He's talking about
combinatory logic, which is a sort of "point-free" version of the lambda calculus.
The primal UNITY is that which is neither two, nor three, nor even truly one.
This is the iota combinator, which I'll write as Q because I is already taken. It can be defined in terms of the famous S and K combinators by the relation Q x = x S K. [EDITED to add:] I'd expected the "two ... three ... one" bit to be some sort of reference to Church numerals, but in fact OP has clarified in chat that he intended this as an allusion to the way that the iota combinator's behaviour can't be specified "equationally" as e.g. that of the S combinator is when one writes S x y z = x z (y z), but has to be written in terms of other combinators like S and K, or in raw lambda calculus.
It reflects upon itself and discovers its IDENTITY.
Let's apply Q to itself. We get Q Q = Q S K = S S K K. Now apply the defining property of S: S S K K = S K (K K). What does this do to, say, x? S K (K K) x = K x (K K x) = x. So indeed Q Q is the identity.
It reflects upon this identity: to know that it is such is to know it is no other. To claim to be other would be FALSE; a new discovery.
The wording here seems wilfully unclear, but what I believe it means is that applying Q to the identity gives (Church's version of) boolean FALSE, which I'll write as F. Its defining property is that F x y = y. (In terms of the usual SKI basis, it's the same as K I.) So, let's give it a go. Q I x y = I S K x y = S K x y = K y (x y) = y. Yup!
When it reflects upon the false, could it do other than arrive at the TRUE?
So now let's apply Q to F and verify that we get Church's boolean TRUE, which I could write as T and which needs to have T x y = x ... but that has a more common name, namely K which I've used several times already. Anyway. Q F x y = F S K x y = K x y = x, as required.
What remains to be learned from reflection? The APPLICATION of these principles, through which all things were made.
Combinatory logic is all about (function/combinator) application. [EDITED to add:] OP says in comments that this was actually meant to indicate another specific combinator, so here we go. Applying Q to the K combinator we just derived gives Q K = K S K = S, and the S combinator is kinda-sorta doing application.
The program is complete, though its end is unknowable. All reduces to one, not two, for truth and its application are inseparable.
Combinatory logic is (Turing-)complete: any computation we know how to formalize has a combinatory-logic equivalent. Now that we've derived all three of S,K,I (which are well known to be a basis for combinatory logic -- given these you can make equivalents for anything else), we have shown that the iota combinator itself is a basis: "all reduces to one". ((Why "not two"? Perhaps because a more commonly used basis is S plus K.)
"Its end is unknowable" is presumably a reference to the undecidability of the halting problem: informally, given a combinator expression, there is no guaranteed-to-terminate procedure that will tell you whether the sort of recursive expansion process we've seen many examples of above will ever come to an end.
I'm not sure whether "truth and its application are inseparable" means anything very precise. Perhaps it's alluding to how Church booleans work -- if b is one, you use it by applying it to two arguments, one of which is the result you want if it's true and one of which is the result you want if it's false.
[EDITED to add:] OP has clarified in chat that "t.a.i.a.a.i." is about the fact that the iota combinator is usually defined in terms of K ("truth") and S ("application").
This is not ay-bee-aitch, it's alpha-beta-eta, and these are the names of the three basic kinds of transformation permitted in the lambda calculus. Alpha-conversion is variable renaming; beta-reduction is substitution of a function's argument for the parameter it's bound to; eta-conversion is the elimination of redundant function applications. (Hmm, so maybe I want lambda calculus rather than combinatory logic as such? Still thinking about that.)
Hint 1: The title is significant.
Yes, there is some logic in the mysticism.
Hint 2: It may help to go back to Church.
That would be Alonzo Church.
Hint 3: The year '94 is not significant, but the year my uncle was born is significant for the bonus.
That year must be either 1888 or 1889. 1889 was Schoenfinkel's year of birth, so I guess your uncle's surname is Finkel or maybe Finkle.
Hint 4: My uncle was fond of birds.
Perhaps a reference to Raymond Smullyan's book about combinatory logic, To Mock a Mockingbird, in which the roles of the combinators are taken by various kinds of birds.