The Logical Problem of the Trinity: Brief Considerations

Introduction

Yesterday, I participated in a three-hour interview with Jake Brancatella, a Muslim, on the Catholic view of the Trinity. During this discussion, I was able to explain various scholastic positions on the Trinity, offer responses to objections against it, and provide some insight into the tradition’s theological development. The scholastic disputes on these questions are extensive, covering many significant issues, so any such treatment in such a medium must remain introductory and cursory.

One topic of particular interest to those studying the Trinity is what has come to be known as the Logical Problem of the Trinity (LPT). For many, the LPT presents one of the most serious intellectual difficulties with the Catholic faith. For others who profess to be Christians, it serves as a stumbling block—tempting them to adopt “the wisdom of the world” and reject the authoritative teachings of the Catholic Church concerning the Trinity. It is therefore natural that the LPT receives disproportionate attention compared to other theological topics. Yet this is a subject that demands the application of numerous principles from both logic and Trinitarian theology. As a result, it is often challenging to express the solution in a way that is at once succinct and simple—doing justice both to the problem’s intricacy and to the need for clarity for those less familiar with theological discourse.

Traditionally, post-Reformation scholastics included at the beginning of their theological manuals a section titled De Veritate Mysterii Sanctissimae Trinitatis, or something similar. In this tract, a number of themes are treated.

First, the Trinity is demonstrated from the loci theologici—that is, from Scripture and Tradition. Of particular interest to modern readers are the sections that argue for the Trinity from the Old Testament and examine the Trinitarian faith of the ante-Nicene Fathers.

Second, the question of the Trinity’s “knowability” by reason is addressed. Here we find debates over whether human reason can grasp the existence and possibility of the Trinity. (I explored this topic in greater depth here). Closely related is the treatment of the conceptual apparatus used to express the Trinity and the role such concepts play in mediating our knowledge of this mystery.

Third, having established that the Trinity is supra rationem (beyond reason), the theologians go on to show that it is not contra rationem (contrary to reason). This is the section in which dozens of objections are raised attempting to demonstrate the Trinity’s incoherence. Chief among these is the classic objection that “things equal to the same thing are equal to each other”—perhaps the most well-known formulation of the LPT.

The General Solution

In logic, there are three acts of the intellect: simple apprehension, judgment, and discursive reasoning. Each of these acts produces a distinct logical product. Simple apprehension yields a term—for example, “Socrates.” Judgment produces a proposition by combining two terms—e.g., “Socrates is a man.” Discursive reasoning produces a syllogism by combining two judgments and drawing a third—e.g., “Socrates is a man; all men are mortal; therefore, Socrates is mortal.”

A central concern of logic is determining which combinations of propositions form valid syllogisms. For instance, if I were to argue: “Some men are white; Tyrone is a man; therefore, Tyrone is white,” I would not be reasoning validly. This is because the premises do not guarantee the conclusion: the first premise allows for the possibility that some men are not white.

To formalize such considerations, classical logic categorizes propositions into four types:

• Universal Affirmative (A): “All men are rational.”

• Particular Affirmative (I): “Some men are white.”

• Universal Negative (E): “No man is a dog.”

• Particular Negative (O): “Some men are not white.”

These classifications—traditionally labeled A, I, E, and O—are foundational to determining valid syllogistic forms and play a crucial role in assessing whether an argument like the Logical Problem of the Trinity is formally coherent.

When we have a syllogism, we are dealing with three propositions—two premises and a conclusion—and each of these propositions typically falls under one of the four categorical types introduced above. (This classification, though widely used, is not without controversy; we will address this issue below.)

Consider the following example: “All men are rational; some animals are men; therefore, some animals are rational.” Here, the major premise (“All men are rational”) is a Universal Affirmative (A); the minor premise (“Some animals are men”) is a Particular Affirmative (I); and the conclusion (“Some animals are rational”) is also a Particular Affirmative (I). Accordingly, this syllogism is said to be in the AII mood, based on the traditional letter coding.

These classifications of propositions within a syllogism are called the moods of the syllogism.

In addition to the mood, syllogisms are also categorized by figure. The figure of a syllogism refers to the position of the middle term—the term that appears in both premises but not in the conclusion. There are four possible figures:

• Figure 1: The middle term is the subject of the major premise and the predicate of the minor premise.

• Figure 2: The middle term is the predicate in both premises.

• Figure 3: The middle term is the subject in both premises.

• Figure 4: The middle term is the predicate of the major premise and the subject of the minor premise.

Each figure allows for a specific set of valid moods, some of which are valid unconditionally and others only under certain conditions. Conversely, certain moods within each figure are invalid and do not yield logically necessary conclusions.

In order to determine which syllogistic moods are valid and which are invalid, we must appeal to what have traditionally been called the “supreme laws” of the syllogism. As Msgr. Henri Grenier explains:

The categorical syllogism depends on the principle of triple identity and the separating third. This principle may be stated thus: two things identical with a same third thing are identical with each other (principle of triple identity); two things, one of which is identical, the other not identical with a same third thing, are not identical with each other (principle of the separating third). In a syllogism, two terms are compared with a third, and their identity is affirmed or denied according as both are united to a third, or as one is separated from it. The principle of identity and the separating third is immediately evident and may be reduced to the principle of contradiction—the most fundamental of all principles: “A thing is or is not, or it is impossible that the same thing be and not be.” However, the principle of triple identity and the separating third cannot, by itself, be directly applied in a syllogism, since the middle term to which the extremes are compared is a universal. It can only be applied through two other principles, which Aristotle designates as the dictum de omni and the dictum de nullo: - Dictum de omni: Whatever is affirmed distributively or universally of a subject is affirmed of all its inferiors. - Dictum de nullo: Whatever is denied distributively or universally of a subject is denied of all its inferiors. Example: Whatever is affirmed or denied of “man” is affirmed or denied of all who fall under the concept “man”—e.g., Peter, Paul, etc. (Thomistic Philosophy, n. 70)

When we apply these rules—particularized into eight traditional logical laws—we derive the following valid moods for each syllogistic figure:

• Figure 1: AAA, EAE, AII, EIO

• Figure 2: EAE, AEE, EIO, AOO

• Figure 3: AAI, EAO, IAI, AII, OAO, EIO

• Figure 4: AAI, EAE, AII, AEO, IEO

All of this, of course, is elementary. Any introductory course in logic that covers categorical syllogisms will address these matters in detail. However, I believe that reviewing these fundamentals is essential for engaging with the question at hand. To evaluate the syllogisms often presented in support of the Logical Problem of the Trinity (LPT), we must understand the very principles by which the validity or invalidity of an argument is judged.

Thus, when we hear the common argument: “The Father is God; the Son is God; therefore, the Father is the Son,” a fundamental question must be asked: What is the distribution of the term “God”? For, the very validity of the syllogism depends essentially on the proper distribution of the middle term.

Theologically, we know that there is not, properly speaking, a distinction of universal and particular in God. Thus, strictly and properly speaking, the term God is not a species, and the Father, Son, and Holy Spirit are not individuals within that species. The union of the divine persons is not merely specific but numerical.

Therefore, it would seem that the term God is perfectly distributed when predicated of the Father, the Son, and the Holy Spirit. In this sense, the dictum de omni applies: if something is affirmed of God, and God is predicated of the three persons, then that predicate would also apply to each of them. For example, if one were to say, “God is the Father,” and God were perfectly distributed, then the predicate “Father” would also apply to the Son and the Spirit—which is false.

Here, precisely, is where the proponent of the Logical Problem of the Trinity (LPT) errs. The LPT assumes that distribution occurs only when there is a proper universal term—such as man—communicated to numerically distinct individuals. In doing so, it mistakes what is essential to distribution for how we commonly experience distribution among creatures. But the fundamental reason for distribution lies not in the individuation of a nature under a species but in the communicability of a term to many supposita.

From this, we are led to a fruitful insight into how the rules governing syllogisms apply to Trinitarian theology. We believe that, due to the infinitude of the divine essence, it is truly communicable to more than one suppositum. As a result, the term God, while not a universal in the created sense, functions as a quasi-universal when employed in theological syllogisms.

As Gonet states in his Manuale Thomistarum: “Such a syllogism is defective due to the lack of perfect distribution of the term ‘God,’ which, although absolutely singular, nevertheless, on account of the real communicability of the divine nature to the three Persons, becomes quasi-universal (aequivalenter universalis).”

This syllogism suffers from the same fundamental defect as the argument: “Peter is a man; John is a man; therefore, Peter is John.”

In both cases, the argument fails due to an absence of perfect distribution, though the reasons differ. In the Trinitarian case, the failure stems from the real communicability of the divine nature, which—while numerically one—is able to subsist in three distinct supposita. In the human example, the failure arises from the communicability of a specific nature (e.g., man) to numerically distinct individuals. In both instances, however, the fallacy lies in the misapplication or misunderstanding of distribution. It is ONLY under this aspect that the comparison is made.

This is why Suárez stated that the proper application of the dictum de omni and dictum de nullo is a fundamental consideration in resolving the Logical Problem of the Trinity:

The figure of reasoning is based on the proper distribution of the middle term, which Aristotle signified by the principles dictum de omni and dictum de nullo. And this form, when correctly applied, also has a place in this mystery [i.e., the Trinity], provided we observe that the same individual reality—by reason of real communicability—is equivalent to a common nature (naturæ communi), that is, it is quasi-universal. Therefore, from this standpoint, distribution must be added in order for the form to be preserved. (De Deo Trino, Bk. 4, ch. 3, n. 9)

Suárez emphasizes that even in the case of the Trinity, where we are dealing with a numerically singular nature and not a proper species, logical form is preserved when distribution is understood in light of real communicability. The divine essence, while not a species, is truly communicable to three supposita; thus, the term God can function analogously to a universal in syllogistic reasoning—provided the logic is carefully applied.

On Identity

From the perspective of distribution, we are better equipped to consider the notion of identity. It is precisely in light of the principles of distribution and communicability that questions concerning identity take on their proper logical and theological significance.

The scholastics distinguished between adequate and inadequate identity. Gonet defines the latter as follows: “A thing is said to be inadequately identified with something when it is not the same as everything with which that thing is identified, nor is that something with which it is identified merely the same as that thing” (nec identificans est idem cum solo identificato).

Here, “identity” is understood in relation to the relative extension of the subject and predicate. If the predicate is really communicated to the subject, this communication may either exclude further predication or allow for it.

For example, when we say, “All men are rational animals,” the extension of both terms is the same, and the communication is exclusive—there is no broader class of rational animals beyond men in this context. It is therefore equally true to say, “All rational animals are men.”

However, not all identity functions this way. When we say, “John is a man,” the predicate man is truly communicated to John, but not exclusively; it does not follow that “All men are John.” The communication is real, but not limiting. Thus, in such cases, reversing subject and predicate is logically invalid.

Now, based on the principles laid out above, are the divine persons adequately or inadequately identified with the divine essence? Clearly, due to the real communicability of the divine nature, each person is only inadequately identified with the divine essence.

As St. Robert Bellarmine explains,

To the second principle I say that that principle is not true universally, except when the two things are equal to a third thing adequately, as in mathematics, where, if two lines are equal to a third, universally they will be equal to each other, because there is found complete adequateness. However, if you say: man and horse are the same thing really with an animal; therefore they are the same between themselves—that is not valid, because man and animal are not the same really adequately; thus also the essence and paternity are not the same adequately, because the essence extends itself to more things. Nor is it true that that principle is the foundation of the whole human discourse, if no limitations are added, otherwise there would be so many figures in vain, and the modes of the syllogisms, whereby it is explained how two extremes are to be joined with the middle term, so that it can be concluded that they are also joined together...if the rational soul, because it is spiritual, naturally is in the many really distinct parts of a body, then this syllogism is not valid: Hands and feet are the same in place with the soul; therefore they are the same in place between themselve, how much greater is the fact that God, who is infinite Spirit, can be at the same time in several suppositums? (Controversy II, Ch. 18)

Further, Gonet explains,

The aforementioned philosophical axiom—on which the art of syllogistic reasoning rests—holds true only in those cases where [the major and minor] are the same with a third [i.e., the middle] adequately, not when they are the same only inadequately with a third. Now, the relations which constitute the divine Persons are not identified with the divine essence adequately, but only inadequately. A thing is said to be inadequately identified with something when it is not the same as everything with which that thing is identified, nor is that something with which it is identical merely the same as that thing (nec identificans est idem cum solo identificato). But each personal relation is so identified with the divine nature, that it is not identified with the other personal relations with which the essence is identified; and likewise, the nature is not so identified with one relation that it is not also identified with the others. Thus, for example, the distinct parts of the body are present to one third thing—namely, to the rational soul—without being present to each other: for the foot and the head are immediately and proximately united to the soul, but are distant from each other, because, evidently, they are not united to the soul adequately, but only inadequately—inasmuch as the soul informs not only those parts but also the other parts of the body and is united to them. Likewise, Christ is present in multiple hosts and altars, which nevertheless are not present to one another: because He is not present to those altars and hosts adequately, since He is present to many others as well. Similarly, the Divine Persons are identified with the essence, without being identified with one another—because they are not identified with [the Divine Essence] adequately, but only inadequately. (Manuale Thomistarum, Tractatus IV, Cap. 2)

But if the persons are only inadequately identified with the essence, why should the Logical Problem of the Trinity (LPT) even arise? How does this argument against the Trinity not result in a fundamental denial of syllogistic reasoning and especially the proper distribution of terms?

Conclusion

It is certainly possible to go further and discuss how the divine essence is really communicable to multiple supposita without multiplying the nature numerically, as happens with species in their individuals. However, that question lies beyond the scope of the present discussion (further, this discussion is more commonly brought up when discussing this issue). The immediate concern has been to demonstrate that, internally, there is no incoherence in the classical explanation of real identity (with the essence) and real distinction (of the persons).

A fuller treatment of the matter—one that may be undertaken in the future—would begin by examining the metaphysical nature of relatio, which is central to the scholastic understanding of divine distinction. But for now, the foregoing should suffice to address the objection posed by the LPT.