CHAPTER 26

Quantification

26.1 Singular Nouns and Noun Phrases (again)

A singular noun is the label for a concept. A concept is associated with:

(a) a class to which entities can belong and with

(b) a set of properties which allow the class of an entity to be recognised.

A singular noun is a label for both these aspects of a concept, which in traditional semantics have been called, respectively, the 'extension' and the 'intension' of a noun. Here we shall use the terms 'concept-class' and 'concept-­properties'. .

In the early chapters of this book we argued that a concept can be represented by a generic structure which can be used as a template to generate any member of its associated concept-class, and we shall adopt that view here. Each noun can be associated with a procedure, and when the noun is encountered in the text being processed the procedure is activated to create the generic structure. As such it is an anonymous structure. None of its formal parameters has been instantiated, and at that stage it has no unique identity.

The processing of adjectives in association with the noun leads to modification of the generic structure by the addition, deletion or amendment of properties. At that stage it is still a generic structure and defines a new composite concept. For example, the expression 'toy dog' is represented as a concept formed by taking the generic structure belonging to 'dog' and amending its properties according to those associated with 'toy'. Where these are incompatible, those of 'toy' prevail, since these are being applied as for an adjective. What we have created is a specialisation of the two concepts 'toy' and 'dog', in the sense that we used that term in section 23.4.

The meaning of a noun phrase is represented as a specific example of the concept-class. The use of the word 'a' for example, takes the generic structure and turns it into an entity with an identity of its own by instantiating the formal parameter associated with its identifier to a specific value. The word 'the' (in the simple case) instantiates it by reference to some existing entity. What we have then is an entity in its own right, as distinct from a concept.

This analysis does have its problems. If we are dealing with a sentence such as 'If I had a car I would keep it in good condition', the noun phrase 'a car' does not refer to a specific car. It lies within the scope of the modal operator 'if', and has therefore been 'possibilated'.



26.2 Plurals

A plural noun can form a noun phrase on its own, without the need for determiners such as 'a' or 'the'. The meaning of a plural noun is therefore not to be represented as a concept, but as a group of entities.

A single entity is represented by a single structural unit which has its own properties and its own unique identifier. At first sight there appears to be no difficulty about extending this method of representation to several entities. Two entities could be represented by two structural units, three by three structural units, and so on.

The problem which is associated with the representation of plural nouns appears when there is no definite number of entities involved, or where the number involved is so great that we are unable to consider the use of structural units on a strict one-to-one association. It is tempting to try to represent a plural entity by a single structure which has a 'number' attribute. This could be an integer or a formal parameter if the number is unknown. This approach is unsatisfactory, however, because we could be faced with a sentence in which there is a need to identify specific units within a collection of indeterminate number. Consider, for example, the following passage: 'A group of men were smashing down the door. One had a sledge hammer and another had an axe.' What is required is a representation which is a single structural unit (for the whole group) and which can be expanded as required by generating exemplars.

A possible solution makes use of the same kind of structure which we used in our analysis of conjunction, with one additional feature. This is illustrated in Figure 26.1.




What we have here is a conjunction of two entities (X and Y) and an indefmite number of other entities which have no explicit representation. These are represented symbolically by the generic structure which is used to generate further examples as required.

The representation is analogous to'the dynamic list structure which is a feature of POP II and is described in the appendix. If a specific number of entities is required, an additional attribute of the conjunction can hold this value.

The representation proposed seems suitable for providing us with a represen­tation of such phrases as 'five men', 'the first man', 'the third man', 'lots of men'.

There does appear to be a problem in dealing with 'the millionth man' but this could be handled by partitioning the conjunction into a conjunction of two such groups. The first ofthese would provide a symbolic representation of one million (minus one) men and the second would be of indefinite length. 'The millionth man' would be identified as the flTSt man of the second group.

For very small numbers, such as one, two or three, it is probably appropriate to expand the definition to produce the actual number of explicit representa­tions. For larger numbers, symbolic representation without expansion (unless required) would be preferable. The watershed number is probably round about seven, which appears to be the limit for the instantaneous perception of number in humans.

26.3 Some, Any and All

The representation of groups of indefinite size requires us to find an appropriate interpretation for the words 'some', 'any' and 'all', and many others. We shall take these three words as a representative sample.

The dynamic list idea described in section 26.2 seems to deal adequately with 'some'. The two sentences 'A group of men was smashing down the door' and 'Some men were smashing down the door' mean the same thing. The syntactical structure is different, however, because the phrase 'group-of-men' is singular whereas 'men' is plural. It might be possible to make a subtle distinction between 'a group of' and 'some'. We could, for example, represent the word 'group' by a single structure which carries a reference pointer to the conjunctive dynamic list structure we described above.

Of more significance is the problem of dealing with an expression such as: , Some of the men... ' Here we have a particular group of men identified (' the men') and we are referring to an indefinite number of men selected from that group. Presumably the representation of the group to which 'the men' refers already exists.

One method of representing the expression 'some X' is to create a dynamic list structure using the generic procedure for X. If X corresponds to the concept 'man', then we use the generic procedure for 'man'. If X corresponds to 'big men', then we must activate the generic procedures for 'big' and for 'man' and combine these to form a genenc structure which will serve as a template for the generation of the derived concept 'big man'.




To be consistent we should do the same for 'some of the men' where X corresponds to 'the men'. What we require, therefore, is not the generic for 'man' but the generic procedure or generic structure for that particular set of men. Figure 26.3 illustrates the suggested arrangement.

If the group referenced by 'the men' has a defmite size then the derived group 'some of the men' will have an indefmite size which is greater than one and less than or equal to the size of the original group. It could be argued that it should be subject to a limit which is smaller than the size of the original group. In normal conversation the use of the expression 'some X' is often interpreted as meaning 'some but not all X'. This is not the case in formal logic.

The expression 'all X' can be represented in the same way as 'some X', with the group size set at 'maximum'.

The expression 'any X' introduces much more complicated issues. 'Any X' is in effect a selection of one item, at random, from the set of all Xs. One way to represent this is to generate one example of X without providing it with a unique identifier or instantiating any of the formal parameters. It is in fact a copy of the generic structure itself. Such a structure has all the properties of a member of the class of entities involved and has no properties peculiar to itself.




Anything which can be said to be true about 'any X' must (potentially) be true about any other X. It follows that the properties described must be a feature of the generic procedure or generic structure, and not of one particular example.

This is why a statement such as 'any triangle has three sides' is thought of as though it meant the same thing as 'all triangles have three sides'. If, however, we say 'any triangle can have a right angle' we mean that it is possible that a selection made at random from the set of all triangles can be found to have a right angle.

That is, the generic procedure or structure does not exclude the possibility of 'right angledness'. The correctness of the statement can be proved ifmanipula­tion of the formal parameters can produce such a triangle.

This is why the word 'any' sometimes appears to mean the same as 'all', and sometimes the same as 'one'. Consider, for example, the sentences:

Sl 'Any student could answer that question'

S2 'Can any student answer that question?'

Sl appears to be synonymous with 'All students can answer that question' and S2 appears to be synonymous with' Can even one student answer that question?'