Semantics: Notes 7

Emmon Bach, SOAS, UMass(Amherst)
Oxford: 26 February, 2008
contact: ebach@linguist.umass.edu
Copyright Emmon Bach 2008. All rights reserved.
Link to Notes:                                            
"http://www.people.umass.edu/ebach/courses/ox08-pl.htm"

(26 February) More on Dynamic Semantics; Adjectives; Structuring the Domain: I

Preclass: Recaps and Discussion

Thought question from last week:
In PTQ there are two ways to derive sentences like this:
The president will live in Texas.
(1) the subject is derived in situ, then the future tense takes scope over the subject; (2) the subject is quantified in, then the subject is evaluated "at now" and takes scope over the future tense of the matrix. So the two readings amount to these:
  1. It will be the case that the president (then) will live in Texas.
  2. The president now will live in Texas (at some future time).

Addendum: More on Dynamic Interpretation

[this secition will be incorporated in last week's Notes 6]
As noted last time, Kamp's Discourse Representation Theory and Heim's File Change theory do not work in a completely compositional way. Chierchia's account does work compositionally and in addition tries to recapture the unitary interpretation of DP's as generalized quantifiers. Chierchia builds essentially on the work of Groenendijk and Stokhof on Dynamic Predicate Logic (1991) and and Dynamic Montague Grammar (1990). The objects that are denoted by their formulas are sets of assignments, more accurately pairs of sets of assignments. An interpretation (first order) in a classical sense consists in a function D that specifies the value of all predicates (sequences of objects) relative to an assignment g of values to variables. A dynamic interpretation takes this one step further and interprets the formulas as functions from sets of assignments to sets of assignments, i.e as ordered pairs of sets of assignments:

As has been observed by several authors, there is a strong correspondence between the dynamic view on meaning, and a basic idea underlying the denotational approach to the semantics of programming languages, viz., that the meaning of a program can be captured in terms of a relation between machine states [longer quote online, Notes 6]. Given the restriction to antecedent-anaphor relations, the observed correspondence comes down to the following. A machine state may be identified with an assignment of objects to variables. The interpretation of a program can then be regarded as a set of ordered pairs of assignments, as the set of all its possible `input-output' pairs. A pair is in the interpretation of a program π, if when π is executed in state g, a possible resulting state is h. For example, the execution of an atomic program consisting of a simple assignment statement `x := a' transforms a state (assignment) g into a state (assignment) h which differs from g at most with respect to the value it assigns to x, and in which the object denoted by the constant a is assigned to x. Another simple illustration is provided by sequences of programs. The interpretation of a sequence of programs `π1 ; π2' is as follows. It can take us from state g to h, if there is some state k such that the program π1 can take us from g to k, and π2 from k to h. Or to put it differently, the second program is executed in a state which is (partly) created by the first. (Groenendijk and Stokhof 1990.)
The two papers just cited are highly recommended if you want to understand the Dynamic Binding in detail: as the titles suggest "Dynamic Predicate Logic" (1991) deals just with first-order formulas, while "Dynamic Montague Grammar" (1990) extends the treatment to deal with intensions. We'll come back to these topics when we make "situations" the focus of our discussion.

  1. Extending the coverage: preliminaries

    2. History: As soon as linguists like Barbara Partee, Michael Bennett started using the ideas they found in the work of Richard Montague, David Lewis, Max Cresswell and others to interpret English and other natural languages, there was much work devoted to extending the coverage of PTQ to reach a better approximation to the whole language. At first, this work was carried out in a very conservative way. For example, the extension of PTQ to cover plurals by Partee's student Michael Bennett used machinery that was already implicit in Montague's model: plurals were interpreted as sets (see Section III Plurals: First Stab below). Where PTQ interpreted the (singular) common noun unicorn as denoting the set of individuals -- actually individual concepts -- that were unicorns, the plural unicorns denoted the set of sets of unicorns, and so on. As time went on, however, people began to make more creative use of the general setup of Montague semantics.

    In this and the next session, we will take a look at some of the developments in the time between Montague's work and the present, particularly those that have to do with modifying the classical model structure inherited from PTQ either by modifying the structure itself or by introducing more structure into the domain of individuals, or both. We will also look at additions to the coverage of PTQ. As a reminder, here is the classical (PTQ) model structure, together with a summary of the modifications we have mentioned or discussed:

    M: Model Structure:
    1. A: a set of individuals, type e
    2. W: a set of worlds, possibly split into:
      3.      I: a set of worlds
           J: a set of times, with (anti-symmetric) ordering ≤
    3. BOOL: a set of truth values, type t
    4. F: family of functions built recursively from M, types <a,b>
    An interpretation of a language L takes such a model structure, and a function D (for Denotation) and assigns elements of M to expressions of L with the help of an assignment g of values to variables. In PTQ D is relativized to a world X time pair and an assignment, and is carried out via a translation of the disambiguated language L' derived from L into an intensional logic IL. An alternative to using a disambiguated language is to assign sets of interpretations to expressions of L (Robin Cooper 1983).

    Note: For Montague, IL is strictly eliminable and is not to be considered as a significant level of representation in the linguist's sense. Some linguists have pursued this line in a consequential way, notably Polly Jacobson in a number of papers under the rubric "direct compositionality" (book by that name about to be published, edited by Chris Barker and Pauline Jacobson). On the other hand, Mark Steedman, who like Jacobson works with a variable free syntax and semantics, posits a linguistically significant level of interpretation, his Logical Form. (Not = LF in Chomsky's sense!?) Steedman (2000) and Jacobson (1999, 2000) and some others use a version of categorial grammar (see -- Combinatory Categorial Grammar -- that deserves a course on its own (See Section VI below).

  2. Adjectives

    3. One of the first categories to be added to Montague's fragment PTQ was that of adjectives (Montague himself included adjectives in another fragment (EFL)). There are two places where expressions traditionally called adjectives occur:
    1. Harry is happy.
    2. The happy fish swims.
    3. The former president is unhappy.
    4. *The president is now former.
    5. The door is ajar.
    6. *The ajar door is the one to take.
    The type of (1) is called predicative, that of (2) attributive. The classic discussion of adjectives in the context of Montague semantics was by Muffy Siegel (1976a, 1976b). Siegel posited two syntactic categories for the two adjective uses, with an account for the large overlap in English (but not necessarily in other languages). For Russian, which has two morphologically distinct forms for adjectives -- socalled short forms and long forms -- she identified the two morphological types with the two categories. Semantically, the predicative type joined common nouns and intransitive verbs as a third syntactic correlate of the type <e,t>, while the attributive category was treated -- like relative clauses -- as a modifier of common nouns (CNP), i.e. CN/CN with type <cn,cn> (according to our "lower case" convention this is short for be <<e,t>,<e,t>>). As part of her analysis or as a spinoff from it, she was able to throw light on a lot of interesting puzzles, including those illustrated in the following examples:
    1. Pat is a beautiful dancer.
    2. That dancer is beautiful.
    3. He is not tall for a basketball player.
    Example (7) can be interpreted two ways: Pat is beautiful and a dancer; or (more immediate for me EB) Pat dances beautifully. (8) may still be ambiguous. (9) could be true of late Dennis Johnson, who was 6' 4".
    In line with PTQ, Siegel simply adopted a third category t///e for predicative adjective and phrases containing them. We can here adopt a new syntactic category Adj (and AdjP), with Siegel's semantics. To generate examples like (1) we need a rule that will combine be with an AdjP to form an IV. I will say no more about predicative adjectives right here and turn instead to the attributive examples. As a first pass, we can add the category CN/CN to the grammar and list some example adjectives:

    CN/CN: old, happy, former, alleged, red, green, large, long,...

    Taking this treatment together with our analysis of relative clauses leads immediately to a lot of questions about the grammaticality and interpretation of English sentences. We'll take up two.

    But let's first note some facts about relative clauses. The relative clause rule, as an optional endocentric modifier allows for stacked relative clauses:

    1. the house that is on the corner that has a gabled roof...
    2. the house that has a gabled roof that is on the corner...
    3. ?The house that is on the corner that has a gabled roof is the only house that is on the corner.
    4. The house that is on the corner that has a gabled roof is the only house that has a gabled roof.
    According to our rules the DP's of (10) and (11) are synonymous, since they just involve intersection of sets. Yet they seem to have a difference of presupposition or focus (or something) as we can see from the continuations in (12) and (13). The function of the restrictive relative clauses in examples like these seems to be to pick out a restricted set from within some set. So in (10) we start with the set of houses, then pick out a set of houses that are on the corner and then from that set the one that has a gabled roof. (12) leads to some kind of a mismatch as it asserts that there is only one house in the second set, and if so what is the restrictive force of the second relative clause? Sentences like those should be compared with parallel sentences with conjoined relative clauses:
    1. the house that is on the corner and that has a gabled roof
    Of course, our rules say nothing about focus or presupposition, and the general theory needs to be extended if we want it to encompass such ideas. But the point here is to sharpen our intuitions and to ask about interactions between the two types of CN modifiers that the rules so far project. Two questions arise immediately:

    i. Do relative clause formation and attributive adjective addition freely interact, as is predicted by our analysis so far?

    ii. Adjective classes: There are strong order restrictions among adjectives. Having a single class of attributive adjectives predicts all can be freely ordered. What to do?

    About question i: consider a simple case:
    1. The big house that is on the corner...
    Here we need to think about whether it makes any difference whether the NP is derived in one or the other of these ways:
    big + house that is on the corner
    big house + that is on the corner
    About question ii: what is the status of examples like these?
    1. big red horse
    2. red big horse
    3. difficult alleged counterexample
    4. alleged difficult counterexample
    5. black fake gun
    6. fake black gun
    7. fake fake gun
    We'll leave both of these questions hanging for now (the second kind of facts has been the focus of considerable research at SOAS: Chao and Mui In preparation, Scott 2002, 2003).

  3. Plurals: First Stab

    4. As mentioned above, Michael Bennett added plurals to the fragment, incorporating a new category for them and interpreting them as set counterparts of the singular terms and common nouns of PTQ. (Bennett 1976 is a somewhat more accessible source than 1974.) A justification for this move might come from the existence of predicates and other items that seem to select for plural arguments:
    1. They are friends.
    2. *They are a friend.
    3. *He is friends.
      4. (But cf. He is friends with me.)
    4. The boys dislike each other.
    5. *The boy dislikes each other.
    6. Arsenal are winning.
    7. ?Arsenal admire each other.
    Note that these new items under this analysis make demands on the type hierachy by proliferating the types all the way up the line. The examples show that some predications want to have plural subjects, some singular. Moreover, some perfectly ordinary English sentences require special treatment or are predicted to be bad. Since plurals are treated directly in the categorial and type theory, it looks like what seems to be a single verb like know has to have four variants, one for singular subjects and singular objects, one for plural subjects and singular objects, and so on... The same goes for attributive adjectives like the ones we just looked at: bigsingular, bigpluraland so on and on...

    1. Kim knows the answer, and so do the other students.
    2. I know the answers.

  4. Sorts

    5. Technically, much of the work that we will consider in this and subsequent sections relies on the idea that A, the domain of individuals, can be taken to be a union of Sorts, perhaps related (as in the case of plural individuals to be considered in a moment), but all logically still first-class members of the A, hence all of type e. For this approach to be useful we need to be able to notate all sorts of expressions according to the Sorts that they go with: individual variables, predicates and so on. We will do this with superscripts, so xpl might indicate a variable that ranges over the Sort of plural entities (whatever that might be in our implementation of the idea). People often save ink by requiring such an indication only on the first occurrence of the expression in a formula. For example:
    1. ∀xpl[x = x]
    Here it is understood the x's in the formula are "really" xpl. But in all such situations you should let clarity and perspicuity be your guiding lights.

    Sorts actually came into semantics for rather different reasons. Consider this sentence:

    1. Caesar is a prime number.
    2. Caesar is not a prime number.
    Is this sentence true or false? It is certainly not true, but if it is false then its negation (33) should be true. But both sentences seem to be strange, suffering from a category mistake of some kind. Many would say that (32) is neither true nor false. But then it has to have some third truth value. And indeed theories of sortal incorrectness often go with a system of three truth values: the third one perhaps "undefined" or zilch or some other "ugly object." (Two references for theories of sortal incorrectness are Thomason 1972 and Waldo 1979.)

  5. Plurals II: Linkian structures

    6. Godehard Link's (1983) paper on mass terms and plurals inspired a lot of work not only on the domains he considered but also event structures (Bach 1986). Like the work on generics that spun off from Greg Carlson's work (special section on Kinds and Generics next time), the innovations did not change the basic model structure per se, but began to build some special structure into the domain of individuals. Link's approach is algebraic, but I will explain it in an intuitive way, without going into too many formal details. We will stick to plurals here, reserving mass terms for later.

    Link's way goes like this: start with the set A of (ordinary) individuals. Allow a join operation to create "plural individuals" from the singulars, and on up. But these new individuals are still of type e. Crucially, the join operation is non-associative.

    So suppose we have a model with three horses: Attila, Balthazar, and Charlemagne. With the join operation (denoted by ⊕ "circle plus") we can make this structure: 

    〚horses〛 =   {        (a ⊕ b ⊕  c)

   (a ⊕ (b ⊕ c)) ((a ⊕ b) ⊕c) ((a ⊕ c) ⊕ b) 


            (a ⊕ b) (b  ⊕ c)  (a  ⊕ c) } 
  ____________________________________________________________
             
〚horse〛 =       { a           b           c }
    The denotation of horse is just the set {a, b, c}, the denotation of horses, on this account, is the seven plural individuals above the line. The structure is atomic, in that there is a lowest level of atoms, the individual horses. This is in contrast to the structures for mass terms, which are not atomic. In addition there is another construct: the set we get by putting together the singulars and the plurals, which Link called horse*.

    Schwarzschild (1991) contrasted two theories about plurals: one a "sets theory" and one a "unions" theory. The sets theory uses structures like those of Link, but using just set formation as the basic operation. In addition singleton sets are equated with individuals. The unions theory gives a much simpler structure, because union formation is associative. (In the diagram above the middle plural individuals above the line would just drop out.) Schwarzschild defends the unions view (see also his 1990 paper "Against Groups").

    A good bit of Schwarzschild's thesis goes to countering arguments based on examples like the following, which seem to favour the sets interpretation (and Link's theory as well). Assume a model where all the animals are just cows and pigs. Look at these sentences (Schwarzschild 1991: 69 -- new numbering here, in the source the numbers are 101 - 103):

    1. a. The cows and the pigs were separated.
      b. The young animals and the old animals were separated.
    2. a. The cows and the pigs talked to each other.
      b. The young animals and the old animals talked to each other.
    3. a. The cows and the pigs were given different foods.
      b. The young animals and the old animals were given different foods.

    On the unions theory the subject phrases in the (a) and (b) versions denote exactly the same sets, on the sets (and groups) theory they do not.

    On the other hand, look at these examples (op.cit. 71: 104 - 105):

    1. a. The animals filled the barn to capacity.
      b. The cows and the pigs filled the barn to capacity.
      c. The young animals and the old animals filled the barn to capacity.
    2. a. The animals were sleeping in the barn.
      b. The cows and the pigs were sleeping in the barn.
      c. The young animals and the old animals were sleeping in the barn.
    On the unions approach, the examples in each triple will have the same truth values, but not in the sets theory. Schwarzschild goes on to explore the further stipulations and shows they reduce to the unions theory.

  6. Combinators I (a different take on quantification and categorial grammars).

    7. There is an important line of research on syntax and semantics that takes off from the thread of categorial grammar that we have been following. It builds on Scho̎nfinkel 1924, Quine 1960, and especially Curry and Feys 1958; an entertaining introduction with many exercises and applications can be found in Smullyan (1982). The general formal framework is called Combinatory Logic. It is an alternative to the lambda calculus and provably equivalent in expressive power. Some linguists who have pursued this line in syntax and semantics are Anna Szabolcsi (1989), Mark Steedman (1988, 1996, 2000), Pauline Jacobson (1999, 2000, 2007 and other papers). David Dowty (2007) compares variable-free and variable-using approaches. There are a number of separate issues here, one being whether the syntax uses something like variables (traces, indexed pronouns) or just the semantics or both.

    To get a bit of a feel for how the combinators work, let us look at two basic combinators used both in combinatorial logic and in Steedman's work:

    • Type Lifting
      • T is a basic combinator that creates from any element function from functions on that type of element to any type:

      Example: generalized quantifier interpretations: e ==> <<e,t>,t>

    • Self operator
      • Example: reflexiviser REFL turns a transitive verb meaning into an intransitive verb meaning true of an individual related to itself by the transitive verb meaning.

    Logicians and mathematicians pursuing the general topic of combinators were concerned with finding the smallest set of such combinators from which all the others can be defined. Many useful combinators then turn out to be reducible to long strings of elementary combinators, exceedingly difficult for humans to process. In expositions of the meanings of the operators writers typically give equivalent lambda expressions which are somewhat easier to process (!). Combinatory logic has had direct applications in computing, as it makes for simpler programmes because it eliminates the need to keep track of the scopes of variables.