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to predicate a term simply of any one instance, and to say that some one particular man is a man or that some one white man is a white man.
Yet this is not always possible: indeed, when in the adjunct there is some opposite which involves a contradiction, the predication of the simple term is impossible. Thus it is not right to call a dead man a man. When, however, this is not the case, it is not impossible.
Yet the facts of the case might rather be stated thus: when some such opposite elements are present, resolution is never possible, but when they are not present, resolution is nevertheless not always possible. Take the proposition ‘Homer is so-and-so’, say ‘a poet’; does it follow that Homer is, or does it not? The verb ‘is’ is here used of Homer only incidentally, the proposition being that Homer is a poet, not that he is, in the independent sense of the word.
Thus, in the case of those predications which have within them no contradiction when the nouns are expanded into definitions, and wherein the predicates belong to the subject in their own proper sense and not in any indirect way, the individual may be the subject of the simple propositions as well as of the composite. But in the case of that which is not, it is not true to say that because it is the object of opinion, it is; for the opinion held about it is that it is not, not that it is.
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As these distinctions have been made, we must consider the mutual relation of those affirmations and denials which assert or deny possibility or contingency, impossibility or necessity: for the subject is not without difficulty.
We admit that of composite expressions those are contradictory each to each which have the verb ‘to be’ its positive and negative form respectively. Thus the contradictory of the proposition ‘man is’ is ‘man is not’, not ‘not-man is’, and the contradictory of ‘man is white’ is ‘man is not white’, not ‘man is not-white’. For otherwise, since either the positive or the negative proposition is true of any subject, it will turn out true to say that a piece of wood is a man that is not white.
Now if this is the case, in those propositions which do not contain the verb ‘to be’ the verb which takes its place will exercise the same function. Thus the contradictory of ‘man walks’ is ‘man does not walk’, not ‘not-man walks’; for to say ‘man walks’ merely equivalent to saying ‘man is walking’.
If then this rule is universal, the contradictory of ‘it may be’ is may not be’, not ‘it cannot be’.
Now it appears that the same thing both may and may not be; for instance, everything that may be cut or may walk may also escape cutting and refrain from walking; and the reason is that those things that have potentiality in this sense are not always actual. In such cases, both the positive and the negative propositions will be true; for that which is capable of walking or of being seen has also a potentiality in the opposite direction.
But since it is impossible that contradictory propositions should both be true of the same subject, it follows that’ it may not be’ is not the contradictory of ‘it may be’. For it is a logical consequence of what we have said, either that the same predicate can be both applicable and inapplicable to one and the same subject at the same time, or that it is not by the addition of the verbs ‘be’ and ‘not be’, respectively, that positive and negative propositions are formed. If the former of these alternatives must be rejected, we must choose the latter.
The contradictory, then, of ‘it may be’ is ‘it cannot be’. The same rule applies to the proposition ‘it is contingent that it should be’; the contradictory of this is ‘it is not contingent that it should be’. The similar propositions, such as ‘it is necessary’ and ‘it is impossible’, may be dealt with in the same manner. For it comes about that just as in the former instances the verbs ‘is’ and ‘is not’ were added to the subject-matter of the sentence ‘white’ and ‘man’, so here ‘that it should be’ and ‘that it should not be’ are the subject-matter and ‘is possible’, ‘is contingent’, are added. These indicate that a certain thing is or is not possible, just as in the former instances ‘is’ and ‘is not’ indicated that certain things were or were not the case.
The contradictory, then, of ‘it may not be’ is not ‘it cannot be’, but ‘it cannot not be’, and the contradictory of ‘it may be’ is not ‘it may not be’, but cannot be’. Thus the propositions ‘it may be’ and ‘it may not be’ appear each to imply the other: for, since these two propositions are not contradictory, the same thing both may and may not be. But the propositions ‘it may be’ and ‘it cannot be’ can never be true of the same subject at the same time, for they are contradictory. Nor can the propositions ‘it may not be’ and ‘it cannot not be’ be at once true of the same subject.
The propositions which have to do with necessity are governed by the same principle. The contradictory of ‘it is necessary that it should be’, is not ‘it is necessary that it should not be,’ but ‘it is not necessary that it should be’, and the contradictory of ‘it is necessary that it should not be’ is ‘it is not necessary that it should not be’.
Again, the contradictory of ‘it is impossible that it should be’ is not ‘it is impossible that it should not be’ but ‘it is not impossible that it should be’, and the contradictory of ‘it is impossible that it should not be’ is ‘it is not impossible that it should not be’.
To generalize, we must, as has been stated, define the clauses ‘that it should be’ and ‘that it should not be’ as the subject-matter of the propositions, and in making these terms into affirmations and denials we must combine them with ‘that it should be’ and ‘that it should not be’ respectively.
We must consider the following pairs as contradictory propositions:
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tbody>
| It may be. | It cannot be. |
| It is contingent. | It is not contingent. |
| It is impossible. | It is not impossible. |
| It is necessary. | It is not necessary. |
| It is true. | It is not true. |
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Logical sequences follow in due course when we have arranged the propositions thus. From the proposition ‘it may be’ it follows that it is contingent, and the relation is reciprocal. It follows also that it is not impossible and not necessary.
From the proposition ‘it may not be’ or ‘it is contingent that it should not be’ it follows that it is not necessary that it should not be and that it is not impossible that it should not be. From the proposition ‘it cannot be’ or ‘it is not contingent’ it follows that it is necessary that it should not be and that it is impossible that it should be. From the proposition ‘it cannot not be’ or ‘it is not contingent that it should not be’ it follows that it is necessary that it should be and that it is impossible that it should not be.
Let us consider these statements by the help of a table:
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tbody>
| A. | B. |
|---|---|
| It may be. | It cannot be. |
| It is contingent. | It is not contingent. |
| It is not impossible that it should be. | It is impossible that it should be. |