Cursus Philosophicus Thomisticus, Secunda Pars Artis Logicae

A demonstrable proposition has two conditions: first, that it concerns necessary matter; second, that it is proved and illuminated by some middle term. By reason of necessity it is opposed to opinion; by reason of the middle term it is opposed to a proposition known through itself, that is, one lacking a middle term. Thus, to distinguish demonstrable from non-demonstrable propositions, these two matters must be explained: namely, how a proposition requires necessary matter, and which propositions are known through themselves.

Concerning the first point, it is certain according to Aristotle, in this first book, chapters 6 and 7¹, that demonstration must proceed from necessary and per se principles.

There are, nevertheless, SOME MODERN AUTHORS who think that a true demonstration can also concern contingent matter, as Vázquez indicates in his commentary on the First Part, disputation 4, chapter 7, and Cabero here, disputation 2, doubt 3; they require only that the acts be certain and evident. Hence, in an imperfect demonstration, such as one from an effect, there could be knowledge of contingents. For this syllogism—“Every animal becomes warm while it runs; Peter runs and is an animal; therefore, he becomes warm”—is neither probable nor erroneous; therefore it is demonstrative.

This can be confirmed in physics, which is a true science and yet deals with many contingents, such as the fact that there is heat in summer and cold in winter, and so on, which could occur otherwise. Finally, it is confirmed in theology, which has many contingents as conclusions, deduced from two propositions of faith or at least one natural proposition, and yet is a science that is absolutely certain and infallible. Likewise, divine faith itself concerns and believes many contingents on the part of its material object, and yet it is infallible.

If it is said that faith has this character by reason of the divine testimony on which it formally rests, not by reason of its material object, we will say the same about science: by reason of its formal character and act it will be certain and infallible, yet it will not have anything necessary as its material object.

BUT THESE POINTS DO NOT OBSTRUCT our assertion, although some ancient philosophers, moved in part by the second argument, denied that true science exists; among them were the Academics, and Heraclitus and Cratylus held the same view. Plato partly agreed with them, thinking that there is no science of corruptible things. Yet, so that he would not deny science altogether, he posited separate Ideas of which there could be science, as we reported above from St. Thomas, q. 3, art. 2. That science must concern a necessary object is clear from this: scientific assent must be certain and firm concerning the conclusions inferred; otherwise it would not be science but opinion, nor would it be an intellectual virtue, since it would be imperfect and weak. A virtue, however, must be a disposition of what is perfect. But a certain and firm assent cannot be had if the connection is not necessary but contingent, since assent must be firm in truth, which is conformity to being or non-being. If, then, someone says with certainty and firmness that something is or is not, while in reality it is contingent, it can change in reality; and then either the assent changes or it does not. If it changes, such science no longer remains, but its opposite does. If it does not change, it is falsified, because reality is otherwise than the assent. Therefore it is incompatible with science for a firm and certain assent to concern a contingent object.

To the first argument, then, the reply is that neither in a priori nor in a posteriori science can there be a certain and evident act concerning contingent matter. In the syllogism given, it is clear that it is not demonstrative but probable, because the minor premise—“Peter is an animal and runs”—is not certain in the part “and runs,” but contingent; and so it infers a probable conclusion. In natural philosophy, however, there are demonstrations about things that could occur otherwise, though not according to a contingent connection, but a necessary one. For example, that there are rains in winter is contingent with respect to the event, yet it is an infallible and necessary connection that at that time rains are natural and generally due; and this is wholly necessary.

To the confirmation, the reply is that theology does not rely on the connections or truths of things as they are in themselves, but on them insofar as they are subject to virtual revelation and a higher light. Although things in themselves have contingent connections, with respect to God and insofar as they are subject to his knowledge they add an infallible connection. Our theology is subalternated to divine science and presupposes that its principles are known there evidently; consequently, there they are considered according to infallible and immutable connections. Hence, formally speaking, theology does not concern contingent connections but rather infallible ones presupposed—that is, insofar as it presupposes that they are evidently proved in divine science—even though they are contingent with respect to their proximate causes, as St. Thomas teaches in Summa theologiae I, q. 14, art. 13, which is usually explained at greater length there. It is proper to a subalternated science to have its evidence and certainty presupposed in the higher, subalternating science, as we shall say in the next question, where we will also discuss theology's subalternation.

As to what is added about faith, the reply is likewise that faith does not derive its infallibility from things themselves but from divine testimony, as is usually taught in Summa theologiae II–II, q. 1. To the further objection, the reply is that there is a difference between faith and science, because science is received from things themselves and uses one truth to illuminate another. Therefore, if the truth is not firm and immutable, its illumination will not be firm and certain either. Thus, from the contingency of the object and the non-necessary connection, one rightly infers a contingent and fallible formal character. Faith, however, does not derive its formal character from the connection of things, but from divine testimony, which is extrinsic to the things themselves but most certain and infallible in itself; therefore it can concern contingent things without doing so contingently.

FROM THESE POINTS IT FOLLOWS what other conditions belong to demonstration on the part of the object demonstrated: namely, that it concern universals and what is perpetual, and proceed from proper principles. It must concern what is perpetual, not positively in existing, but objectively, by grounding a truth that does not depend on a mutable connection. It must also concern universals, because science concerns necessities; a necessary connection is found only between essential predicates, and between a subject and its properties. Essential predicates belong to the definition of a thing; but a definition is not given of singulars as singulars, but of the thing's essence itself, which is common and belongs to the species. A proper property, however, flows from the principles of the essence, not from the principles of individuation; thus the connection of proper properties is not with singulars as singulars. Finally, demonstration must proceed from proper principles and not extraneous ones, because it must proceed from things necessarily and per se connected. But it is incompatible for something to be per se connected with another and yet extraneous to it, since nothing is more proper than that on which something per se depends and with which it has a per se, that is, intrinsic, connection. Even if demonstrations in mathematics sometimes proceed according to extrinsic relations, a proportion or adaptation of one thing to another that is wholly intrinsic is presupposed there; thus, by reason of the relation included, they do not proceed wholly from extrinsic principles, although that relation is not an effect or cause but a proportion. And what is demonstrated by supernatural light, although extrinsic, is nevertheless derived from God, in whom there are immutable and infallible connections with things, however contingent they may be. Therefore that manifestation is ultimately resolved into something intrinsic, at least on the part of the relation that all creatures have to the first cause and its ordering.

Concerning the second point to be explained in this article, namely, which propositions lack a middle term and are called known through themselves, two matters seem to require explanation. First, must propositions known through themselves lack every a priori and intrinsic middle term, or also an extrinsic and a posteriori one? Likewise, although they lack a middle term in one science, can they nevertheless be proved by another, for example, by metaphysics or another higher science? Second, specifically concerning propositions in which a first property is predicated of a subject, are they absolutely indemonstrable?

To the first difficulty the reply is that, absolutely speaking, it suffices for a proposition known through itself to lack an intrinsic middle term; it need not also lack an extrinsic middle term, whether through effects, through an extrinsic cause (efficient or exemplary), or through a reduction to impossibility. The reason is that, by the very fact that something is united to another immediately through itself, it has within itself a sufficient principle for the connection of the extremes to be known immediately once the terms are known and understood, since they are united and connected immediately. Thus their conjunction can become known without any middle term other than the terms themselves. Whatever is known, from the explanation of the terms alone, to be connected with another is the object of a proposition known through itself—that is, a proposition whose connection is knowable from the explanation of the terms. Therefore, an extrinsic middle term does not prevent a proposition from being known through itself, since it is knowable from its own principles and intrinsically, by knowing the terms alone; thus it is called per se known in itself. Only accidentally and with respect to us can it be called not per se known, if the terms themselves are not commonly known to us. By “terms” we do not mean words and concepts, but the things signified by them, insofar as the connection of the extremes becomes known through their simple explanation, without proof. I say “simple explanation” because of angelic cognition, which by a simple intuition penetrates both principles and conclusions; but this is so because, by its eminence, that simple mode is equivalent to our scientific mode of knowing through discourse and proof. For a proposition to be known through itself, however, it must become known through a simple explanation of the terms, without any middle term for proving or inferring it, either formally or virtually.

From this it is inferred that all propositions containing a definition or an essential predicate are absolutely indemonstrable, even if they are not first principles containing the most common predicates. For, by the very fact that something is immediately connected with another and belongs to its concept, once the terms alone are explained, the connection of the extremes is knowable in itself, without any other circuit of reasoning or proving middle term.

It is inferred, secondly, that there is no incompatibility in propositions known through themselves being proved by an extrinsic middle term or by a reduction to impossibility; for this does not oppose the immediate and intrinsic connection of the predicates. For this reason metaphysics explains and defends all the other principles, not ostensively, but by reducing them to impossibility and to that highest principle: “Everything either is or is not.”

You will say: A reduction to impossibility is equivalent to an ostensive demonstration of the truth, because a reduction to impossibility shows the falsity of the opposite proposition and therefore the truth of its own principle. For whoever asserts the falsity or impossibility of one side asserts the truth of its opposite, since the same thing cannot be affirmed and denied.

The reply is that a reduction to impossibility proceeds only from the supposition that some conclusion is denied and some premise granted; thus, from the opposite of the denied conclusion, it infers the opposite of what was granted. But this does not show the truth of the conclusion itself determinately and in itself. Rather, it shows that if the truth of the inferred proposition is denied, one must grant two contradictories among the granted premises. It does not demonstrate how or from what the proposition has its truth.

It is inferred, thirdly, that when we take as premises or principles something proved by another science through intrinsic proof—as occurs in a subalternated science with respect to its subalternating science—those are not principles absolutely, but conclusions demonstrated by another science. This is not absolutely to lack an intrinsic and a priori middle term, but for that middle term not to be reached by this lower science, just as in the same science one demonstrated conclusion can be taken as a principle for demonstrating another conclusion. How a subalternated science uses principles proved by its subalternating science will be discussed in the next question. Thus Aristotle is understood when he teaches that some principles are absolutely indemonstrable, while others are indemonstrable in a qualified sense. Those are absolutely indemonstrable that lack an intrinsic middle term; those are indemonstrable in a qualified sense that are not demonstrated by a lower science but by a higher one. Nevertheless, it is also possible for propositions absolutely indemonstrable to be proved through some extrinsic middle term. Hence it happens that in the sciences their subjects are sometimes demonstrated a posteriori, through extrinsic causes, or from another science to which they are subalternated. Therefore, it belongs to a science only to presuppose of its subject what it is—which is to lack an intrinsic middle term pertaining to its quiddity—not an extrinsic middle term that does not pertain to what it is.

You will say: What if a middle term demonstrating a proposition known through itself a posteriori and its knowledge from the terms occur at the same time? How will the intellect stand?

The reply is that these two kinds of knowledge can coexist in the intellect, because they proceed through different middle terms, just as one object could be known through different sciences. They cannot, however, belong to the same act, because one act is of only one species and, consequently, of one middle term and one specifying principle. But it is not one specifying principle to know something from a simple explanation of the terms and to know it a posteriori through an effect. For even propter quid knowledge and quia knowledge differ in species, as we shall say in article 4. Nor can one species contain within itself lower differences unless it belongs to a higher order, which does not appear to be the case here.

Concerning the second difficulty, note only that a first property flowing from a subject can be compared either with the subject itself, really and in itself, or with its definition, through which it is explained to us. If it is compared with the subject in itself, it lacks a real middle term, because no middle term is given between it and the subject; but it does not lack a middle term distinct by reason, namely, the definition, which is distinguished by reason from the subject and through which the property is proved of the subject. For example, it is proved of a human being that he is teachable because he is a rational animal. If, however, the property is compared with the definition itself, to which it is immediately connected, it renders the proposition indemonstrable, that is, devoid of a middle term. Nor does it matter that the property flows from the essence and so can be known through the essence itself. It has this immediately, that is, without an additional real middle term, and thus it lacks a middle term, although it has a subject from which it flows. The middle term that is the definition is not really distinct from the defined thing, which means that the middle term is not distinct from the subject.