P1. If x and y share the same essence, then x and y share the same set of essential properties.
P2. If the Father and the Son share the same essence, then the Father and the Son share the same set of essential properties.
P3. If the Father is essentially paternal and the Son is not essentially paternal, then the Father has the essential property of paternity and the Son does not have the essential property of paternity.
P4. The Father is essentially paternal, and the Son is not essentially paternal.
P5. Therefore, the Father has the essential property of paternity, and the Son does not have the essential property of paternity.
P6. If the Father has the essential property of paternity and the Son does not, then the Father and the Son do not share the same set of essential properties.
P7. Therefore, the Father and the Son do not share the same set of essential properties.
P8. Therefore, the Father and the Son do not share the same essence.
Here is your revised argument, simplified and written with clear theological language while keeping the formal structure:
If the Father is God and the Son is God, then both the Father and the Son are fully and equally God in essence. That means they are ontologically identical to the one divine essence, and according to the Law of Identity, there should be no real distinction between them.
However, Trinitarian theology also teaches that the Son is not the Father. The Son is a distinct person from the Father.
This means the Son cannot be absolutely identical to the Father, even though both are said to be fully God. So, if both persons are fully God, yet distinct from one another, then they cannot be completely identical to the one divine essence. This creates a contradiction.
To illustrate: If I say, “I am a father, a son, and a brother,” all these roles belong to one person, me. There is no real ontological difference between them. They are simply different relationships, but they do not represent three distinct persons.
If we apply that to the Trinity, and say:
Father = God (divine essence)
Son = God (divine essence)
Then, by the Law of Identity, Father = Son.
But orthodox Trinitarianism denies this, insisting that the Father and Son are not the same person. So, either we accept that there are three separate beings (which is tritheism), or that the three persons are just different roles or modes of one being (which is modalism). Both options are rejected as heresies.
Therefore, the doctrine of the Trinity cannot be logically defined without falling into contradiction or heresy.
Christian Trinitarian doctrine affirms these seven statements:
- The Father is God
- The Son is God
- The Holy Spirit is God
- The Father is not the Son
- The Father is not the Holy Spirit
- The Son is not the Holy Spirit
- There is exactly one God
The Logical Problem of the Trinity is the claim that these seven statements, when taken together, lead to a contradiction.
Why These Statements Seem to Contradict
If “is God” in P1–P3 means is identical to God (a classical identity claim), then:
- The Father = God
- The Son = God
- The Holy Spirit = God
But identity is transitive, so this leads to:
- The Father = The Son = The Holy Spirit
That directly contradicts P4–P6, which say:
- The Father ≠ The Son
- The Father ≠ The Holy Spirit
- The Son ≠ The Holy Spirit
So, if we treat these statements with classical logic, and interpret “is” and “is not” in the normal way (identity and non-identity), the set becomes formally inconsistent. This is the LPT1 version of the problem.
How Philosophers Classify the Problem
The problem can be framed in three levels:
- Explicit inconsistency: One statement directly denies another
- Formal inconsistency: A contradiction can be derived using formal logic
- Implicit inconsistency: A contradiction arises only when you add extra necessary truths (e.g. about identity, divinity, or numbers)
The LPT is claimed by critics to be at least formally inconsistent, not implicit. That still makes it incoherent.
What Makes LPT1 Invalid or Unacceptable to Trinitarians?
Philosophers group responses to the LPT into families, depending on how they try to escape the contradiction:
LPT1 Family (Fails to Solve the Problem)
This family:
- Uses classical identity logic
- Interprets “is God” as identity with God
- Interprets “is not” as classical non-identity
- Then derives the contradiction: Father = Son (from divinity claims) vs. Father ≠ Son (from personhood claims)
Any solution that keeps all of these together cannot solve the problem, because the contradiction is built in. So LPT1-family responses are invalid.
Possible Ways to Escape the Contradiction
To avoid contradiction, one of the assumptions must change. That leads to different families of proposed solutions:
1. Non-PLI Family
Rejects classical identity altogether (i.e. changes the logic used).
Example: Relative Identity theory – says each person is the same God but not the same person.
2. Naïve Modalist Family
Softens or avoids the claim that the persons are really distinct. This leads toward modalism (The heresy that Father, Son, and Spirit are just modes of one person).
3. Equivocation1 Family
Changes the meaning of “is God” across P1–P3. For example, “The Son is God” might not mean the same thing as “The Father is God”. This leads toward subordinationism or Arianism, where only the Father is truly God, and the Son and Spirit are lesser.
4. Social Trinitarianism and Counting
Some Social Trinitarians try to avoid the contradiction by using “is God” in different ways. They say each of the Father, Son, and Holy Spirit “is God” in one sense (for example, each has the attribute of divinity), but only together they are “the one God” in another sense (for example, forming a collective or divine community). This is called equivocation, using the same phrase with different meanings.
But if “is God” has one meaning when applied to each Person, and a different meaning when stating “there is one God,” then the reasoning no longer works formally. It avoids contradiction only by changing the terms mid-argument, which breaks the logical consistency.
If instead “is God” is used in the same sense throughout, and each Person is fully God, and the Persons are not each other, then the conclusion must follow that there are three distinct Gods. That directly contradicts the claim that there is only one God.
Therefore, either:
- You commit to one sense of “is God” and face logical contradiction, or
- You avoid contradiction by using two meanings of “is God,” but then your solution becomes logically invalid due to equivocation.
This is why only two possible paths avoid formal contradiction without heresy:
- Reject classical identity and use a non-standard way of counting (Relative Identity view).
- Equivocate on “is God” (Social Trinitarian view), but then clarity and precision are lost, and the claim of logical coherence weakens.
All other attempts either fall into logical inconsistency or theological heresy.
5. Pure Relative Identity Trinitarianism (PRIT)
PRIT claims that the Father, Son, and Holy Spirit are each numerically identical to God. It defines “God” as the divine essence and treats the identity statements as literal. However, this view leads to problems. According to classical logic, if A = B and A = C, then B = C. So, if the Father = God and the Son = God, then the Father = the Son. But this is denied by Trinitarians, leading to contradiction. PRIT therefore avoids this by rejecting classical identity or denying that it applies universally.
Rejecting classical identity is controversial because it is a core metaphysical principle. According to Leibniz’s Law, if two things are identical, then they must share all the same properties. But the Son is said to be begotten, while the divine essence is not.
Therefore, they cannot be identical without violating Leibniz’s Law. PRIT also contradicts the transitivity of identity, which states that if two things are each identical to a third thing, then they must be identical to each other.
This leads to the problem of modalism, where the three persons collapse into one. PRIT avoids modalism only by denying classical identity altogether, but that move makes the theory unappealing to most philosophers. It also does not avoid the worry of tritheism, as each person is said to be God but not each other. To avoid this, PRIT must deny that we count gods by identity, but then it fails to provide a principled method for counting.
6. Impure Relative Identity Trinitarianism (IRIT)
IRIT tries to solve this by interpreting the phrase “is God” in a different way. Instead of treating it as identity, it treats it as predication, meaning that each person possesses the attribute of divinity. IRIT still rejects counting gods by classical identity. Branson’s model, for example, suggests counting by “division” rather than identity. He claims that as long as the persons are not divided by time or space, they can be counted as one God.
However, this approach has problems. If Branson uses division to count gods, he should also use it to count persons. Since the persons are inseparable, he should count them as one person. But this leads back to modalism. If he instead counts persons by identity and gods by division, the method becomes inconsistent. This inconsistency makes the theory appear ad hoc and unprincipled.
Branson also claims that the ancients counted by division, but this is shown to be false. Classical theologians and even Branson himself count divine persons and attributes by identity. If identity is used for persons, it should also be used for gods. Otherwise, the theory lacks coherence.
Other responses to the Logical Problem of the Trinity (LPT) include J.C. Beall’s view that the Trinity is a true contradiction. This requires rejecting the law of noncontradiction, which is unacceptable to most people. Another response is mysterianism, where thinkers like James Anderson argue that no solution is needed. They claim that belief in the Trinity is justified by independent reasons, even if it is logically incoherent.
In conclusion, interpreting the Trinity as identity leads to modalism, while interpreting it as predication leads to tritheism. PRIT avoids modalism by denying identity, but this denies a basic principle of logic. IRIT avoids tritheism by redefining counting methods, but this introduces inconsistency. Other options like contradiction or mystery are also deeply problematic. Therefore, the LPT remains unresolved, and those who find all proposed solutions unsatisfactory may consider alternatives such as Unitarianism.
Where Does That Leave Us?
If one insists on all the following:
- The three persons are distinct (classical non-identity)
- Each person is God in the same full sense
- There is only one God
- And all of this must be expressed using classical logic (with identity and non-identity)
Then the doctrine becomes formally inconsistent.
To solve the problem, one must give up or revise at least one of these assumptions. Every major theory tries to do just that, but no option avoids all problems. For example:
- Giving up classical identity leads to problems with defining personhood or unity
- Giving up real distinction collapses into modalism
- Giving up full divinity for Son or Spirit becomes Arianism
- Giving up monotheism becomes polytheism
