The sixth century Christian philosopher-theologian Boethius was one of the first to engage the obvious difficulty of the Trinitarian doctrine in a logical and philosophical way. His solution to the difficulty is here. I shall briefly discuss it.
His solution rests on a presumed distinction between predication which ascribes real properties that belong to a subject 'of itself', and predication which is merely circumstantial, and which "is not grounded in that which it is for a thing to be". A real property is one which belongs to some thing in respect of that 'which it truly is' (in eo quod ipsa est). A circumstantial property is what no way belongs to a thing itself (minime vero ex sese). Circumstantial predication in no way "augments, diminishes or changes the thing itself of which it is said". He gives the following example of circumstantial predication:
[…] let someone stand. If I approach on his right he will be 'left' inBut certain forms of predication, in particularly the category of relation (such as father to son, son to father) are merely circumstantial. He says:
comparison to me, not because he is himself left (ille ipse sinister sit), but
because I will have approached him on his right. Again, I approach on his left:
he will likewise be to my right. not because he is 'right’ in himself (non quod
ita sit per se dexter), as something might be white or tall, but because he
becomes right by my approaching, and that which he is by me or from me is in no
wise from him himself.
Therefore those things which do not produce a predicate in respect to a property of some thing, in that which it truly is, are able to alter or change nothing and can vary no essence in any way. Thus if 'Father' and 'Son' are predicated in relation, and they differ in no respect but this relation alone, as was stated, and if [this] relation is not predicated of that of which it is predicated as though that relation were also on the side of the thing [secundum rem] of which it is said*, then this predicate does not produce a difference of things in that of which it is spoken, but indeed -if it can be said- it produces something that can scarcely be understood: a difference of persons.
[…] Thus substance holds together unity, while relation brings number to the Trinity: therefore those things which are brought forth in isolation and separately are of relation. For the Father is not the same as the Son, nor is the Holy Ghost the same as either of them. Yet God is the same as the Father, the Son and the Holy Ghost. He is the same as justice, goodness, greatness and all the things which can be predicated of Him Himself.This is not at all easy to follow! The argument seems to be that the relation of God the father to God the son is circumstantial to God himself, and so is consistent with the identity of the Father and the Son. But where does that leave us with respect to Leibniz principle that things which are truly identical are indiscernible, and that things which are discernible are not identical? Boethius appears to be saying that
(*) father(a, b) and not father(b, a) and b = a.
on the grounds that the relation of fatherhood of a to b is merely circumstantial to a, and circumstantial to b, and so is consistent with the essential, non-circumstantial identity of a and b. This clearly violates Leibniz' principle, which does not make any distinction between the two kinds of predication (if indeed the distinction is intelligible in the first place).
Boethius, if he were around, might argue that the principle fails for certain forms of predication: indeed, that his distinction between real and circumstantial predication is precisely the distinction between predicates for which Leibniz holds, and those for which it doesn't. But that doesn't seem to work. You might argue that the distance between me and some fly buzzing around in China is not real, but merely circumstantial. But it nonetheless holds that if I am now 8,000 miles from that person, and some person X is now 8,001 miles from that fly, then I am not identical with X. Leibniz principle ought to be valid for any form of predication. But in that case, Boethius' argument fails.
* The italicised portion is my my translation as I cannot believe Kenyon is right. I am translating relatio vero non praedicatur ad id de quo praedicatur quasi ipsa sit et secundum rem de qua dicitur. Kenyon has "and if this relation is predicated neither relative to that of which it is predicated, as though it were the same, nor according to the thing itself of which it is said". It may be that Moreschini's edition has a different Latin, but I doubt it.