Cursus Philosophicus Thomisticus, Prima Pars Artis Logicae

Since every opposition and equipollence of propositions results from their quality and quantity, wherever a special difficulty is found concerning quantity and quality, one is also found concerning opposition and equipollence. Therefore, to explain the difficulty concerning opposition and equipollence in modals, we must first explain their quantity and quality insofar as they are modal propositions.

These four modes—impossible, necessary, possible, contingent—therefore, in themselves, signify a quantity like that of the signs “every,” “no,” “some,” etc. For necessary and impossible signify universality: what belongs necessarily belongs always and to all, whereas what belongs impossibly belongs to none and never. Possible and contingent, however, signify particularity, because to establish that something is possible it suffices that it be possible at some time or for some subject. Here “contingent” is taken as the same as “possible,” not as “liable to fail,” in which sense it is opposed to the necessary; rather, it means what can be at some time, leaving aside whether it is liable to fail.

It is clear from this that since a modal proposition consists of a dictum and a mode, and the mode itself has quantity while the dictum too can have quantity, two quantities sometimes concur in the same proposition, one belonging to the dictum and the other to the mode. These must be changed in order to form oppositions and equipollences. This twofold quantity is evident in divided modals, in which the quantity of the dictum operates in the same way as in propositions de inesse. For example, if you say, “Every man can run,” “every” distributes without qualification. In compound modals, however, since the whole dictum becomes as it were one extreme and the mode is predicated of it, the quantity of the dictum seems to operate less, while the quantity of the mode stands out more. Nevertheless, the quantity of the dictum must also be changed in opposition, because of the sense of the proposition; for a compound modal is reduced to a divided one, and consequently the truth of one must be regulated like the truth of the other. To avoid this, and to see the proper opposition and equipollence of modals precisely as modals, examples are usually given in propositions whose dictum concerns a singular term, as “That Peter run is possible,” in which only the quantity of the mode is considered. If the dictum concerns a common term, we shall give examples below in tables of divided modals.

Likewise, as regards quality, since a compound modal has a twofold copula—that of the dictum and that of the mode—both may be affirmed or both denied; or the dictum may be affirmed and the mode denied, or the mode affirmed and the dictum denied. To make these four combinations of affirmation and negation easier to remember and use, dialecticians established the four vowels A, E, I, U. A signifies a proposition in which both the mode and the dictum are affirmed; E signifies one with the dictum denied and the mode affirmed; I, one with the mode denied and the dictum affirmed; U, one with both the dictum and mode denied. Hence the verse:

E denies the dictum, I the mode; A leaves nothing out, but U denies all.

Therefore, to form opposition in these modals, always observe that just as quantity is considered in propositions de inesse, so the quantity of the modes is considered in modals. We suppose that “necessary” and “impossible” are universal, like “every” and “no,” and that “possible” and “contingent” are particular, like “some,” “any,” etc. Thus, just as “every” and “no” are contraries, so are “necessary” and “impossible”; just as “some” and “some not” are subcontraries, so are “possible” and “possibly not”; and just as “no” and “some,” and “every” and “some not,” are contradictories, so are “impossible” and “possible,” and “necessary” and “possibly not.”

And just as equipollences are formed by negation placed before or after, etc., so equipollences in modals are formed through the same negation placed before or after.

Next, one must attend to the change of negation and affirmation, so that in oppositions an affirmation and a negation of the dictum are always given. For the mode, negation or affirmation is required only insofar as it is necessary to change its quantity or preserve it. Sometimes the mode’s quantity is changed by denying the mode itself, just as the quantity of “every” is changed by saying “not every.” An example may be taken from this proposition: “That Peter run is impossible.” If you want to form its contrary, say, “That Peter run is necessary”; if its contradictory, say, “That Peter run is possible,” since “impossible” denies the dictum. To form subcontraries, say, “That Peter run is possible,” and “That Peter not run is possible.” To form equipollents, say, “That Peter run is impossible,” and “It is not possible that Peter run,” or “It is necessary that Peter not run,” depending on whether you want to form equipollences from contraries by postposing negation (“necessary” and “impossible”), or from contradictories by prefixing it, etc.

To make all this easier to remember, dialecticians devised four words or tables composed of those four vowels A, E, I, U, in each of which equipollence and opposition are distinctly displayed according to all possible combinations, whether one affirms or denies the mode, the dictum, or both. They are Amabimus, edentuli, illiace, and purpurea, beginning with the letter A in amabimus, E in edentuli, and so on. Proceed as follows: in each word, let the first letter indicate the mode “possible,” the second the mode “contingent,” the third the mode “impossible,” and the fourth the mode “necessary.” Examples should be given with a singular term in the dictum for compound modals, and with a common term for divided modals. In this way you can readily understand both and pass from one to the other.

[Note on the layout: the printed form arranges four headings in a square: “Equipollent propositions” appears at the margin of each column; “Contraries” lies between the two above; “Subcontraries” between the two below; and “Contradictories” along the diagonals. This geometric arrangement is represented here.]

The title on each side is a set of four equivalent propositions; the syllables make up the mnemonic:

Position	Mnemonic	Syllables
upper left	Purpurea	Pur- Pu- Re- A
lower left	Amabimus	A- Ma- Bi- Mus
upper right	Illiace	Il- Li- A- Ce
lower right	Edentuli	E- Den- Tu- Li

                    CONTRARIES
         Purpurea  <------------->  Illiace
             |     \             /     |
             |       \         /       |
 EQUIPOLL.   |         X CONTRAD.      |  EQUIPOLL.
             |       /         \       |
             |     /             \     |
         Amabimus  <------------->  Edentuli
                   SUBCONTRARIES

• Contraries: Purpurea — Illiace (upper position).
• Subcontraries: Amabimus — Edentuli (lower position).
• Contradictories (transversal): Purpurea — Edentuli; Amabimus — Illiace.
• Equipollents: the four propositions in each mnemonic compared with one another.

Purpurea (upper left)

• Pur-: It is not possible for Peter not to run.
• Pu-: It is not contingent that Peter does not run.
• Re-: It is impossible for Peter not to run.
• A: It is necessary for Peter to run.

Illiace (upper right)

• Il-: It is not possible for Peter to run.
• Li-: It is not contingent that Peter runs.
• A-: It is impossible for Peter to run.
• Ce: It is necessary for Peter not to run.

Amabimus (lower left)

• A-: It is possible for Peter to run.
• Ma-: It is contingent that Peter runs.
• Bi-: It is not impossible for Peter to run.
• Mus: It is not necessary for Peter not to run.

Edentuli (lower right)

• E-: It is possible for Peter not to run.
• Den-: It is contingent that Peter does not run.
• Tu-: It is not impossible for Peter not to run.
• Li: It is not necessary for Peter to run.

We began the mnemonic with Purpurea, not Amabimus, so that contrary opposition would be placed above, subcontrary opposition below, and contradictory opposition across the diagonals. Contrariety, then, lies between Purpurea and Illiace; subcontrariety between Amabimus and Edentuli; contradiction between Purpurea and Edentuli, and between Amabimus and Illiace; and equipollence among the four propositions in each mnemonic.

Simple universals with respect to both dictum and mode:

• Every man necessarily disputes.
• Every man impossibly disputes.

Simple particulars with respect to both dictum and mode:

• Some man possibly disputes.
• Some man possibly does not dispute.

The first two in the upper place are simply contraries; the two in the lower place are simply subalterns of those first two, and are simply subcontraries to one another. Considered transversely, they are simply contradictories; and by prefixing or postposing negation they become equipollent according to the rules given above for equipollents.

Universals with respect to dictum, but not with respect to mode:

• Every man possibly disputes.
• Every man possibly does not dispute.

Particulars with respect to both dictum and mode:

• Some man possibly disputes.
• Some man possibly does not dispute.

These are subalterns only with respect to the subject, but not with respect to the mode.

For this reason we have placed them here on account of subalternation. The particulars with respect to dictum and the universals with respect to mode, which seemed to belong here, will be placed more appropriately in the following table. The first two are not contraries to one another, because although they require contrariety with respect to the subject, the particular mode expressed by “possibly” prevents contrariety: the same universality and particularity remain. The two lower propositions are simply subcontraries, but when considered transversely they are not contradictories, because the same particularity remains through “possibly”; rather, they are subcontraries with respect to mode. The first two nevertheless contradict the modals with a particular dictum and common mode, which we shall list immediately.

Universals with respect to mode, and particulars with respect to dictum:

• Some man necessarily disputes.
• Some man impossibly disputes.

Particulars with respect to mode, but not with respect to dictum:

• Every man possibly disputes.
• Every man possibly does not dispute.

The two above are disparate with respect to one another, as are the two below, because the same particularity and universality remain in each pair; but considered transversely they have simple contradiction. To make them contrary only with respect to the subject or only with respect to the mode, observe this: always place a universal dictum and mode in one of the propositions. Otherwise, if one is universal and the other particular in both respects, they will be disparate. For example, these are contrary with respect to mode: “Every man necessarily disputes,” “Some man impossibly disputes”; and these with respect to dictum: “Every man possibly disputes,” “Every man impossibly disputes.” Similarly, in subcontraries with respect only to the subject or only to the mode, observe that one of the propositions must always have particularity of both dictum and mode.