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  Now if this is the case, (5) 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’ is merely equivalent to saying ‘man is walking’.

  If then this rule is universal, (10) the contradictory of ‘it may be’ is ‘it 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. (15)

  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, (20) 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, (25) 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, (30) 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 ‘it cannot be’. Thus the propositions ‘it may be’ and ‘it may not be’ appear each to imply the other: for, (35) 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. [22a]

  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’. (5)

  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. (10)

  We must consider the following pairs as contradictory propositions:

  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.

  13 Logical sequences follow in due course when we have arranged the propositions thus. (15) 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. (20) 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: (25)

  A. It may be. B. 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.

  It is not necessary that it should be. It is necessary that it should not be.

  C. It may not be. D. It cannot not be.

  It is contingent that it should not be. It is not contingent that it should not be.

  It is not impossible that it should not be. It is impossible that it should not be.

  It is not necessary that it should not be. It is necessary that it should be.

  Now the propositions ‘it is impossible that it should be’ and ‘it is not impossible that it should be’ are consequent upon the propositions ‘it may be’, (30) ‘it is contingent’, and ‘it cannot be’, ‘it is not contingent’, the contradictories upon the contradictories. But there is inversion. The negative of the proposition ‘it is impossible’ is consequent upon the proposition ‘it may be’ and the corresponding positive in the first case upon the negative in the second. (35) For ‘it is impossible’ is a positive proposition and ‘it is not impossible’ is negative.

  We must investigate the relation subsisting between these propositions and those which predicate necessity. That there is a distinction is clear. In this case, contrary propositions follow respectively from contradictory propositions, and the contradictory propositions belong to separate sequences. For the proposition ‘it is not necessary that it should be’ is not the negative of ‘it is necessary that it should not be’, for both these propositions may be true of the same subject; for when it is necessary that a thing should not be, it is not necessary that it should be. [22b] The reason why the propositions predicating necessity do not follow in the same kind of sequence as the rest, lies in the fact that the proposition ‘it is impossible’ is equivalent, when used with a contrary subject, to the proposition ‘it is necessary’. For when it is impossible that a thing should be, it is necessary, not that it should be, (5) but that it should not be, and when it is impossible that a thing should not be, it is necessary that it should be. Thus, if the propositions predicating impossibility or non-impossibility follow without change of subject from those predicating possibility or non-possibility, those predicating necessity must follow with the contrary subject; for the propositions ‘it is impossible’ and ‘it is necessary’ are not equivalent, but, as has been said, inversely connected.

  Yet perhaps it is impossible that the contradictory propositions predicating necessity should be thus arranged. (10) For when it is necessary that a thing should be, it is possible that it should be. (For if not, the opposite follows, since one or the other must follow; so, if it is not possible, it is impossible, and it is thus impossible that a thing should be, which must necessarily be; which is absurd.)

  Yet from the proposition ‘it may be’ it follows that it is not impossible, and from that it follows that it is not necessary; it comes about therefore that the thing which must necessarily be need not be; which is absurd. (15) But again, the proposition ‘it is necessary that it should be’ does not follow from the proposition ‘it may be’, nor does the proposition ‘it is necessary
that it should not be’. For the proposition ‘it may be’ implies a twofold possibility, while, if either of the two former propositions is true, the twofold possibility vanishes. For if a thing may be, it may also not be, but if it is necessary that it should be or that it should not be, (20) one of the two alternatives will be excluded. It remains, therefore, that the proposition ‘it is not necessary that it should not be’ follows from the proposition ‘it may be’. For this is true also of that which must necessarily be.

  Moreover the proposition ‘it is not necessary that it should not be’ is the contradictory of that which follows from the proposition ‘it cannot be’; for ‘it cannot be’ is followed by ‘it is impossible that it should be’ and by ‘it is necessary that it should not be’, (25) and the contradictory of this is the proposition ‘it is not necessary that it should not be’. Thus in this case also contradictory propositions follow contradictory in the way indicated, and no logical impossibilities occur when they are thus arranged.

  It may be questioned whether the proposition ‘it may be’ follows from the proposition ‘it is necessary that it should be’. If not, (30) the contradictory must follow, namely that it cannot be, or, if a man should maintain that this is not the contradictory, then the proposition ‘it may not be’.

  Now both of these are false of that which necessarily is. At the same time, it is thought that if a thing may be cut it may also not be cut, if a thing may be it may also not be, and thus it would follow that a thing which must necessarily be may possibly not be; which is false. (35) It is evident, then, that it is not always the case that that which may be or may walk possesses also a potentiality in the other direction. There are exceptions. In the first place we must except those things which possess a potentiality not in accordance with a rational principle, as fire possesses the potentiality of giving out heat, that is, an irrational capacity. Those potentialities which involve a rational principle are potentialities of more than one result, that is, of contrary results; those that are irrational are not always thus constituted. [23a] As I have said, fire cannot both heat and not heat, neither has anything that is always actual any twofold potentiality. Yet some even of those potentialities which are irrational admit of opposite results. (5) However, thus much has been said to emphasize the truth that it is not every potentiality which admits of opposite results, even where the word is used always in the same sense.

  But in some cases the word is used equivocally. For the term ‘possible’ is ambiguous, being used in the one case with reference to facts, to that which is actualized, as when a man is said to find walking possible because he is actually walking, and generally when a capacity is predicated because it is actually realized; in the other case, (10) with reference to a state in which realization is conditionally practicable, as when a man is said to find walking possible because under certain conditions he would walk. This last sort of potentiality belongs only to that which can be in motion, the former can exist also in the case of that which has not this power. Both of that which is walking and is actual, and of that which has the capacity though not necessarily realized, it is true to say that it is not impossible that it should walk (or, in the other case, that it should be), but while we cannot predicate this latter kind of potentiality of that which is necessary in the unqualified sense of the word, (15) we can predicate the former.

  Our conclusion, then, is this: that since the universal is consequent upon the particular, that which is necessary is also possible, though not in every sense in which the word may be used.

  We may perhaps state that necessity and its absence are the initial principles of existence and non-existence, and that all else must be regarded as posterior to these. (20)

  It is plain from what has been said that that which is of necessity is actual. Thus, if that which is eternal is prior, actuality also is prior to potentiality. Some things are actualities without potentiality, namely, the primary substances; a second class consists of those things which are actual but also potential, whose actuality is in nature prior to their potentiality, though posterior in time; a third class comprises those things which are never actualized, (25) but are pure potentialities.

  14 The question arises whether an affirmation finds its contrary in a denial or in another affirmation; whether the proposition ‘every man is just’ finds its contrary in the proposition ‘no man is just’, or in the proposition ‘every man is unjust’. Take the propositions ‘Callias is just’, ‘Callias is not just’, ‘Callias is unjust’; we have to discover which of these form contraries. (30)

  Now if the spoken word corresponds with the judgement of the mind, and if, in thought, that judgement is the contrary of another, which pronounces a contrary fact, in the way, for instance, in which the judgement ‘every man is just’ pronounces a contrary to that pronounced by the judgement ‘every man is unjust’, the same must needs hold good with regard to spoken affirmations. (35)

  But if, in thought, it is not the judgement which pronounces a contrary fact that is the contrary of another, then one affirmation will not find its contrary in another, but rather in the corresponding denial. We must therefore consider which true judgement is the contrary of the false, that which forms the denial of the false judgement or that which affirms the contrary fact.

  Let me illustrate. There is a true judgement concerning that which is good, (40) that it is good; another, a false judgement, that it is not good; and a third, which is distinct, that it is bad. [23b] Which of these two is contrary to the true? And if they are one and the same, which mode of expression forms the contrary?

  It is an error to suppose that judgements are to be defined as contrary in virtue of the fact that they have contrary subjects; for the judgement concerning a good thing, that it is good, and that concerning a bad thing, (5) that it is bad, may be one and the same, and whether they are so or not, they both represent the truth. Yet the subjects here are contrary. But judgements are not contrary because they have contrary subjects, but because they are to the contrary effect.

  Now if we take the judgement that that which is good is good, and another that it is not good, and if there are at the same time other attributes, which do not and cannot belong to the good, we must nevertheless refuse to treat as the contraries of the true judgement those which opine that some other attribute subsists which does not subsist, (10) as also those that opine that some other attribute does not subsist which does subsist, for both these classes of judgement are of unlimited content.

  Those judgements must rather be termed contrary to the true judgements, in which error is present. Now these judgements are those which are concerned with the starting points of generation, and generation is the passing from one extreme to its opposite; therefore error is a like transition.

  Now that which is good is both good and not bad. (15) The first quality is part of its essence, the second accidental; for it is by accident that it is not bad. But if that true judgement is most really true, which concerns the subject’s intrinsic nature, then that false judgement likewise is most really false, which concerns its intrinsic nature. Now the judgement that that which is good is not good is a false judgement concerning its intrinsic nature, the judgement that it is bad is one concerning that which is accidental. (20) Thus the judgement which denies the truth of the true judgement is more really false than that which positively asserts the presence of the contrary quality. But it is the man who forms that judgement which is contrary to the true who is most thoroughly deceived, for contraries are among the things which differ most widely within the same class. If then of the two judgements one is contrary to the true judgement, but that which is contradictory is the more truly contrary, then the latter, it seems, (25) is the real contrary. The judgement that that which is good is bad is composite. For presumably the man who forms that judgement must at the same time understand that that which is good is not good.

  Further, the contradictory is either always the contrary or never; therefore, if it must necessarily be so
in all other cases, our conclusion in the case just dealt with would seem to be correct. (30) Now where terms have no contrary, that judgement is false, which forms the negative of the true; for instance, he who thinks a man is not a man forms a false judgement. If then in these cases the negative is the contrary, then the principle is universal in its application.

  Again, the judgement that that which is not good is not good is parallel with the judgement that that which is good is good. Besides these there is the judgement that that which is good is not good, parallel with the judgement that that which is not good is good. Let us consider, (35) therefore, what would form the contrary of the true judgement that that which is not good is not good. The judgement that it is bad would, of course, fail to meet the case, since two true judgements are never contrary and this judgement might be true at the same time as that with which it is connected. For since some things which are not good are bad, both judgements may be true. Nor is the judgement that it is not bad the contrary, for this too might be true, since both qualities might be predicated of the same subject. It remains, therefore, that of the judgement concerning that which is not good, (40) that it is not good, the contrary judgement is that it is good; for this is false. [24a] In the same way, moreover, the judgement concerning that which is good, that it is not good, is the contrary of the judgement that it is good.

  It is evident that it will make no difference if we universalize the positive judgement, for the universal negative judgement will form the contrary. For instance, the contrary of the judgement that everything that is good is good is that nothing that is good is good. (5) For the judgement that that which is good is good, if the subject be understood in a universal sense, is equivalent to the judgement that whatever is good is good, and this is identical with the judgement that everything that is good is good. We may deal similarly with judgements concerning that which is not good.