Fr. Réginald Garrigou-Lagrange, O.P., says, in Le Sens du Mystère et le Clair-Obscur Intellectuel: Nature et Surnaturel p. 128 fn. 1 (Engl. transl. p. 142 fn. 41):

St. Thomas never would have admitted like Suarez that the principle of contradiction is not applied in the case of the Trinity. It is applied there according to an eminent mode that remains hidden to us, and nothing can show that this mystery implies a contradiction.

Jamais saint Thomas n'aurait admis comme Suarez que le principe de contradiction ne s'applique pas dans la Trinité; il s'y applique selon un mode éminent qui nous reste caché, et nul ne peut montrer que ce mystère implique contradiction.

Where did Suárez write that the principle of non-contradiction does not apply in the Trinity?

  • The principle of non-contradiction negates itself when the identity laws are applied to themselves: (P = P) v (P =/ P). Under such contexts the trinity does not require these laws...however even so ths trinity ard three different contexts of the same phenomenon.
    – Eodnhoj7
    Jan 8 '20 at 3:16
  • The Trinity is a description of Unity, as is explained by the Doctrine of Divine Simplicity. The Three would reduce to a non-numerical One. In this case there are no contradictions. The three terms of the Trinity do not stand for three different things that could give rise to contradictions. I can't help with the source of the quote.
    – user20253
    May 4 '20 at 14:52

Suarez has commented on Aquinas's Summa Prima Pars, and he also wrote a theological summa called De Deo Uno et Trino (cf. Suarez in Latin online).

The image below shows that

(1) Pater est Deus

(2) Filius est Deus

(3) Pater non est Filius

However, from (1) and (2) ( with the symmetry and the transitivity property of equality) one can infer that

(4) Pater est Filius .

Consequently, the principle of non contradiction does not apply here.

Should we conclude that Trinity is contrary to reason, or that it is above reason?

enter image description here

  • I believe we should conclude that logic has limits, and those limits are exceeded by Reality. Logic requires that we divide the world by the use of distinctions and categories. Reality requires that we sublate them all for Unity and non-duality. There is no evidence that Reality is contrary to reason, but a great deal to suggest that logic must be transcended for any understanding. Indeed Nagarjuna use logic to prove exactly this. . .
    – user20253
    Jan 6 '20 at 13:19
  • Yes, what I'm looking for must be in one of those works.
    – Geremia
    Jan 6 '20 at 18:11

Perhaps we can here begin to show, without specifically answering the question posted on this page, that a regular use of two-valued Aristotelian logic, useful though it is in its own right and realm of specific operation, fails to apply to the powerful doctrine of the Most Blessed Trinity.

The intrinsic relationality of the One divine essence is self-subsistent and necessarily consubstantial, yielding the following properties, which will give us expanded logical tools to express the theological uniqueness of the inner life/dynamics of the Triune God:

  1. The divine relations are truly distinct.

  2. Each relation, shorthanded Ri, has all the others as its terms

  3. For each subsistent Ri, the esse in is concomitantly “induced” by its terms. Thus, each Ri is equivalently constituted by the n – 1 esse ad of its concomitant terms

I will unpack and illustrate 3) a bit further down.

The matter pertaining to the application or non-application of the principle of non-contradiction to the case of the Blessed Trinity can be addressed by expanding the two-valued logic inherited from the Aristotelian mode of thinking. Notice that Saint Thomas himself had to partially depart from this mode of thinking in order to articulate, as he so carefully did, the Trinitarian doctrine of the Church, coming up with the paramount notion of subsistent relationships.

The ad intra relationality of God comprises n relations Ri implying the subsisting esse ad of all the others. If n = 2, there actually cannot be any real relational distinction in God, but a mere duality of extrinsic terms, which ruins together the unitas essentiæ and the internal principium distinctivum in divinis. An everyday-like application of the Aristotelian principle of non-contradiction (i.e. to casual cases in which two contradictory assertions cannot be true at the same time) to basic theological assertions appertaining to the essence of God would seem here to yield intelligible fruits, as believe the Mohammedans: God is either one or plural/divided, in which case he is not actually God.

However, when n = 3, the divine essence’s internal relationality comprises three binary relations (as opposed to three passive terms logically interconnected and conditioned by “the principle gathered from Aristotle, Metaphysics, Bk. IV, text. 3,56 that things which are the same as a third thing are the same as each other”).

We will simply label them A, B, C (to keep the notation used by Geremia in describing Suárez’s discussion of “the applicability to the Persons of the Trinity of this form of the principle of non-contradiction”), such that B and C are the terms of A, C and A the terms of B, and A and B the terms of C.

The indivisible divine unity of the Triune God therefore consists of three real subsistent relations, the dynamics of which yields six respectively “active” (⇒) and “passive” (→) esse ad. Each of the three esse in consists of two esse ad from two other concomitant relations.

A ⇒ B  B → A             B ⇒ C   C → B               C ⇒ A  A → C                  
B ⇒ C  C → B             C ⇒ A   A → C               A ⇒ B  B → A
A ⇒ C  C → A             B ⇒ A   A → B               C ⇒ B  B → C   

Notice, therefore, the NON-contradictory equivalency of 1 ≡ 3 ≡ 6 in this indivisibility (1) of three divine Persons (3) actively and passively (6) communicating the One Godhead as ipsæ relationes subsistentes.

We can finally admire the fact that God’s revealed Thrice Holy Name, יהוה/Y-H-W-H (a consonantical Hebraic Tetragrammaton), provides the original and hieroglyphic expression of this very doctrine of the four divine relations (three of which are real, distinct Hypostases). Thus the Tetragrammaton alone, given to Moses in the theophany of the burning bush (Exodus 3:1–15), attests to the unmistakable truth and unity of the Catholic religion spanning inerrantly through the 73 Books of Holy Scripture:

Y ≡ R1 ≡ Paternity (a relation of origination)

W ≡ R2 ≡ Filiation (a relation of generative procession)

H ≡ R0 ≡ Active Spiration (a relation of origination equivalent to R1 & R2)

H ≡ R3 ≡ Passive Spiration (a relation of spirated procession)


Francisco Suárez

Suárez discusses the applicability to the Persons of the Trinity of this form of the principle of non-contradiction,

  • A = C
  • B = C
  • ∴ A = B

in On the Various Kinds of Distinctions p. 59 (Disputationes Metaphysicæ, Disputatio VII, De Variis Distinctionum Generibus),

  • §2 The Signs or Norms for Discerning Various Grades of Distinction in Things
    Quibus signis seu modis discerni possint varii gradus distinctionis rerum
    • A Doubt Concerning the Separability of Distinct Things
      Dubium occurrens de separabilitate rerum distinctarum
        1. Third exception: the divine Persons
          Tertia de personis divinis ad invicem

The objection might be made that this argument is based on the principle gathered from Aristotle, Metaphysics, Bk. IV, text. 3,56 that things which are the same as a third thing are the same as each other — a principle which has no place in the Trinity; otherwise we should infer not only that one relation cannot exist without another, but that it is this other.

Dices, hoc argumentum fundari in illo principio Aristotelis, IV Metaph., text. 3: Quæ sunt eadem uni tertio, sunt eadem inter se, quod in Trinitate locum non habet; alioqui non solum inferretur unam relationem non posse esse sine alia, sed etiam esse aliam

56. The axiom is nowhere explicitly formulated in Metaphysica, IV, text 3. Junta VII fol. 31 vb. 32–58. Cf. Metaphysica, IV, 2, 1003b 22–34. But it is an obvious application of the principle of contradiction, which Aristotle discusses throughout this book.

(courtesy: Quaestiones Disp.)

and in the last section of this disputatio:

  • §3 Comparison of “The Same” and “Other” Both with Each Other and with Being
    Quomodo idem et diversum tum inter se, tum ad ens comparentur
      1. Explanation of an Aristotelian dictum (p. 67)
        Expositio pronuntiati aristotelici

Finally, we can gather from the foregoing doctrine the meaning of an axiom enunciated by Aristotle in the fourth book of the Metaphysics:72 Things which are the same as a third thing, are the same as each other. Due proportion must be observed in applying this principle. If two things are in reality identical with a third thing, they will also be identical with each other in reality, although they may be diverse in concept. And if they are identical with a third thing both in reality and in concept, they will be identical with each other in the same way. In creatures and in finite things this principle avails absolutely. But in an infinite thing, such as is the divine essence, the maxim is not verified, absolutely speaking, since on account of its infinity the divine essence can be identical with opposite relations which, because of this opposition, cannot be identical with one another, except in the essence alone. But we have dealt with this problem on another occasion.73

Ultimo potest ex dictis colligi, quomodo sit intelligendum illud axioma quod Aristoteles posuit IV Metaph.: Quæcumque sunt eadem uni tertio, sunt eadem inter se; intelligendum est enim cum proportione, nam si sunt eadem re uni tertio, simili modo erunt eadem re inter se, poterunt autem esse ratione diversa; si autem re et ratione sint uni tertio eadem, erunt eodem modo eadem inter se. Sed hoc principium in creaturis, et in rebus finitis simpliciter tenet; in re autem infinita, qualis est divina essentia, non verificatur illa maxima absolute loquendo, quia propter suam infinitatem potest esse idem relationibus oppositis quæ propter oppositionem inter se idem esse non possunt nisi tantum in essentia; de quo alias.

72. See note 56.
73. De Sanctissimo Trinitatis mysterio, Lib. IV, cap. 3; in Vol. I of the Vivès edition.

St. Thomas Aquinas

In De Sanctissimo Trinitatis mysterio, Lib. IV, cap. 3, Suárez discusses St. Thomas's solution in Summa Theologica I q. 28 a. 3 "Whether the relations in God are really distinguished from each other?" arg./ad 1. St. Thomas writes:

Objection 1: It would seem that the divine relations are not really distinguished from each other. For things which are identified with the same, are identified with each other. But every relation in God is really the same as the divine essence. Therefore the relations are not really distinguished from each other.

Videtur quod relationes quæ sunt in Deo, realiter ab invicem non distinguantur. Quæcumque enim uni et eidem sunt eadem, sibi invicem sunt eadem. Sed omnis relatio in Deo existens est idem secundum rem cum divina essentia. Ergo relationes secundum rem ab invicem non distinguuntur.

Reply to Objection 1: According to the Philosopher (Phys. iii), this argument holds, that whatever things are identified with the same thing are identified with each other, if the identity be real and logical; as, for instance, a tunic and a garment; but not if they differ logically. Hence in the same place he says that although action is the same as motion, and likewise passion; still it does not follow that action and passion are the same; because action implies reference as of something "from which" there is motion in the thing moved; whereas passion implies reference as of something "which is from" another. Likewise, although paternity, just as filiation, is really the same as the divine essence; nevertheless these two in their own proper idea and definitions import opposite respects. Hence they are distinguished from each other.

Ad primum ergo dicendum quod, secundum philosophum in III Physic., argumentum illud tenet, quod quæcumque uni et eidem sunt eadem, sibi invicem sunt eadem, in his quæ sunt idem re et ratione, sicut tunica et indumentum, non autem in his quæ differunt ratione. Unde ibidem dicit quod, licet actio sit idem motui, similiter et passio, non tamen sequitur quod actio et passio sint idem, quia in actione importatur respectus ut a quo est motus in mobili, in passione vero ut qui est ab alio. Et similiter, licet paternitas sit idem secundum rem cum essentia divina, et similiter filiatio, tamen hæc duo in suis propriis rationibus important oppositos respectus. Unde distinguuntur ab invicem.

John of St. Thomas

John of St. Thomas (João Poinsot), O.P., 1589-1644, compares Suárez's and St. Thomas's solutions in Cursus theologicus d. 12 a. 3 n. 31 arg./ad 2, noting their differences in defining the aforementioned logical principle:

This Mystery is not Contrary to the Principles of the Natural Light of Reason.
Non Contrariatur Principiis Luminis Naturalis Hoc Mysterium.

[Objection:] 28. […] The second principle known by the natural light [of reason], which destroys this mystery, is that: Whatever are similarly one in a third, are similarly between themselves, upon which the entire rationale of the syllogistic art and deduction is based. But this principle is denied in this mystery because three Persons are one and the same in the divine essence, but between themselves they differ and are not one.
Secundum principium naturali lumine notum, quod destruit hoc mysterium, est illud: Quæcumque sunt eadem uni tertio, sunt eadem inter se, super quod tota ratio syllogisticæ artis fundatur, et omnis firmitas illationis, et consequentiæ. Hoc autem principium negatur in hoc mysterio, quia tres Personæ sunt unum et idem in essentia divina: inter se autem differunt, et unum non sunt.


31. St. Thomas replies to the second [objection] in [Summa Theologica I] q. 28 a. 3 ad 1, that the principle: Whatever are similarly one in a third, etc. is understood by the philosopher in III Physics text 21 to apply to things that are similarly one in a third, in the thing and in reason, but not of things that differ in reason. The holy doctor explains the same principle in another way [Super Sent.] I d. 33 q. 1 a. 1 ad 2, which is understood: Those which are similarly one in a third, are the same between themselves in the same way by which they are the same to the third, and not otherwise nor in another manner. And so three Persons are the same one by the third, i.e., in the divine substance, in the absolute rationale, and between themselves they are also one in the same absolute and substantial rationale; however, they are not one between themselves relatively, because in that third they are not one relatively, but only absolutely. Therefore, that natural principle—that whatever are similarly one in a third are similarly between themselves, i.e., in the same way by which they are similarly one in the third—saves this mystery; but, in another way it is not necessary that they be similarly between themselves, but by one only, by which in that third they are one.
Ad secundum respondetur ex D. Thoma infra q. 28 art. 3 ad 1, quod illud principium: Quæcumque sunt eadem uni tertio, etc. intelligitur a philosopho III physic. textu 21 de his, quæ sunt eadem uni tertio, re, et ratione, non autem de his quæ differunt ratione. Idem principius explicat aliter s. doctor in I, distinct. 33, q. 1, art. 1 ad 2, quod intelligitur: Ea quæ sunt eadem uni tertio, sunt idem inter se, eo modo, quo sunt eadem illi tertio, et non aliter, nec alio modo. Et sic tres Personæ sunt idem uni tertio, scilicet substantiæ divinæ, in ratione absoluti, et inter se etiam sunt unum in eadem ratione absoluta et substantiali, non tamen sunt inter se unum relative, qua nec in illo tertio sunt unum relative, sed solum absolute. Salvat ergo mysterium hoc illud principium naturale quod: Quæ sunt eadem uni tertio, sunt eadem inter se, videlicet eo modo, quo sunt eadem uni tertio; alio autem modo non est necesse quod sint eadem inter se, sed eo solum, quo in illo tertia unum sunt.

Others explain that principle thus: Whatever are similarly one in a third, are similarly between themselves, if they be similarly one in the third equally and simply, but not if they be similarly one in the third unequally. But the divine relations are unequally related with respect to the essence, because whichever of them so affects the essence leaves place for the other; and so all the three Persons equal it, not one only. This explanation coincides entirely with the first explanation of St. Thomas.
Alii explicant illud principium sic: Quæ sunt eadem uni tertio, sunt eadem inter se, si sint eadem uni tertio adæquate, et simpliciter, non vero si sint eadem uni tertio inadæquate. Relationes autem divinæ inadæquate se habent respectu essentiæ, quia quælibet illarum ita afficit essentiam, quod alteri locum relinquit; et sic omnes tres Personæ adæquant illam, non una tantum. Quæ explicatio fere coincidit cum priori explicatione D. Thomæ.

Others say that principle is to be understood thus, that: Whatever are similarly one in a third, are similarly between themselves, without any opposition intervening between those extremes, that they be one between themselves more than that unity by which are the same in one by a third. In fine, Fr. Suárez lib. IV de Trinitate, cap. 3 denies that principle to be true, except in created things according to their limitation, even false if it be extended to divine things, according to their infinitude, in which place it also has absolute perfection, in which all are one, and relative [perfection] between what is opposed. Fr. Vasquez also teaches this in tome II, in the first part, disp. 123, c. 2, and others who teach that principle only holds where there is limited power, not where there is infinite eminence embracing the absolute genus with supreme unity, and relative [eminence] with correlative opposition. From which all agree how that natural principle is not destroyed in this mystery, but remains unharmed, understanding it in its truth and legitimate sense.
Alii dicunt illud principium sic esse intelligendum, quod: Quæ sunt eadem uni tertio, sunt eadem inter se, nisi obstet aliqua oppositio inter illi extrema, ut sint unum inter se magis, quam illa unitas, qua sunt idem in uno tertio. Denique p. Suarez lib. IV de Trinitate, cap. 3 negat illud principium esse verum, nisi in rebus creatus propter suam limitationem, falsum tamen si extendatur ad divina, propter suam infinitatem, in qua locum habet et perfectio absoluta, in qua omnia sunt unum, et relativa inter quæ est oppositio. Quod etiam docet p. Vasquez II tomo, in prima parte, disp. 123, c. 2, et alii, qui docet illud principium tenere solum, ubi est limitata virtus, non ubi est eminentia infinita complectens genus absolutum cum summa unitate, et relativum cum oppositione correlativa. Ex quibus omnibus constat quomodo illud naturale principium in hoc mysterio non destruatur, sed maneat illæsum, intelligendo illud in sua veritate, et legitimo sensu.



John of St. Thomas’ following remark, introducing his comparative treatment of Suárez and St. Thomas, captures a distinctly Catholic principle:

“Non Contrariatur Principiis Luminis Naturalis Hoc Mysterium.”

We may also say of the doctrine of the Most Blessed Trinity—inspired by the same principle:

Supra luminis naturalis rationis sed non adversus eam.

Relying on such a sound basis, let us precisely return to the matter of applying some expanded logical tools to that sublime doctrine. The relations denoted A, B, and C constitute a self-subsistent and consubstantial third-order relational unity. We will call it Θ, the One Godhead (אֵל) that is the Father, the Son, and the Holy Ghost (יהוה).

If we seek to establish an order between the three mutually opposed relations of Θ (to logically account for relative opposition being inferable on the basis of origin), we then have to permute the binary terms in each of the dynamic sets of three pairs. This yields height different ways (vias) of binarily permuting the three relations that are eternally and dynamically consubstantial to one another as undivided communication of Θ. But two orders of such permutations/rotations of pairs (literally per περι-/circum- χώρησις/verto or incessio) do not logically reflect the distinction of the three real divine Hypostases respective of origins (origines) and processions (processiones):

A ⇒ B       B → A    
B ⇒ C       C → B    
C ⇒ A       A → C    

Contrariwise, the other six orders of mutually opposed pairs of subsistent relations do reflect and express it in terms of active (⇒) and passive (→) esse ad in Θ, and are again the following:

A ⇒ B       B → A       B ⇒ C       C → B       C ⇒ A       A → C        
B ⇒ C       C → B       C ⇒ A       A → C       A ⇒ B       B → A  
A ⇒ C       C → A       B ⇒ A       A → B       C ⇒ B       B → C

As St. Thomas puts it (ST Ia q. 28 a. 4 ad 5): “… via est eadem ab uno termino ad alterum, et e converso; sed tamen respectus sunt diversi.” That is what the notion of permuting the ways A, B, and C are each the n – 1 esse ad of its concomitant two terms is here intended to show.

Moreover, the interchange of A ≡ R₁ and B ≡ R₂, which together are R₀ but NOT C ≡ R₃, concomitantly shows the intrinsic importance of not reducing to three (as Greek anti-filioquists ruinously do) the four mutually opposed relations in Θ, i.e. paternitas, filiatio, spiratio (communis), processio (passivae). In Θ, by virtue of the nature of its dynamic ternary structure, R₀ is necessarily present as a distinction pertaining to its relational ad intra order established upon two processions. These concomitantly establish four relations, given that each procession concomitantly yields two mutually opposed relations.

There is more to A, B, and C than meets the eyes of their syllogistic application to an argument designed to illustrate the principle of non-contradiction...

R₁ ⇒ R₂     R₂ → R₁     R₂ ⇒ R₃     R₃ → R₂     R₃ ⇒ R₁     R₁ → R₃
R₂ ⇒ R₃     R₃ → R₂     R₃ ⇒ R₁     R₁ → R₃     R₁ ⇒ R₂     R₂ → R₁
R₁ ⇒ R₃     R₃ → R₁     R₂ ⇒ R₁     R₁ → R₂     R₃ ⇒ R₂     R₂ → R₃     
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