Leibniz’s Law and Identity?
Question 56: Leibniz’s Law and Identity?
“How can one apply the modern Leibnizian laws of identity to the Trinity in a way that preserves both divine simplicity (“one God”) and real personal distinction (“three hypostases”) without collapsing the Persons or violating his type of logic?
The following Piece is Inspired by Dr. Sijuwade’s use of identity in this debate, which I found very suspect, so this piece corrects his misappropriation of identity theory as he presupposes it (in error).
*Terms and Definitions for this Piece*
Dr. Sijuwade’s model is heavily dependent on a very problematic use of identity, among other issues1. So, I thought I’d help clarify how one should technically appropriate the theory of identity he is trying to peddle for his case. To that end, I had AI organize a list of definitions and basically give an outline how to use said theory if a classical theist wants Leibniz. This should serve as a helpful reference for holding this type of theorizing accountable (that is if one so chooses). If you want a list of the terms and definitions, I had AI generate skip pass the piece proper to the end where the dictionary of terms are listed in alphabetical order and defined.
God and Identity
When the tradition says that in God “to be is to X,” (any attribute claim) the claim is not identity in the later Leibnizian-Fregean sense (x = y if and only if every predicate true of x is true of y). Scholastic writers are working with a pre-modern, metaphysical notion of “real identity”:
- In creatures we distinguish act from essence, goodness from existence, intellect from will.
- In God, because He is pure act with no potentiality, those aspects are really one and the same reality, though we grasp them under different “notional” or “conceptual” aspects (Aquinas calls this a distinction of reason, Scotus calls it formal).
So “identity” here means transcendental convertibility: whatever is truly said of God (esse, verum, bonum, love, self-gift) converts with the simple divine reality without introducing composition. It is neither a numeric identity between two objects nor an extensional principle governed by Leibniz’s Law.
Instructions for Appropriating Specific So Called Laws of Identity
The coherence of Trinitarian doctrine within a modern (now dated) logical framework begins with a strict separation of two semantic fields or domains or categories, if you will. I’ll stick with field to keep it consistent. “Field” here helps accentuate the problem with the dated Leibnizian worldview that thought an idealized view of math could solve and be appropriated for all of identity via a math-like set of laws. It can’t. But the class distinctness of fields, the language is helpful here. So, in this case, when talking about the Trinity, one field concerns what God is and the other concerns who God is. In the field of “what,” the single divine essence is named and predicated; in the field of “who,” the three hypostases are identified and compared. Confusion only arises when predicates or identity signs are allowed to drift from one field into the other. (Something Sijuwade wants to misappropriate in the video above.)
Essential predication uses the ordinary copula “is.” When a creed says “Father is God,” the term God denotes the one divine essence and the verb simply links the personal subject to that nature. Logically we can capture this with a unary predicate “G” applied in the personal domain: G(F), G(S), and G(H) each mean that Father, Son, and Spirit fully possess the single essence. Predication is many-one; it allows several subjects to share the same property without implying those subjects are numerically identical. For that reason specifically, no classical law of identity is challenged by Father is God or Son is God taken in this sense.
Personal identity, by contrast, belongs to a separate domain that contains only the three hypostases. Here we employ the usual equality sign. Reflexivity, symmetry, and transitivity all hold: Father equals Father, Son equals Son, Spirit equals Spirit, and Father is not equal to Son. To keep the bookkeeping clear we can label this relation P₁. Because P₁ is confined to the personal sort, even the strongest forms of Leibniz’s Law hold inside it. If x equals y, every personal predicate true of x is true of y; if two persons differ by any relational predicate of origin they cannot be identical. The relations of being unoriginated, begotten, and proceeding provide precisely such discriminating predicates, so Father and Son never collapse into one person.
This clear typing of variables and predicates protects the Post Leibnizian era (the so called “classical laws”) of identity, and inserts accountability into a discussion where Sijuwade mispurposes this kind of theorizing. Transitivity, for instance, cannot be triggered across the line between predication and personal identity because no term belonging to the essence field ever appears on the left or right of the personal equality sign. Substitution likewise never migrates a personal constant into an essential statement or vice versa. Thus, there is no valid route from Father is God and Son is God to Father equals Son. The inference is blocked before it starts.
A deeper metaphysical ordering underlies the formal one. First comes the divine essence itself, conceived as pure act without parts or potentiality. Because it is absolutely indivisible, the word one applied to God is analogical. It denies all real composition in God yet does not treat God as a lone member of a set to be counted alongside others. Only within this horizon of analogical oneness do we meaningfully speak of three personal subsistences. Each hypostasis is the entire essence under a distinct relational stance: unoriginated source, filial reception, or personal procession. Because the essence is limitless, no competition arises when three hypostases each possess it totally.
Leibniz’s indiscernibility of identicals and the so-called law of identity of indiscernibles remain sound once the range of quantifiers is fixed. They apply rigorously within the personal domain and fail automatically across the boundary to the essence domain, because there is only one entity in the essence sort and no binary identity relation defined there at all. The same typed discipline covers modern concerns about absolute discernibility: sharing the predicate G never forces numerical identity, for no identity predicate ranges over cases where only essential predication is at stake.
When all these distinctions are held in place, every classical Trinitarian formula proves coherent. One may say without contradiction that the Father is God, the Son is God, and the Spirit is God, yet the Father is not the Son and the Son is not the Spirit. One may also add that the Father is by identity the one God, so long as “by identity” is understood as essential predication and not as personal collapse. The creeds, the councils, and someone like Aquinas (doctor’s and theologians who get technical enough in this type of theology) presuppose exactly this disciplined grammar. Modern logic, far from undermining their claims, even being invented after classical philosophy, shows why the pattern works: two senses of is, two logical domains, a single infinite essence (infinite in the sense of unbounded, undivided, and without any principle of limitation), and three irreducibly distinct hypostases.[i]
AI Generated Summary and Answer to the Question at hand:
By working in two strictly typed domains.
• In the essence domain a unary predicate G(x) states “x fully possesses the one, infinite, indivisible divine act,” so Father is God and Son is God are statements of essential predication, not equations.
• In the person domain the ordinary identity sign (=) compares only hypostases; reflexivity, symmetry, transitivity, and Leibniz’s two maxims hold here, ensuring Father ≠ Son because they differ by relations of origin.
No identity statement ever crosses the domain boundary, so the inference from G(Father) and G(Son) to Father = Son is blocked. Thus, modern identity theory and orthodox Trinitarian formulae remain fully compatible.
Essential Terms and Definitions
Act of self-gift: The single, infinite actuality that God is in Himself, understood as pure giving without loss. It is the theological way of naming the one divine essence in dynamic, self-communicating mode.
Analogical oneness: The affirmation that God’s unity is without division or parts while not reducing Him to a mere member in a countable class. The term safeguards that “one God” means absolute indivisibility, not numerical singleton in the creaturely sense. A theological statement that God is “one” where the term one is used neither in the strict numerical sense of a countable unit nor in the Platonic sense of an abstract monad. It signifies absolute indivisibility and the impossibility of any real composition within the divine essence. Because the oneness is analogical, it does not and CANNOT submit to the rigid law of identity that presupposes discrete units capable of being counted or compared. When the creed says one God it affirms that nothing in God is other than God, not that the divine reality matches the mathematical concept of the number 1 in a one-to-one fashion. Hence the predicate one guards simplicity without implying that God can be placed in a set and tallied; it transcends arithmetic categories while still ruling out any second deity or division of substance. Personal identity—Father, Son, and Spirit—is appropriated only within the horizon of this analogical unity, so each hypostasis is confessed as truly God without reducing the divine oneness to a numerical label.
Anathema: A conciliar condemnation that marks the outer boundary of orthodox belief. In Trinitarian contexts it excludes any claim that divides the divine essence or collapses the personal distinctions.
Category leap: A logical error that treats statements about essence as if they were statements about personal identity, or vice versa, thereby violating the coherent grammar of Trinitarian language.
Consubstantiality: The doctrine that Father, Son, and Spirit each possess the whole divine essence without partition or hierarchy, guaranteeing full equality of power and glory.
Copula: The verbal link “is” when used to join a subject to a predicate of nature or essence, as in “Father is God.” It differs from the equality sign that joins two tokens within the same personal domain.
Divine essence: The unlimited, simple act of being that is God. It contains no composition and is identical with God’s existence.
Domain: A formally delimited set of objects over which quantifiers and identity signs range, such as the Person domain or the Essence domain in a typed logical language.
Essence sort: The logical type whose only inhabitant in this theology is the unique divine act, ensuring that talk of “what God is” stays distinct from talk of “who God is.”
Father (F): The hypostasis who is un-originated source, fully identical with the divine essence while personally distinguished by paternity.
Hypostasis: A subsisting personal reality. In Trinitarian theology it denotes Father, Son, or Spirit, each being the entire essence under a distinct relational identity.
Hypostatic stance: The irreducible relational orientation by which a hypostasis subsists as source, begotten, or proceeding. It expresses “how” each Person fully possesses the one essence.
Identity predicate P1(x,y): The binary relation of strict numerical identity restricted to the Person domain. It yields true when x and y are the same hypostasis and false otherwise.
Identity of indiscernibles: Leibniz’s principle that if two entities share every predicate true in their domain, they are numerically identical. Applied within the Person domain it helps show that distinct relations of origin suffice to keep the Persons numerically diverse.
Indiscernibility of identicals: The converse Leibnizian rule that whatever is true of one entity is true of another if those entities are numerically identical. It underwrites standard substitution within the Person domain.
Leibniz’s Law: The paired principles of indiscernibility of identicals and identity of indiscernibles. Both hold inside a single, well-typed logical domain but never across the Essence–Person boundary.
Logical bridge: Any inference rule that tries to move from essential predication to personal identity or from one logical sort to another. Guarding against illegitimate bridges preserves Trinitarian coherence.
Metaphysical foundation: The level at which the divine essence as pure act grounds every further affirmation. Personal distinctions and logical predicates are posterior in the order of explanation, though not in temporal sequence.
Non-competition principle: The claim that because the divine act is limitless, each Person can possess it wholly without rivalry or division, allowing full equality alongside real distinction.
Numerical identity: The relation that obtains when two tokens are in fact one and the same entity. In Trinitarian logic it is confined to comparisons within the Person sort.
Personal identity: Numerical identity as applied specifically to hypostases. It answers the question “which Person is which” without reference to how each possesses the essence.
Predicate: A linguistic or logical function that attributes a property or relation to a subject. Predicates carry the descriptive weight of doctrinal affirmations.
Predicate G(x): The unary predicate defined as “x fully possesses the divine essence.” It is true of Father, Son, and Spirit alike, expressing consubstantiality without implying personal identity.
Pure act: Another term for the divine essence, emphasizing that in God actuality is not mixed with potentiality. It excludes temporal becoming and metaphysical composition.
Relation of origin: The constitutive personal property by which a hypostasis is distinguished. Paternity, filiation, and procession are the classic relations of origin.
Relational predicate: A logical predicate that expresses how one subject is related to another, such as “is begotten by.” In Trinitarian logic these mark what differentiates the Persons.
Sort: A logical type that restricts quantifiers, variables, and predicates to a specific class of objects, preventing illicit inference across heterogeneous categories.
Son (S): The hypostasis eternally begotten of the Father, numerically distinct from Father and Spirit while identical with them in essence.
Spirit (H): The hypostasis eternally proceeding from the Father and, in Latin theology, through the Son, likewise fully consubstantial yet personally distinct.
Substitutivity: The rule that identical terms may replace one another in any true statement without loss of truth. Its use is confined to a single logical domain to avoid category leaps.
Transitivity: The property of an identity relation whereby if x equals y and y equals z, then x equals z. True within the Person domain but never employed between Person and Essence.
Typed language: A formal system that assigns every constant, variable, predicate, and function to a specific sort, thereby enforcing grammatical distinctions that mirror theological ones.
Unary predicate: A one-place predicate that conforms to a single argument, such as G(x). It states a property without invoking a binary relation.
Variable range: The set of objects that a given variable is allowed to denote. Separate ranges for Person variables and Essence variables keep logical operations properly sorted.
Defending Divine Simplicity – RobertDryer
[i] Because pure act has no internal composition or external perimeter, nothing can circumscribe it or set it side by side with something else. In that sense the essence is both boundless and perfectly unified: there is no distinction between “parts of God” that could be counted, and there is no external reality that could fence God in. Infinity here therefore signifies plenitude of actuality and absolute simplicity at once.