The Logic of the Trinity
Trinitarian Set Up
Symbol key
DP, DE = types; f, s, sp, e = constants; N = nature function; R = “is from”; ∀ “for all”; ∃ “there exists”; ¬ “not”; ∧ “and”; = / ≠ identity / non-identity.
- DP
The type “Divine Persons.” This is the kind or domain in which f, s, and sp live. - DE
The type “Divine Essence.” This is the kind or domain in which e lives. - f
A constant of type DP. Stands for the Father. - s
A constant of type DP. Stands for the Son. - sp
A constant of type DP. Stands for the Holy Spirit. - e
A constant of type DE. Stands for the one divine essence (the single divine nature). - N
A function from DP to DE. N(x) means “the nature of x,” where x is one of the divine Persons. - R
A binary relation on DP, read “is from.” R(x, y) means “x is from y” (a relation of origin between Persons). - ∀
The universal quantifier, read “for all” or “for every.” - ∃
The existential quantifier, read “there exists” or “for some.” - ¬
The negation sign, read “not.” - ∧
The logical conjunction sign, read “and.” - =
The identity sign within a type, read “is identical to.” - ≠
The non-identity sign within a type, read “is not identical to.”
Let’s separate “person” and “essence” into different types, so we do not treat a hypostasis and the essence as the same kind of thing. “The Father is God” is written as N(f) = e, where N picks out the nature of a person. We also build in f ≠ s, f ≠ sp, s ≠ sp, and that there is one divine essence and three divine Persons.
Types (the kinds of things)
• DP = Divine Persons
The person type, with domain {f, s, sp}.
• DE = Divine Essence
The essence or nature type, with domain {e}, the one divine essence.
We only compare things of the same type with “=”.
Constants (named things)
• f : DP the Father
• s : DP the Son
• sp : DP the Holy Spirit
With f ≠ s, f ≠ sp, s ≠ sp.
• e : DE the one divine essence
Function (picking out a nature)
• N : DP → DE
N(f) = “the nature of the Father”
N(s) = “the nature of the Son”
N(sp) = “the nature of the Holy Spirit”
We require: ∀x ∈ DP, N(x) = e. So:
• N(f) = e
• N(s) = e
• N(sp) = e
Each Person has, or is of, the one divine essence. e is not a fourth person, but what the three Persons wholly are.
Relations of origin (real distinctions)
We add a relation R(x, y) on DP, read “x is from y”:
• ¬∃y R(f, y) the Father is from no one
• R(s, f) the Son is from the Father
• R(sp, f) ∧ R(sp, s) the Spirit is from the Father and the Son as one principle
Equality and inequality
• “=” is identity within a type (for example, N(f) = e).
• “≠” is non-identity within a type (for example, f ≠ s).
Expressions like f = e are ill formed, since they mix types. This “typed safety” forces us to say “has the divine essence” via N and keeps us from collapsing person and essence into a single undifferentiated logical item.
How simplicity shows up here
Simplicity means there is one divine what, not built out of parts, and all perfections are one act of being. In the system:
• DE has only one element, {e}. There are no multiple divine natures.
• N is constant on DP: for all x in DP, N(x) = e. There are not three essences or three instances of divinity, only one essence fully belonging to each Person.
You are not picturing “essence plus three things.” You have one essence type with one item e, and a function N that always picks out that same e. At the level of what God is, this is simple and strictly one.
How Trinity shows up here
The Trinity requires three really distinct Persons, distinct by origin, not by three essences. In the system:
• DP = {f, s, sp} with f ≠ s, f ≠ sp, s ≠ sp. There are exactly three Persons.
• R encodes origin: the Father is from no one, the Son is from the Father, the Spirit from Father and Son as one principle.
The Persons differ, not by different natures, but by the pattern of R.
How Trinity and simplicity are one reality
• Simplicity: one essence e, with N(f) = N(s) = N(sp) = e.
• Trinity: three distinct Persons f, s, sp related by R.
So:
• There is one what, e.
• There are three who, f, s, sp.
• What each Person is in nature is identical, the one divine essence e.
• How they are distinct is given by origin, R.
Nothing in DE is split or added to. The plurality lies on the DP side, in how the one simple essence subsists as three relations of origin. There is not really “simplicity and Trinity” as two items, but one simple divine being seen under the aspect of what (DE) and under the aspect of who (DP with R).
Why the post-Leibnizian era “classical” logic objection does not apply
The objection runs: f = God, s = God, so f = s by transitivity. In this framework we never write f = e or s = e at all. Those are not just false, they are not well formed, since f and e have different types. Instead we write N(f) = e and N(s) = e. Identity applies only within a type, so the transitivity move that collapses the Persons is blocked at the level of syntax.
At the same time, you are not introducing a hidden fourth thing behind the Persons. The one simple essence is e in DE, the three subsisting relations are f, s, sp in DP with R, and N connects them as “the same divine being, three origin-distinct Persons.” You can then say: one will, one power in e, and all external works are one undivided act of this single essence, from the Father, through the Son, in the Holy Spirit.
How classical metaphysics of identity shows up here
Seen through classical metaphysics, this whole setup is just a careful way of writing in symbols what the tradition already said in prose. In God, identity is not a bare logical relation. It is the fact that there is one simple act of being and one undivided essence. That is what DE and the single element e capture. The three Persons in DP are not three substances with three acts of being, but three subsisting relations of origin in that one simple act. That is what R encodes. The function N is the formal way of saying what the Fathers meant when they said “the Father is God” and “the Son is God”: each hypostasis is wholly the one essence, not a part of it and not a bearer of a shared genus. So the identity of God as the one simple “what” lies at the level of e, while the real distinctions of “who” are carried by f, s, sp and their origin structure. Classical identity talk about God as “one simple substance without composition” appears here as the fact that the only real identity claim at the level of nature is N(f) = N(s) = N(sp) = e, and that everything else, the non-identity of the Persons and their relations of origin, is how that same indivisible being subsists in three distinct ways.
How identity is downstream in the older classical metaphysics here
In this setting, the identity symbol “=” is not doing the deepest work. The real work is done by the metaphysical structures we have already built in: the one essence e in DE, the three hypostases f, s, sp in DP, the origin pattern R that distinguishes the Persons, and the nature map N that sends each Person to that one essence. Once those are fixed, the true identity and non-identity statements are just the logical shadows of that deeper structure. For example, N(f) = e is not a mysterious logical fact about two featureless items. It is a compact way of saying that the Father is subsistent in, and nothing other than, the simple divine act of being that e names. Likewise, f ≠ s is not a brute logical stipulation, but a short way of recording that the relations of origin encoded by R for f and s are really distinct. In Gilson’s terms, what is first in reality are act, essence, and origin. The identity and non-identity signs simply report, at the level of notation, how that underlying classical metaphysics is already arranged.