LESSON X

The Logical Consequents of the Modals

             22a 14 Logical sequences result from modals ordered thus. From "possible to be" follows "contingent to be" and the latter is convertible with the former; "not impossible to be" and "not necessary to be" also follow from "possible to be." From "possible not to be" follows "contingent not to be," "not necessary not to be," and "not impossible not to be." From "not possible to be" and "not contingent to be" follows "necessary not to be" and "impossible to be." From "not possible not to be" and "not contingent not to be" follows "necessary to be" and "impossible not to be." Let us consider these with the help of a table.

             22a 24

  possible to be    not possible to be

  contingent to be   not contingent to be

  not impossible to be  impossible to be

  not necessary to be   necessary not to be

  possible not to be   not possible not to be

  contingent not to be  not contingent not to be

  not impossible not to be  impossible not to be

  not necessary not to be  necessary to be

             22a 32 Now the impossible and the not impossible follow contradictorily upon the contingent and the possible and the not contingent and the not possible, but inversely. The negation of "impossible to be" follows upon "possible to be" and the affirmation of the former follows upon the negation of the latter, i.e., "impossible to be" follows upon "not possible to be"; for "impossible to be" is an affirmation, "not impossible to be" a negation.

             22a 38 Now we must consider how enunciations predicating necessity are related to these. It is evident that the case here is not the same, for the contraries follow, but their contradictories are separated.

             22a 39 For the negation of "necessary not to be" is not "not necessary to be," since both may be true of the same subject, for the necessary not to be is not necessary to be.

             22b 3 Now the reason why enunciations predicating necessity do not follow in the same way as the others is that the impossible expresses contrarily the same thing as the necessary. For if it is impossible that this be, it is necessary, not that it be, but necessary that it not be; and if it is impossible that it not be, it is necessary that it be. So, if the impossible and not impossible follow in like manner from the possible and not possible, the necessary and not necessary follow contrarily, since the necessary and the impossible signify the same thing, but as has been said, inversely.

             22b 10 Or is it impossible to arrange the contradictions of enunciations predicating necessity in this way? For what is necessary to be, is possible to be (for if not, the negation will follow, since it is necessary either to affirm or deny; and if it is not possible to be, it is impossible to be; therefore, that which is necessary to be is impossible to be, which is absurd). But from "possible to be" "not impossible to be" follows, and from this, "not necessary to be"; and thus what is necessary to be is not necessary to be, which is absurd.

             22b 17 But in fact neither "necessary to be" nor "necessary not to be" follow upon "possible to be"; for "to be possible" admits of two possibilities, whereas if either "necessary to be" or "necessary not to be" is true both possibilities will no longer be true. For a thing is at once possible to be and not to be, but if it is necessary to be or not to be, the two alternatives will not be possible. It remains, therefore, that "not necessary not to be" follows upon "possible to be";

             22b 23 for this is true also with respect to "necessary to be." For "not necessary not to be" is the contradictory of what follows upon "not possible to be," for "not possible to be" is followed by "impossible to be" and by "necessary not to be," and the negation of this is "not necessary not to be."

             22b 23 Thus, these contradictions also follow in the way indicated, and nothing impossible follows when they are thus arranged.

             1. Having established the opposition of modals, Aristotle now intends to determine their consequents. He first presents the true doctrine; then, he raises a difficulty where he says, But it may be questioned whether "possible to be" follows upon "necessary to be," etc. In presenting the true doctrine, he first posits the consequents of the opposition of modals according to the opinion of others; secondly, he determines the truth by examining and correcting their opinion, where he says, Now the impossible and the not impossible follow contradictorily upon the contingent and the possible and the not contingent and the not possible, but inversely, etc.

             2. Before we consider these consequents according to the opinion of others, we must first note that since any mode makes two affirmations and there are two negations opposed to these, there will be four enunciations according to any one mode, two affirmatives and two negatives. And since there are four modes, there will be sixteen modals. Among these sixteen, anyone of each mode, from wherever you begin, has only one of each mode following upon it. Hence, to assign the consequents of the modals, we have to take one from each mode and arrange them among themselves to form an order of consequents.

             3. The modals were ordered in this way by the ancients. They disposed them in four orders placing together in each order those that were consequent to each other. Aristotle speaks of this order when he says, Logical consequents follow according to the order in the table below, which is the way in which the ancients posited them.

             Henceforth, however, to avoid confusion let us call the affirmative of dictum and mode in any one mode, the simple affirmative, as it is by Averroes, among others; affirmative of mode and negative of dictum, the declined affirmative; negative of mode and not of dictum, the simple negative; negative of both mode and dictum, the declined negative. Hence, simplicity of mode designates affirmation or negation, and so, too, does declination of dictum.

             The ancients said, then, that simple affirmation of the contingent, i.e., "contingent to be" follows upon simple affirmation of the possible, i.e., "possible to be" (for the contingent is converted with the possible); the simple negative of the impossible also follows upon this, i.e., "not impossible to be"; and the simple negative of the necessary, i.e., "not necessary to be." This is the first order of modal consequents.

             In the second order they said that the declined negatives of the necessary and impossible, i.e., "not necessary not to be" and "not impossible not to be," follow upon the declined affirmative of the possible and the contingent, i.e., "possible not to be" and "contingent not to be."

             In the third order, according to them, the declined affirmative of the necessary, i.e., "necessary not to be," and the simple affirmative of the impossible, i.e., "impossible to be," follow upon the simple negatives of the possible and the contingent, i.e., "not possible to be" and "not contingent to be."

             Finally, in the fourth order, the simple affirmative of the necessary, i.e., "necessary to be," and the declined affirmative of the impossible, i.e., "impossible not to be," follow upon the declined negatives of the possible and the contingent, i.e., "not possible not to be" and "not contingent not to be."

             4. To make this ordering more evident, let us consider it with the help of the following table.

CONSEQUENTS OF MODAL ENUNCIATIONS IN THE FOUR ORDERS POSITED AND ORDERED BY THE ANCIENTS

 FIRST ORDER     THIRD ORDER

It is possible to be   It is not possible to be

It is contingent to be   It is not contingent to be

It is not impossible to be  It is impossible to be

It is not necessary to be  It is necessary not to be

 SECOND ORDER     FOURTH ORDER

It is possible not to be   It is not possible not to be

It is contingent not to be  It is not contingent not to be

It is not impossible not to be It is impossible not to be

It is not necessary not to be  It is necessary to be

             5. When he says, Now the impossible and the not impossible follow contradictorily upon the contingent and the possible and the not contingent and the not possible, but inversely, etc., he determines the truth by examining the foresaid opinion. First, he examines the consequents of enunciations predicating impossibility; secondly, those predicating necessity, where he says, Now we must consider how enunciations predicating necessity are related to these, etc.

             From the opinion advanced, then, he concludes with approval that the impossible and the not impossible follow upon the contingent and the possible and the not contingent and the not possible, contradictorily, i.e., the contradictories of the impossible follow upon the contradictories of the possible and the contingent, but inversely, i.e., not so that affirmation follows upon affirmation and negation upon negation, but inversely, i.e., negation follows upon affirmation and affirmation upon negation.

             He explains this when he says, The negation of "impossible to be" follows upon "possible to be," i.e., the negation of the impossible, i.e., "not impossible to be," follows upon the affirmation of the possible, and the affirmation of the impossible follows upon the negation of the possible. For the affirmation, "impossible to be" follows upon the negation, "not possible to be." In the latter the mode is negated, in the former it is not. Therefore, the ancients were right in saying that in any order, the consequences of enunciations predicating impossibility are as follows: from affirmation of the possible, negation of the impossible is inferred; and from negation of the possible, affirmation of the impossible is inferred. This is apparent in the diagram.

             6. When he says, Now we must consider how enunciations predicating necessity are related to these, etc., he proposes an examination of the consequents of enunciations predicating necessity in order to determine the truth about them. First he examines what was said by the ancients; secondly, he determines the truth, where he says, But in fact neither "necessary to be" nor "necessary not to be" follow upon "possible to be," etc. In his examination of the ancients, Aristotle makes four points. First, he shows what was well said by the ancients and what was badly said.

             It must be noted in regard to this that, as we have said, there are four enunciations predicating necessity, which differ among themselves in quantity and quality, and hence they make up a diagram of opposition in the manner of the absolute enunciations. Two of them are contrary to each other, and two are contradictory to these contraries, as is clear in the diagram below.

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Diagram #12 (Image 12)]]

             Now the ancients correctly inferred the universal contraries from the possibles, contingents, and impossibles, but incorrectly inferred their contradictories, namely, particulars. This is the reason Aristotle says that it remains to be considered how enunciations predicating necessity are related consequentially to the possible and not possible. From what Aristotle says, it is clear that those predicating necessity do not follow upon the possibles in the same way as those predicating impossibility follow upon the possibles, for all of the enunciations predicating impossibility were correctly inferred by the ancients, but those predicating necessity were not. Two of them, the contraries, "necessary to be" and "necessary not to be," follow, i.e., correct consequents were deduced by the ancients in the third and fourth orders; the remaining two, "not necessary not to be" and "not necessary to be," which are contradictories of the contraries, are outside of the consequents of these, i.e., in the second and first orders. Hence, the ancients represented everything correctly in the third and fourth orders, but in the first and second they erred, not with respect to all things, but only with respect to enunciations predicating necessity.

             7. Secondly, he says, For the negation of "necessary not to be" is not "not necessary to be," since both may be true of the same subject, etc. Here he replies to a tacit objection. This reply could be used to defend the consequent of the enunciation of the necessary made by the ancients in the first order. The tacit objection is this: "not possible to be" and "necessary not to be" follow convertibly in the third order which has already been shown to be correct; therefore, "possible to be" and "not necessary to be" ought to follow upon each other in the first order. The consequent holds; for the contradictories of two that convertibly follow upon each other, mutually follow upon each other; but those two follow upon each other convertibly in the third order and these two in the first order are their contradictories; therefore, those of the first order, i.e., "possible to be" and "not necessary to be," mutually follow upon each other.

             Aristotle replies here to this objection by destroying what was assumed in the minor, i.e., that the necessary of the first order and the necessary of the third order are contradictories. He says, For the negation of "necessary not to be" (which is in the third order) is not "not necessary to be" (which has been placed in the first order). He also gives the reason: it is possible for both to be true at once of the same subject, which is repugnant to contradictories. For the same thing which is necessary not to be, is not necessary to be; for example, it is necessary that man not be wood and it is not necessary that man be wood. Notice, as will be clear later, that these two which the ancients posited in the first and third orders, are subalterns and therefore are at once true, whereas they should be contradictories; hence the ancients were in error.

             8. Boethius and Averroes read both this and the preceding part of the text, not reprovingly, but as explanatorily joined together. They say Aristotle explains the quality of the above table with respect to the consequents of enunciations predicating necessity after he has explained in what way those predicating impossibility are related. What Aristotle is saying, then, is that those of the necessary do not follow those of the possible in the same way as those of the impossible follow upon the possible. For contradictories of the impossible follow upon contradictories of the possible, although inversely; but contradictories of the necessary are not said to follow the contradictories of the possible, but rather the contraries of the necessary follow upon them. It is not the contraries among themselves that follow, but contraries in this way: the negation of the necessary is said to follow upon the affirmation of the possible; but what follows on the negation of this possible is not the affirmation of the necessary contradictory to that negative of the necessary following upon the possible, but the contrary of such an affirmation of the necessary. That this is the case is evident in the first and third orders. The sources are negation and affirmation of the possible, and the extremes are "not necessary to be" and "necessary not to be." But these are not contradictories, for the negation of "necessary not to be" is not "not necessary to be," for it is possible for them to be at once true of the same thing. "Necessary not to be" is the contrary of the contradictory of "not necessary to be," which contradictory is "necessary to be."

             In my judgment, however, the first exposition should be accepted and this portion of the text taken as a reproof of the ancients, because the contraries seem to be explained in a forced way by others, whereas our introduction is more in accord with what follows in the next part of the text; in addition, it agrees with Albert's interpretation.

             9. Thirdly, he says, Now the reason why enunciations predicating necessity do not follow in the same way as the others, etc. Here Aristotle shows why enunciations predicating impossibility and necessity do not follow in a similar way upon those predicating possibility. This was the error made by the ancients in both the first and second orders, for in the first order they posited the simple negative of the impossible, and in a similar way the simple negative of the necessary, and in the second order their declined negatives, the reason being that they inferred those predicating impossibility and necessity in a similar way. The cause of this error, then, and the reason why enunciations predicating necessity do not follow the possible in the same way, i.e., in a similar mode, as the others, i.e., as the impossibles, is that the impossible expresses the same meaning as the necessary, i.e., is equivalent to the necessary, contrarily, i.e., taken in a contrary mode, and not in the same mode. For if something is impossible to be, we do not infer, therefore it is necessary to be, but it is necessary not to be. Since, therefore, the impossible and necessary mutually follow each other when their dictums are taken in a contrary mode--and not when their dictums are taken in a similar mode--it follows that the impossible and necessary are not related in the same way to the possible, but in a contrary way. For the negated dictum of the necessary follows upon that possible which follows the affirmed dictum of the impossible, and contrarily. Why this is so will be explained later. Therefore, the ancients erred when they located similar enunciations of the impossible and necessary in the first and in the second orders.

             10. Hence it appears that our exposition is more in conformity with Aristotle. For he introduced this text to manifest these words: It is evident that the case here is not the same, etc. By taking this meaning, then, these words are made clear through the cause. Moreover, it is evident that here the cause is given of a true dissimilitude between necessaries and impossibles in following the possibles, and not of a dissimilitude falsely held by the ancients, for from a true cause only the truth is concluded. Therefore in reproving the ancients it must be understood that a true dissimilitude between the necessary and impossible in following the possible, which they did not heed, has been proposed, and now has been made manifest. It will be clear from what will be said later that the dissimilitude posited by the ancients between the necessary and impossible is falsely posited, for it will be shown that contradictories of the necessary follow contradictories of the possible inversely, and that in this they do not differ from enunciations predicating impossibility. They do differ, however, in the way we have indicated, i.e., the dictum of the possibles and of the impossibles following on them is similar, but the dictum of the possibles and of the necessaries following on them is contrary, as will be seen clearly later.

             11. Fourthly, when he says, Or is it impossible to arrange the contradictions of enunciations predicating necessity in this way? he manifests another point he had proposed, namely, that contradictories of enunciations predicating necessity were badly placed according to consequence by the ancients when they ordered them thus: the contradictory negation to "necessary to be," i.e., "not necessary to be," in the first order, and the contradictory negation to "necessary not to be," i.e., "not necessary not to be," in the second.

             Aristotle only proves that this mode of consequence is incorrect in the first order, for when this is known the mistake in the second order is readily seen. He does this by an argument leading to an impossibility. "Possible to be" follows upon "necessary to be"; otherwise "not possible to be" would follow, which it manifestly implies. "Not impossible to be" follows upon "possible to be" as is evident, and, according to the ancients, in the first order, "not necessary to be" follows upon "not impossible to be." Therefore, from first to last, "not necessary to be" follows upon "necessary to be," which is inadmissible because there is an obvious implication of contradiction. Therefore, it is erroneous to say that "not necessary to be" follows in the first order.

             He says, then, that in fact it is impossible to posit contradictions of the necessary according to consequence as the ancients posited them, i.e., in the first order the contradictory negation of "necessary to be," i.e., "not necessary to be" and in the second the contradictory negation of "necessary not to be," i.e., "not necessary not to be." For "possible to be" follows upon "necessary to be"; if not, i.e., if you deny this consequence, the negation of the possible follows upon "necessary to be," since the possible must either be asserted of the necessary or denied, the reason being that of anything there is a true affirmation or a true negation. Therefore, if you say that "possible to be" does not follow upon "necessary to be," but "not possible to be" does follow, then, since the latter is equivalent to the former, i.e., "not possible to be" to "impossible to be," "impossible to be" follows upon "necessary to be" and the same thing will be "necessary to be" and "impossible to be," which cannot be admitted. Consequently, the first inference was good, i.e., "It is necessary to be, therefore it is possible to be."

             But again, "possible to be" follows upon "not impossible to be," as is evident in the first order, and according to the ancients, "not necessary to be" follows upon "not impossible to be" in the same first order. Therefore, from first to last we arrive at this: "not necessary to be" follows upon "necessary to be," which is unlikely, not to say impossible.

             12. There is a doubt about this, for in I Priorum, it is said that the not necessary follows upon the possible, while here the opposite is said.

             The possible, however, is taken in two ways: commonly, and thus it is superior to the necessary and the contingent to either of two alternatives, as is the case with animal in relation to man and cow; taken in this way, the not necessary does not follow upon the possible, just as not-man does not follow upon animal. In another way the possible is taken for one part of the possible commonly, i.e., for the possible or contingent to either of two alternatives, namely, for what can be and not be. The not necessary follows upon the possible taken in this way, for what can be and not be is not necessary to be, and likewise is not necessary not to be.

             In the Prior Analytics, then, Aristotle is speaking of the possible in particular; here of the possible commonly.

             13. When he says, But in fact neither "necessary to be" nor "necessary not to be" follow upon "possible to be," etc., he determines the truth. First he determines which enunciation of the necessary follows upon the possible; secondly, he orders the consequents of all of the modals, where he says, Thus, these contradictions also follow in the way indicated, etc.

             Aristotle has reproved the ancients in two ways; on the basis of these two he now proves which enunciation of the necessary follows upon the possible. What he intends to show is that "not necessary not to be" follows upon "possible to be." The first argument is taken from a locus of division. "Not necessary to be" does not follow upon "possible to be" (as has been proved), but neither does "necessary to be" nor "necessary not to be." Therefore, "not necessary not to be" follows upon "possible to be," since there are no more enunciations of the necessary.

             He first proposes the remaining two members that are to be excluded from this common division: But in fact neither "necessary to be" nor "necessary not to be" follow upon "possible to be." Then he proves this: no formal consequent diminishes its antecedent, for if it did, the opposite of the consequent would stand with the antecedent; but both of these, namely, "necessary to be" and "necessary not to be," diminish "possible to be"; therefore, etc.

             The major is therefore implied and he gives the proof of the minor when he says that "possible to be" admits of two possibilities, namely, "to be" and "not to be"; but of these, namely, "necessary to be" and "necessary not to be" (whichever should be true), these two, "to be" and "not to be," will not be true at the same time in potency. He explains the first point thus: when I say "possible to be" it is at once possible to be and not to be. With respect to the second, he adds: if you should say, "necessary to be" or "necessary not to be," both do not remain, i.e., possible to be and not to be do not remain, for if a thing is necessary to be, possibility not to be is excluded, and if it is necessary not to be, possibility to be is removed. Both of these, then, diminish the antecedent, possible to be, for it is extended to "to be" and "not to be," etc.

             Thirdly, he concludes: it remains, therefore, that "not necessary not to be" accompanies "possible to be," and consequently will have to be placed in the first order.

             14. A difficulty arises at this point with respect to his saying that the necessary does not follow upon the possible, since he has also said that the not necessary does not follow upon it. For the necessary and the not necessary are opposed contradictorily, and since of anything there is a true affirmation or negation, it seems impossible to avoid the conclusion that either the necessary or the not necessary follows upon the possible; and since the necessary does not follow, the not necessary must follow, as the ancients said. Furthermore, the difficulty is augmented by the fact that Aristotle just used such a mode of argumentation when, to prove that the possible follows upon the necessary, he said, for if not, the negation will follow; for it is necessary either to affirm or deny.

             15. In order to resolve this, we must recall the relationship between the possible and the necessary, namely, that the possible is superior to the necessary. Now the superior potentially contains its own inferior and the opposite of it in such a way that neither of them is actually appropriated by the superior, but each is possible to it; as in the case of man and not-man in relation to animal. We must also consider that the proportion of the superior as related to the affirmation and negation of one inferior is the same (which is the proportion of some subject to the affirmative and negative of a future contingent), for it is had by neither of the two, and the potency to either is kept. Accordingly, as in future contingents neither the affirmation nor the negation is determinately true, but under disjunction one is necessarily true (as was concluded at the end of the first book), so neither the affirmation nor negation of the inferior follows upon the affirmation or negation of the superior determinately, but under disjunction one follows necessarily. This, for instance, is not valid: "It is animal, therefore it is man," nor is "therefore it is not man" valid, but, "therefore it is man or it is not man."

             Since, then, the possible is superior to the necessary, Aristotle has correctly determined that neither part of the contradiction of the necessary determinately follows upon the possible. However, he has not said that under disjunction neither follows; for this would be opposed to the first principle, that of anything there is a true or false affirmation.

             The response to what was added, beginning with "Furthermore, the difficulty is augmented," etc., is based upon the same point. Since the necessary is inferior to the possible, and the inferior does not include its superior in potency but in act, the superior must follow determinately upon the inferior; otherwise the contradiction of it would follow determinately. Hence, because of the dissimilar relationship between the necessary and the possible and not possible on the one hand, and between the possible and the necessary and not necessary on the other, the movement of the earlier argument to one part of the contradiction determinately was quite right, and the movement here to neither determinately was quite right.

             16. There is another slight difficulty, for it seems that Aristotle takes the possible in a different way in the preceding text and in this. There he takes it commonly as it follows upon the necessary; here he seems to take it specifically for the possible that is indifferent to alternatives, since he says that the possible is at once possible to be and not to be.

             But in fact Aristotle has used the possible uniformly. Nor are his words at variance, for it is also true to say of the possible as common that it admits of both possibilities, i.e., of "to be" and "not to be"; first, because whatever is verified of its inferior is verified also of its superior, although not in the same mode; secondly, because the possible as common determines neither part of the contradiction to itself and consequently admits of either happening, although it does not affirm a potency to each part, as does the possible to either of two alternatives.

             17. The second grounds for proving the same thing corresponds to the tacit objection of the ancients he excluded above: For this, he says, is true also with respect to "necessary to be," etc. It should be noted here that Aristotle subsumes under the major cited as a proof for the position of the ancients (namely, contradictories of consequences convertibly following each other mutually follow upon each other) this minor: but the contradictories of those following upon each other convertibly in the third order (i.e., of "not possible to be" and "necessary not to be") are "possible to be" and "not necessary not to be" (for they are opposed to them by negation of mode); therefore, these two (i.e., "possible to be" and "not necessary not to be") follow upon each other and are to be placed in the first order.

             Hence, with respect to the basis of the above argument, he says, For this, i.e., what has been said, is true, i.e., is shown to be true, also with respect to "necessary not to be," i.e., of the opposite of "not necessary not to be," i.e., "necessary not to be." Or, For this, namely, "not necessary not to be," is true, namely, is the true contradictory of "necessary not to be." He gives the minor when he says, For "not necessary not to be" is the contradictory of what follows upon "not possible to be."

             Then he states this explicitly: for "not possible to be," which is the source of the third order is followed by this impossible, namely, "impossible to be," and by this one of the necessary, namely, "necessary not to be," of which the negation or contradictory is "not necessary not to be." And since, other things being equal, the mode is negated, and, "possible to be" is (it is understood) the contradictory of "not possible to be," therefore, these two mutually follow upon each other, namely, "possible to be" and "not necessary not to be," as contradictories of the two mutually following upon each other.

             18. When he says, Thus, these contradictions also follow in the way indicated, etc., he orders all of the consequents of modals according to his own opinion. He says, then, that these contradictions, namely, of the necessary, follow those of the possible, according to the foresaid and approved mode of those of the impossible. For just as contradictories of the impossible follow upon contradictories of the possible, although inversely, so contradictories of the necessary follow contradictories of the possible inversely. In the latter, however, as has been said, there is a dissimilarity in that the dictum of the contradictories of the possible and impossible is similar, but the dictum of the contradictories of the possible and necessary is contrary. This can be seen in the following table.

CONSEQUENTS OF MODAL ENUNCIATIONS POSITED AND ORDERED BY ARISTOTLE ACCORDING TO FOUR ORDERS

 FIRST ORDER     THIRD ORDER

It is possible to be   It is not possible to be

It is contingent to be   It is not contingent to be

It is not impossible to be  It is impossible to be

It is not necessary not to be  It is necessary not to be

 SECOND ORDER     FOURTH ORDER

It is possible not to be   It is not possible not to be

It is contingent not to be  It is not contingent not to be

It is not impossible not to be It is impossible not to be

It is not necessary to be  It is necessary to be

             Here you see that there is no difference between Aristotle and the ancients except in the first two orders with respect to those of the necessary. The ancients inverted the position of these, placing the necessary that should have been placed in the first order in the second order, and the one that should have been in the second in the first.

             Notice, too, that he has ordered them in such a way that the contradictories of those following upon each other convertibly, always follow each other, for each one in the first order is the contradictory of each one in the third order, and similarly, each of the fourth order the contradictory of each in the second. This the ancients did not observe.