universal Grammar as well as Keenan and Faltz (1985), Bach (1980) this relationship is reversed so that subjects are treated as arguments of tensed IVP's. This is another topic worthy of discussion, but will be better done after we have tangled with intensionality a bit.

    Syntax and Semantics

    The method we have been following about how to relate the syntax and semantics is what has been called the rule-to-rule or rule-by-rule assumption or hypothesis (Bach 1976). It is opposed to the assumption made in much generative grammar, which lays down a configurational approach. In the latter it is assumed that semantic interpretation is defined on structures at some syntactic level or levels of representation: Deep Structure, Surface Structure (perhaps decorated with indexed traces etc), Logical Form, or the like. The two ways impose different restrictions and allow different options. It is an interesting exercise to try and see whether there is any empirical evidence that would dictate a choice, or whether one or the other is more restrictive in principle.

    Here is an alternative representation of analysis trees (used, for example, in Steedman 2000): 

         

              John loves Mary, S : [love'(m)](j)
     -------------------------------------------FA<[+inflection]
                           S
                                      love Mary : love'(m)
    John, DP : λM[M(j)]                  
-----------------------------------FA>[+inflection]
    || (TL)                                S/DP
  John : j                   love   : love'               Mary  : m
 ---------LEX              --------------------LEX     ------------LEX
    N                            (S\DP)/DP                    DP

    LEX stands for the selection of an item from the lexicon. TL stands for the type lifting operation that takes N's to DP's.

    Note on inflectional morphology

    The addition [+inflection] is shorthand for a system of inflectional government and agreement, spelled out in some such manner as in Bach 1983, or a system of features of the sort used in various extended phrase-structure grammars or LFG. This is a worthy topic in its own right. The basic ideas in a categorial context are these:

    I believe these understandings are in line with traditional uses of the terms.

    Implementation of these notions can follow two plans: (1) the lexicon delivers fully inflected forms with feature values already specified; (2) forms come uninflected as base forms, the syntax is keyed to inflectional operations that have to find the crucial bases and dress them up with morphological realizations. In the first option, government and agreement are checking operations. In the second option, the grammar invokes morphological operations on the appropriate forms. In either case, you need to have a system of percolation worked out. I rather favour the second option (see Bach 1983).

    Practice:

    I. Write out a few derivations or analysis trees from our grammar so far, using either style of presentation.

    II. More practice with lambdas:

    1. Show the equivalence of
      2. λM[(M(j)](sleep') and sleep'(j).
    2. Show the equivalence of
      3. λx[Koala(x)] and λy[Koala(y)] and Koala.
    3. Show the non-equivalence of
      4. λy[Love(x)(y)] and λx[Love(x)(x)]
    4. Show the equivalence of
    5. λM[λN[∀x[M(x) ⇒ N(x)]]](Fish)(Swim) and Fish ⊆ Swim
    References