Modelling intensions

In Chapter 9, an interpretation of the modal adverbs necessarily and possibly was presented in terms of possible worlds, but the other contexts in which extensional entailments fail were not discussed. As proposed in Section 9.2, the general solution to the problem of referentially opaque contexts lies in the concept of intensionality, but the interpretation of modality given above was not stated in terms of this concept and it has not been made clear how possible worlds enable a formal definition of intension to be made. Let us now remedy the situation and provide a general semantic theory for opaque contexts, thus completing our survey of formal semantic theory.

The definitions for the interpretation of modal formulae given in Chapter 9 embody the idea that formulae may be true in some worlds but not in others, i.e. that the extensions of formulae (i.e. truth values) may vary from world to world. Furthermore, it was suggested at the end of Section 9.2 that an intension is something that picks out the extension of an expression in any state of affairs. The intension of a formula may thus be defined as something that specifies its truth value in every state of affairs. Equating states-of-affairs with possible worlds, we interpret the intensions of formulae as functions that map possible worlds onto truth values: functions that map a possible world onto 1 if the formula is true in that world and onto 0, otherwise.

The variety of noun phrases

We have so far in this book been looking at the meaning of sentences primarily in terms of the properties that entities have and the relations that hold between them. This has meant a concentration on the interpretations of verbs and verb phrases, with the meanings of the phrases that serve as their arguments, noun phrases, taking second place. Indeed, only two sorts of English noun phrase have been analysed in the grammar fragments so far: proper names and simple definite noun phrases. But there are, of course, many other types of noun phrase in English, including indefinite noun phrases like those in (1.a & b), quantified noun phrases as in (1.c & d), noun phrases containing adjectives or relative clauses (1.e & f), noun phrases with possessive modifiers (1.g), and many more.

1. a. a book.

2. b. some cat.

3. c. every dog.

4. d. each person.

5. e. the happy student.

6. f. any student who gives a good report.

7. g. Ethel's friend's dog.

The noun phrases in (1) all require a more sophisticated interpretation than the one supplied for proper names and definite descriptions in earlier chapters. Both of these expressions have been translated as expressions of type e, denoting entities in the model. Such an analysis is, however, not ultimately tenable for proper nouns, is suspect for definite noun phrases and cannot be sustained at all for the other types of noun phrase in (1).

Although billed as an introduction to formal semantics in general, this textbook is concerned primarily with what has come to be called Montague Semantics and is therefore based primarily on Montague (1970a; 1970b; 1973). A good deal of research within Montague's general framework has been carried out since the 1970s and this has led to many changes in, and many variations of, the original theory. Other research has also led to reactions to Montague's programme and the development of rival theories. Only a few of these revisions and extensions to Montague's theory have, however, found their way into the text of the book. This may seem retrogressive, but it is my conviction that many of the questions being asked in formal semantics and the directions of research are best understood by learning about the more radical elements of Montague's original approach, particularly the semantic analysis of noun phrases and the theory of intensionality. Once these have been grasped, later developments can be understood more easily. For this reason, the exposition develops an account of the now classical version of Montague's theory, but references are given for the major revisions and extensions at the end of each chapter for readers to pursue as their interests dictate. Furthermore, there is no attempt in this book to give more of the logical and mathematical background than is necessary to understand how such things can help in the analysis of the semantics of natural languages.

Compound sentences

The grammar presented in Chapter 2 generates some very basic sentences of English and the translation procedure enables each one to be associated with at least one representation in Lp (more than one, if it contains a homonym). While the number of sentences generated by the grammar, G1 is relatively large (and can be made larger if more words are added to the lexicon), the language it generates is still finite. One of the properties that all natural languages are assumed to have is that they contain an infinite number of sentences. Since one of the goals of a theory of semantics, as we saw in Chapter 1, is to pair each sentence in a natural language with an interpretation, the theory must contain the means to provide an infinite number of interpretations. Furthermore, because natural languages are here being interpreted indirectly via a logical representation, the logical translation language must itself be infinite.

The reason that natural languages are infinite is that they are recursive. This means that expressions in certain categories may contain other expressions of the same category. For example, sentences may contain other sentences conjoined by the expressions and or or, or they may be connected by if…then, or a sentence may contain one or more repetitions of the expression it is not the case that. Some examples based on the grammar G1 are given in (1).

Temporal contingency

The grammar fragment developed in previous chapters generates sentences only in the past tense form. This has been a matter of expediency to allow more natural English to be used for the examples. However, the temporal properties of the sentences implied by the use of this tense have, in fact, been completely absent from their interpretation. The models we have been working with only contain a set of basic entities, A, and a function F which assigns an extension to each lexeme in the language. The denotations assigned by the latter are, however, static and no notion of change or development is (or could be) incorporated. This means that really the model theory treats all formulas as universal truths or universal falsehoods, as if they were all of the same sort as sentences like e = mc2, All humans are mortal, No bachelors are unmarried, The square root of nine is seventeen, and so on. The propositions expressed by such sentences have the same truth value at all times and so may be thought of as ‘timeless’. Most of the sentences generated by the grammar fragment, however, translate into formulae that could vary in truth value according to time and place, and other contextual factors. For example, A lecturer screamed may be true if uttered today but false if uttered the day before yesterday. Such sentences do not denote universal truths or falsehoods but contingent ones, ones whose truth depends on what is happening or has happened at a particular time.

The passive

In discussing the translation from English into Ltype in Chapter 4, rules for generating and interpreting simple passives were omitted. Although it is possible to define the extension of a passive verb phrase like kicked by Jo as the characteristic function of the set of things that Jo kicks, it is not possible with the apparatus we currently have to link this function directly with the extension of the active verb kick, kicks, kicked. The appropriate relationship between the two voices is that, in the relation denoted by the passive, the entity denoted by the object of the preposition by corresponds to the entity denoted by the subject of the active and the entity denoted by the passive subject corresponds to that denoted by the object in the active. This correspondence was handled in Chapter 2 directly in the translation for the passive rule by switching around the individual constants translating the two noun phrases in the passive rule to yield an identical translation to that of the active. So, for example, Jo kicked Chester and Chester was kicked by Jo are both translated into Lp as kick' (jo', Chester'). Unfortunately, this simple expedient is no longer open to us because of the existence in G2 of a verb phrase constituent. This prevents subject and complement NPs from being ordered with respect to each other in a translation rule because they are no longer introduced by the same syntactic rule.

Making inferences

The semantic theory developed up to Chapter 6 has concentrated mainly on the interpretation of sentences and phrases in isolation from each other, but one of the criteria for assessing the adequacy of a semantic theory set out in Chapter 1 is that it should account for the meaning relations that hold between different expressions in a language. This means, amongst other things, that the semantic theory proposed here ought to guarantee that, where reference and context are kept constant, the sentences in (1.b) and (1.c) are paraphrases of (1.a) while (1.d) and (1.e) are entailments of it and (1.f) and (1.g) are contradictions of it.

1. a. Jo stroked the cat and kicked the dog.

2. b. Jo kicked the dog and stroked the cat.

3. c. The cat was stroked by Jo and the dog was kicked by Jo.

4. d. Jo stroked the cat.

5. e. Someone kicked the dog.

6. f. The dog wasn't kicked.

7. g. No-one stroked anything.

The intuitively identified relations between the sentences in (1) derive from the interpretations of the conjunction and, the negative not and the quantifier pronouns no-one and someone. Such relations are generally referred to as logical entailments, paraphrases or contradictions. (Note that these terms are used ambiguously between the relation that holds amongst sentences, as here, and the product sentences themselves, as in the first paragraph above.)

Verb phrases and other constituents

One of the conditions of adequacy for a semantic theory set up in Chapter 1 is that it conform to the Principle of Compositionality. This principle requires the meaning of a sentence to be derived from the meaning of its parts and the way they are put together. The interpretation procedure for the grammar fragment set up in the last two chapters adheres to this principle insofar as the translations of sentences, and thereby their interpretations, are derived from the translations of their parts and the syntactic rules used to combine them. Thus, for example, the translation of the sentence Ethel kicked the student is derived from the translations of the two noun phrases Ethel and the student and the verb kicked. These are combined using the translation rule for transitive sentences to give kick'(ethel',the-student'). The truth or falsity of the resulting formula can then be directly ascertained by checking whether the ordered pair of entities denoted by the subject and object in that order is in the set of ordered pairs denoted by the predicate, kick'.

Unfortunately, in the theory of Chapters 2 and 3, compositionality is maintained only at the expense of the syntax. The ‘flat’ structure of the predicate-argument syntax of Lp and its interpretation requires a flat sentence structure in the English syntax in order to maintain a direct correspondence between syntax and translation, and thus a transparent relation between elements in the interpretation and constituents of the English sentence.

Translating English into a logical language

In this chapter and the next, we will lay the foundations on which a good deal of logical semantics is built. In accordance with the discussion in Chapter 1, we first define a logical language into which sentences of English are translated in order to circumvent the problems of ambiguity and underdeterminacy found in the object language. Having defined the translation language, and specified the procedure for translating simple English sentences into it, our attention will turn to the interpretation of these logical expressions in terms of their truth-conditions, thus providing an indirect interpretation of the corresponding English sentences.

The syntax of LP

Like all languages, natural or artificial, logical languages have a syntax, i.e. a set of rules for constructing composite expressions from simpler ones. The logical language described in this chapter, called LP, contains expressions that fall into one of four logical categories: individuals, predicates, formulae and operators (or connectives). Expressions in each of the first three categories can be further subdivided into two sorts: constants, which have a fixed interpretation, and variables, which do not. These two sorts of expression correspond, roughly, to content words (e.g. table, run, Ethel) and pronominal expressions (e.g. she, they) in natural languages, respectively. This chapter deals only with constants, but variables will become increasingly important in later chapters.

Sentences in natural languages translate into formulae in LP which have the logical category t (as sentences have the syntactic category S).

Semantics and semantic theory

In its broadest sense, semantics is the study of meaning and linguistic semantics is the study of meaning as expressed by the words, phrases and sentences of human languages. It is, however, more usual within linguistics to interpret the term more narrowly, as concerning the study of those aspects of meaning encoded in linguistic expressions that are independent of their use on particular occasions by particular individuals within a particular speech community. In other words, semantics is the study of meaning abstracted away from those aspects that are derived from the intentions of speakers, their psychological states and the socio-cultural aspects of the context in which their utterances are made. A further narrowing of the term is also commonly made in separating the study of semantics from that of pragmatics. Unfortunately, the nature of the object of inquiry of the discipline (what constitutes semantic meaning, as opposed to pragmatic meaning) and the domain of the inquiry (what aspects of meaning should be addressed by the discipline) remain difficult and controversial questions. There are, however, three central aspects of the meaning of linguistic expressions that are currently accepted by most semanticists as forming the core concern of linguistic semantics. These central concerns of semantic theory, adapted from Kempson (1977:4), are stated in (1) and may be adopted as criteria for ascertaining the adequacy of semantic theories which apply in addition to the general conditions on scientific theories of falsifiability and rigour.

Where entailments fail

In previous chapters (particularly Chapter 7), we looked at certain types of entailment relations that are guaranteed by the theory of interpretation set out in the earlier part of this book. Certain contexts exist, however, where expected entailments do not hold. Consider, for example, the inference pattern in (1).

1. a. The Morning Star is the planet Venus.

2. b. The Evening Star is the Morning Star.

3. c. Therefore, the Evening Star is the planet Venus.

The validity of this inference pattern illustrates a general rule that holds in the extensional semantic theory developed in Chapters 2 to 6 of this book. This rule is called Leibniz's Law or the Law of Substitution and it allows the substitution of extensionally equivalent expressions for one another in a formula while maintaining the truth value of the original formula. Thus, in (1), since the Morning Star and the Evening Star denote the same entity, the latter expression may be substituted for the former in the first premiss to give the conclusion. Indeed, because all three terms in (1) have the same extension all of them may be substituted for each other salva veritate (the Latin phrase used by Leibniz meaning ‘with truth unchanged’). The Law of Substitution can be formally defined as in (2) which, in words, says that if an expression a is extensionally equivalent to another expression b, then a formula φ is truth-conditionally equivalent to the formula formed from φ by substituting an instance of b for every instance of a.