Semantics: Notes 1

Emmon Bach, SOAS, UMass(Amherst)
Oxford: 15 January, 2008
contact: ebach@linguist.umass.edu
Copyright Emmon Bach 2008. All rights reserved.
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"http://www.people.umass.edu/ebach/courses/ox08-pl.htm"

Basics: model theoretic semantics, preliminary assumptions, some formal tools

Plan of course:

1. Review/exposition of basic model-theoretic semantics: roughly, Montague semantics.
2. Selected topics of recent and current research.

Preliminaries:

Semantics is the study of the meaning of linguistic expressions:
basic questions of linguistic semantics:
• What meanings, kinds of meanings, relations among them are needed to interpret expressions of natural languages?
• Interface? How is the semantics related to the grammar of a language?
• There are different kinds of meanings. How are they related?

Meanings of "meaning":

The word meaning is used in a number of different ways. We look first at meanings of "meaning" we will not focus on here in the first instance.
1. Meaning as translation:
"Hut" means "hat." : In German, the word pronounced /hu:t/ means the same as the English word represented by the letters "hat." Is this helpful? Only if we already know what "hat" means. Maybe this will be helpful if we think of translating into the Language of Thought.

Problem: How to identify the expression in question: For example: there are several words pronounce /hu:t/ in German, only one translates as "hat."

Moral: a linguistic expression is not to be identified with its phonological form. What we are dealing with are linguistic signs, complexes of properties, feature-values, etc. Among these properties: specification of language.

2. Meaning as concept:
A concept is a psychological entity, something in the head, an idea. Different heads have different concepts. Is there a kind of meaning that is independent of people's heads?

I say: "My cat is hungry." I am not talking about concepts of cats or hunger, I am talking about a real cat and a real state of affairs.

3. Meaning as intention:

Frege's essay Der Gedanke ("the thought" or "thoughts," Frege 1918-1919) is a basic and still relevant source for arguments against the identification of denotations and psychological entities of whatever sort, as well as a clear delineation of various other kinds of "meanings," including connotations etc. (next).

4. Meaning as connotation:

"A house is not a home." Etc.

5. Meaning as reference:
Model theoretic semantics: basic assumption:

The meaning of a linguistic expression is something that is in general* not a linguistic expression. When we want to be careful we will call this meaning of meaning denotation or semantic value.

Why "in general"? Because linguistic expressions can refer to linguistic expressions. Compare (i) above: "Cesar" is a four-letter word. Cesar is not a four-letter word. Cesar was stabbed by Brutus. "Cesar" was not stabbed by Brutus. Cesar said: "Et tu, Brute."

A trap for linguists: linguists are very good at manipulating symbols. So it is generally easy for them to learn to translate expressions of some language into expressions of some other language, for example, logical form. We will try to avoid this trap by focussing as much as we can on the things that language refers to.

What is "formal semantics"? Not to be confused: "formal" means here more less the same as explicit, precise, etc., not "expressed in a (strange) formalism." Cf. the trap for linguists.

Here is a perfectly formal meaning rule for an English sentence:

"John runs" is true if and only if John runs.

The use of a special language for showing denotations is a solely a matter of perspicuity and convenience. In this instance: translating the sentence just given into a formula Run(j) doesn't help unless we have a clearer idea of that expression than of the original English sentence.

In the meaning rule just given English is the object language, the language we are focussing on, interpreting etc. English is also the metalanguage, the language we are using to explain, interpret etc.

We could make this clearer by using another language as a meta-language. But we still have to give an interpretation of that language. Etc.

So why do model-theoretic semanticists generally resort to using "heavy" symbolisms?

6. The question of "Logical Form" (LF)

Excursus: There are two positions that have been taken with respect to a language of "logical form." One is (as in Montague) that an intermediate language or logic like Montague's intensional logic IL is merely a way of making the "real interpretation" perspicuous. The other is that LF is a linguistically significant level of representation. We will return to this question from time to time. There is another concept that comes in: the idea of a "disambiguated language" that is associated with the object language in a definite way. This language is given a direct interpretation. This would be another candidate for the idea of LF logical form, and it comes closer perhaps to the LF of Chomskyan treatments of natural language.

The goal of generative grammar, as originally conceived in the last century, was to treat natural languages as formal languages. The goal of formal semantics is to treat natural languages as interpreted formal languages. (In Bach, 1989, and elsewhere I called these two goals "Chomsky's thesis" and "Montague's thesis." Note that the second subsumes or presupposes the first.)

Compare this (formerly often quoted!) passage from Chomksy (1957: 5, quoted by Carpenter):

Precisely constructed models for linguistic structure can play an important role, both negative and positive, in the process of discovery itself. By pushing a precise but inadequate formulation to an unacceptable conclusion, we can often expose the exact source of this inadequacy, and consequently, gain a deeper understanding of the linguistic data.... I think that some of those linguists who have questioned the value of precise and technical development of linguistic theory have failed to recognize the productive potential in the method of rigorously stating a proposed theory and applying it to strictly linguistic material with no attempt to avoid unacceptable conclusions by ad hoc adjustments or loose formulation.

A bit of history: Linguistic semantics in the model-theoretic tradition has roots in modern logic and philosophy of language. Gottlob Frege was probably the most important early thinker in the modern period. We still return to his work for fundamental insights and ideas. We will take up Compositionality as a special topic, as it was first clearly stated by Frege. A necessary step in the modern development was the introduction of the very idea of a generative grammar in the work of Chomsky and others. All model-theoretic semantics (as well as other formal approaches) presupposes a precise specification of a language which is to be interpreted, that is, treating a natural language as a formal system. Up until Montague's work it was assumed by most logicians and philosophers that natural languages were too messy and ambiguous to be given a precise interpretation. (Some main figures in the development of model-theoretic semantics were Tarski, Carnap, Church.)

Montague denied that there was any principled difference between the artificial languages studied by logicians and natural languages. A famous quote:

I reject the contention that an important theoretical difference exists between formal and natural languages. (Montague 1970a: Montague 1974: 188)

(And a famous continuation: "On the other hand I do not regard as successful the formal treatments by certain contemporary linguists." Unnamed linguists, but compare in Universal Grammar a famous footnote disparaging Chomsky and transformational grammarians, on grounds of "adequacy, mathematical precision, and elegance." A challenge to produce substantial "fragments" continues. See discussion of the method of "fragments" below.)

Meanwhile back at the farm: generative grammarians of both the generative and interpretive semantics camps were paying more and more attention to data about quantification and other semantic aspects of language. Jerrold J. Katz and Jerry A. Fodor (1963) had proposed a semantic theory to go along with a generative grammar (famously criticized by David Lewis, 1972, but see now defence by Katz in Lappin 1996).

Montague's semantics was taken up by linguists in the seventies, under the leadership of Barbara H. Partee. For a sketch of developments in formal semantics up until a decade ago see her paper in Lappin 1996 (Partee 1976).

In recent years, introductory courses in semantics have broken away from the early tradition of working through Montague's papers, especially PTQ (Montague 1973). In this course, we will break somewhat from this break and go back to Montague for a good bit of the basic material, including Montague's general theory (metatheory) about syntax and semantics. (Why? Because I think the general theory deserves more of a hearing than it has gotten in recent years, plus it anticipates a lot that has gone on in recent work on syntax and semantics.)

"The compleat fragment"[!]

A noteable feature of Montague's work was the idea of a complete fragment. What is a complete fragment?

A complete fragment is a an explicit grammar and model-theoretic interpretation for some portion of a language, usually chosen so as to throw light on some set of problems.

PTQ is the best known example of a complete fragment. The problem areas treated in the fragment include -- as the title indicates -- quantification, and a number of related areas such as intensional and extensional verbs (see,seek, dual interpretations of such expressions as the president and so on. The main virtues of giving complete fragments are just those given by Chomsky in his first major work (Syntactic Structures for explicit -- i.e. generative -- grammar. You can't know what you are doing and what follows from your work unless you are reasonably explicit.

This does not mean that there is no value to informal work. Without some pretheoretic insights into structures and meanings there would be no basis for doing any kind of descriptive or theoretical work. But the generative ideal, as we may call it demands that such informal description be ultimately cashed out in explicit treatments.

Montague's rejection of MIT linguistics was based largely (imho) on the fact that there was no interpretation in the generative grammars of the day. (And probably on the rather arcane formalisms developed for them.) Compare:

I regard the construction of a theory of truth -- or rather, of the more general notion of truth under an arbitrary interpretation -- as the basic goal of serious syntax and semantics; and the developments emanating from the Massachusetts Institute of Technology offer little promise toward that end. (Montague's EFL, quote from Montague 1974, p. 188.)

Some basic ideas that we will draw on

Model theoretic semantics makes use of a lot of ideas from modern logic and mathematics. We will try to go easy on this aspect of the general theories we will look at. For orientation here is a simple list of some of the ideas that we draw on.

Sets and Relations

Example: Determiner interpretations.

The meanings of determiners like every, some, no can be explicated as denoting relations between sets:

1. Every koala is a marsupial. The set of koalas is a subset of the set of marsupials.
In symbols: let K denote the set of koalas, and M the set of marsupials: K ⊆ M

Some useful symbols for dealing with sets, plus a few logical symbols:

Symbol	Label	Meaning
∊	Membership	is a member of set
∅	empty set	set with no members
⊆	subset	is a subset of
⊂	proper subset	is a subset of and not equal to
∩	intersection	set of things that are both... and ...
∪	union	set of things that either...or...
¬	not	negation
∧	AND	both...and...
∨	OR	either...or...or both
⇒	implies	if... then...
⇔	iff	if and only if

With liberal use of parentheses and brackets we can write expressions for quite complex sets.

Exercises:

2. Do the same for some and no. Using either words or symbols OK.
3. Using capitalized English words for sets (Koala, Fish, Number) and lowercase words for individuals (john, max, chicago), write expressions translating these English phrases:
1. Kikki is a koala.
2. People who are British
3. John is either a Londoner or a Cantabridgian.
4. 11 is a prime number

Functions, lambda expressions (1)

The word function is used in a number of different ways in linguistics. In our deliberations here we will use the word in a strictly mathematical way:

Given two sets M and N, a function f from M -- the domain -- to N -- the range -- is a relation or mapping that associates at most one member -- f(a) of N with a given member a of M.

There are many ways to say this in a precise way. You can think of functions in a basically static way as simply sets of ordered pairs, or more dynamically as procedures or mappings. What is crucial is the uniqueness property. Writing f(a) for the second member of the ordered pair we have:

If f(a) = b and f(a) = c, then b = c.

We will continue to explore functions throughout this course. In anticipation of next time: a standard way of giving names for functions is the lambda-notation, often used in linguistics for naming predicates, but much more general in application:

λx[x ∊ M ∧ x ∊ N]

This is an expression for the notion of intersection as given above.

If every member a of the first set (M above) has a correspondent f(a in the second set, then f is total, otherwise partial. If f(a) does not exist for some a we say that f(a) is undefined. Note that a partial function f can be extended to a total function f' by adding a new element, say zilch to N, making N', and letting f'(a) = zilch.

More generally, functions can be defined over more than one set as the domain: a n-place function is an n+1-place relation between sequences of n arguments chosen from the n sets and such that the n+1-th element is unique.

5. Which of the following are functions? Discuss why or why not. Name the domains and ranges of the functions. For those that are not functions think of a closely related function that is a function. For example, "brother-of" is not a function from individuals to individuals, as I can several brothers, but there is a function "brothers-of" that goes from individuals to sets of individuals.
1. mother-of
2. biological mother-of
3. child-of

Mathematical examples:

4. square-of
5. successor-of
(on natural numbers)
6. square-root of

Syntactic examples:

7. immediately dominated by
(normal phrase markers)
8. commands
9. precedes
10. immediately precedes
Next session we will go carefully through the lambda notation and its interpretation and look at more examples of functions, with a lot of examples.

References

References are posted in a separate file. They will be added to in subsequent notes, and repeated in the general reference file.