Modality and Excluded Middle

The principle of excluded middle is a cornerstone of classical logic and states that for any sentence, either that sentence or its negation is true. This principle, though intuitively appealing, turns out to be systematically violated by natural language in a variety of ways. For example, neither Peter read the books nor Peter didnt read the books are quite true in a situation where Peter read exactly half of the books. The identification and classification of such excluded middle violations has recently been a lively area of research in the study of natural language meaning. The project Modality and Excluded Middle will investigate phenomena in the domain of the so-called language of modality which violate the principle of excluded middle. The language of modality is what we use to speak not about what actually is, but about what is possible or necessary, permissible or required. Verbs like must, can, and should are therefore known as modal verbs. This area of natural language is of cross-disciplinary interest, as it has the potential to tell us something about the nature of, and the way human beings represent and conceptualise, such things as obligations, permissions, and hypothetical possibilities. There are, in particular, three constructions of interest. The first is so-called free choice sentences. An example is Mary may eat the apple or the pear, which conveys that Mary is free to choose between the two fruits. Its negation Mary may not eat the apple or the pear, however, says that Mary cannot take either. If May can take the apple, but cannot take the pear, then neither sentence is true, and so excluded middle is violated. Second are conditional sentences. If Nina appears, Adam will be happy is true if, disregarding perhaps some extraordinary eventualities, Ninas appearance guarantees Adams happiness. Its negation If Nina appears, Adam wont be happy, however, is true only if Ninas appearance guarantees Adams unhappiness. Nei- ther conditional is appropriate when Adam may or may not be happy regardless of Ninas appearance. Finally, there are the so-called weak necessity modals like should, which are used to express what is ideal or advisable. Mary should read the book means that it is advisable for Mary to read the book, while Mary shouldnt read the book expresses that it is advisable for her to abstain from reading the book. Neither sentence is true when it simply doesnt make a difference whether or not Mary reads the book. The project aims to improve our understanding of these constructions by investigating in detail the nature of these excluded middle violations, thereby bringing to bear new evidence that will help consolidate the currently very fragmented theoretical literature.

The principle of excluded middle is a cornerstone of classical logic and states that for any sentence, either that sentence or its negation is true. This principle, though intuitively appealing, turns out to be systematically violated by natural language in a variety of ways. For example, neither 'Peter read the books' nor 'Peter didn't read the books' are quite true in a situation where Peter read exactly half of the books. The identification and classification of such excluded middle violations has recently been a lively area of research in the study of natural language meaning. The project Modality and Excluded Middle was concerned with tracing this phenomenon in other areas of linguist meaning which are not obvious connected to plural noun phrases like 'the books'. The project focused on what is called free-choice sentences, that is, sentences that present a plurality of options as equally viable. For example, 'Mary may eat the apple or the pear' is true if Mary is free to choose between the two fruits, whereas 'Mary may not eat the apple or the pear' is true only if neither option is available to her. If only one of the two options is available to Mary, then neither the original sentence nor its negation is true and the principle of excluded middle is violated. What the project found was that there is a deep-seated analogy between such free-choice sentences and sentences with plural predication. The disjunction 'or' in the free-choice sentence functions like the conjunction 'and' in 'Peter read Anna Karenina and War and Peace'. The disjunction 'either or' in a free-choice sentence functions like 'both and' in 'Peter read both Anna Karenina and War and Peace'. The indefinite 'a book' in 'Peter may take a book' functions like the plural 'the books' in 'Peter read the books' - they both allow a certain number of exceptions that are considered irrelevant, and they both violate excluded middle. And finally, the so-called free-choice indefinite 'any book' in 'Peter may take any book' functions like 'all the books' in 'Peter read all the books', in that no exceptions are allowed and excluded middle is not violated. This underscores that the excluded middle violations in both types of sentences, plural sentences and free-choice sentences, are of the same nature and arise at some common level of description, cementing the view that they represent a fundamental feature of human language.