PTQChris Potts, Ling 230b: Advanced semantics and pragmatics, Spring 2012

April 3

1 On Montague Grammar

The term Montague grammar refers primarily to the three papers Montague 1970a (EFL), Mon-tague 1970b (UG), and Montague 1973 (PTQ), listed here in increasing order of influence. Some-times the term is applied to all the language-related papers in Montague 1974. And of course it isoften used as a broad label for any kind of formal semantics.

It is here that Montague made his biggest contribution. To most logicians (like the first au-thor) trained in model-theoretic semantics, natural language was an anathema, impossiblyvague and incoherent. To us, the revolutionary idea in Montagues PTQ paper (and earlierpapers) is the claim that natural language is not impossibly incoherent, as his teacher Tarskihad led us to believe, but that large portions of its semantics can be treated by combin-ing known tools from logic, tools like functions of finite type, the -calculus, generalizedquantifiers, tense and modal logic, and all the rest. (Barwise and Cooper 1981:204)

Montague had a certain job that he wanted to do and used whatever tools he had at handto do it. If the product looks a bit like a Rube Goldberg machine, well, at least it works prettywell. (Barwise and Cooper 1981:204)

Montague grammer is a very elegant and a very simple theory of natural language se-mantics. Unfortunately its elegance and simplicity are obscured by a needlessly baroqueformalization. (Muskens 1995:7)

Montague revolutionized the field of semantic theory. He introduced methods and toolsfrom mathematical logic, and set standards for explicitness in semantics. Now all semanti-cists know that logic has more to offer than first order logic only. Finally, recall that BarbaraPartee said: lambdas really changed my life; in fact lambdas changed the lives of all seman-ticists. (Janssen 2011:4.1)

To really understand the intellectual tradition that Montague grew out of, one must read Fefermanand Feferman 2004, a biography of Alfred Tarski. Indeed, the Fefermans biography providesinsights into all of West Coast linguistics, logic, philosophy, and computer science, probably amongother fields. Barbara Partee is presently working on a history of formal semantics that will shedmore light on the origins and development of the ideas.

There are a number of extremely rich and useful primers on Montague semantics (broadlyconstrued): Halvorsen and Ladusaw 1979, Dowty et al. 1980, Gamut 1991, Partee 1997.1

1Montague in popular culture: a reference to Dowty et al. 1980 in Wallace 1996, and the novel Campbell 2009.

Ling 230b, Stanford (Potts) PTQ

2 The interpreted grammar

The next few subsections consist of a series of questions about PTQ. Well encounter a few majorchallenges as we work through them and the paper:

i. In the theory, noun phrases dont denote entities, but rather have the same type as quantifiers.This affects many of the other types.

ii. Very little of the intensionality in the system is reflected in its syntax.

iii. Properties are sets of individual concepts, rather than being functions from entities to propo-sitions or functions from worlds to sets of entities.

2.1 Syntax

(1) The full lexicon is defined on page 19. Annotate the definition of BA with the un-abbreviatedsyntactic categories. (For example, BIV becomes Bt/e.)

(2) What is going on with the clause BA = ? What linguistic claim might this embody, andwhat consequences might it have for the theory?

(3) How many lexical items are there in the PTQ fragment?

(4) Write an English paraphrase for

ACat BA.

(5) Write an English paraphrase for

ACat PA.

(6) PTQ syntactic categories are not mere symbols. What are they?

(7) What is the purpose of distinguishing t/e from t//e?

(8) In S2, the quantificational determiners are introduced as part of the syntactic rules. Why?Could we put them in the lexicon instead?

(9) Illustrate rule S3 using items from the lexicon. Present one case where the gender specifi-cation matters and one where it doesnt.

(10) Annotate rules S3-S10 with intuitive labels.

(11) How general are the rules for conjunction and disjunction, S11-S13? Are there intuitivelywell-formed phrases built from items in the fragment that it doesnt allow?

(12) Provide annotated derivation trees for the following phrases:

a. Bill finds a unicorn

b. every man dates every woman

c. man such that Mary saw him0

(13) On page 22, we see that the string John seeks a unicorn has two different derivations.How does this happen? (Which rules conspire to create this possibility?)

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Ling 230b, Stanford (Potts) PTQ

2.2 Intensional logic and models

Note: the character A is called Fraktur A, so we can just call it A (German accent optional).

(14) What is the set of basic types for the intensional logic? How does it correspond to the setof basic syntactic categories?

(15) What is unusual about the recursive definition of the set of all semantic types?

(16) Compare clause 5 with the lexicon and syntactic rules. Are there discrepancies in expres-sivity that we might want to address?

(17) The operator defined in clause 6 is sometimes called up, and the operator defined inclause 7 is sometimes called down? This makes sense given how they look. It also makesa deeper sense. How?

(18) What special limitation does Montague place on ?

(19) What are the types of and assuming MEa?(20) What are the components of an interpretation (intensional model)?

(21) How are senses, extensions, and intensions defined in general terms?

(22) What are the denotations of and assuming MEa? (We need to refer to footnote10 to obtain the denotations in the general case.)

(23) Montague sneaks in a definition of truth right below the definition of the interpretationfunction. What does the definition say?

(24) The final paragraph of this section (on p. 25) defines some abbreviations that are used in anumber of subsequent papers. Lets define them in both symbols and words if possible:

a. ( ,)

b. {}c. {,}d. u

e. u

f.

g. (p. 28-29)

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Ling 230b, Stanford (Potts) PTQ

2.3 Translation: English (the syntax) into the intensional logic

(25) Provide the types that correspond to the following categories:

a. IV = t/e

b. CN = t//e

c. T = t/(t/e)

d. TV = (t/e)/(t/(t/e))

(26) What kind of functions do IV and CN objects denote?

(27) The translation of be, in (T1b), looks extravagant, but it hides a simple idea. What is it?

(28) The translations of proper names (T1d) and pronouns (T1e) are also obscure, but there isa guiding intuition. What is it?

(29) The operator is used systematically throughout T4-T10. What is it doing?

(30) What is the type and denotation of necessarily ?

(31) What is the value of the man runs where there is more than one man?

(32) The rules in T14-T16 pull a neat trick with lambda abstraction. How does it work, andwhat does it accomplish?

(33) What is the force of meaning postulate 2 (p. 28)?

(34) What is a logically possible interpretation for Montague?

(35) The editors point out that the first equivalence on page 29 does not hold for BCN. Constructa counterexample, i.e., a legitimate CN-type meaning that obeys the meaning postulatesbut nonetheless fails this equivalence.

(36) Unpack all the abbreviations in the following formulae so that the truth conditions areevident:

a.p[2 p]fish(m)

b.

u[man(u)walk(u)]

c.

u[unicorn(u) seek( j, u)]

d. seek j, Pu[unicorn(u) P{u}]

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Ling 230b, Stanford (Potts) PTQ

3 Sub-topics

Section 4, Examples, discusses two phenomena in detail. Each has given rise to its own literature:

Intensional objects: Quine 1960; Partee 1974; Dowty 1979; Zimmermann 1993; Larsonet al. 1997; Moltmann 1997; van Geenhoven and McNally 2005; Moltmann 2008; Schwarz2006

Temperature-type objects: Partee 1974; Bennett 1975; Jackendoff 1979; Dowty et al. 1980;Lbner 1981; Lasersohn 2005

4 Variables for possible worlds

One of the central innovations/simplifications of Gallin 1975 was to have variables of type s. Thisresults in a system that is considerably more transparent than PTQs. For a system of this sort, seeCarpenter 1997:2.

5 Perspectives from Lewis 1970

The logic of Lewis 1970 is very similar to that of PTQ (and simpler; I dont actually know why itisnt more famous than PTQ), and the exposition is more expansive Lewis provides insights andperspectives on the approach, its place within linguistics, and its limitations. A few excerpts:

Dissing generative semantics

My proposals regarding the nature of meanings will not conform to the expectationsof those linguists who conceive of semantic interpretation as the assignment to sen-tences and their constituents of compounds of semantic markers or the like. (Katzand Postal, 1964, for instance.) Semantic markers are symbols: items in the vocabularyof an artificial language we may call Semantic Markerese. Semantic interpretation bymeans of them amounts merely to a translation algorithm from the object languageto the auxiliary language Markerese. But we can know the Markerese translation ofan English sentence without knowing the first thing about the meaning of the Englishsentence: namely, the conditions under which it would be true. Semantics with notreatment of truth conditions is not semantics. (p. 18)

Narrowly grammatical

My proposals will also not conform to the expectations of those who, in analyzingmeaning, turn immediately to the psychology and sociology of language users: tointentions, sense-experience, and mental ideas, or to social rules, conventions, andregularities. I distinguish two topics: first, the description of possible languages orgrammars as abstract semantic systems whereby symbols are associated with aspectsof the world; and second, the description of the psychological and sociological facts

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Ling 230b, Stanford (Potts) PTQ

whereby a particular one of these abstract semantic systems is the one used by a per-son or population. Only confusion comes of mixing these two topics. This paper dealsalmost entirely with the first. (I discuss the second elsewhere: Lewis, 1968b and 1969,Chapter V.) [These works are cited here as Lewis 1969 and Lewis 1975 CP.] (p. 19)

Intensions vs. meanings

Intensions, our functions from indices to extensions, are designed to do part of whatmeanings do. Yet they are not meanings; for there are differences in meaning unac-companied by differences in intension. It would be absurd to say that all tautologieshave the same meaning, but they have the same intension; the constant function hav-ing at every index the value truth. Intensions are part of the way to meanings, however,and they are of interest in their own right. (p. 25)

We have already observed that intensions for sentences cannot be identified with mean-ings since differences in meaning for instance, between tautologies may not carrywith them any difference in intension. The same goes for other categories, basic or de-rived. Differences in intension, we may say, give us coarse differences in meaning.For fine differences in meaning we must look to the analysis of a compound into con-stituents and to the intensions of the several constituents. (p. 31)

References

Barwise, Jon and Robin Cooper. 1981. Generalized quantifiers and natural language. Linguisticsand Philosophy 4(4):159219.

Bennett, Michael. 1975. Some extensions of a Montague fragment of English. Ms., Indiana Uni-versity Linguistics Club.

Campbell, Aifric. 2009. The Semantics of Murder. London: Serpents Tail.Carpenter, Bob. 1997. Type-Logical Semantics. Cambridge, MA: MIT Press.Dowty, David. 1979. Word Meaning and Montague Grammar. Dordrecht: D. Reidel.Dowty, David R.; Robert E. Wall; and Stanley Peters. 1980. Introduction to Montague Semantics.

Dordrecht: Kluwer.Feferman, Anita Burdman and Solomon Feferman. 2004. Alfred Tarski: Life and Logic. Cambridge:

Cambridge University Press.Gallin, David. 1975. Intensional and Higher-Order Modal Logic. Amsterdam: North-Holland.Gamut, L. T. F. 1991. Logic, Language, and Meaning, volume 2. Chicago: University of Chicago

Press.van Geenhoven, Veerle and Louise McNally. 2005. On the property analysis of opaque comple-

ments. Lingua 115:885914.Halvorsen, Per-Kristian and William A. Ladusaw. 1979. Montagues Universal grammar: An intro-

duction for the linguist. Linguistics and Philosophy 3(2):185223.Jackendoff, Ray S. 1979. How to keep ninety from rising. Linguistic Inquiry 10:172177.Janssen, Theo M. V. 2011. Montague semantics. In Edward N. Zalta, ed., The Stanford Encyclopedia

of Philosophy. Stanford, CA: CSLI, winter 2011 edition. URL http://plato.stanford.edu/archives/win2011/entries/montague-semantics/.

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Ling 230b, Stanford (Potts) PTQ

Katz, Jerrold J. and Paul M. Postal. 1964. An Integrated Theory of Linguistic Descriptions. Cam-bridge, MA: MIT Press.

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Philosophy of Science, volume VII, 335. Minneapolis: University of Minnesota Press. Reprintedin Lewis 1983, 163188. Page references are to the reprinting.

Lewis, David. 1983. Philosophical Papers, volume 1. New York: Oxford University Press.Lbner, Sebastian. 1981. Intensional verbs and functional concepts: More on the rising tempera-

ture problem. Linguistic Inquiry 12:471477.Moltmann, Friederike. 1997. Intensional verbs and quantifiers. Natural Language Semantics

5(1):152.Moltmann, Friederike. 2008. Intensional verbs and their intentional objects. Natural Language

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Montague, Richard. 1973. The proper treatment of quantification in ordinary English. In JaakkoHintikka; Julius Matthew Emil Moravcisk; and Patrick Suppes, eds., Approaches to Natural Lan-guage, 221242. Dordrecht: D. Reidel. Reprinted in Montague (1974), 247270.

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On Montague GrammarThe interpreted grammarSyntaxIntensional logic and modelsTranslation: English (the syntax) into the intensional logic

Sub-topicsVariables for possible worldsPerspectives from Lewis70GS