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.
7. There is exactly one God.
In the comments I put forward my (among philosophers) idiosyncratic view that this problem was entirely solved by the Church Fathers, and given especially lucid treatment by Basil and by Gregory of Nyssa:
To see this, think of Gregory of Nyssa's "On Not Three Gods". Gregory takes the standard (eespecially Eastern) view that while the metaphysics of the Trinity is unique, the logic of it is entirely ordinary. For, says Gregory, take three human beings. The 'is God' in (1)-(3) are usually understood as meaning 'has a divine nature'. Thus we get the parallel:
(1a) Peter has a human nature.
(2a) Paul has a human nature.
(3a) John has a human nature.
Now, Peter, Paul, and John are different people, so:
(4a) Peter is not Paul.
(5a) Paul is not John.
(6a) Peter is not John.
And says, Gregory, this is also true:
(7a) There is exactly one nature that is human nature.
For, Gregory points, out, this follows if you hold (as he does) that everything that is human shares (in whatever metaphysical way you prefer) one human nature.
Now, we know that (1a)-(7a) are not logically inconsistent. (Even if you think Gregory is stretching on (7a), it's not obvious that he's making a logical error.) The difference in the two cases, Gregory insists, is not logical, but metaphysical. The difference between the Father, the Son, and the Holy Spirit on the one hand, and Peter, Paul, and John on the other, is just that they are related to their natures in a different way: Whereas human beings share a human nature by (as it were) dividing it up materially, the Persons of the Trinity share it by way of the traditional Trinitarian processions.
There are additional questions that might be raised, particularly about whether the parallel to (7a) (There is exactly one nature that is divine nature) is an adequate translation of (7). Gregory deals with such questions (indeed, "To Ablabius, on Not Three Gods" is primarily devoted to exactly these questions), but I thought the above was enough for a comment.
I've dealt briefly with this issue a long time ago. Perhaps this discussion will allow me to update my discussion then by helping me to get more clear about other people's objections and questions.
UPDATE: Added (7) to the heptad.