The entry for "Trinity" in the Standford Encyclopedia of Philosophy has recently (20 Nov, 2020) been updated. One can find it here. It is a very useful overview of the multitude of problems of various Trinity theologies.
The following is a common image used to explain/defend the Trinity (the "Trinity Shield"):
Under section 1.4 Trinity as Incoherent, we find the following discussion of the problems therein:
If
each occurrence of “is” here expresses numerical identity, commonly expressed
in modern logical notation as “=” then the chart illustrates these claims:
1.
Father = God
2.
Son = God
3.
Spirit = God
4.
Father ≠ Son
5.
Son ≠ Spirit
6.
Spirit ≠ Father
But
the conjunction of these claims, which has been called “popular Latin
trinitarianism”, is demonstrably incoherent (Tuggy 2003a, 171; Layman 2016,
138–9). Because the numerical identity relation is defined as transitive and
symmetrical, claims 1–3 imply the denials of 4–6. If 1–6 are steps in an
argument, that argument can continue thus:
7.
God = Son (from 2, by the symmetry of =)
8.
Father = Son (from 1, 4, by the transitivity of =)
9.
God = Spirit (from 3, by the symmetry of =)
10.
Son = Spirit (from 2, 6, by the transitivity of =)
11.
God = Father (from 1, by the symmetry of =)
12.
Spirit = Father (from 3, 7, the transitivity of =)
This
shows that 1–3 imply the denials of 4–6, namely, 8, 10, and 12. Any Trinity
doctrine which implies all of 1–6 is incoherent. To put the matter differently:
it is self-evident that things which are numerically identical to the same
thing must also be numerically identical to one another. Thus, if each Person
just is God, that collapses the Persons into one and the same thing. But then a
trinitarian must also say that the Persons are numerically distinct from one
another.
But
none of this is news to the Trinity theorists whose work is surveyed in this
entry. Each theory here is built with a view towards undermining the above
argument. In other words, each theorist discussed here, with the exception of
some mysterians (see section 4.2), denies
that “the doctrine of the Trinity”, rightly understood, implies all of 1–6.
Related to this is the following (from 1.6 Analogy to an Extended Simple):
The prima facie trinitarian claims
which generate a contradiction are as follows, where each “is” means numerical
(absolute, non-relative) identity.
1. The Father is God.
2. The Son is God.
3. The Holy Spirit is God.
4. The Father is not the Son.
5. The Father is not the Holy
Spirit.
6. The Son is not the Holy Spirit.
If the three Persons are
numerically distinct (4–6), then it can’t be that all of them are
numerically identical to some one God. (1–3) Pickup proposes to understand
these using person space concepts. Using p1, p2, and p3 for different points in
person space, the points which correspond to being, respectively, the Father,
Son, and Spirit, claims 1–3 are read as:
1. The occupant of p1 is God.
2. The occupant of p2 is God.
3. The occupant of p3 is God.
These claims entail the 1–3 we
started with, understood as claims of numerical identity. But this method of
interpretation transforms 4–6, which become:
4. p1 is not p2.
5. p1 is not p3.
6. p2 is not p3.
In other words, “4. The Father is
not the Son” is not understood as asserting the numerical distinctness of the
Father and the Son, but rather, as asserting the distinction of the Father’s
person space from the Son’s person space. (And similarly with 5, 6.) The point
of all of this is that the six interpreted claims seem to be coherent, such
that possibly, all of them are true. Notably, this account accepts what we can
call “Person-collapse”, the implication of 1-3 that the Father just is the Son,
the Father just is the Spirit, and the Son just is the Spirit. (In other words,
those are numerically identical.) Thus, unlike many Trinity theories, this one
is arguably compatible with a doctrine of divine simplicity.