Contradictories, contraries and subcontraries
A contradiction is a pair of propositions with opposite truth values. We will distinguish three types of contradictions, contradictories, contraries and subcontraries.
The most basic type of contradiction is formed by a proposition and its negation.
(80a) and (80b) are contradictories. It is impossible to conceive of a world in which (80a) and (80b) are both true. Nor can we conceive of a world in which they are both false: if one is true, the other must be false, and if one is false, the other must be true.
Contradictory pairs of propositions do not always contain negations:
These propositions cannot both be simultaneously true or both simultaneously false: they always must have opposite truth-values. They are therefore contradictory, but they do not contain any negation.
QUESTION Think of ten pairs of propositions which are contradictories.
The contradiction of a proposition is, of course, a denial of that proposition. It is not the only type of denial, however. Some contradictions cannot be simultaneously true, but can be simultaneously false. (82a) and (82b) are cases in point:
(82) (a) and (b) cannot be simultaneously true, but they can be simultaneously false. José Bové might be neither happy nor sad: he might, for example, be neutral, or asleep. Pairs of propositions like this, which cannot both be true but can both be false, are called contraries.
QUESTION Think of ten pairs of propositions which are contraries.
For completeness, let’s finally look at subcontraries, pairs of propositions which cannot be simultaneously false, but can be simultaneously true:
These can be true at the same time: in (83), it is perfectly possible for some people to be happy while others are not, while in (84), (a) and (b) are both true if the Eiffel tower is 175 meters high. They cannot both, however, be false, since the two propositions in each pair exhaust the possibilities. If (83a) is false, i.e. if some people are not happy, then (83b) must be true: there is no other possibility. Hence, (83) (a) and (b) cannot be simultaneously false. As for (84), if (a) is false and the Eiffel tower is not over 150 meters high, then (b) must be true: the two propositions do not exclude each other, but they exhaust all the possibilities.
QUESTION Think of five pairs of propositions which are subcontraries. We can show the different relations of opposition between propositions using the traditional square of opposition (Figure 6.1), which goes back to Aristotle and the mediaeval logical tradition. This diagram allows a concise representation of the relationships between the different quantificational operators (‘all’ and ‘some’) and negation.
The letters A E I and O are used to mark the four corners of the square (they are taken from the vowels of the Latin verbs affirmo ‘I affirm’ and nego ‘I deny’). They represent the fact that the left-hand propositions are positive, and the right-hand ones negative. The square can be used to rep resent both propositional and quantificational operators, but here we will only discuss the traditional quantificational version (see Girle 2002: 24 for the propositional square of opposition), as shown in Figure 6.2. Reading B as ‘book’ and Was ‘white’, the square represents the following relationships:
• Every book is white and No book is white are contraries. Both cannot be true, but both may be false: this would be the case if some books are white and some are grey.
• Every book is white and Some book is not white are contradictories. They always have the opposite truth value.
• Some book is white and No book is white are also contradictories. They always have the opposite truth value.
• Some book is white and Some book is not white are subcontraries. They can both be true (as they are when some books are white and some are grey), but cannot both be false.
QUESTION Give the contradictories of the following propositions:
a. No monarchy is democratic.
b. All horses detest cobblestones.
c. No Italian twilight is not beautiful.
d. All Italian twilights are not beautiful.
QUESTION Label the following pairs of propositions as contrary, contradictory or subcontrary:
a. All magicians are ignorant. Some magicians are not ignorant.
b. Some beavers don’t build dams. Some beavers build dams.
c. Some painters are not surrealists. Some painters are surrealists.
d. No mushrooms grow on mountains. No mushrooms do not grow on mountains.
The organization of quantified expressions into the square of opposition reveals some interesting facts about the way that natural languages express quantification. English is entirely representative in this respect, in that it has individually lexicalized words for the A, I and E squares, but not for the O square, as shown in Figure 6.3:
In order to express the O corner of the square, English, like many other languages, must resort to the expression some. . . not or not all (these two are logically equivalent, as can be seen by comparing the expressions Some book is not white and Not all books are white). This is true of many unrelated languages. Furthermore, the pattern extends to a whole range of related negative notions, as shown in Table 6.6 (adapted from Horn 1989: 254):
Let’s examine each column in turn. The first column we have already seen: English has pronouns and pronominal adjectives for each of the quantifiers in the square of opposition except O. There is no hypothetical pronoun neverybody with a use like that in (85):
The second column concerns adverbs of time, which can be seen as quantifying over the domain of time. The A corner is occupied by always, the I corner by sometimes and the E corner by never. These represent universal, existential and the negation of universal quantification respectively. But the O corner, again, has no monolexemic expression in English: not always has to be used instead.
Lastly, let’s turn our attention to the case of ‘binary quantifiers’, in other words quantifiers which apply to pairs of objects. Once again, it is only the O corner of the square of opposition which cannot be expressed by a single word in English:
(Note that (88) would be true where only one, or neither of them, jumped overboard.)
This generalization holds true in many languages. In Hungarian (Finno Ugric, Hungary), for instance, there is no monolexemic O quantifier. The O corner of the square is expressed as in English by combining the words for some and not:
Why should this be the case? Horn (1989) offers the following explanation. The subcontrary I tends to implicate the other subcontrary O: in other words, the use of the I-subcontrary some in a sentence like some Xs are Y invites the inference that the O subcontrary some . . . not also holds: some Xs are not Y/not all Xs are Y. If I say (93a), you will conclude that (93b) is also true.
The subcontrary quantifiers I and O are thus informationally equivalent. As a result, Horn argues, languages do not need separate words for both the I and the O quantifiers: given the informational equivalence between them, just lexicalizing one is enough. (This does not explain why it is the I corner and not the O corner that is lexicalized: for some discussion see Horn 1989: 264.)
