CHAPTER 25

Conjunction, Disjunction and Negation

25.1 Simple Negation

In order to avoid complications we will confine this part of the discussion to the use of the word 'not', and to very simple examples of its use. We will ignore negation which can creep into sentences through the use of words beginning 'un-' or 'non-'. Even so we shall find that the topic is a difficult one.

Let us assume that we know how to construct a representation of the meaning of a sentence such as 'Jack went up the hi/f. Let us represent that representation symbolically by a box with the words 'Jack went up the hill inside it.

The problem of negation then resolves itself into finding the most convenient way of appending a negation flag to that representation.

It is a relatively simple matter to append a true/false flag to any structure within our representational scheme, but the very multiplicity of possible locations for such a flag is one of the main sources of difficulty. If the representational structure has internal elements or sub-units, then the negation flag could be appended to any of these, all of them, or to the complete structure. If I say 'John did not go up the hi/f then I may mean that he did nothing of the kind, or I may mean that he did not go of his ownfn;e will- he was dragged by the hair, perhaps. Then again he may have gone somewhere else, and so on.

In spoken English we often indicate the element of the sentence which we wish to negate by stressing the word concerned, e.g.:

Jack  did  not  GO  up  the  hill.
Jack  did  not  go  up  the  HILL.
JACK  did  not  go  up  the  hill.
Jack  did  not  go  UP  the  hill.



But if such emphasis is missing there is no way to tell which part (or parts) of the sentence is associated with the element(s) which are being negated. In the idiom of the 'who-done-it' we have a room full of suspects and no clear evidence as to which one is the culprit. We do know, however, that at least one is the culprit.

If we look at the structure of the sentence being negated it is usually assumed that the negation applies to everything from the word 'not' to the end of the grammatical unit in which 'not' occurs. The area over which the negation applies is called the 'scope' of the negation. The scope can extend to the end of a sentence. It never extends beyond the end of a sentence.

There is no way that a computer system can decide (without additional evidence) what elements are to be negated within the scope. Humans also have difficulty with this aspect oflanguage. Our best policy is to ensure that our form of representation is compatible with any or all of the possible interpretations. To this end it is sensible to apply the negation flag to the representational structure as a whole (e.g. to a rep-box) and not to any of its constituent parts. The representational structure will be one which corresponds to the grammatical unit covered by the scope of the negation. That way the finger of suspicion points at all constituent parts pending further evidence. That in turn means that the structure must have an identity of its own which is distinct from the list of its constituent parts.

Our simplified representation is illustrated in Figure 25.1




We should note, however, that there are certain aspects of the sentence and its representation which are not subject to possible negation.

On reading the sentence 'Jack did not go up the hill' no one is likely to suspect that 'Jack' does not exist, or that 'the hill' does not exist. In contrast, when reading the sentence 'Jack did not go up a hill', we cannot be sure that' a hill' exists for Jack to have climbed.

One explanation for this phenomenon runs as follows. The reference implicit in the noun phrase' the hill' has as its target of reference some hill of which the existence is known from other evidence. That reference is therefore pointing to something which lies outside the scope of the negation, and which cannot therefore be suspect. In terms of the 'who-done-it' it has a perfect alibi. So also has the word Jack, which is an implicit reference to some person known to both writer and reader. Therefore in the sentence 'It was not Jack who went up the hill' (in which Jack lies within the scope) the existence of Jack is not doubted. What is negated is his role in the scenario. This explanation brings into question what we mean by existence, and we may need to remind ourselves that for our purposes 'existence' implies presence in someone's mental model. It should not be associated with some so-called 'real world' existence.

In the sentence 'It was not Jack who went up the hill' the scope of the conditional extends over, but not into, the subordinate clause beginning 'who'. What is negated is the reference between Jack and the agent of the action described by the subordinate clause. Consider also the sentence 'I did not like the book because Jack wrote it'. This sentence could be interpreted as meaning the same as (a) 'I did not like the book. The reason for my dislike of the book is that Jack wrote it'. or (b) '] liked the book. However, this is not because Jack wrote it'. Two constituent parts lie within the scope of negation - the liking of the book and the subordinate clause in the sentence beginning' because... '. If the first is negated the interpretation corresponds to (a). If the second is negated then the interpretation corresponds to (b). The scope of negation does not extend into the subordinate clause. It would be perverse, therefore, to interpret the sentence as casting doubt on Jack's authorship of the book.

And there we shall leave our discussion of negation for the time being.

25.2 Conjunction

A conjunction is a 'thing' formed by combining two or more 'things' which are of the same kind as each other. It is usually accomplished by use of the conjunction operator 'and', but such an operator is not always required. We will represent a conjunction in rather the same way as we represented the constituent parts of an entity in Chapter 23. We shall introduce an additional 'entity' state and an additional 'consists of' state, which will refer to the entities participating in the conjunction. Figure 25.2 illustrates this arrangement for the sentence 'Jack and Jill went up the hill'.




When we need to compare such a structure with another, which may be a representation of the sentence 'Jack went up the hill', it is necessary that the matching function can detect the fact that the representation of the second sentence is embedded inside the representation of the first. Therefore the first sentence implies the second. This seemingly innocent step takes us into rather deep water, because we have now introduced the modal operator 'possibly'.

25.3 Conjunction and Negation

The need for a separate entity to represent a conjunction is made evident when we consid r the combination of conjunction and negation. Consider the sentence 'Jack and Jill did not go up the hill'. Here what is negated is the suggestion that both Jack and Jill went up the hill. Either could have gone alone. Therefore when we compare the representation of this sentence with that of the sentence 'Jack did not go up the hill' it is important that the matching function is able to recognise that the second is not embedded in the first.

25.4 Disjunction

Disjunction is usually associated with the use of the word 'or', but other constructions are possible. We shall confine our discussion to simple 'or' sentences. The operators 'and', 'or' and 'not' are related by the well known De Morgan's laws:

not(X  and  Y)  <=>  not(X)  or  not(Y) 
not(X  or  Y)  <=>  not(X)  and  not(Y)



where  <=>  means  'is  equivalent  to'



It follows that only one of the two operators 'and' and 'or' needs to be included in the primitive set of elements in our representational scheme. We could handle 'or' by substituting for the expression (X or 1? the equivalent expression not( not(X) and not(J?) and we could equally well defme 'and' in terms of 'or' and 'not'. These rules, however, are found by most people to be counter­intuitive. Many programmers have fallen foul of conditional clauses which use a complex combination of 'and's, 'or's and 'not's and which do not mean what they think they mean. It is unlikely, therefore, that this type of substitution is normally used by humans (without prior training) as a means of dealing with the logical connectors.

We could handle the representation of disjunction in much the same way as we are proposing to deal with conjunction, that is by introducing a new, higher structure which contains the disjoined elements as its constituent parts. The difference is that in a disjunction only one of the subordinate elements is required to be true in order to satisfy the truth condition of the statement as a whole. The exact method of representation is relatively unimportant, provided the functions which match structures and work out implications do so in an appropriate way. Briefly, when we have an 'and' list, which might be denoted and(a,b,c,d,...) we are able to say that such a statement implies any of its individual components (a, b, c, ...). If we have an 'or' list such as or(a,b,c,d,...) we can only say that it implies and(possibly(a), possibly(b), possibly(c), )

25.5 Modality and Negation

In modal logic a distinction is made between contingent truths and necessary truths. A statement which is necessarily true is one which must hold in all possible circumstances. A contingent truth is one which just happens to be true but could have been otherwise. Formal modal logic makes use of the predicates 'necessarily' and 'possibly' which we shall denote 'nee' and 'pos'. Only one needs to be considered as a primitive, because it is possible to express each in terms of the other if negation is also allowed.

nec(X)  <=>  not(pos(not(X))) 
pos(X)  <=>  not(nec(not(X)))



In the sentence: 'Jack MA Y have gone up the hill' the word 'may' acts in very much the same way as the word 'not'. It has a scope. Each element within the scope may be the culprit which is causing the entire structure to be 'possibilated' (if we can be allowed to coin such a dreadful word by analogy with the word 'negated'). The representational structure which corresponds to the part of the sentence within the scope of 'possibilation' can be given the condition value 'pos'. When two structures are being compared or matched we should use the rule x => > pos(x).

The combination of negation and modality produces further problems. There is a clear difference of meaning between 'possibly not X' and 'not possiblyX'. The order of application of the two operators is therefore of great importance. We can use the rules pos(not(x)) => > pos(x) and not(pos(x)) = > > not(x). This rule works when applied to an expression such as not(pos(not(x))) because we then have not(pos(not(x))) =>> not(not(x)) =>> x. We noted above that not(pos(not(x))) also implies nec(x).

25.6 May and Must: the other Meaning

An analysis of modality involving the use of the words 'may' and 'must' always introduces an element of ambiguity, due to the fact that these words have an alternative interpretation which is concerned with 'permission' and 'require­ment'. The sentence 'John may go up the hill' could be said by someone giving permission to John. The sentence '[ must go up the hill' usually indicates some feeling of social or psychological pressure rather than a logical conclusion. A proper interpretation of these sentences will therefore involve the representation of hidden motivation, in rather the same way as the examples shown in Chapter 21.