A Schema of the Various Kinds of Demonstration

Translator’s note

The text presented below is taken from note τ in the original Leonine volume on the Posterior Analytics (bk. 1, ch. 13, lect. 25, pp. 239–240). I have seen it cited by several authors and thought, upon reviewing it, that it would be of use as a resource for others. My goal in posting this is not to provide a full account of the various data needed for understanding the whole of scientific demonstration (the task of the entire tradition of material logic in the wake of the Posterior Analytics). Rather, I am only looking to provide the reader with an outline of the types of demonstration which are possible within such a scientific structure. To understand what follows, it is important to note, briefly, the way that scientific demonstration is tied up with the distinction between the per se and the per accidens and, more specifically, within the per se between that which is per se primo and that which is per se secundo. The “convertibility” of predicates and subjects will play a very important role in what follows precisely because of the various ways that one can predicate something per se of a subject.

In general, abstracting from the matter of the statement, we cannot immediately say that an A statement simply converts to an A statement. Thus, “every man is an animal” does not simply convert to “every animal is a man” (though it can convert per accidens to the I statement “some animal is a man”). However, things are different for certain cases of per se attribution. When a simple conversion of a per se predication is possible, such predication is said to be per se primo in that mode of attribution. Thus, for the first and second modes of per se attribution, we can say:

• Mode 1 Per se: Predicate is definition or part of the definition of the subject

  • Primo (definition or specific difference attributed as predicate): e.g., Every man is a rational animal. ⬄ Every rational animal is a man. Or, Every man is rational. ⬄ Every rational [being] is a man.

  • Secundo (genus attributed as predicate): e.g., Every man is an animal cannot simply convert to Every animal is a man.

• Mode 2 Per se: Predicate is property of subject.

  • Primo (Property flows strictly from the species of the subject): e.g., Every man is a political being. ⬄Every political being is a man. Or, Every triangle has three angles equal to two right angles. ⬄ Every figure having three angles equal to two right angles is a triangle.

  • Secundo (Property flows from the genus of the subject): e.g., Every man is capable of emotion cannot simply convert to Every being capable of emotion is a man. Or, Every isosceles triangle has three angles equal to two right angles cannot simply convert to Every figure having three angles equal to two right angles is an isosceles triangle.

Classically, the 3^rd mode is concerned with a mode of existence (as a substance, not as an accident) and is not a predication at all. Following Woodbury, I will be setting it aside. Thus, what St. Thomas and others call the 4^th mode will become the 3^rd for us.

This 3^rd mode of per se predication is the case of an effect which is the proper effect of a subject. Thus, the sculpture is the proper effect of a sculptor as a sculptor. Likewise, being created ex nihilo is the proper effect of God alone. In the order of extrinsic formal causality, the practical intellect is the per se cause of the choice in the will; in the order of exercise, the will is the per se cause of the practical intellect’s judgment. One usually does not speak of convertibility here, given that effects and causes don’t convert the same way that per se primo definitions or properties do. However, as we will see below (in no. V), we can speak of a kind of convertible and non-convertible effect.¹

The following diagram summarizes, briefly, the division discussed below:

Divisions of Demonstration: Schema

Demonstration comes about through a remote cause.—Based on what we have said thus far, it should be abundantly clear just how important it is for the philosopher to understand the nature of propter quid and quia demonstrations, the difference between them, and their force in settling disputed questions. Hence, it seems quite worthwhile for us to schematically present all the modes of demonstration, so that the beginner may, in a single glance, take in everything pertaining to this matter. This will make it easy to interpret the conclusion of St. Thomas’s lectio on this text.

I. Demonstration a priori and demonstration a posteriori.

Demonstration a priori proceeds from causes to effects—that is, it takes as its principle the cause of the effect to be demonstrated, which effect is accordingly inferred in the conclusion. Conversely, demonstration a posteriori takes effects as its principles and from them infers the cause in the conclusion.

II. Demonstration propter quid and demonstration quia.

This is the first subdivision of demonstration a priori, which, as we said, proceeds from causes (i.e., from the principles of the thing demonstrated). For the cause taken as principle either is or is not convertible with the effect. When it is convertible with the effect, we have an immediate or proximate cause, and the a priori demonstration will be propter quid. When the cause is not convertible with the effect, what we have is a remote cause. In such a case, the a priori demonstration will be quia.—Example: “This is a rational animal; therefore, it is able to laugh.” The ability to laugh is attributed to the rational animal as to the cause whose property it is. But, just as “every rational animal is able to laugh,” so too “everything that is able to laugh is a rational animal.” The nature of the causality being preserved, then, this cause is converted with its effect. Therefore, it is a proximate cause (inasmuch as, when it is posited or removed, the effect is posited or removed), and the demonstration is a priori and propter quid.—Another example: “This is an animal.” Now, this cause does not entail this effect: “therefore it is able to laugh.” However, from the denial of that same cause—namely, to say, “this is not an animal”—the denial of the effect is legitimately and necessarily inferred: “therefore it is not able to laugh.” In other words, the denial of the effect (ability to laugh) can be concluded from the denial of its remote cause (animal), if it is such as to be only virtually the cause of the effect. However, the affirmation of this remote cause does not enable us to conclude to this effect affirmatively. Therefore, the demonstration is a priori, though quia, not propter quid.—I deliberately said “if it is such,” etc., on account of what is to be said below in no. IV.

III. Propter quid demonstration of the cause of the effect (i.e., propter quid only) and propter quid demonstration of the cause and of the being of the effect (i.e., propter quid and without qualification [propter quid et simpliciter]).

In his commentary on the Posterior Analytics,² Cajetan affirms that this division of demonstration propter quid is handed down implicitly by Aristotle. Averroes sets it out explicitly in the Prologue to his commentaries on Aristotle’s Physics, in comment 22 on book II of the same work, and likewise in comment 35 on book II of De caelo et mundo. For example, I see an eclipse of the moon and ask about the cause, the “why,” of the eclipse. When I am told that this cause is the interposition of the earth between the sun and the moon, a demonstration propter quid is assigned for what I am seeking. However, I am not asking whether the eclipse is (esse)—for I see it with my own eyes—but, rather, why it is (through what cause, causam); and the demonstration manifests not the being or existence of the eclipse, but its cause. This demonstration propter quid, which manifests the cause but not the being of the effect, was called demonstration “propter quid only” by Averroes.—The same Averroes called a demonstration propter quid that manifests not only the cause but also the being of the effect a demonstration “propter quid and absolute.” For example, if I did not know that a triangle has three angles equal to two right angles, a mathematician would demonstrate this to me by means of a demonstration that both shows that (esse) the three angles of the triangle are equal to two right angles and assigns the cause of their being so (causam). Mathematicians make the greatest use of this mode of demonstrating.—The rationale for this division is assigned by Averroes himself: “As has been made clear in logic, demonstrations are those in which one proceeds from what is more known to what is more hidden. Therefore, if what is posterior is more known, the demonstrations will be quia. If what is prior is more known, and if the thing’s being (esse) is more hidden, while its cause is more known, this will be an absolute demonstration, which gives both cause and being (esse). However, if the thing’s cause is more hidden than its being (esse), the demonstration will be propter quid only” (In II De caelo et mundo, comm. 35 [Venice 1489]).

IV. A priori demonstration through a remote cause by reason of predication; a priori demonstration through a remote cause by reason of position or order.

Above, we said that a priori demonstration quia proceeds through a remote cause. Now (as we said in note β on the preceding lectio), the remote cause is such either by reason of predication or by reason of order or position. The first [kind of remote cause] belongs to the logical order, insofar as it stands, as it were, as a genus toward the effects of the species, and therefore, just as the genus is such-and-such a species only as contracted through the specific difference, so too, the remote cause is the cause of the specific effect only as contracted through such a difference. Moreover, just as the genus is required for constituting the species, as a potential part (cf. In I Perih., lect. 8, note ν), so too, under the same aspect, the remote cause is required for the effect of the species. Nonetheless, just as the genus is not convertible with the species—so that from the affirmation of the genus [e.g., animal] the affirmation of the species [e.g., man] does not follow, whereas the affirmation of the genus [e.g., animal] does follow from the affirmation of the species [e.g., man]—so it stands with the remote cause by reason of predication with respect to the specific effect, which accordingly does not exist without this remote cause, but is not from it.

But the cause which is remote by reason of position or order differs greatly from the cause which is remote by reason of predication. For the latter can never properly be a cause, whereas the former [i.e., the cause that is remote by reason of position or order] truly is a cause of the effect. That is: the cause which is remote by reason of predication is contracted through a [specific] difference and, therefore, the effect is formally from the difference; the cause which is remote by reason of position is determined by something which is not a difference but, rather, is something that either 1˚ proceeds from that cause and is needed by it in order that it cause, or 2˚ is a certain material condition. Take an example of 1˚: “Every spiritual substance acts from free choice.” A spiritual substance does not act immediately of itself but, rather, does so through the mediation of faculties (namely, its intellect and will). However, these faculties are from the spiritual substance. They are not its [specific] differences but, rather, its necessary concomitants [necessariae resultantiae]. Hence, the proposition converts: “Everything that acts from free choice is a spiritual substance.” Consequently, the demonstration of free choice from spiritual substance is a demonstration propter quid, even though from a cause which is remote by reason of order or position. Another example is the one St. Thomas brings forward concerning the circular wound, which is not easily healed. The circle is healable neither by reason of itself nor by reason of some imaginary difference, nor is wound a difference contracting circle to a determinate species—for golden circles, silver circles, and the like differ not in species but only in matter. Therefore, if a circular wound heals with difficulty, this attribute belongs to the circular by reason of the matter, that is, of the wound. Hence, the argument, “The wound is circular; therefore it heals with difficulty,” is an a priori demonstration propter quid, and it is a demonstration in which what is assigned is not a cause contracted through a difference, but a cause existing in determinate matter.

I consider this division of a priori demonstration to be of no small importance for the sciences. This will be evident if we consider the differences that hold between the one demonstration and the other (i.e., between these two modes of causing).

First difference. The effect of a remote cause that must be contracted through a difference, in order for the effect to be had, formally emanates only from the lower cause, that is, from the contracted cause [i.e., as this or that species, thus constituted by the specific difference]. The effect is only radically from the remote cause [i.e., the genus in question], in the way the properties of species are radically from their genera³—indeed, all the more radically, to the degree that the genera in question are distant from the subordinate species.— Conversely, the effect of a cause which is remote by reason of order or position truly and formally emanates from that cause itself. To continue with the aforementioned example from St. Thomas: being difficult to heal is truly and formally an effect of the circular nature of this matter, namely of the wound.

Second difference. The scientific resolution of an effect into its cause need not be carried to the remote cause from which the effect arises only radically. For example, if we are asked why something breathes, it suffices to say, “because it has a lung” (presuming that the lung is the immediate cause of breathing).—However, when considering an effect whose cause is remote by reason of order, we must arrive at that cause if we wish to know the propter quid of the effect itself. For no true reason is given why a circular wound heals more slowly except by assigning the circular nature, in virtue of which the wound lacks parts that could more easily be knit together. Likewise, we would fail to provide the ultimate reason for free choice if we did not rise up to the spiritual substance itself, which is the reason for the [dominating] indifference of the intellect’s judgment and of the will’s election [thus causing potential for free choice].

Third difference. A remote cause that must be contracted through a difference is not assumed in an affirmative demonstration. It is assumed in a negative one, but in that case, it only makes known the quia of the effect. [Thus, to return to an example used earlier: though we cannot affirmatively argue, “This is an animal; therefore, it is able to laugh,” we can negatively argue, “This is not an animal; therefore, it is not able to laugh.”]

By contrast, a cause which is remote either by reason of an intermediary [ratione medii] or by reason of determinate matter is assumed in an affirmative demonstration and makes known the propter quid [of the effect], as is clear from what was said about the second difference above. The reason is that when the first sort of cause is posited [i.e., one that is remote by reason of predication], the effect is not posited. Therefore, it cannot be assumed for concluding affirmatively. However, when that remote cause is denied, the effect is denied; therefore, it can be assumed for concluding negatively. Accordingly, it makes known that [quia] a thing is not this or that, though not why [propter quid] the thing is constituted in itself, which must be something positive.—Conversely, when the second sort of cause is posited [i.e., one that is remote by reason of position or order], the effect is posited, and when it is negated, the effect is negated. Therefore, it serves for concluding both negatively and affirmatively.—[Note regarding knowledge through subalternated sciences:⁴] Nevertheless, because the cause which is remote by reason of matter is the object of another, higher science (to know the circle and its parts, for example, belongs not to the physician but to the geometer), when the physician assumes the aforementioned principle about the circle, then precisely as a physician he knows only the quia and not the propter quid of the assumed principle.

Therefore, when St. Thomas says (at the end of the lectio) that this difference between quia and propter quid according to diverse sciences is contained under the other of the aforesaid modes, this should be understood, with Cajetan,⁵ as meaning that, among diverse sciences, the lower science knows the quia of the effect, while the higher science knows the propter quid, that is, the remote cause which, when applied to the matter, produces the effect.

V. Demonstration a posteriori through a convertible effect and demonstration a posteriori through a non-convertible effect.

Demonstration a posteriori always proceeds from effects to their cause, from what is “principled” to its principles. This is, in fact, the opposite of demonstration propter quid. Now, in such demonstration a posteriori the assumed effect is either convertible with the cause to be demonstrated, or non-convertible. Example of the first: “It is able to laugh; therefore it is a rational animal.” [Or: “Cultural artifacts are the effect of discursive, embodied intelligence; this is a cultural artifact; therefore, this cultural artifact is the effect of a discursive, embodied intelligent being.”] Example of the second: “It is able to laugh; therefore it is an animal.” [Or: “Cultural artifacts are the effect of a kind of animal; this is a cultural artifact; therefore, this cultural artifact is the effect of an animal.”] The convertibility or non-convertibility of the effect gives rise to the first division of demonstration a posteriori.

VI. Univocal a posteriori demonstration and analogical a posteriori demonstration.

An effect, of whatsoever nature, presents to us the being of its cause, on account of the necessary connection that holds between effect and cause: for an effect cannot be given without a cause. However, the same cannot be said as regards our knowledge of the nature of the cause. For if the effect is univocal with its cause, then, once its nature has been examined, it can—so far as lies in it, namely as an objective medium of knowledge—lead us to knowledge, whether specific or generic, concerning the nature of the cause, inasmuch as it agrees univocally with the cause in species or in genus. Thus, having considered the nature of a son, I legitimately infer not only the existence but even the specific nature of the father, given their univocal agreement. However, if the effect agrees with its cause not univocally but only analogically, then only analogous concepts concerning the nature and perfections of the cause may be gathered from knowledge of the effect’s nature and perfections. Such is the case in Theology.⁶ For creatures are infinitely distant from God in their perfections. Consequently, knowledge of the perfections of creatures does indeed infer the existence of perfections in God. However, because the perfections in God are infinite in nature, whereas the perfections of creatures are absolutely finite, and the finite cannot agree univocally with the infinite, it follows that from creatures we can know concerning that (quia) God’s perfections are, but not what [quid] they are in themselves.

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1. On this division of per se propositions, which appears to my eyes to be a correct reworking of the traditional schemata, see Austin Woodbury, Logic, The John N. Deely and Anthony Russell Collection, St. Vincent College, Latrobe, PA, nos. 229–232. AI was used for the initial drafting and editing of translation, though with significant human intervention.↩︎

2. See Cajetan, In I Poster., ch. 13, § Tria demonstrationum genera, etc. — Circa divisionem, etc., fol. 59, col. 3.↩︎

3. See the opening translator’s note for this point regarding the properties that come from the genera (per se mode 2, secundo).↩︎

4. Translator’s note: I am setting aside, however, the fact that the physician (medicus) is the possessor of an art, not, strictly speaking, a science.↩︎

5. In I Poster., ch. 13, § Circa particulam, etc., fol. 58, col. 3.↩︎

6. Translator’s note: One must, however, be careful here, due to the nature of supernatural analogy in supernatural theology. In the metaphysical consideration of God—if one desires to use the problematic expression, “natural theology”—such natural analogizing is indeed very important. But, the analogy in revealed matters involves not only such natural analogy but, above all, supernatural analogy. On this, see my article in Nova et Vetera, as well as the notes that I put together on To Be a Thomist.↩︎