Semantics: Notes 5

Emmon Bach, SOAS, UMass(Amherst)
Oxford: 12 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"

(12 February)Intensionality, language levels, multimodalities;
Fragment III: relative clauses, quantifying in

Preclass: Recaps

From last class:
  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

The three kinds of "reductions" that are used in the lambda-calculus are: α-reduction, β-reduction, η-reduction. The first allows us to change variables (ii, iii) to give socalled "alphabetic variants," the second is just function-argument application or substitution (i, ii), the third eliminates vacuous lambda-abstraction. All require keeping careful track of free variables to avoid inadvertent capture (cf. iii).

There are three valid reductions that are used in the lambda-calculus:

  1. α-reduction
    λx[φ] ==> λy[φ'] where φ' is the result of replacing free x in φ by y
  2. β-reduction
    λx[φ](β) ==> φ' where φ' x is the result of replacing free x in φ by β
  3. η-reduction
    λx[φ](x) ==> φ where x is not free in &phi
Reference: Carpenter 50 and thereabouts. A careful statement of these rules has to make sure that variables don't get inadvertently "captured." The first rule licenses replacement of variables in such a way as to make sure that such unintended consequences don't happen. For careful statements about this see, for example, the relevant pages of Carpenter.

The way that goes is this (based on Carpenter):

We define a set FREE(φ) of free variables for each expression φ in the language of the lambda-calculus recursively like this

  1. if φ is a variable x, then x ∊ FREE(φ)
  2. if φ is a constant, then FREE(φ) = the empty set ∅
  3. if φ is α(β), then FREE(φ) = FREE(α) ∪ FREE(β)
  4. if φ = λx[ψ], then FREE(φ) = FREE(ψ) - {x}.
Now if x occurs in an expression φ and is not in FREE(φ), x is bound in φ.

In addition we need to define the idea of an expression α being free for a variable x in an expression β, if none of the free variable in α become bound when substituting α for x in β. A clause to prevent this inadvertent binding must be added to the definitions of α- and β-reduction. (See Carpenter: 44-45.)

Intensions

So far we have been restricting ourselves to extensional interpretations of our grammars. Before going on, let's amend this situation by introducing some intensional ideas. This whole issue goes back in modern times to puzzles about identity brought to the fore by Frege. Frege (1892) starts from the observation that expressions that refer to the same thing should be substitutable freely salva veritate (preserving truth). But there are some contexts where this principle fails. Frege's most famous example used the Morning Star and Evening Star, both assumed to refer to the same object, the planet Venus. Compare these examples:

  1. 2+3 = 5
  2. 5 is a prime number.
  3. 2+3 is a prime number.
  4. Meinong believes that 5 is a prime number.
  5. Meinong believes that 2+3 is not a prime number.
  6. Meinong doesn't believe that the Morning Star is the Evening Star.
  7. Meinong doesn't believe that the Morning Star is the Morning Star.

(i) is true as "2+3" refers to the same thing as "5". (ii) and (iii) are both true, and it seems reasonable to think that (iv) and (v) would both be true. But with (vi) and (vii) it seems that the latter could be true and the former false. Frege concluded from examples like these that expressions in language could have two kinds of meaning: Sinn and Bedeutung, usually translated as "sense" and "reference" or expressed as a difference between "intensions" and "extensions" (NB: intension with an "s"). A number of philosophers and logicians explicated this difference in various ways. Montague followed one stream of this discussion, using the apparatus of possible worlds and times. The sense or intension of an expression is a function from worlds (really worlds X times in PTQ). (Montague's development is a little different from what I am following here. There is a technical difference for him between senses and intensions, which I ignore.) So the recursive definition of Type is supplemented by this clause:

If α is a member of Type, so is <w,α>

But Montague uses s -- think sense -- rather than w, and includes a set of times in the model, ordered by ≤. More on this later. We can make our model exactly equivalent by reinterpreting W as the Cartesian product of the set of worlds I and the set of times J (I × J) ordered by ≤ and letting w be the type for this Cartesian product. But, to deal with tenses and modalities we need to be able to access worlds and times separately.

Montague followed the practice of interpreting English at the "highest" or most complex level, and relying on meaning postulates (constraints on proper interpretations) to climb back down to simpler meanings like the ones we have been using: for example, sets of entities rather than properties of individual concepts. Let's note explicitly these types:



TypeNameExample
<w,t>propositionthat the earth is flat
<w,e>individual conceptthe president, Kripke
<w,<<w,e>,t>> propertythe property of being a horse
<w,<<w,e>,<w,e>t>>>relations-in-intensionsseek: intensional version
of a transitive verb

Applications

In giving interpretations (mediated by the intensional logic IL), Montague interprets functional application uniformly as the application of the denotation of the functor to the sense of the denotation. This is notated by using the operator ˄ (pronounced "up") which gives senses of denotations: senses are functions from world-time pairs to the extensions of the denotation at that index. There is a corresponding "down" operator ˅ that gets us down to extensions at an index.

Notice that the type assigned to propositons gives us the exact idea that a proposition is a set of worlds.

What is an individual concept? In PTQ, it is a function from world-time pairs to individuals. So if we interpret the president as an individual concept we can understand sentence (a) as referring to the office of president rather than as a statement about the individual who is president at a given index. But in the following sentence (b) we understand the DP as referring to a particular individual: the value of the individual concept at a given index:

  1. The president is gaining power.
  2. The president should be impeached.

Another example from PTQ (due to Barbara H. Partee): How do you block the following reasoning: Appeal to the individual concept of temperature as a function from times and worlds (perhaps better: contexts, but more on that later).

The explication of the notion of a property in Montague's PTQ makes it possible to go part way toward solutions to some problems of extensional identity. Suppose there are no unicorns and also no chimaeras in our world. This means that the set of unicorns and the set of chimaeras are identical, namely, the empty set. This seems unsatisfactory for our understanding of natural language. When we have possible worlds in our interpretation, the property of being a unicorn and the property of being a chimaera are different, they will yield different sets when applied to various different possible worlds.

Levels of representation

Montague's PTQ works with the following levels:

English -- Disambiguated English (DE) -- Intensional Logic (IL) -- Denotations

As we've seen, the language of analysis trees is the disambiguated language corresponding to English. There is a functional mapping from DE to IL and from IL to Denotations. Hence, there must be a function relating DE directly to Denotations. Therefore, IL is strictly speaking eliminable. A linguistic question arises: what are the significant levels of representations that give a good fit to natural languages. Different frameworks have made different claims about this question.

Multimodalities

As we've noted a number of times in class, the programme of Montague Grammar and its various extensions and descendants follows a rule-to-rule assumption in specifying the relation between syntax and interpretation. In a fuller substantive theory of language, this picture would be amplified by adding further elements to the n-tuples generated by the grammar: phonological, phonetic (an interpretive component very like the semantic component of interpretations), syntactic, denotations,...

This assumption -- what we've called the H+-model -- has become popular in a number of different frameworks. (Question: which?) Here, let's just note that a natural way of looking at a grammar is as a multimodal categorial grammar.


Fragment III: L3: Extensions of L2,1

In this extension, we will add two new possibilities: relative clauses and "quantifying in." The latter is accomplished by several rules that allow for quantifying in to several different kinds of expressions: sentences, common noun phrases, and intransitive verb phrases. Both relative clauses and the quantifying in rules depend crucially on Montague's treatment of pronouns. Pronouns are represented by hei (i = 0, 1, 2, ...) which are members of the basic set indexed by T (Terms). Here we might use BN, in which case they could like names be type-lifted to DP.

Both constructions depend on finding the first occurrence of an indexed proform, and doing something to it and all subsequent occurrences of the proforms with the same index, some of which have been changed to accusative or objective forms when they are objects of verbs or prepositions. In the case of relative clauses they are all changed to {he, she, it} or {him, her,it} depending on the gender of the head noun. This works because PTQ just produces relative clauses with such that. Here we will instead work with relative clauses with that. In the case of quantifying in, there is a substitution of the quantifying DP for the first occurrence of the relevant proform and the same operation as just mentioned changing any other occurrences of same-indexed proforms in the manner specified.

Montague's rules are not very finessed. You can probably think of problems already coming from the mechanical stipulation of the linear first condition.


Relative clauses

We will depart from the treatment of relative clauses in PTQ, which generates only relative clauses of the form such that S. We will make a similarly simplifying assumption and deal just with clauses of the form that SDP, where SDP is a sentence that is missing a DP. Moreover, we will do this in a very sketchy fashion.

We make the assumption that the end result of the interpretation of a DP like (1) requires a meaning for a CNP like (2) as represented in (3):


  1. every fish that Bill catches
  2. fish that Bill catches
  3. λx[fish'(x) ∧ catch'(x)(b)

    4. Accordingly, we can take the relative clause itself to belong to the category CN\CN, that is an optional modifier looking for a CN to its left to make a CN. Letting vcn stand for variables of the type of a CN, we want the relative clause to have some such meaning as

  4. λvcn[λx[vcn(x) ∧ catch'(x)(b)]]


    5. This is a simplified account again, using an extensional interpretation. So the special variable vcn I have used is just the same as the variable M we have used before. The notation is used here just to illustrate a method used widely for tagging variables with their types. Note I am following the "case-lowering" convention introduced in Notes 4.


DP-S or NP-S

Are relative clauses sisters of a DP, as in (a), or sisters of NP (Nom) as in (b)? 

 
a.        DP                            b.       DP
          /\                                     /\
         /  \                                   /  \ 
        /    \                                 /    \
       DP     REL                             /      \
       /\      /\                           /        NP
      /  \    :  :                         Det       /\
     /    \   :   :                         |       /  \
    Det  NP   :    :                        |      /    \
     |    |   :     :                       |     NP    REL
     |    |   :      :                      |     |     :  :
     |    |   :       :                     |     |     :   :
   every fish  that Bill catches            every fish   that Bill catches

In (b) we can think of REL as short for NP\NP (CNP\CNP).
Both views -- as well as a third view in which relative clauses are introduced under Det and there is a "wrap" operation or transformation to get them into their surface order -- have been defended in the literature.

Exercise: Can you think of a way of preserving the relationships implicit in (a) but achieving the interpretation implicit in (b)?

How to do the syntax. First stab: just add that to the front of the relative clause (sentence) and delete the first instance of relevant indexed proform.

What syntactic problems will come up from this operation?

Quantification

It is probably not worth going in great detail into the way the quantification rules operate in PTQ. The interpretation is straight-forward. Since all DP's are interpreted as sets of properties, all we need to do is to bind the variable corresponding to the crucially indexed pronoun that is the substituend of the DP with a lambda and we will have the proper kind of denotation to fold into the generalized quantifier. Here are the last stages of two possible derivations of the following sentence presented schematically, using quantifying in:

Every unicorn seeks a fish

(a)      every unicorn seeks a fish  :
              /  \
             /    \
every unicorn      he0 seeks a fish
                           /  \                                      
                          /    \
                   a fish      he0 seeks him1


(b)      every unicorn seeks a fish
          /          \          
         /            \
  a fish             every unicorn seeks him1
                             /      \
                            /        \
                  every unicorn    he0 seeks him1
In addition to these two possibilities for quantifying in, there are derivations where the DP is simply introduced in situ. When this happens with the object DP, there is a good result, namely the result we want for an intensional verb like seek. For subject position, given PTQ's treatment of subjects as functors, there will be no difference in interpretation between derivations where the subject DP is there in situ and ones where it is quantified in as long as negation, modalities, or tenses are not involved. In the next extension, we will provide for different tenses. For now, note that Montague's Subject - Predicate rules put the interpretations of negation and the two tenses of PTQ as having scope over the subject DP.

The operation presupposed in the above examples is a simple substitution of the DP for the first occurrence of the relevant indexed proform and a through-substitution of appropriate pronouns for subsequent occurrences of that proform (as in the relative clause rule).


Optional Exercise: discuss seek, compare with treatments in other frameworks.

Optional Exercise: show what happens with and without tenses:

Example: Every president runs.
Example: Every president will run.

History: classical transformation grammar and Montague Grammar

There are quite a few affinities between Montague Grammar and classical form of Transformational Grammar, as exemplified in Chomsky, 1957, and work of the following few years. Some discussion of this in Bach 1976 and other papers from around then. Discussion of this point in class if we have time.

Excursus: classical transformational grammar, i.e. pre-Aspects. Recall (or learn!) about this.

The model of Chomsky's Syntactic Structures (1957) posited two syntactic components: a phrase-structure grammar G and a set T of transformations. G defined a set of kernel-structures (phrase-markers) which were the objects underlying simple active declarative sentences. T extended this set to various optional structures: questions, negations by Singulary Transformations, and embedded complement and other subordinate clauses appearing in complex sentences, as well as conjunctions by means of socalled Generalized Transformations. The syntactic objects generated from these sources were "phrase markers" and "transformation markers": the latter have a close affinity to the derivation trees of Montague grammar. For comparison of Montague grammar with classical transformational grammar, see Bach 1976, 1979. Cooper and Parsons (1976) is a formal comparision of later post-Aspects versions of transformational grammar. Stabler (1997) is a formal reconstruction of a minimalist treatment of quantification, and thus gives a basis for a more thorough formal examination of a formal semantics for minimalism.

Desideratum: comparison of multimodal MG and the minimalist programme of Chomsky, 1995 and on. A start: Merge can be identified with simple function-argument application. MG has no movements, so what about Move? It is probably best thought of as attachment of the "moved" constituent to a structure with a suitable missing argument. (See Steedman, 2000, for treatment of such structures in a combinatorial categorial grammar, of which more anon.)

Exercise: lambda-abstractions (or some equivalent) have played a crucial role in explicating the two readings of sentences like these:

  1. Sally washed her car and so did Nancy.
  2. Harry loves his house more than Ned does.
Discuss.

References: